NUMBER AND OPERATIONS: 3-5

Introduction

Number sense gives students confidence to use mathematics in everyday life. In grades

3 – 5, students’ understanding of the base-ten number system is extended to larger numbers and decimals. Using benchmark values, common fractions are compared to each other and to whole numbers.

Computational fluency is essential and may be accomplished using various methods. The focus at this level is multiplication and division. This fluency should be developed with an understanding of arithmetic operations and problem solving.

Estimation is encouraged to judge the reasonableness of an answer. A range of strategies should be employed and students should be able to explain their thinking both orally and in writing.

When appropriate, calculators and computers can enhance and extend mathematical understanding at this level.

The Research Base Benchmarks, pages 350, 358-360

Whole numbers. “Elementary and middle school students may have limited ability with place value (Sowder, 1992a). Sowder reports that middle school students are able to identify the place values of the digits that appear in a number, but they cannot use the knowledge confidently in context (for example, students have trouble determining how many boxes of 100 candy bars could be packed from 48,638 candy bars).”

Operations with whole numbers. “Students make a variety of errors in multi-digit addition and subtraction calculations (Brown & Van Lehn, 1982). Given traditional instruction, a substantial number of 4^th and 5^th graders are not able to subtract some whole numbers successfully. (Fuson, 1992). Student errors suggest students interpret and treat multi-digit numbers as single-digit numbers placed adjacent to each other, rather than using place-value meanings for the digits in different positions (Fuson, 1992). With specially designed instruction, 2^nd graders are able to understand place value and to add and subtract four-digit numbers more accurately and meaningfully than 3^rd graders receiving traditional instruction (Fuson, 1992). Research also suggests students interpret multiplication of whole numbers mainly as repeated addition. This interpretation is inadequate for many multiplication problems and can lead to restrictive intuitive notions such as ‘multiplication always makes larger’ (Greer, 1992).”

Operations with fractions and decimals. “Elementary and middle school students make several errors when they operate on decimals and fractions (Benander & Clement, 1985; Kouba et al., 1988; Peck & Jencks, 1981; Wearne & Hiebert, 1988). For example, many middle school students cannot add 4 + 0.3 correctly or 7 1/6 + 3 ½ (Kouba et al., 1988; Wearne & Hiebert, 1988). These errors are due, in part, to the fact that students lack essential concepts about decimals and fractions, and they have memorized procedures that they apply incorrectly. Interventions to improve concept knowledge can lead to increased ability by 5^th graders to add and subtract decimals correctly (Wearne & Hiebert, 1988).”

“Students of all ages misunderstand multiplication and division (Bell et al., 1984; Graeber & Tirosh, 1988; Greer, 1992). Commonly held misconceptions include ‘multiplication always makes larger,’ ‘division always makes smaller,’ ‘the divisor must always be smaller than the dividend.” Students may correctly select multiplication as the operation needed to calculate the cost of gasoline when the amount and unit cost are integers, then select division for the same problem when the amount and unit cost are decimal numbers (Bell et al., 1981).”

Rational numbers: “Upper elementary and middle school students often do not understand that decimals and fractions represent concrete objects that can be measured by units, tenths of units, hundredths of units, and so on (Hiebert, 1992). For example, students have trouble writing decimals for shaded parts of rectangular regions divided into 10 or 100 equal parts (Hiebert & Wearne, 1986). Other students have little understanding of the value represented by each of the digits of a decimal number or know the value of the number is the sum of the value of its digits. Students of all ages have problems choosing the largest or the smallest in a set of decimals with different numbers of digits to the right of the decimal points (Carpenter et al., 1981; Hiebert & Wearne, 1986; Resnick et al., 1989). Upper elementary school students can establish rich meanings for decimal symbols and do a variety of decimal tasks well after specially designed instruction using base-10 blocks (Wearne & Heiberts, 1988, 1989).”

Converting between fractions and decimals. “Instruction that focuses on the meaning of fractions and decimals forms a basis on which to build a good understanding of the relationship between fractions and decimals. Instruction that merely shows how to translate between the two forms does not provide a conceptual base for understanding the relationship (Markowits & Sowder, 1991).”

Number comparison. “Lower elementary students do not have procedures to compare the size of whole numbers. By 4^th grade, students generally have no difficulty comparing the sizes of whole numbers up to four digits (Sowder, 1992). Students are less successful when the number of digits is much larger or when more than two numbers are to be compared. This might be due to increased memory requirements of working with more or larger numbers (Sowder, 1988). Upper elementary and middle school students taught traditionally cannot successfully compare decimal numbers (Sowder, 1988, 1992). Rather, they overgeneralize the features of the whole number system to the decimal numbers (Resnick et al., 1989). They apply a ‘more digits make bigger’ rule (according to which .1814 > .385). After specially-designed instruction which develops good meanings for decimal symbols, many students are able to compare decimal numbers with understanding by 5^th grade (Wearne & Hiebert, 1988). Upper elementary and middle school students taught traditionally, cannot compare fractions successfully (Sowder, 1988). Students’ difficulties here indicate they treat the numerator and the denominator separately. Specially-designed instruction to teach meanings for fractions can help to improve ordering fractions by as early as the end of the 5^th grade (Behr et al., 1984).”

Calculators. “The use of calculators in K-12 mathematics does not hinder the development of basic computation skills and frequently improves concept development and paper-and-pencil skills, both in basic operations and in problem solving (Hembree & Dessart, 1986; Kaput, 1992). The use of calculators in testing produces higher scores than paper-and pencil efforts in problem solving as well as in basic operations (Hembree & Dessart, 1986).”

Estimation. “Middle school and even high school students may have limited understanding about the nature and purpose of estimation. They often think it is inferior to exact computation and equate it with guessing (Sowder, 1992b), so that they do not believe estimation is useful (Sowder & Wheeler, 1989). Students who see estimation as a valuable tactic for obtaining information use estimation more frequently and successfully (Threadgill-Sowder, 1984).”

“Good estimators use a variety of estimating tactics and switch easily between them. They have a good understanding of place value and the meaning of operations, and they are skilled in mental computation. Poor estimators rely on algorithms that are more likely to yield the exact answer. They lack an understanding of the notion and value of estimation and often describe it as ‘guessing’ (Sowder, 1992b). Before 6^th grade, students develop very few estimation skills from their natural experiences (Case & Sowder, 1990; Sowder, 1992b). As a result, some researchers caution that teaching estimation to young children may have, as its single effect, that they master specific procedures in a superficial manner (Sowder, 1992b).

MEASUREMENT: 3-5

Introduction

Measurement is a link that connects ideas within areas of mathematics and bridges mathematics to other disciplines. Using measurement, students in grades 3-5 explore questions related to their environment. They investigate real-world situations involving measurement of temperature, perimeter, angles, area, and volume.

Students should select appropriate tools and units of measurement and recognize factors that affect precision. In addition, it is important that students realize that all measurements are approximations.

The Research Base Benchmarks,

Teaching and Learning Mathematics: p.36-40

Math Matters: page 177, 195

Adding it Up, pages 88, 282

In order to realize that arbitrary measures are not reliable, a child must reconcile the varying lengths and numbers of arbitrary units and reason transitively. On the other hand, to use a standard device such as a 30cm or meter ruler to make direct comparisons of lengths of objects is a less demanding task. It also has the advantage that it appears to be a real-world meaningful activity. (Boulton-Lewis et al., 1994, p.130).

It was concluded…the use of drawing in the development of area concepts helps children to develop abstractions and to recognize the units that go to make up a shape….(Wheatley and Reynold, 1996).

“Most researchers agree that there are three components of measuring: conservation, transitivity, and units and unit iteration.” (Chapin & Johnson, page 177). “Students in the United States must become proficient in using both the English system and the metric system of measurement.” (Chapin & Johnson, page 195).

“Tools…help children reason about the mathematically important components of an activity so that invariants like unit are represented physically and then mentally.” (Lehrer & Schauble, page 282). “Students find it very difficult to decompose and then recompose shapes or even to see one shape as a composition of others, an idea that is fundamental to conservation.” (Lehrer, Jenkis, and Osana, page 88).

GEOMETRY: 3-5

Introduction

As students investigate the attributes and properties of geometric shapes, they develop more precise descriptions of the relationships they discover. They are learning to reason and to make, test, and justify conjectures about these relationships. Students need to extend geometric knowledge and develop spatial reasoning ability by visualizing geometric relationships. Spatial understanding is necessary for interpreting, understanding, and appreciating our inherently geometric world.

The Research Base Benchmarks, page 352

NCTM Principles and Standards, page 41

Students advance through levels of thought in geometry. Van Hiele has characterized them as visual, descriptive, abstract/relational, and formal deduction (Van Hiele, 1986; Clements & Battista, 1992). At the first level, students identify shapes and figures according to their concrete examples. For example, a student may say that a figure is a rectangle because it looks likes a door. At the second level, students identify shapes according to their properties, and here a student might think of a rhombus as a figure with four equal sides. At the third level, students can identify relationships between classes of figures (e.g., a square is a rectangle) and can discover properties of classes of figures by simple logical deduction.

Progress from one of Van Hiele’s levels to the next is more dependent upon instruction than age. Given traditional instruction, middle school students perform at levels one or two (Clements & Battista, 1992). Some evidence suggests it is possible for students to understand the abstract properties of geometric figures by 5^th grade (Clements & Battista, 1989, 1990, 1992; Wirszup, 1976).

With well-designed activities, appropriate tools, and teachers’ support, students can make and explore conjectures about geometry and can learn to reason carefully about geometric ideas from the earliest years of schooling. Geometry is more than definitions; it is about describing relationships and reasoning. The notion of building understanding in geometry across the grades, from informal to more formal thinking, is consistent with the thinking of theorists and researchers (Burger and Shaughnessy 1986; Fuys, Geddes, and Tischler, 1988; Senk 1989; Van Heile, 1986).

ALGEBRA: 3-5

Introduction

Algebra is a style of thinking where students study patterns and relationships and learn to use them in daily life. Patterns are the basis for reasoning about regularity and consistency. As students move into upper elementary, they need to generalize these patterns and express the relationships using language symbols, tables, and graphs.

Change is an important mathematical idea that can be studied using the tools of algebra. Research indicates that this is not an area that students typically understand with much depth. Using graphs and tables, student in grades 3-5 start to notice and describe change. As they look at sequences, they can begin to distinguish between arithmetic growth and geometric growth.

The Research Base Benchmarks, pages 334, 351-352

NCTM Principles and Standards, pages 40, 163

Preliminary research hints that students have difficulty making connections between mathematical expressions, sentences, and sequences that share common structural patterns. They focus instead upon incidental similarities or differences (Ericksen, 1991).

Students of all ages often do not view the equality sign of equations as a symbol of the equivalence between the left and the right side of the equation, but rather interpret it as a sign to begin calculation (Kieran, 1992). Students who are encouraged initially to use trial-and-error substitution develop a better notion of the equivalence of the two sides of the equation and are more successful in applying more formal methods later on (Kieran, 1988, 1989).

DATA ANALYSIS AND PROBABILITY: 3-5

Introduction

The analysis of data helps students begin to understand the world around them. Books, newspapers, the Internet, and other media are filled with graphical displays. With such widespread use, data analysis becomes very critical. Hence it is important that students in grades 3-5 progress from reading data to interpreting tables and graphs.

Moreover, students should formulate questions to investigate relevant issues in their lives. Furthermore, they must develop the skills of collecting valid data, organizing it, describing its central tendency and variability, and creating meaningful representations that can be used to make predictions and inferences.

Students at this level will also begin to investigate the concepts of probability. Through experiments, students will explore the frequency of various outcomes and use the results to make predictions.

The Research Base Benchmarks, pages 353-354

Research suggests that a good notion of representativeness may be a prerequisite to grasping the definitions for measure of location like mean, median, or mode. Students can acquire notions of representativeness after they start seeing data sets as entities to be described and summarized rather than as “unconnected” individual values. This occurs typically around 4^th grade (Mokros & Russell, 1992).

Research suggests students should be introduced first to location measures that connect with their emerging concept of the “middle,” such as the median, and later in the middle school grades, to the mean. Premature introduction of the algorithm for computing the mean divorced from a meaningful context may block students from understanding what averages are for (Mokros & Russell, 1992; Pollatsek et al., 1981).

The concept of the mean is quite difficult for students of all ages to understand even after several years of formal instruction. Several difficulties have been documented in the literature: students of all ages can talk about the algorithm for computing the mean and relate it to limited contexts, but cannot use it meaningfully in problems (Mokros & Russell, 1992; Pollatsek, Lima, & Well, 1981); upper elementary and middle school students believe that the mean of a particular data set is not one precise numerical value but an approximation that can have one of several values (Mokros & Russell, 1992).

Research presents somewhat contradictory results on elementary children’s understanding of probability. Piagetian research says lower elementary children have no conception of probability (Piaget & Inhelder, 1975; Shayer & Adey, 1981), but other studies indicate that even lower elementary school children have probabilistic intuitions upon which probability instruction can build. Falk et al. (1980) presented elementary school students with two sets, each containing blue and yellow elements. Each time, one color was pointed out as the payoff color. The students had to choose the set from which they would draw at random a “payoff element” to be rewarded. From the age of six, children began to select the more probable set systematically. The ability to choose correctly precedes the ability to explain these choices.

Upper elementary students can give correct examples for certain, possible, and impossible events, but cannot calculate the probability of independent and dependent events even after instruction on the procedure (Fischbein & Bazit, 1984). That is partly because students at this age tend to create “part to part” rather than “part to whole” comparisons (e.g., 9 men and 11 women rather than 15% of men and 10% of women).

Extensive research points to several misconceptions about probabilistic reasoning that are similar at all age levels and are found even among experienced researchers (Kahneman, Slovic, & Tversky, 1982; Shaughnessy, 1992). One common misconception is the idea of representativeness, according to which an event is believed to be probable to the extent that is “typical.” For example, many people believe that after a run of heads in coin tossing, tails should be more likely to come up. Another common error is estimating the likelihood of event based on how easily instances of it can be brought to mind.

NUMBER AND OPERATIONS: 6-8

Introduction

Students in grades 6-8 must develop number sense, computational estimation, mental computation, and number size in order to thoroughly understand the real number system. The primary focus in the middle grades should be on fractions, decimals, percents, integers, and rational numbers. Students should apply their understanding of factors, multiples, and prime factorization to problems involving fractions. Students need to develop an understanding of decimals as fractions whose denominators are powers of 10. The concept of fractions should be extended to include rates, ratios, and proportionality. Percents can be thought about in ways that combine aspects of both fractions and decimals, paying particular attention to percents less than 1 or greater than 100. Applications with integers will develop the notation that they represent relative changes in values. As a result of the studies in numbers and operations, students will be able to judge the advantages and disadvantages of various representations of numbers.

Students in middle grades must also understand the meaning of operations and how they relate to one another. In addition to developing proficiency with fraction, decimal, percent, integer, and rational number computations, students should be able to determine the reasonableness of their answers. Technologies such as calculators and computers can aid in connecting basic skills and calculation procedures to a deeper mathematical understanding. Students should also have experiences solving problems in context, choosing the appropriate computational method, and deciding whether to use approximate or exact values.

The Research Base Benchmarks, page 350

NCTM Principles and Standards, pages 216,218

Science for All Americans, page 131

Middle school students are able to identify the place values of the digits that appear in a number, but they cannot use the knowledge confidently in context (Sowder, 1992a). Upper elementary- and middle-school students often do not understand that decimal fractions represent concrete objects that can be measured by units, tenths of units, hundredths of units, and so on (Hiebert, 1992). Other students have little understanding of the value represented by each of the digits of a decimal number or know the value of the number is the sum of the value of its digits. Students of all ages have problems choosing the largest or the smallest in a set of decimals with different numbers of digits to the right of the decimal points (Carpenter et al., 1981; Hiebert & Wearne, 1986; Resnick et al., 1989). Upper elementary- and middle-school students may exhibit limited understanding of the meaning of fractional numbers (Kieren, 1992).

From their experience with whole numbers, many students appear to develop a belief that “multiplication makes bigger and division makes smaller.” When students solve problems in which they need to decide whether to multiply or divide fractions or decimals, this belief has negative consequences that have been well researched (Greer, 1992). Also, a mistaken expectation about the magnitude of a computational result is likely to interfere with students’ making sense of multiplication and division of fractions or decimals (Gaeber & Tannenhaus, 1993). For example, fewer than one-third of the thirteen-year-old U.S. Students tested in the National Assessment of Education Progress (NAEP) in 1988 correctly chose the largest number from 3/4, 9/16, 5/8, and 2/3 (Kouba, Carpenter, and Swafford, 1989). Students’ difficulties with comparison of fractions have also been documented in more recent NAEP administrations (Kouba, Zawojewski, and Strutchens, 1997).

Students are allowed much more flexibility in mathematics with the use of integers, which can be thought of in terms of a number line (AAAS, p. 131). They can now analyze numbers in terms of below sea level, debt, and left of zero on the real number line.

Middle-school and even high-school students may have limited understanding about the nature and purpose of estimation. They often think it is inferior to exact computation and equate it with guessing (Sowder, 1992b), so they do not believe estimation is useful (Sowder & Wheeler, 1989). Students who see estimation as a valuable tactic for obtaining information use estimation more frequently and successfully (Threadgill-Sowder, 1984).

MEASUREMENT: 6-8

Introduction

It should be recognized that students bring to the middle grades many diverse experiences from prior classroom instruction and life experiences. Important aspects of measurement at this level should include choosing and using appropriate units for attributes being measured, estimating measurements, solving problems involving perimeter, area, surface area, and volume. In addition, students should become proficient in using measurement tools while working within both metric and customary measurement systems.

Students should become proficient in composing and decomposing two- and three-dimensional shapes in order to find lengths, areas, and volumes of complex objects. Through these investigations, students can discover formulas and use them to solve problems involving perimeter, area, and volume. Student should explore the effect on perimeter and area when dimensions are proportionately changed.

Measurement concepts should be used throughout the school year by providing connections to other mathematics strands. Many measurement topics are closely related to what students learn in geometry.

The Research Base Research Ideas for the Classroom,

Middle Grades Mathematics, Chapter 5

Most students’ estimation skills are not well developed, especially for metric units; only 30% of 13-year-olds could estimate the length of a segment to the nearest centimeter (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981). About 70% of seventh graders could choose the best estimate of the height of a tall man in feet, but only about 35% could do so in meters. More experience with estimation in both systems of measurement appears to be needed.

Development of measurement formulas is an important part of middle grade mathematics. Formulas should be a product of exploration and discovery. This is appropriate at middle grades for concepts like area and perimeter, if students have spent time measuring these in their own ways. Among seventh graders, 56% could compute the area of a rectangle with length and width given, but only 46% could compute the area of the same rectangle drawn on grid paper without the dimensions written in. Only 33% of students could compute the volume of a rectangular solid with dimensions written in (Lindquist & Kouba, 1989).

GEOMETRY: 6-8

Introduction

Students come to the middle grades with an informal knowledge about geometric concepts. They have had experience in visualizing and drawing lines, angles, triangles, and other polygons. Moreover, they have developed intuitive notions about geometry from interacting with objects in their daily lives.

Middle school geometry programs should allow students to investigate relationships by drawing, measuring, visualizing, comparing, transforming and classifying two-dimensional and three-dimensional geometric objects. Geometry provides opportunities for developing mathematical reasoning (inductive and deductive), making and validating conjectures, and investigating properties that lead to the classification of geometric shapes. In the middle grades, students begin to use the coordinate plane to investigate transformations, congruence, similarity and symmetry.

Many topics included in the measurement strand are closely connected to the study of geometry.

The Research Base Research Ideas for the Classroom, Chpt. 11

Benchmarks, p. 352

Students advance through levels of thought in geometry. Van Hiele has characterized them as visual, descriptive, abstract/relational, and formal deduction (Van Hiele, 1986; Clements & Battista, 1992). At the first level, students identify shapes and figures according to their concrete examples. For example, a student may say that a figure is a rectangle because it looks like a door. At the second level, students identify shapes according to their properties, and here a student might think of a rhombus as a figure with four equal sides. At the third level, students can identify relationships between classes of figures (e.g., a square is a rectangle) and can discover properties of classes of figures by simple logical deduction.

Progress from one of Van Hiele’s levels to the next is more dependent upon instruction than age. Given traditional instruction, middle-school students perform at levels one or two (Clements & Battista, 1992). Some evidence suggests it is possible for students to understand the abstract properties of geometric figures by 5^th grade (Clements & Battista, 1989, 1990, 1992; Wirszup, 1976).

Since learning geometry requires student to recognize figures, their properties, and their relationships, spatial visualization skills are essential and contribute in an important way to the learning process. Professional educators point out the need to equip students with mathematical methods that support the full range of problem solving. These methods include the use of imagery, visualization, and spatial concepts and emphasize activities that use concrete representations to improve the perception of spatial relationships (Lappan & Schram, 1989; Young, 1982). Many researchers have hypothesized the differences in students’ spatial visualization skills are one cause of their problem solving difficulties. Major findings indicated a strong correlation between spatial ability and problem-solving performance, suggesting that spatial visualization skill is a good predictor of general problem solving (Tillotson, 1985).

Transformations bring a spatial-visual aspect to geometry that is as important as logical-deductive aspects. Also, transformation geometry has important real-world applications such as fabric patterns, mirrors, symmetry in nature, photos and enlargements (Sanok, 1987).

ALGEBRA: 6-8

Introduction

Algebra and algebraic thinking are fundamental to the basic education of all middle school students. Algebraic thinking is a natural extension of arithmetical thinking, but while arithmetic is effective in describing static pictures of the world, algebra is dynamic and a necessary vehicle for describing a changing world. Students in middle grades should investigate patterns expressed in tables, graphs, words, or symbols, with the emphasis on patterns that exhibit linear relationships (constant rate of change). They should explore notions of dependence and independence as the values of variables change in relation to one another. Students should connect this rate of change to the slope of a line and be able to interpret its meaning. In addition, they should develop facility in recognizing the equivalence of mathematical representations they can use to transform expressions, to solve problems, and to relate graphical, tabular, and symbolic representations. Students should be given the opportunity to solve linear equations as well as inequalities. Whenever possible, the teaching and learning of algebra in the middle grades should be integrated with other topics in the curriculum.

The Research Base Benchmarks, pp. 351-352

Research Ideas for the Classroom –

Middle Grades Mathematics, p. 94

Students have difficulty translating between graphical and algebraic representations, especially moving from a graph into an equation (Leinhardt et al., 1990). Results from the second study of the National Assessment for Educational Progress showed, for instance, that given a line with indicated intercepts, only 5% of 17-year-olds could generate the equation (Carpenter, et al., 1981). Students sometimes resist dealing with multiple representations because they do not find them helpful in solving problems (Dufour-Janvier et al., 1987). Rather, they see generating any representation as an end in itself, demanded by the requirements of the teacher of the text rather than by the needs of the problem. In addition, students confound the slope of a graph with the maximum or the minimum value and do not know that the slope of a graph is a measure of the rate (McDermott et al., 1987; Clement, 1989).

Beginning algebra students use various intuitive methods for solving algebraic equations (Kieran, 1992). Some of these methods may help their understanding of equation and equation solving. Students who are encouraged initially to use trail-and-error substitution develop a better notion of the equivalence of the two sides of the equation and are more successful in applying more formal methods later on (Kieran, 1988). By contrast, students who are taught to solve equations only by formal methods may not understand what they are doing. Students who are taught to use the method of “transposing” are found to only mechanically apply the change side/change sign rule (Kieran, 1988, 1989).

Additional research on classroom practices can be found in Research Ideas for the Classroom – Middle Grades Mathematics, Chapter 10.

DATA ANALYSIS AND PROBABILITY: 6-8

Introduction

Students in grades 6-8 should build on past experiences with data analysis to answer questions about populations and samples. To do this, students should begin to develop and conduct more complex studies. Data collection can be extended to other resources such as websites, and spreadsheets can facilitate data collection and organization. Acquiring new techniques to express the distribution of data will aid in the analysis and interpretation. Interpretation of results should contain appropriate uses of measures of central tendency and spread and construction of lines of best fit. Furthermore, the use of box plots, scatter plots, as well as histograms, circle graphs, and stem-and-leaf plots, will facilitate the representation of relationships between two populations.

In the middle grades, students should have numerous opportunities to engage in activities that promote probabilistic thinking. The resulting observations and inferences should be discussed using appropriate terminology. Comparison of theoretical and experimental probability should be undertaken. In addition, students can use and further develop their emerging understanding of proportionality to make predictions about future experiments.

The Research Base Benchmarks, pp. 353-354

Science for All Americans, p. 19

The concept of the mean is quite difficult for students of all ages to understand even after several years of formal instruction. It is important for students to be able to express ideas about the results of their research in order to see whether it says something useful about the original data (AAAS, p. 19). Several difficulties have been documented in the literature: students of all ages can talk about the algorithm for computing the mean and relate it to limited contexts, but cannot use it meaningfully in problems (Mokros & Russell, 1992; Pollatsek, Lima & Well, 1981); upper elementary and middle school students believe that the mean of a particular data set is not one precise numerical value but an approximation that can have one of several values (Mokros & Russell, 1992); some middle school students cannot use the mean to compare two different-sized sets of data (Gal et al., 1990).

Students of all ages often interpret graphs of situations as literal pictures rather than as symbolic representations of the situations (Leinhardt, Zaslavsky & Stein, 1990). “In addition, students confound the slope of a graph with the maximum or the minimum value and do not know that the slope of a graph is a measure of rate” (McDermott, et al., 1987; Clement, 1989). When constructing graphs, middle school students have difficulties with the notions of interval scale and coordinates even after traditional instruction (Kerslake, 1981). “Finally, students read graphs point-by-point and ignore their global feature. This has been attributed to the fact that they are rarely asked questions about maximum and minimum values; intervals over which a function increases, decreases, or levels off; or rates of change” (Herscovics, 1989).

Extensive research points to several misconceptions about probabilistic reasoning that are similar at all age levels and are found among experienced researchers (Kahneman, Slovic, & Tversky, 1982; Shaughnessy, 1992). One common misconception is the idea of representativeness, according to which an event is believed to be probable to the extent that it is “typical.” For example, many people believe that after a run of heads in coin tossing, tails should be more likely to come up.

NUMBER AND OPERATIONS: 9-12

Introduction

Students in grades 9-12 should see number and operations from a more global perspective. Their understanding of numbers is the foundation for their understanding of algebra, the core of all mathematics. High school students should understand more fully the concept of a number system, how different number systems are related, and whether the properties of one system hold in another. Students will develop an increased ability to estimate the results of an arithmetic computation and judge the reasonableness of results obtained through technology.

Students in high school will understand the meaning of exponents and how to apply their properties in computations. The real number system will be explored for its use in matrices and vectors. High school students will relate complex numbers to problems for which there are no real solutions. Students will apply these concepts in a variety of problem-solving situations.

The Research Base NCTM Principles and Standards, page 32

Regardless of the particular method used, students should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general. Students also need to be able to estimate and judge the reasonableness of results. Computational fluency should develop in tandem with understanding of the role and meaning of arithmetic operations in number systems (Hiebert, et al., 1997; Thornton, 1990).

MEASUREMENT: 9-12

Introduction

Opportunities to use and understand measurement arise naturally during high school in various disciplines. By ninth grade, students will have a good understanding of measurement concepts and well-developed measurement skills. Electronic measurement instruments aid the students in collecting, storing, and analyzing real-time measurement data. High school students should be able to make reasonable estimates and sensible judgments about the precision and accuracy of the values they obtain.

Students in high school will distinguish between precision and accuracy of measurements. With the widespread use of calculator and computer technologies for gathering and displaying data, students will understand that selections of scale and viewing window become important choices. Through logarithmic scaling, students will graphically represent some naturally occurring phenomena. Students will understand how unit analysis can be used to make decisions about which units are most appropriate.

The Research Base Benchmarks, pages 350, 351, 360

Students who can use measuring instruments and procedures when asked to do so often do not use this ability while performing an investigation. Typically, a student asked to undertake an investigation and given a set of equipment that includes measuring instruments, will make a qualitative comparison even though she might be competent to use the instruments in a different context (Black, 1990). It appears students often know how to take measurements, but not what or when.

Middle-school and even high-school students may have limited understanding about the nature and purpose of estimation. They often think it is inferior to exact computation and equate it with guessing (Sowder, 1992b), so that they do not believe estimation is useful (Sowder & Wheeler, 1989). Students who see estimation as a valuable tactic for obtaining information use estimation more frequently and successfully (Threadgill-Sowder, 1984).

Students of all ages often interpret graphs as literal pictures rather than as symbolic representations of the situations (Leinhardt, Zzaslavsky, & Stein, 1990; McDermott, Rosenquist, & Van Zee, 1987). Many students interpret distance/time graphs as the paths of actual journeys (Kerslake, 1981).

GEOMETRY: 9-12

Introduction

High school students should develop capacity with several ways of representing geometric ideas. These representations will allow multiple approaches to solve geometric problems. Geometry offers a means of describing, analyzing, and understanding the world; its ideas can be useful both in other areas of mathematics and in applied settings. By the ninth grade, students will have explored and discovered relationships among two- and three-dimensional geometric shapes. The students’ high school experiences in geometry will enhance their ability to discover patterns and formulate conjectures. Technology will be a useful tool for accomplishing this task.

The Research Base Research Ideas for the Classroom,

High School, page 151

Research Ideas for the Classroom,

Middle School, page 219

Probably the most comprehensive study of an alternative to traditional Euclidean geometry instruction is Usiskin’s investigation of the feasibility of a transformation approach (Cosford & Usiskin, 1971; Usiskin, 1969). Neither approach was clearly superior overall. On some measures, particularly attitudinal, the transformational approach seemed more successful on some measures of achievement.

“It is not enough….to learn about properties of shapes and the vocabulary of geometry; they [students] must understand what geometry is and how it relates to the real world and other topics in mathematics. Research has shown that our students must be active learners engaged in the process of discovering, conjecturing, and thinking at higher levels” (Fortunato, 1993).

ALGEBRA: 9-12

Introduction

Algebra is the core of mathematics. High school students’ experiences in mathematics should provide insights into algebraic abstractions and structures. These insights can help students develop a deeper understanding of real-world phenomena. By the ninth grade, students will have explored various ways of representing linear and non-linear qualities. At the same, time, working in real-world contexts may help students make sense of the underlying mathematical concepts and may foster an appreciation of those concepts. Using technology, students can model and analyze a wide range of phenomena.

Students in high school will become competent with their use of algebra. They will create models that satisfy applications of exponential and other non-linear functions. The development of function notation will assist students to better understand the effects of translations on graphs. In addition, students will recognize the effects of parameter changes. Functions notation will also help students identify how a relation might be represented through the use of parametric equations. Having gained deeper insight into the applications of mathematics, students will be able to use technology to solve a variety of problems and to identify the reasonableness of the answers that they obtain.

The Research Base Benchmarks, page 351

Research Ideas for the Classroom –

High School, pp. 202, 204-205

Students have difficulty understanding how symbols are used in algebra (Kieran, 1992). They are often unaware of the arbitrariness of the letters chosen to represent variables in equations (Wagner, 1981). Middle-school and high-school students may regard the letters as shorthand for single objects, or as specific but unknown numbers, or as generalized numbers before they understand them as representations of variables (Kieran, 1992).

Another study focused on the mathematical performance of upper secondary students who had regular and prolonged access to graphing calculators (Ruthven, 1990). These students developed specific calculator techniques for finding symbolic rules for graphically represented functions. Interestingly, the graphing-calculator group outperformed students who did not have such access on tasks that required symbolization.

A study involving beginning high school students in learning mathematical modeling while using a computer for symbolic manipulation also suggested conceptual gains without noticeable skill loss (Heid, Sheets, et al., 1988). In the Heid study, distinctive patterns of classroom interaction were noted in the experimental course. The activities that characterized the experimental course included: making, defending, and debating mathematical conjectures; interpreting and reasoning about mathematical representations; and suggesting and justifying mathematical models (Heid, 1988).

DATA ANALYSIS AND PROBABILITY: 9-12

Introduction

Upon entering high school, students should be familiar with designing simple surveys and experiments; gather data through the use of tables, charts, and graphs; and summarizing that data in various ways. Students will have computed probabilities of simple and some compound events, and will have performed simulations, comparing the results of the simulations to predicted probabilities.

In grades 9-12, students should gain a deep understanding of the issues entailed in drawing conclusions in light of variability. They should learn to ask questions that will help them evaluate the quality of survey, observational studies, and controlled experiments. Students can use their skills in algebra to model and analyze data, with increasing understanding of what it means to fit data well.

High school students should link probability to other topics in mathematics, especially counting techniques, area concepts, and relationships between functions and the area under their graphs. Students should learn to determine the probability of a sample statistic for a known population and draw simple inferences about a population from randomly generated samples.

The Research Base Benchmarks, page 361

NCTM Principles and Standards, page 50

Research Ideas for the Classroom,

High School, page 188

Research has shown that students in grades 5-8 expect their own judgment to be more reliable than information obtained from data (Hancock, Kaput, and Goldsmith, 1992). In the later middle grades and high school, students should address the ideas of sample selection and statistical inference and begin to understand that there are ways to quantifying how certain one can be about statistical results.

Even researchers trained in the use of statistics entertain statistical misconceptions. For example, they may erroneously believe that when conducting a replication studys’ [sic] even smaller sample sizes than the first study’s are sufficient, since sample should be “representative” of the population regardless of its size (Tversky & Kahneman, 1971). If trained researchers have trouble with statistical concepts, it should not surprise us that students have misconceptions of some of the most elementary concepts, such as mean and variance.

A basic problem appears to be understanding the distinction between a variable making no difference and a variable that is correlated with the outcome in the opposite way than the students initially conceived (Kuhn, et al., 1988).

References:

American Association for the Advancement of Science. (1993). Benchmarks For

Science Literacy. New York, New York: Oxford University Press.

American Association for the Advancement of Science. (1990). Science For All

Americans. New York, New York: Oxford University Press.

Chapin, S. H., and A. Johnson. (2000). Math Matters: Understanding The Math You

Teach, Grades K-6. Sausalito, CA: Math Solutions Publications.

Ma, L. (1999). Knowing and Teaching Elementary Mathematics. Mahwah, N.J.:

Lawrence Erlbaum Associates.

National Council of Teachers of Mathematics. (2000). Principles and Standards for

School Mathematics. Preston, VA: National Council of Teachers of Mathematics.

National Research Council. (2001). Adding It Up: Helping Children Learn

Mathematics. Washington, D.C.: National Academy Press.

Van de Walle, John. (2001). Elementary and Middle School Mathematics: Teaching

Developmentally, Addison Wesley Longman, Inc.

Wagner, Sigrid, ed. (1993). Research Ideas for the Classroom. National Council of

Teachers of Mathematics.

Kindergarten

Number, Number Sense and Operations Standard

Students develop number sense, understand number and number systems, understand the meaning of operations and how they relate to one another, and gain fluency in computation and estimation. Students estimate and compute using a variety of strategies including technology-supported methods.

Number and 1. Compare and order whole numbers up to 10.

Number Systems

2. Determine “how many” in sets (groups) of 10 or fewer objects.

3. Construct multiple sets containing the same number of objects.

4. Explain rules of counting such as that each object should be counted once and that order does not change the number.

5. Count to twenty by rote (e.g., in play situations or while reading number books).

6. Relate, read and write numerals for single digit numbers (0 to 9).

7. Compare the number of objects in two or more sets (up to 10) when one set has one to two more, or one to two less.

8. Represent and use whole numbers in flexible ways, including relating, composing and decomposing numbers (e.g., five marbles can be 2 red, and 3 green or 1 red, 4 green can be 5 marbles etc.)

9. Identify and state the value of pennies, nickels and dimes.

Meaning of Operations 10. Model and represent single digit addition as

combining sets and counting on, and single digit subtraction as take-away, comparison.

a) Combine and separate small sets of objects (e.g., add or subtract one, two, or another small amount) in contextual situations.

b) Count on (forward) and count back (backward) on a number line between 0 and 10.

11. Demonstrate multiplication as repeated joining (addition) of groups of equal size up to 10.

12. Demonstrate division as sharing of or partitioning into groups of equal size in contextual situations (e.g., sharing 6 stickers equally among 3 children) with an initial set of 10 or less.

Computation 13. Recognize the number or quantity of sets up to 5 and Estimation without counting (e.g., recognize without

counting (e.g., recognize without counting the

square dot arrangement on a domino as 5).

Technology 14. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMS.

15. Use draw and paint applications.

16. Use electronic resources to practice skills and re-mediate deficits.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Recognize and compare calendar elements (day,

week, month and year).

Use Measurement 2. Compare and order objects of different lengths,

Techniques and Tools weights, and/or capacities and use relative terms

like longer, shorter, bigger, smaller, heavier,

lighter, more or less.

3. Measure length and volume (capacity) using uniform objects in the environment.

a) Determine how many links long is a box.

b) Determine how many small containers it takes to fill one big container using sand, rice, beans, etc.

4. Order events based on time. For example:

a) Activities that take a long time or a short time.

b) Review what we do first, next, last.

c) Recall we did or plan to do yesterday, today, tomorrow.

Technology 5. Use electronic resources to practice skills and re-mediate deficits.

6. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board on measurement, and/or posted work on the Internet.

Geometry and Spatial Sense Standard

Students identify, classify, compare and analyze characteristics, properties, and relationships of one-, two-, and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects and transformations to analyze mathematical situations and solve problems.

Characteristics and 1. Identify and sort two-dimensional shapes and

Properties three-dimensional objects

a) Name two-dimensional figures (e.g., circle, square, triangle, rectangle, pentagon, hexagon) and 3-dimensional objects (e.g., cube, sphere, cone, cylinder).

b) Describe two-dimensional figures and three-dimensional objects from the environment using the child’s own vocabulary.

c) Sort shapes and objects into groups based on student defined categories.

d) Select all shapes or objects of one type from a group

e) Build two-dimensional figures; build simple three-dimensional objects.

Spatial Relationships 2. Name and demonstrate the relative position of

objects.

a) Place objects over, under, inside, outside, on, beside, between, above, below, on top of, upside-down, behind, in back of, in front of.

b) Describe placement of objects with terms such as on, inside, outside, above, below, over, under, beside, between, in front of, behind.

Technology 3. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMs.

4. Use draw and paint applications.

5. Use electronic resources to practice skills and re-mediate deficits.

Patterns, Functions, and Algebra Standard

Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations such as tables, graphs and equations.

Use Patterns, Relations 1. Sort, classify, and order objects by size, color,

and Functions shape, number, and other properties

a) Identify how objects are alike and different.

b) Order three events and objects according to a given attribute such as time or size.

c) Recognize and explain how objects can be classified in more than one way.

d) Identify what attribute was used to sort groups of objects that have already been sorted.

2. Identify, create, extend and copy sequences of sounds (such as musical notes), shapes (such as buttons, leaves or blocks), motions (such as hops or skips), and numbers from 1 to 10 (e.g.: AB, AAB, ABB, AABB).

3. Describe orally the pattern of a given sequence (e.g.: AB, AAB, ABB, AABB).

Use Algebraic 4. Model a problem situation using concrete

Representations materials (e.g.: graphs, tables).

Technology 5. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMs.

6. Use electronic resources to practice skills and re-mediate deficits.

Data Analysis and Probability Standard

Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data.

Data Collection 1. Gather and sort data in response to a question

posed by teacher and students, such as shoe

color or number of brothers and sisters, or their

surroundings.

2. Sort and classify objects according to attributes such as use, size, color, or shape, and arrange in a floor or table graph.

Statistical Methods 3. Arrange three objects by attributes such as

length, width, or height and identify objects by

position (e.g., first, middle, last).

4. Select the category or categories that have the most objects or the fewest objects in a floor or

table graph.

Technology 5. Use a variety of input and output devices such as

keyboards, cameras, microphones, printers, CD-ROMs.

6. Use draw and paint applications.

7. Use electronic resources to practice skills and re-mediate deficits.

Grade 1

Number, Number Sense and Operations Standard

Number and 1. Use ordinal numbers (e.g., first, second, third)

Number Systems to compare.

2. Order objects and count objects by twos, fives and tens.

3. Recognize and generate equivalent forms for the same number using physical models, words and number expressions (e.g., concept of ten is described by “10 blocks”, full tens frame, numeral 10, 5+5, 15-5, one less than 11, my brother’s age).

4. Count, read, and write the numerals for numbers to 100.

5. Count forward to 100, count backwards from 100, and count forward or backward starting at any number between 1 and 100.

6. Identify and state the value of a penny, nickel, dime, quarter and dollar.

7. Determine the value (up to a dollar) of a small collection of coins (containing 1 or 2 different type coins) including pennies, nickels, dimes and/or quarters.

8. Show different combinations of coins that have the same value (up to one dollar).

9. Use place value concepts to represent whole numbers using numerals, words, expanded notation, and physical models with ones and tens.

a) Develop a system to group and count by fives and tens.

b) Make and explore patterns and grouping in a 100’s chart.

c) Recognize the first digit of a two digit number as the most important to indicate the size of a number and the nearness to 10 or 100.

10. Represent commonly used fractions using words and physical models for halves, thirds, and fourths recognizing fractions are represented by equal size pieces of a whole.

Meaning of Operations 11. Model, represent and explain addition as

combining sets (part + part = whole) and counting on.

a) Model to explain addition performed with physical materials or pictures in contextual situations.

b) Draw pictures to model addition.

c) Write number sentences to represent addition.

d) Explain that adding two whole numbers yields a larger whole number.

12. Model, represent and explain subtraction as take-away, and comparison.

a) Model to explain subtraction with physical materials or pictures in contextual situations.

b) Draw pictures to model subtraction.

c) Write Number Sentences to represent subtraction.

d) Explain that subtraction of whole numbers or taking away yields an answer smaller than the original number.

13. Use conventional symbols to represent the operations of addition and subtraction

14. Model and represent multiplication as repeated addition and rectangular arrays in contextual situations (e.g.: Four people will be at my party. If I want to give 3 balloons to each, how many will I need to buy?)

15. Model and represent division as sharing equally in contextual situation (e.g.: sharing cookies).

16. Understand the equal sign to mean “the same as” using concrete materials and visual representation.

Computation and 17. Develop strategies for basic addition facts to

Estimation twelve

a) Counting all.

b) Counting on.

c) One more, two more.

d) Doubles.

e) Doubles plus or minus one.

f) Make ten.

g) Using tens frames.

h) Identify property (adding zeros).

18. Develop strategies for subtraction of basic facts such as:

a) Relating to addition (i.e.: 7 – 3 = :think “3 plus? Equals 7”).

b) One less, two less.

c) All but one (for example, 8-7, 5-4).

d) Using tens frames.

e) Missing addends.

19. Double-digit addition and subtraction without regrouping.

Technology 20. Use a variety of input and output devices such as keyboard, cameras, microphones, printers, CD-ROMs.

21. Use draw and paint applications.

22. Use electronic resources to practice skills and re-mediate deficits.

23. Use word processing applications to write numerals to 100.

24. Print, post, publish, and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board on measurement, and/or posted work on the Internet.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Recognize and explain the need for fixed units

and tools for measuring length and weight with

standard units.

2. Tell time to the hour and half hour on digital and analog timepieces.

Use Measurement 3. Order sequence of events as related to time

Techniques and Tools (e.g., Summer, Fall, Winter and Spring or

morning, afternoon, and night).

4. Estimate and measure lengths using non-standard and standard units (e.g., centimeters, inches and feet).

5. Estimate and measure weight using non-standard units (e.g., blocks of uniform size.)

Technology 6. Use electronic resources to practice skills and re-mediate deficits.

7. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board on measurement, and/or posted work on the Internet.

Geometry and Spatial Sense Standard

Characteristics and 1. Identify, compare and sort two-dimensional

Properties shapes (e.g., square, circle, ellipse, triangle,

rectangle, rhombus, trapezoid, parallelogram,

pentagon and hexagon).

a) Recognize and identify triangles and rhombuses independent of position, shape or size.

b) Describe two-dimensional shapes using attributes such as number of sides and number of vertices (corners, or angles).

2. Create new shapes by combining or cutting apart existing shapes.

3. Identify cone, cylinder, sphere, and cube.

4. Identify the shapes of the faces of three-dimensional objects.

Spatial Relationships 5. Extend the use of location words to include

distance (near, far, close to) and directional

words (left, right).

6. Copy figures and draw simple two-dimensional shapes from memory.

Technology 7. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMs.

8. Use draw and paint applications

9. Use electronic resources to practice skills and re-mediate deficits.

10. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board, and/or posted work on the Internet.

11. Access, print, save and retrieve resources using the network.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Sort, classify, and order objects by two or more

and Functions attributes, such as color and shape, and explain

how objects were sorted.

2. Extend sequences of sounds, shapes or simple number patterns, and create and record similar patterns.

a) Analyze and describe patterns with multiple attributes using numbers and shapes (e.g., AA,B,aa,b,AA,B,aa,b,…).

b) Continue repeating and growing patterns with materials, pictures, and geometric items (e.g., XOXOOXOOOXOOOO).

3. Describe orally the basic unit or general plan of a repeating or growing pattern.

Use Algebraic 4. Solve open sentences by representing an

Representations express in more than one way using the

commutative property, (e.g., 4+5 = 5+4, or the number of blue balls plus red balls is the same as the number of red balls plus blue balls: R+B=B+R).

5. Describe orally and model a problem situation using a number phrase, symbols, sentences and/or concrete materials.

Technology 6. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMs.

7. Use electronic resources to practice skills and re-mediate deficits.

Data Analysis and Probability Standard

Data Collection 1. Identify multiple categories for sorting data.

2. Collect and organize data into charts using tally marks.

3. Display data in picture graphs with units of 1 and bar graphs with intervals of 1.

4. Read and interpret charts, picture graphs and bar graphs as sources of information to identify main ideas, draw conclusions and make predictions.

5. Construct a question that can be answered using information from a graph.

Statistical Methods 6. Arrange five objects by an attribute such as size

or weight, and identify the ordinal position of each object.

7. Identify most, fewest, and number of objects represented in two or more categories in a graph (picture graph, bar graph or table graph).

Probability 8. Describe the likelihood of simple events (e.g.,

informal activities with spinners or number cubes) as impossible/possible and more likely/less likely.

Technology 9. Use a variety of input and output devices such as

keyboards, cameras, microphones, printers, CD-ROMs.

10. Use draw and paint applications.

11. Use electronic resources to practice skills and re-mediate deficits.

12. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, print document/drawing for class book or bulletin board on measurement, and/or posted work on the Internet.

13. Collect data and input into a template spreadsheet to produce a graph.

Grade 2

Number, Number Sense and Operations Standard

Number and 1. Use place value concepts to represent, compare

Number Systems and order whole numbers using physical

models, numerals, and words with ones, tens

and hundreds. For example:

a) Recognize ten can mean “ten ones” or a single entity (1 ten) through concrete models and trading games.

b) Read and write 3 digit numerals (e.g.: 243 as two hundred forty three, 24 tens and 3 ones, or 2 hundreds and 43 ones, etc.), construct models to represent each.

2. Recognize and classify numbers as even or odd.

3. Count money and make change using coins and a dollar bill.

4. Represent and write the value of money using the ¢ sign and in decimal form when using the $ sign.

5. Represent fractions (halves, thirds, fourths, sixths, and eights) using words, numerals and physical models. For example:

a) Recognize that a fractional part can mean different amounts depending on the original quantities

b) Recognize that a fractional part of a shaded two-dimensional shape is not necessarily touching (contiguous).

c) Identify and illustrate parts of a whole and parts of sets of objects (using manipulatives, candy, classmates).

d) Compare and order physical models of halves, thirds, and fourths in relationship to 0 and 1.

Meaning of Operations 6. Model, represent and explain addition and

subtraction.

a) Solve missing addend problems by counting up or subtracting (e.g., “I had six baseball cards, my sister gave me some more, I now have ten. How many did she give me?” 6 + ? = 10 or 10 – 6 = ?).

b) Solve addition problems by combining sets and counting on.

7. Model, represent and explain multiplication as repeated addition, rectangular arrays and skip counting.

8. Model, represent and explain division as sharing equally, and repeated subtraction.

9. Model and use the commutative property for addition (e.g., 8+2=10 and 2+8=10).

Computation and 10. Demonstrate multiple strategies for addition and

Estimation subtraction of 2 and 3 digit numbers with or

without regrouping.

11. Compatible numbers (tens and doubles)

a) Compensatory numbers (9+6 = 15 is like (9+1) + (6-1) = 15).

b) Informal use of commutative and associative properties of addition (6+4=10, 10-4=6).

12. Demonstrate fluency of addition and subtraction facts with addends through 9 and corresponding subtractions.

13. Estimate the results of whole number addition and subtraction problems using front-end estimation and judge the reasonableness of the answers.

14. Add and subtract multiples of 10.

Technology 15. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMs.

16. Use draw and paint applications.

17. Use electronic resources to practice skills and re-mediate deficits.

18. Use word processing applications to write and classify numbers as even and odd.

19. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board on measurement, and/or posted work on the Internet.

20. Use draw and paint applications to model and represent addition and subtraction.

21. Use word processing applications to explain addition, subtraction and money representation.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Identify and select appropriate units of measure.

a) For length use centimeters, meters, inches, feet or yards.

b) For volume (capacity) use liters, cups, pints or quarts.

c) For weight use grams, ounces or pounds.

d) For time use hours, half-hours, quarter-hours, or minutes and time designations a.m. or p.m.

2. Describe and compare the relationships among the same type of units like centimeters and meters; inches, feet and yards; cups, pints and quarts; ounces and pounds; and hours, half-hours, and quarter-hours (e.g., how many inches in a foot?)

3. Establish personal or common referents for units of measure to make estimates and comparisons; (e.g., the width of a finger is a centimeter, a large bottle of soda is 2 liters, a small paperclip weighs about one gram.)

4. Tell time to the nearest five minute interval on digital and analog timepieces.

Use Measurement 5. Estimate, measure, and compare the length and

Techniques and Tools weight of common objects using metric and

U.S. Standard units.

6. Select and correctly use appropriate measurement tools (e.g., ruler, scale, etc.)

7. Make and test predictions about measurements using different units to measure the same length or volume.

Technology 8. Use electronic resources to practice skills and re-mediate deficits.

9. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board on measurement, and/or posted work on the Internet.

10. Use draw and paint applications to model units of measure.

Geometry and Spatial Sense Standard

Characteristics and 1. Identify, describe, compare and sort three-

Properties dimensional objects (i.e., cubes, spheres, prisms,

cones, cylinders, and pyramids) according to the shape of the faces or the numbers of faces, edges, or vertices.

2. Predict what new shapes will be formed combining or cutting apart existing shapes.

3. Recognize two-dimensional shapes (circle, rectangle, square, triangle, hexagon, trapezoid, parallelogram, and rhombus) and three-dimensional objects from different positions.

4. Identify and determine whether two-dimensional shapes are congruent (same shape and size) or similar (same shape different size) by copying or using superposition (lay one thing on top of another).

Spatial Relationships 5. Describe location, using comparative (before

and after), direction (above and below), and

positional (first, last) words.

6. Create and identify two-dimensional figures with line symmetry (e.g., what letter shapes are symmetrical?)

Technology 7. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMs.

8. Use draw and paint applications to model geometric figures.

9. Use electronic resources to practice skills and re-mediate deficits.

10. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, print document/drawing for class book or bulletin board, and/or posted work on the Internet.

11. Access, print, save and retrieve resources using the network.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Extend simple number patterns (both repeating

and Functions and growing patterns) and create similar

patterns using different objects such as using

concrete objects or shapes to represent a

numerical pattern.

2. Use patterns to make generalizations and predictions determined by a rule (e.g., determine a missing element in a pattern).

3. Create new patterns with consistent rules or plans and describe the rule or general plan of existing patterns.

Use Algebraic 4. Model problem situations using objects,

Representations Pictures, tables, numbers, letters, and other

symbols (e.g., Determine the rule and identify missing numbers in a table of number pairs).

5. Understand equivalence and extend the concept to situations involving symbols (e.g., 4+5=9 and 9=4+5; and 4+5 = 3+6 = Δ+χ…).

6. Use symbols to represent unknown quantities in an expression or equation using addition and subtraction (e.g. □ + 8 = 10; Δ – 2 = 4).

Analyze Change 7. Describe quantitative and qualitative changes,

especially those involving addition and

subtraction (e.g., a student growing two inches in one year).

Technology 8. Use a variety of input and output devices such as keyboard, cameras, microphones, printers, CD-ROMs.

9. Use electronic resources to practice skills and re-mediate deficits.

10. Use draw and pint applications to create simple patterns.

Data Analysis and Probability Standard

Data Collection 1. Pose questions, use observations, interviews,

surveys to collect data, organize data in charts,

picture graphs and bar graphs.

2. Read, interpret, and make comparisons and predictions from data represented in charts, line plots, picture graphs and bar graphs.

3. Read and construct simple time lines to sequence events.

4. Sort and classify objects by attributes.

Statistical Methods 5. Write a few sentences to describe and compare

categories of data represented in chart or graph and make statements about the data as a whole.

6. Identify untrue or inappropriate statements about a given set of data.

7. Recognize that data may vary from one population to another (e.g., favorite TV shows of students and of parents).

8. Organize sorted objects into a simple table or chart.

Probability 9. List some of the possible outcomes of a simple

experiment and predict whether given outcomes

are more, less or equally likely to occur.

10. Use physical models and/or pictures to represent possible arrangements of 2 to 3 objects.

Technology 11. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMS.

12. Use draw and paint applications.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board on measurement, and/or posted work on the Internet.

15. Collect data and input into a spreadsheet and produce a graph.

Data Analysis and Probability Standard

BENCHMARKS:

By the end of the A. Gather and organize data from surveys and

3-4 program: classroom experiments, including data collected over a period of time.

B. Read and interpret tables, charts, graphs (bar, picture, line, line plot), and timelines as sources of information, identify main idea, draw conclusions, and make predictions.

C. Construct charts, tables and graphs to represent data, including picture graphs, bar graphs, line graphs, line plots and Venn diagrams.

D. Read, interpret and construct graphs in which icons represent more than a single unit or intervals greater than one: e.g., each ♥ = 10 hearts or the intervals on an axis are multiples of 10.

E. Describe data using mode, median and range.

F. Conduct a simple probability experiment and draw conclusions about the likelihood of possible outcomes.

G. Identify and represent possible outcomes, such as arrangements of a set of up to four members and possible combinations from several sets, each containing 2 or 3 members.

H. Use the set of possible outcomes to describe and predict events.

Grade 3

Number, Number Sense and Operations Standard

Number and 1. Identify and generate equivalent forms of whole

Number Systems numbers (e.g., 36, 30+6, 9x4, 46-10, number of

inches in a yard).

2. Use place value concepts to represent whole numbers and decimals using numerals, words, expanded notation and physical models

a) Recognize 100 means “10 tens” as well as a single entity (1 hundred) through physical models and trading games.

b) Describe the multiplicative nature of the number system (e.g., the structure of 3205 as 3x1000 plus 2x100 plus 5x1).

c) Model the size of 1000 in multiple ways (e.g., packaging 1000 objects into 10 boxes of 100; modeling a meter with centimeter and decimeter strips, or gathering 1000 pop-can tabs).

d) Explain the concept of tenths and hundredths using such physical models as metric pieces, base ten blocks, decimal squares or money.

3. Use mathematical language and symbols to compare and order; e.g., less than, greater than, at most, at least, <,>,=,<, >.

4. Count money and make change using coins and paper bills to ten dollars.

a) By subtracting

b) By counting up (e.g., ($19.65 + .10 =

$19.75) + .25 = $20.00)

5. Represent fractions and mixed numbers using words, numerals, and physical models.

6. Compare and order commonly used fractions and mixed numbers using a number line, models (such as fraction circles or bars), points of reference (such as more or less than ½), and equivalent forms found using physical or visual models.

7. Recognize and use decimal and fraction concepts and notations as related ways of representing parts of a whole or a set; e.g., 3 of 10 marbles are red can also be described as 3/10 and 3 tenths are red.

Meaning of Operations 8. Model, represent and explain multiplication (e.g., repeated addition, skip counting, rectangular arrays, and area model). For example:

a) Use conventional mathematical symbols to write equations from word problems involving multiplication.

b) Understand that unlike addition and subtraction, the factors in multiplication and division may have different units (e.g., 3 boxes of 5 cookies each).

9. Model, represent and explain division (e.g., sharing equally, repeated subtraction and rectangular arrays and area model). For example:

a) Translate contextual situations involving division into conventional mathematical symbols.

b) Explain how a remainder may impact an answer in a real-world situation (e.g., 14 cookies being share by 4 children).

10. Explain and use relationships between operations, such as:

a) relate addition and subtraction as inverse operations;

b) relate multiplication and division as inverse operations;

c) relate addition to multiplication (repeated addition).

d) relate subtract to division (repeated subtraction);

11. Model and use the commutative and associative properties for addition and multiplication.

Computation and 12. Add and subtract whole numbers with and

Estimation without regrouping.

13. Demonstrate fluency of multiplication facts through 10 and corresponding division facts.

14. Multiply and divide 2 and 3 digit numbers by a single-digit number, without remainders for division.

15. Evaluate the reasonableness of computations based upon operations and the numbers involved; e.g., considering relative size, place value and estimates.

Technology 16. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMs.

17. Use draw and paint applications.

18. Use electronic resources to practice skills and re-mediate deficits.

19. Use word processing applications to explain the concept of tenths and hundredths.

20. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board on measurement, and/or posted work on the Internet.

21. Use draw and paint applications to model and represent place value concepts to represent whole numbers and decimals.

22. Integrate two or more applications.

23. Use the computer calculator.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Identify and select appropriate units for

measuring.

a) Length- use mile, kilometer and other units of measure as appropriate;

b) volume (capacity)- use gallons;

c) weight- use ounce, pound, gram or kilogram.

d) Temperature- use degrees (Fahrenheit and Celsius).

2. Establish personal or common referents to include additional units; e.g., a gallon container of milk; a postage stamp is about a square inch.

3. Tell time to the nearest minute and find elapsed time using a calendar or clock.

4. Read thermometers in both Fahrenheit and Celsius scales.

Use Measurement 5. Estimate and measure length, weight, and

Techniques and Tools volume (capacity), using metric and U.S.

Customary units (measured to the nearest ½ or ¼ unit as appropriate).

6. Use appropriate measurement tools and techniques to construct a figure or approximate an amount of specified length, weight , or volume (capacity); e.g., construct a rectangle with length 2 1/2 inches and width 3 inches, fill a measuring cup to the ¾ cup mark.

7. Make estimates for perimeter, area, and volume using links, tiles, cubes, and other models.

Technology 8. Use electronic resources to practice skills and re-mediate deficits.

9. Print, post, publish, and/or distribute technology products. Show or explain completed work.

10. Use draw and paint applications to model units of measure.

Geometry and Spatial Sense Standard

Characteristics and 1. Analyze and describe properties of two-

Properties dimensional objects using terms such as vertex

and edge, angle, side, and face.

2. Identify and describe the relative size of angles with respect to right angles as follows:

a) Use physical models, like straws, to make different sized angles by opening and closing the sides, not by changing the side lengths.

b) Identify, classify and draw right, acute, obtuse and straight angles.

3. Find and name locations on a labeled

grid or coordinate system; e.g., a map or graph.

Transformations and 4. Draw lines of symmetry to verify

Symmetry symmetrical two-dimensional shapes.

Visualization and 5. Build a three-dimensional model of an object

Geometric models composed of cubes; e.g., construct a model

based on an illustration of an actual object.

Technology 6. Use a variety of input and output devices such as keyboard, cameras, microphones, printers, CD-ROMs.

7. Use draw and paint applications to model geometric figures.

8. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board, and/or posted work on the Internet.

9. Access, print, save and retrieve resources using the network.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Extend multiplicative and growing patterns, and

And Functions describe the pattern or rule in words.

2. Analyze and replicate arithmetic sequences with and without a calculator.

3. Use patterns to make predictions, identify relationships, and solve problems.

Use Algebraic 4. Model problem situations using objects,

Representations pictures, tables, numbers, letters, and other

symbols.

5. Write, solve, and explain simple mathematical statements such as 7 + ! > 8 or ∆ + 8 = 10.

6. Express mathematical relationships as equations and inequalities.

Analyze Change 7. Create tables to record, organize and analyze

data to discover patterns and rules.

8. Identify and describe quantitative changes,

subtraction; e.g., the height of water in a glass

becoming 1 centimeter lower each week due to

evaporation.

Technology 9. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMs.

10. Use electronic resources to practice skills and re-mediate deficits.

11. Use draw and paint applications to create simple patterns.

12. Use word processing applications to explain simple patterns and analyze change.

Data Analysis and Probability Standard

Data Collection 1. Collect and organize data from an experiment,

such as recording and classifying observations

or measurement in response to a question

posed.

2. Draw and interpret picture graphs in which the symbol or picture represent more than one object.

3. Read, interpret, and construct bar graphs with intervals greater than one.

4. Support a conclusion or prediction orally and in writing, using information in a table or graph.

5. Match a set of data with a graphical representation of the data.

6. Translate information freely among charts, tables, line plots, picture graphs, and bar graphs; e.g., create a bar graph from the information in a chart.

7. Analyze and interpret information represented on a timeline.

Statistical Methods 8. Identify the mode (most frequently identified data) of a data set and describe the information it gives about a data set.

Probability 9. Conduct a simple experiment or simulation of

simple event, record the results in a chart, table or graph, and use the results to draw conclusions about the likelihood of possible outcomes.

10. Use physical models, pictures, diagrams, and lists to solve problems involving possible arrangements or combinations of two to four objects.

Technology 11. Use a variety of input and output devices such as

keyboards, cameras, microphones, printers, CD-ROMs.

12. Use draw and paint applications to create graphs.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, print document/drawing for class book or bulletin board on measurement, and/or posted work on the Internet.

15. Collect data and input into a spreadsheet to produce a graph.

16. Create multimedia presentations explaining a probability experiment. The presentation should contain a graph of the results.

Grade 4

Number, Number Sense and Operations Standard

Number and 1. Represent fractions greater than one, including

Number Systems mixed numbers using words, numerals and

physical models.

2. Identify and generate equivalent forms of fractions, and decimals.

a) Connect physical, verbal and symbolic representations of commonly used fractions, decimals and whole numbers (e.g., ½, 5/10, “five tenths”, 0.5, shaded rectangles with half, and five tenths).

b) Understand and explain that ten-tenths is the same as one whole in both fraction and decimal form.

3. Use place value structure of the base-ten number system to read, write represent and compare whole numbers through millions and decimals through thousandths.

4. Model and identify all factors of whole numbers through 100.

5. Recognize and classify numbers as prime or composite and list factors and multiples.

6. Round whole numbers to a given place value to millions.

7. Use models and points of reference to compare commonly used fractions.

Meaning of Operations 8. Use associative and distributive properties to

simplify and/or perform computations (e.g. use left to right multiplication and the distributive property to find and exact answer without pencil and paper, 5x47=5x40+5x7=200+35=235).

9. Recognize that division may be used to solve different types of problem situations, and interpret the meaning of remainders (e.g., situations involving measurement and money).

Computation and 10. Solve problems involving counting money and

Estimation making change using both coins and paper bills (up to $20).

11. Estimate the results of computations involving whole numbers to millions, fractions, and decimals using a variety of strategies.

a) Addition and subtraction computations involving whole numbers to millions.

b) Multiplication computations involving whole numbers to 2-digits.

c) Division computation involving whole numbers through 1-digit divisors.

12. Use concrete models, points of reference, visual representation, paper and pencil, and equivalent forms to add and subtract decimals and compare commonly used fractions with like denominators.

13. Develop and explain strategies for performing computations mentally.

14. Analyze and solve multi-step problems involving addition, subtraction to millions, multiplication (3 digit x 2 digit), and division (2-digit divisors) using an organized approach; verify and interpret results with respect to the original problem.

15. Use a variety of methods and appropriate tools, (e.g., mental math, paper and pencil, and calculator) for computing with whole numbers.

16. Demonstrate fluency for adding, subtracting, multiplying and dividing whole numbers by 1 and 2 digit numbers and multiples of ten.

Technology 17. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-Roms.

18. Use draw and paint applications.

19. Use electronic resources to practice skills and re-mediate deficits.

20. Print, post, publish and/or distribute technology products. Show or explain completed work.

21. Use draw and paint applications to model and represent place value concepts to represent whole numbers and decimals.

22. Integrate two or more applications.

23. Use the computer calculator.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Relate the number of units that measure an

object to the size of the unit as well as the size

of the object (e.g., compare the number of cups

to fill a pitcher to the number of quarts to fill

the same pitcher).

2. Identify and select appropriate units to measure.

a) For length or perimeter use links, inches to the nearest ¼ inch, or centimeters to the nearest centimeter

b) For area use tiles, square inches, or square centimeters.

c) For volume (capacity) use cubes, cups, liters, cubic inches, or cubic centimeters.

Use Measurement 3. Develop and use strategies as appropriate to

Techniques and Tools determine the perimeter, area, and volume of

squares, rectangles and rectangular prisms.

4. Develop and use strategies for estimating the area and perimeter of irregular shapes.

5. Make simple unit conversions within a measurement system (e.g. inches to feet, kilograms to grams, & quarts to gallons).

6. Use appropriate organization, strategies, and technology as needed to solve problems involving length, weight, capacity, perimeter, area, volume, time and temperature.

7. Write and solve meaningful, multi-step problems involving elapsed time (to the nearest minute), perimeter, area and temperature. Verify solutions.

8. Demonstrate and describe perimeter as surrounding and area as covering a two-dimensional shape, and volume as filling a three-dimensional object.

Transformation and 9. Create and identify two-dimensional figures

Symmetry with line symmetry (e.g., what letter shapes,

logos, polygons are symmetrical.)

Technology 10. Use electronic resources to practice skills and re-mediate deficits.

11. Print, post, publish and/or distribute technology products. Show or explain completed work.

12. Use draw and paint applications to model units of measure.

Geometry and Spatial Sense Standard

Characteristics and 1. Identify, describe and build models that

Properties illustrate intersecting, parallel and perpendicular

lines.

a) Sketch lines on paper.

b) Use manipulatives to model lines.

2. Describe, classify, compare and model two dimensional figures and three-dimensional objects using their attributes.

3. Recognize and compare similar shapes within a class of figures (e.g., triangles, rectangles, etc.) and find patterns among the relationships of their angle and of their side measures.

4. Identify similarities and differences of quadrilaterals (e.g., squares, rectangles, parallelograms and trapezoids).

5. Identify and define triangles by their angle measures (e.g., equiangular, right, acute and obtuse triangles) and side lengths (e.g., isosceles, equilateral and scalene triangles).

6. Describe points, lines and planes. Identify examples in the environment.

Spatial Relationships 7. Specify locations and plot ordered pairs on a

coordinate plane, using first quadrant points.

Transformations and 8. Identify, describe and use reflections (flips),

Symmetry rotations (turns), and translations (slides) in

solving geometric problems (e.g., to determine if 2 shapes are congruent).

Visualization and 9. Use geometric models to solve problems in

Geometric Models other areas of mathematics, such as number

(multiplication/division) and measurement (area, perimeter, border).

Technology 10. Use a variety of input and output devices such as keyboards, cameras, microphones, printers, CD-ROMs.

11. Use draw and paint applications to model geometric figures.

12. Use electronic resources to practice skills and re-mediate deficits.

13. Print, post, publish and/or distribute technology products. Show or explain completed work (with assistance). Examples of published pieces might be a slide show with voice, printed document/drawing for class book or bulletin board, and/or posted work on the Internet.

14. Access, print, save and retrieve resources using the network.

Data Analysis and Probability Standard

BENCHMARKS:

By the end of the A. Read, create and use line graphs, histograms,

5-7 program: circle graphs, box-and-whisker plots, stem-and-leaf plots, and other representations when appropriate.

B. Interpret data by looking for patterns and relationships, draw and justify conclusions, and answer related questions.

C. Evaluate interpretations and conclusions as additional data are collected, modify conclusions and predictions, and justify new findings.

D. Compare increasingly complex displays of data, such as multiple sets of data on the same graph.

E. Collect, organize, display and interpret data for a specific purpose or need.

F. Determine and use the range, mean, median and mode to analyze and compare data, and explain what each indicates about the data.

G. Evaluate conjectures and predictions based upon data presented in tables and graphs, and identify misuses of statistical data and displays.

H. Find all possible outcomes of simple experiments or problem situations, using methods such as lists, arrays and tree diagrams.

I. Describe the probability of an event using ratios, including fractional notation.

J. Compare experimental and theoretical results for a variety of simple experiments.

K. Make and justify predictions based on experimental and theoretical probabilities.

Grade 5

Number, Number Sense and Operations Standard

Number and 1. Use models and visual representation to

Number Systems develop the concept of ratio as part to part and

part to whole, and the concept of percent as part

to whole.

2. Use various forms of “one” to demonstrate the equivalence of fractions; e.g., x x .

3. Identify and generate equivalent forms of fractions, decimals and percents.

4. Round decimals to a given place value and round fractions (including mixed numbers) to the nearest half.

5. Recognize and identify perfect squares and their roots.

Meaning of Operations 6. Represent and compare numbers less than 0 by

extending the number line and using familiar applications; e.g., temperature, owing money.

7. Use commutative, associative, distributive, identity and inverse properties to simplify and perform computations.

8. Identify and use relationships between operations to solve problems.

9. Use order of operations, including use of parentheses, to simplify numerical expressions.

10. Justify why fractions need common denominators to be added or subtracted.

11. Explain how place value is related to addition and subtraction of decimals; e.g., 0.2 + 0.14; the two tenths is added to the one tenth because they are both tenths.

Computation and 12. Use physical models, points of reference, and

Estimation equivalent forms to add and subtract commonly

used fractions with like and unlike denominators and decimals.

13. Estimate the results of computations involving whole numbers, fractions and decimals, using a variety of strategies

Technology 14. Use the Internet and other electronic resources

for research and digital media retrieval.

15. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

16. Integrate two or more applications.

17. Use electronic resources to practice skills and re-mediate deficits.

18. Print, post, publish and/or distribute technology products.

19. Make appropriate technology resource choices according to learning purposes and outcomes.

20. Demonstrate an understanding of terminology related to technology.

21. Access, print, save and retrieve resources using the network.

22. Use basic operating system features (e.g.: use help menus and control panels.)

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Identify and select appropriate units to measure

angles; i.e., degrees.

2. Identify paths between points on a grid or coordinate plane and compare the lengths of the paths; e.g., shortest path, paths of equal length.

3. Demonstrate and describe the differences between covering the faces (surface area) and filling the interior (volume) of three-dimensional objects.

4. Demonstrate understanding of the differences among linear units, square units and cubic units.

Use Measurement 5. Make conversions within the same measurement

Techniques and Tools system while performing computation.

6. Use strategies to develop formulas for determining perimeter and area of triangles, rectangles and parallelograms, and volume of rectangular prisms.

7. Use benchmark angles (e.g.; 45º, 90º, 120º) to estimate the measure
of angles, and use a tool to measure and draw angles.

Technology 8. Use the Internet and other electronic resources

for research and digital media retrieval.

9. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

10. Use spreadsheet applications.

11. Use draw and paint applications.

12. Integrate two or more applications.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Print, post, publish and/or distribute technology products.

15. Demonstrate an understanding of terminology related to technology.

16. Access, print, save and retrieve resources using the network.

17. Use basic operating system features (e.g.: use help menus and control panels.)

18. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Geometry and Spatial Sense Standard

Characteristics and 1. Draw circles and identify and determine

Properties relationships among the radius, diameter, center

and circumference; e.g., radius is half the

diameter, the ratio of the circumference of a circle to its diameter is an approximation of π.

2. Use standard language to describe line, segment, ray, angle, skew, parallel and perpendicular.

3. Label vertex, rays, interior and exterior for an angle.

4. Describe and use properties of congruent figures to solve problems.

5. Use physical models to determine the sum of the interior angles of triangles and quadrilaterals.

Spatial Relationships 6. Represent and compare numbers less than 0 by

extending the number line and using familiar

applications; e.g., temperature, owing money.

Visualization and 7. Understand that the measure of an angle is

Geometric Models determined by the degree of rotation of an angle side rather than the length of either side.

8. Predict what three-dimensional object will result from folding a two-dimensional net, then confirm the prediction by folding the net.

Technology 9. Use the Internet and other electronic resources

for research and digital media retrieval.

10. Evaluate and critique the quality and credibility

of electronic information.

11. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

12. Use spreadsheet applications.

13. Use draw and paint applications.

14. Integrate two or more applications.

15. Use electronic resources to practice skills and re-mediate deficits.

16. Create multimedia and/or online projects.

17. Present multi-media and/or online projects to audience inside and outside the classroom.

18. Print, post, publish and/or distribute technology products.

19. Demonstrate an understanding of terminology related to technology.

20. Access, print, save and retrieve resources using the network.

21. Use basic operating system features (e.g.: use help menus and control panels.)

22. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Justify a general rule for a pattern or a function

and Functions by using physical materials, visual

representations, words, tables or graphs.

2. Use calculators or computers to develop patterns, and generalize them using tables and graphs.

Use Algebraic 3. Use variables as unknown quantities in general

Representations rules in describing patterns and other

relationships.

4. Create and interpret the meaning of equations and inequalities representing problem situations.

5. Model problems with physical materials and visual representations, and use models, graphs and tables to draw conclusions and make predictions.

Analyze Change 6. Describe how the quantitative change in a

variable affects the value of a related variable; e.g., describe how the rate of growth varies over time, based upon data in a table or graph.

Technology 7. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

8. Use word processing applications.

9. Use spreadsheet applications.

10. Use draw and paint applications.

11. Integrate two or more applications.

12. Use electronic resources to practice skills and re-mediate deficits.

13. Make appropriate technology resource choices according to learning purposes and outcomes.

14. Demonstrate an understanding of terminology related to technology.

15. Access, print, save and retrieve resources using the network.

16. Use basic operating system features (e.g.: use help menus and control panels.)

17. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Data Analysis and Probability Standard

Data Collection 1. Read, construct and interpret frequency tables,

circle graphs and line graphs.

2. Select and use a graph that is appropriate for the type of data to be displayed; e.g., numerical vs. categorical data, discrete vs. continuous data.

3. Read and interpret increasingly complex displays of data, such as double bar graphs.

4. Determine appropriate data to be collected to answer questions posed by students or teacher, collect and display data, and clearly communicate findings.

5. Modify initial conclusions, propose and justify new interpretations and predictions as additional data are collected.

Statistical Methods 6. Determine and use the range, mean, median and

mode, and explain what each does and does not indicate about the set of data.

Probability 7. List and explain all possible outcomes in a given

situation.

8. Identify the probability of events within a simple experiment, such as three chances out of eight.

9. Use 0,1 and ratios between 0 and 1 to represent the probability of outcomes for an event, and associate the ratio with the likelihood of the outcome.

10. Compare what should happen (theoretical/expected results) with what did happen (experimental/actual results) in a simple experiment.

11. Make predictions based on experimental and theoretical probabilities.

Technology 12. Use the Internet and other electronic resources for research and digital media retrieval.

13. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

14. Use spreadsheet applications.

15. Integrate two or more applications.

16. Make appropriate technology resource choices according to learning purposes and outcomes.

17. Demonstrate an understanding of terminology related to technology.

18. Access, print, save and retrieve resources using the network.

19. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Grade 6

Number, Number Sense and Operations Standard

Number and 1. Decompose and recompose whole numbers

Number Systems factors and exponents

(e.g., 32 = 2 x 2 x 2 x 2 x 2 = 2⁵), and explain why “squared” means “second power” and “cubed” means “third power.”

2. Find and use the prime factorization of composite numbers. For example:

a) Use the prime factorization to recognize the greatest common factor (GCF).

b) Use the prime factorization to recognize the least common multiple (LCM).

c) Apply the prime factorization to solve problems and explain solutions.

3. Explain why a number is referred to as being “rational,” and recognize that the expression can mean a parts of size each, a divided by b, or the ratio of a to b.

4. Describe what it means to find a specific percent of a number, using real-life examples.

5. Use models and pictures to relate concepts of ratio, proportion and percent including percents less than 1 and greater than 100.

Meaning of Operations 6. Use the order of operations, including the use of

exponents, decimals and rational numbers, to simplify numerical expressions.

7. Use simple expressions involving integers to represent and solve problems; e.g., if a running back loses 15 yards on the first carry but gains 8 yards on the second carry, what is the net gain/loss?

8. Represent multiplication and division situations involving fractions and decimals with models and visual representations; e.g., show with pattern blocks what it means to take .

9. Give examples of how ratios are used to represent comparisons; e.g., part to part, part to whole, whole to part.

10. Recognize that a quotient may be larger than the dividend when the divisor is a fraction; e.g., 6 ¸ = 12.

Computation and 11. Perform fraction and decimal computations and

Estimation justify their solutions; e.g., using manipulatives,

diagrams, mathematical reasoning.

12. Develop and analyze algorithms for computing with fractions and decimals, and demonstrate fluency in their use.

13. Estimate reasonable solutions to problem situations involving fractions and decimals; e.g., + » 2 and 4.23 x 5.8 » 25.

14. Use proportional reasoning, ratios, and percents to represent problem situations and determine the reasonableness of solutions.

15. Determine the percent of a number and related problems; e.g., find the percent markdown if the original price was $140, and the sale price is $100.

Technology 16. Use a variety of input and output devices such as keyboard, scanners, cameras, microphones, printers, projectors, CD-Roms.

17. Use spreadsheet applications.

18. Use draw and paint applications.

19. Integrate two or more applications.

20. Use electronic resources to practice skills and re-mediate deficits.

21. Create multimedia and/or online projects.

22. Present multi-media and/or online projects to audience inside and outside the classroom.

23. Print, post, publish and/or distribute technology products.

24. Demonstrate an understanding of terminology related to technology.

25. Access, print, save and retrieve resources using the network.

26. Use basic operating system features (e.g.: use help menus and control panels.)

27. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Understand and describe the difference between

surface area and volume.

Use Measurement 2. Use strategies to develop formulas for finding

Techniques and Tools circumference and area of circles and determine

the area of sectors; e.g., ½ circle, ⅔ circle, ⅓ circle, ¼ circle.

3. Estimate perimeter or circumference and area for circles, triangles, and quadrilaterals, and surface area and volume for prisms and cylinders by:

a) estimating lengths using string or links, areas using tiles or grid, and volumes using cubes;

b) measuring attributes (diameter, side lengths, or heights) and using established formulas for circles, triangles, rectangles, parallelograms and rectangular prisms.

4. Determine which measure (perimeter, area, surface area, volume) matches the context for a problem situation; e.g., perimeter is the context for fencing a garden, surface area is the context for painting a room.

5. Understand the difference between perimeters and area and demonstrate that two of the same shapes may have the same perimeter, but different areas or they may have the same area, but different perimeters.

6. Describe what happens to the perimeter and area of a two-dimensional shape when the measurements of the shape are changed; e.g. length of sides are doubled.

Technology 7. Use a variety of input and output devices such s keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

8. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

9. Use database applications.

10. Use draw and paint applications.

11. Integrate two or more applications.

12. Use electronic resources to practice skills and re-mediate deficits.

13. Create multimedia and/or online projects.

14. Present multi-media and/or online projects to audience inside and outside the classroom.

15. Print, post, publish and/or distribute technology products.

16. Demonstrate an understanding of terminology related to technology.

17. Access, print, save and retrieve resources using the network.

18. Use basic operating system features (e.g.: use help menus and control panels.)

19. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Geometry and Spatial Sense Standard

Characteristics and 1. Classify and describe two-dimensional and

Properties three-dimensional geometric figures and objects

by using their properties; e.g., interior angle measures, perpendicular/parallel sides, congruent angles/sides.

2. Use standard language to define geometric vocabulary: vertex, face, altitude, diagonal, isosceles, equilateral, acute, obtuse, and other vocabulary as appropriate.

3. Use multiple classification criteria to classify triangles; e.g., right scalene triangle.

4. Identify and define relationships between planes; i.e., parallel, perpendicular and intersecting.

Spatial Relationships 5. Predict and describe sizes, positions and

orientations of two-dimensional

shapes after transformations such as reflections, rotations, translations and dilations.

Transformation and 6. Draw similar figures that model proportional

Symmetry relationships; e.g., model similar figures with a

1 to 2 relationship by sketching two of the same figure, one with corresponding sides twice the length of the other.

Visualization and 7. Build three-dimensional objects with cubes and

Geometric Models sketch the two-dimensional representations of

each side; i.e., projection sets.

Technologies 8. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

9. Use draw and paint applications.

10. Demonstrate an understanding of terminology related to technology.

11. Access, print, save and retrieve resources using the network.

12. Use basic operating system features (e.g.: use help menus and control panels.)

13. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Represent and analyze patterns, rules and

and Functions functions, using physical materials, tables

and graphs.

2. Use words and symbols to describe numerical and geometric patterns, rules and functions.

Use Algebraic 3. Recognize and generate equivalent forms of

Representations algebraic expressions, and explain how the

commutative, associative and distributive properties can be used to generate equivalent forms; e.g., perimeter as 2(1 + w) or 21 + 2w.

4. Solve simple linear equations and inequalities using physical models, paper and pencil, tables and graphs.

5. Produce and interpret graphs that represent the relationship between two variables.

6. Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations.

Analyze Change 7. Identify and describe situations with constant or

varying rates of change, and compare them.

8. Use technology to analyze change; e.g., use computer applications or graphing calculators to display and interpret rate of change.

Technology 9. Use the Internet and other electronic resources

for research and digital media retrieval.

10. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

11. Use electronic resources to practice skills and re-mediate deficits.

Data Analysis and Probability Standard

Data Collection 1. Read, construct and interpret line graphs, circle

graphs and histograms.

2. Select, create and use graphical representations that are appropriate for the type of data collected.

3. Compare representations of the same data in different types of graphs, such as a bar graph and circle graph.

Statistical Methods 4. Understand the different information provided

by measures of center (mean, mode and median) and measures of spread (range).

5. Describe the frequency distribution of a set of data, as shown in a histogram or frequency table, by general appearance or shape; e.g., number of modes, middle of data and level of symmetry, outliers.

Probability 6. Make logical inferences form statistical data.

7. Design an experiment to test a theoretical probability and explain how the results may vary.

Technology 8. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones, printers, projectors, CD-Roms.

8. Use spreadsheet applications.

9. Use draw and paint applications.

10. Integrate two or more applications.

11. Use electronic resources to practice skills and re-mediate deficits.

12. Demonstrate an understanding of terminology related to technology.

13. Access, print, save and retrieve resources using the network.

14. Use basic operating system features (e.g.: use help menus and control panels.)

15. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

General Mathematics Grade 7

Number, Number Sense and Operations Standard

Number and 1. Demonstrate an understanding of place value

Number Systems using powers of 10 and write large numbers in

scientific notation.

2. Explain the meaning of exponents that are negative or 0.

3. Describe differences between rational and irrational numbers; e.g., use technology to show that some numbers (rational) can be expressed as terminating or repeating decimals and others (irrational) as non-terminating and non-repeating decimals.

Meaning of Operations 4. Use order of operations and properties to

simplify numerical expressions involving integers, fractions and decimals.

5. Explain the meaning and effect of adding, subtracting, multiplying and dividing integers; e.g., how adding two integers can result in a lesser value.

Computation and 6. Simplify numerical expressions involving

Estimation integers, and use integers to solve real-life

problems.

7. Solve problems using the appropriate form of a rational number (fraction, decimal or percent).

8. Develop and analyze algorithms for computing with percents and integers, and demonstrate fluency in their use.

9. Represent and solve problem situations that can be modeled by and solved using concepts of absolute value, exponents, and square roots (for perfect squares).

Technology 10. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

11. Use spreadsheet applications.

12. Integrate two or more applications.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Create multimedia and/or online projects.

15. Present multimedia and/or online projects to audience inside and outside the classroom.

16. Print, post, publish and/or distribute technology products.

17. Make appropriate technology resource choices according to learning purposes and outcomes.

18. Demonstrate an understanding of terminology related to technology.

19. Access, print, save and retrieve resources using the network.

20. Use basic operating system features (e.g.: help menus and control panels).

21. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Select appropriate units for measuring derived measurements; e.g., miles per hour, revolutions

per minutes.

2. Convert units of area and volume within the same measurement system using proportional reasoning and a reference table when appropriate; e.g., square feet to square yards, cubic meters to cubic centimeters.

Use Measurement 3. Estimate measurement to a greater degree of

Techniques and Tools precision than the tool provides.

4. Solve problems involving proportional relationships and scale factors; e.g., scale models that require unit conversions within the same measurement system.

5. Analyze problem situations involving measurement concepts, select appropriate strategies, and use an organized approach to solve narrative and increasingly complex problems.

6. Use strategies to develop formulas for finding area of trapezoids, and volume of cylinders and prisms.

7. Develop strategies to find the area of composite shapes using the areas of triangles, parallelograms, circles, and sectors.

8. Understand the difference between surface area and volume and demonstrate that two of the same objects may have the same surface area, but different volumes or they may have the volume, but different surface areas.

9. Describe what happens to the surface area and volume of a three-dimensional object when the measurements of the object are changed; e.g., length of sides are doubled.

Technology 10. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

11. Use spreadsheet applications.

12. Integrate two or more applications.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Demonstrate an understanding of terminology related to technology.

15. Access, print, save and retrieve resources using the network.

16. Use basic operating system features (e.g.: help menus and control panels).

17. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Geometry and Spatial Sense Standard

Characteristics and 1. Use proportional reasoning to describe and

Properties express relationships between parts and

attributes of similar and congruent figures.

2. Determine sufficient (not necessarily minimal) properties that define a specific two-dimensional figure or three-dimensional object. For example:

a) Determine when one set of figures is a subset of another; e.g., all squares are rectangles.

b) Develop a set of properties that eliminates all but the desired figure; only squares are quadrilaterals with all sides congruent and all angles congruent.

3. Use and demonstrate understanding of the properties of triangles. For example:

a) Use Pythagorean Theorem to solve problems involving right triangles.

b) Use triangle angle sum relationships to solve problems.

4. Determine necessary conditions for congruence of triangles.

5. Apply properties of congruent or similar triangles to solve problems involving missing lengths and angle measures.

Spatial Relationships 6. Determine and use scale factors for similar

figures to solve problems using proportional

reasoning.

Transformation and 7. Identify the line and rotation symmetries of two-

Symmetry dimensional figures to solve problems.

8. Perform translations, reflections, rotations and dilations of two-dimensional figures using a variety of methods (paper folding, tracing, graph paper).

Visualization and 9. Draw representations of three-dimensional

Geometric Models geometric objects from different views.

Technology 10. Use draw and paint applications.

11. Use electronic resources to practice skills and re-mediate deficits.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Represent and analyze patterns, rules and

and Functions functions with words, tables, graphs and simple

variable expressions.

2. Generalize patterns by describing, in words, how to find the next term.

a) Identify arithmetic and geometric

sequences

3. Recognize and explain when numerical patterns are linear, or nonlinear progressions; e.g., 1,3,5,7... is linear (common interval) and 1,3,4,8,16... is nonlinear (irregular interval).

Use Algebraic 4. Create visual representations of equation-

Representations solving processes that model the use of inverse

operations.

5. Represent linear equations by plotting points in the coordinate plane.

6. Represent inequalities on a number line or a coordinate plane.

7. Justify that two forms of an algebraic expression are equivalent, and recognize when an expression is simplified; e.g., 4m=m+m+m+m or a · 5 + 4 = 5a + 4.

8. Use formulas in problem solving situations.

9. Recognize a variety of uses for variables; e.g., placeholder for an unknown quantity in an equation, generalization for a pattern, formula.

Analyze Change 10. Analyze linear and simple nonlinear

relationships to explain how a change in one variable results in the change of another.

11. Use graphing calculators or computers to analyze change; e.g., distance-time relationships.

Technology 12. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

13. Use spreadsheet applications.

14. Use electronic resources to practice skills and re-mediate deficits.

15. Demonstrate an understanding of terminology related to technology.

16. Access, print, save and retrieve resources using the network.

17. Use basic operating system features (e.g.: help menus and control panels).

18. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Data Analysis and Probability Standard

Data Collection 1. Read, create and interpret box-and-whisker

plots, and other types of graphs, when

appropriate.

2. Analyze how decisions about graphing affect the graphical representation; e.g., scale, size of classes in a histogram, number of categories in a circle graph.

Statistical Methods 3. Analyze a set of data using and comparing

combinations of measures of center (mean, mode, median) and measures of spread (range, quartile, interquartile range), and describe how the inclusion or exclusion of outliers affects those measures.

4. Construct opposing arguments based on analysis of the same data, using different graphical representations.

5. Compare data from two or more samples to determine how sample selection can influence results.

6. Identify misuses of statistical data in articles, advertisements, and other media

Probability 7. Compute probabilities of compound events; e.g.,

multiple coin tosses or multiple rolls of number cubes, using such methods as organized lists, tree diagrams and area models.

8. Make predictions based on theoretical probabilities, design and conduct an experiment to test the predictions, compare actual results to predicted results, and explain differences.

Technology 9. Use a variety of input and output devices such

As keyboards, scanners, cameras, microphones,.

Printers, projectors, CD-Roms.

10. Use spreadsheet applications.

Pre-Algebra Grade 7

Number, Number Sense and Operations Standard

Number and 1. Demonstrate an understanding of place value

Number Systems using powers of 10 and write large numbers in

scientific notation.

2. Explain the meaning of exponents that are negative or 0.

4. Recognize that natural numbers, whole numbers, integers, rational numbers and irrational numbers are subsets of the real number system.

Meaning of Operations 5. Use order of operations and properties to

simplify numerical expressions involving integers, fractions, decimals and radicals.

6. Explain the meaning and effect of adding, subtracting, multiplying and dividing integers; e.g., how adding two integers can result in a lesser value.

7. Explain and use the inverse and identity properties and use inverse relationships (addition/subtraction, multiplication/division, squaring/square roots) in problem solving situations.

Computation and 8. Simplify numerical expressions involving

Estimation integers, and use integers to solve real-life

problems.

9. Solve problems using the appropriate form of a rational number (fraction, decimal or percent).

10. Develop and analyze algorithms for computing with percents and integers, and demonstrate fluency in their use.

11. Represent and solve problem situations that can be modeled by and solved using concepts of absolute value, exponents, and square roots (for perfect squares).

12. Estimate, compute and solve problems involving rational numbers (including ratio, proportion and percent), and judge the reasonableness of solutions.

Technology 13. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

14. Use spreadsheet applications.

15. Integrate two or more applications.

16. Use electronic resources to practice skills and re-mediate deficits.

17. Demonstrate an understanding of terminology related to technology.

18. Access, print, save and retrieve resources using the network.

19. Use basic operating system features (e.g.: help menus and control panels).

20. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Select appropriate units for measuring derived

measurements; e.g., miles per hour, revolutions

per minute.

Use Measurement 3. Estimate measurement to a greater degree of

Techniques and Tools precision than the tool provides.

4. Solve problems involving proportional relationships and scale factors; e.g., scale models that require unit conversions within the same measurement system.

5. Analyze problem situations involving measurement concepts, select appropriate strategies, and use an organized approach to solve narrative and increasingly complex problems.

6. Use strategies to develop formulas for finding area of trapezoids, and volume of cylinders and prisms.

7. Develop strategies to find the area of composite shapes using the areas of triangles, parallelograms, circles, and sectors.

9. Describe what happens to the surface area and volume of a three-dimensional object when the measurements of the object are changed; e.g., length of sides are doubled.

Technology 10. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

11. Use spreadsheet applications.

Geometry and Spatial Sense Standard

Characteristics and 1. Use proportional reasoning to describe and

Properties express relationships between parts and

attributes of similar and congruent figures.

2. Determine sufficient (not necessarily minimal) properties that define a specific two-dimensional figure or three-dimensional object. For example:

a) Determine when one set of figures is a subset of another; e.g., all squares are rectangles.

b) Develop a set of properties that eliminates all but the desired figure; only squares are quadrilaterals with all sides congruent and all angles congruent.

3. Use and demonstrate understanding of the properties of triangles. For example:

a) Use Pythagorean Theorem to solve problems involving right triangles.

b) Use triangle angle sum relationships to solve problems.

4. Determine necessary conditions for congruence of triangles.

5. Apply properties of congruent or similar triangles to solve problems involving missing lengths and angle sizes.

Spatial Relationships 6. Determine and use scale factors for similar

figures to solve problems using proportional

reasoning.

Transformation and 7. Identify the line and rotation symmetries of two-

Symmetry dimensional figures to solve problems.

8. Perform translations, reflections, rotations and dilations of two-dimensional figures using a variety of methods (paper folding, tracing, graph paper).

Visualization and 9. Draw representations of three-dimensional

Geometric Models geometric objects from different views.

Technology 10. Create multimedia and/or online projects.

11. Present multimedia and/or online projects to audience inside and outside the classroom.

12. Print, post, publish and/or distribute technology products.

13. Demonstrate an understanding of terminology related to technology.

14. Access, print, save and retrieve resources using the network.

15. Use basic operating system features (e.g.: help menus and control panels).

16. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Represent and analyze patterns, rules and

and Functions functions with words, tables, graphs and simple

variable expressions.

2. Generalize patterns by describing, in words, how to find the next term.

a) Identify arithmetic and geometric sequences

3. Recognize and explain when numerical patterns are linear, or nonlinear progressions; e.g., 1,3,5,7... is linear (common interval) and 1,3,4,8,16... is nonlinear (irregular interval).

Use Algebraic 4. Create visual representations of equation-

Representations solving processes that model the use of inverse

operations.

5. Represent linear equations by plotting points in the coordinate plane.

6. Represent inequalities on a number line or a coordinate plane.

7. Justify that two forms of an algebraic expression are equivalent, and recognize when an expression is simplified; e.g., 4m=m+m+m+m or a · 5 + 4 = 5a + 4.

8. Use formulas in problem solving situations.

9. Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve problems.

10. Use symbolic algebra (equations and inequalities), graphs, and tables to represent situations and solve problems.

11. Recognize a variety of uses for variables; e.g., placeholder for an unknown quantity in an equation, generalization for a pattern, formula.

Analyze Change 12. Analyze linear and simple nonlinear

relationships to explain how a change in one

variable results in the change of another.

13. Use graphing calculators or computers to analyze change; e.g., distance-time relationships.

Technology 14. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

15. Use spreadsheet applications.

16. Use electronic resources to practice skills and re-mediate deficits.

17. Demonstrate an understanding of terminology related to technology.

18. Access, print, save and retrieve resources using the network.

19. Use basic operating system features (e.g.: help menus and control panels).

20. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Data Analysis and Probability Standard

Data Collection 1. Read, create and interpret box-and-whisker

plots, and other types of graphs, when

appropriate.

2. Analyze how decisions about graphing affect the graphical representation; e.g., scale, size of classes in a histogram, number of categories in a circle graph.

Statistical Methods 3. Analyze a set of data using and comparing

Combinations of measure of center (mean,

mode, median) and measures of spread (range, quartile, interquartile range), and describe how the inclusion or exclusion of outliers affects those measures.

4. Construct opposing arguments based on analysis of the same data, using different graphical representations.

5. Compare data from two or more samples to determine how sample selection can influence results.

6. Identify misuses of statistical data in articles, advertisements, and other media

Probability 7. Compute probabilities of compound events; e.g.,

multiple coin tosses or multiple rolls of number cubes, using such methods as organized lists, tree diagrams and area models.

8. Make predictions based on theoretical probabilities, design and conduct an experiment to test the predictions, compare actual results to predicted results, and explain differences.

Technology 9. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

10. Use spreadsheet applications.

Data Analysis and Probability Standard

BENCHMARKS:

By the end of the A. Create and analyze tabular and graphical

11-12 program: displays of data using appropriate tools, including spreadsheets and graphing calculators.

B. Use descriptive statistics to analyze and summarize data, including measures of center, dispersion, correlation and variability.

C. Design and perform a statistical experiment, simulation or study; collect and interpret data; and use descriptive statistics to communicate and support predictions and conclusions.

D. Connect statistical techniques to applications in workplace and consumer situations.

General Mathematics Grade 8

Number, Number Sense and Operations Standard

Number and 1. Use scientific notation to express large numbers

Number Systems and small numbers between 0 and 1.

2. Recognize that natural numbers, whole numbers, integers, rational numbers and irrational numbers are subsets of the real number system.

a) Given any real number, categorize the number as natural, whole number, integer, rational, or irrational.

b) Draw and explain a Venn diagram representing the relationships of subsets in the real number system.

Meaning of Operations 3. Extend the understanding of the order of

operations to include exponents or square roots

when evaluating numerical expressions.

4. Use associative and commutative properties and distributive property of multiplication over addition to simplify computations involving integers, fractions, and decimals.

a) Identify the use of the associative, commutative, and distributive properties.

Computation and 5. Estimate, compute and solve problems

Estimation involving rational numbers (including ratio,

proportion and percent) and judge the

reasonableness of solutions.

a) Mastery of conversion between fractions, decimals, and percents.

b) Compare and order fractions, decimals, and percents.

c) Explain the meaning of percents greater than 100.

d) Use proportions to solve word problems using rates, similar figures, scale drawings, and indirect measurements.

e) Solve the 3 types of simple percent problems.

f) Solve word problems involving percents such as simple and compound interest, discounts, sales, taxes

g) Mastery of computations with rational numbers (decimals, fractions, integers)

6. Find the square root of perfect squares, and approximate the square root of non-perfect squares as consecutive integers between which the root lies; e.g., is between 11 and 12.

7. Add, subtract, multiply, divide and compare numbers written in scientific notation.

Technology 8. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

9. Use spreadsheet applications.

10. Use electronic resources to practice skills and re-mediate deficits.

11. Demonstrate an understanding of terminology related to technology.

12. Access, print, save and retrieve resources using the network.

13. Use basic operating system features (e.g.: help menus and control panels).

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Compare and order the relative size of common

U.S. customary units and metric; e.g., mile and

kilometer, gallon and liter, pound and kilogram.

2. Use proportional relationships and formulas to convert units from one measurement system to another; e.g., degrees Fahrenheit to degrees Celsius.

Use Measurement 3. Use appropriate levels of precision when

Techniques and Tools calculating with measurements.

4. Solve and determine the reasonableness of the results for problems involving rates and derived measurements such as velocity and density, using formulas, models and graphs.

5. Demonstrate understanding of the concepts of perimeter, circumference and area.

a) Master finding the perimeter and area of rectangles, parallelograms, triangles, trapezoids, and figures formed by combinations of these.

b) Master finding the circumference and area of circles

c) Solve word problems involving perimeter circumference and area.

6. Understand the meaning of surface area and volume and calculate using formulas.

a) Define surface area and volume of prisms, cylinders, pyramid, cones and spheres

b) Calculate surface area of prisms, pyramids, and cylinders

c) Calculate the volume of prisms, pyramids, cylinders cones, and spheres

7. Apply proportional reasoning to solve problems involving indirect measurements or rates.

8. Find the sum of the interior and exterior angles of regular convex polygons with and without measuring the angles with a protractor.

Technology 9. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

10. Use spreadsheet applications.

11. Integrate two or more applications.

12. Use electronic resources to practice skills and re-mediate deficits.

13. Demonstrate an understanding of terminology related to technology.

14. Access, print, save and retrieve resources using the network.

15. Use basic operating system features (e.g.: help menus and control panels).

16. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Geometry and Spatial Sense Standard

Characteristics and 1. Make and test conjectures about characteristics

Properties and properties of two-dimensional and three-

dimensional objects and angles.

a) Identify distinct characteristics of two-dimensional objects such as quadrilaterals, parallelograms, rectangles, squares, kites, and trapezoids.

b) Find the measure of an angle using estimation and a protractor.

c) Classify angles as acute, right, obtuse, or straight.

d) Know the names and parts of 3-dimensional objects – prisms, cylinders, prisms, cones, and spheres.

2. Recognize the angles formed and the relationships between the angles when two lines intersect and when parallel lines are cut by a transversal.

a) Given one angle in a diagram when parallel lines are cut by a transversal, find the measure of the other angles formed.

b) Recognize and know the relationships of vertical and corresponding angles.

3. Use proportions in several forms to solve problems involving similar figures (part-to-part, part-to-whole, corresponding sides between figures).

Spatial Relationships 4. Represent and analyze shapes using coordinate

geometry.

a) Given three vertices and the type of quadrilateral, find the coordinates of the fourth vertex.

b) Find the perimeter and area of a two-dimensional shape in a coordinate plane.

Transformation and 5. Draw the results of translations, reflections,

Symmetry rotations and dilations of objects in the

coordinate plane, and determine properties

that remain fixed; e.g., lengths of sides remain

the same under translations.

Visualization and 6. Draw nets for a variety of prisms and

Geometric Models pyramids, cylinder and cones.

Technology 7. Use electronic resources to practice skills and re-mediate deficits.

8. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Relate the various representations of a

and Functions relationship; i.e., relate a table to graph,

description and symbolic form.

2. Generalize patterns and sequences by describing, in words, how to find the nth term.

3. Identify functions as linear or nonlinear based on information given in a table, graph or equation.

Use Algebraic 4. Extend the uses of variables to include co-

Representations variants where y depends on x

5. Add and subtract monomials and polynomials, and multiply a polynomial by a monomial.

6. Describe the relationship between the graph of a line and its equation, including being able to explain the meaning of slope as a constant rate of change and y-intercept in real world problems.

a) Graph an equation of a line in slope-intercept form.

b) Find the slope of a line given a line or a pair of points.

c) Find the x and y intercepts of a line given a graph or equations.

7. Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems.

8. Write, simplify, and evaluate algebraic expressions (including formulas) to generalize situations and solve problems.

9. Solve linear equations and inequalities graphically, symbolically and using technology.

10. Solve 2 by 2 systems of linear equations graphically and by simple substitution.

11. Interpret the meaning of the solution of a 2 by 2 system of equations; i.e., point, line, no solution.

12. Solve simple quadratic equations graphically; e.g., y = x²-16.

a) Graph quadratic equations using a table.

13. Compute and interpret midpoint and distance given a set of ordered pairs.

Analyze Change 14. Differentiate and explain types of changes in

in mathematical relationships, such as linear vs. nonlinear, continuous vs. non-continuous, direct variation vs. inverse variation.

15. Describe and compare how changes in an equation affects the related graphs; e.g., for a linear equation changing the co-efficient of x affects the slope and changing the constant affects the intercepts.

16. Use graphing calculators and computers to analyze change; e.g., interest compounded over time as a nonlinear growth pattern.

Technology 17. Use a variety of input and output devices such

as keyboards, scanners, cameras, microphones,.

printers, projectors, CD-Roms.

18. Use word processing applications.

19. Use spreadsheet applications.

20. Integrate two or more applications.

21. Use electronic resources to practice skills and re-mediate deficits.

Data Analysis and Probability Standard

Data Collection 1. Use, create, and interpret scatter plots and other

types of graphs as appropriate.

a) Evaluate different graphical representations of the same data to determine which is the most appropriate representation for an identified purpose; e.g., line graph for change over time, circle graph for part-to-whole comparison, scatter plot for relationship between two variants.

b) In a scatter plot, determine what type of relationship exists and approximate line of best fit.

c) Differentiate between discrete and continuous data and appropriate ways to represent each.

Statistical Methods 2. Compare two sets of data using measures of

center (mean, mode, median) with measures of

spread (range, quartiles, interquartile range, percentiles).

3. Explain the mean’s sensitivity to extremes and explain its use in comparison with the median and mode.

4. Make conjectures about possible relationship in a scatter plot and approximate line of best fit.

5. Identify different ways of selecting samples, such as survey response, random sample, representative sample and convenience sample.

6. Describe how the relative size of a sample compared to the target population affects the validity of predictions.

7. Construct convincing arguments based on analysis of data and interpretation of graphs.

Probability 8. Use the Counting Principle as a basis of

determining the number of possible outcomes

in a given situation.

a) Calculate the number of possible outcomes for a situation, recognizing and accounting for when items may occur more than once or when order is important.

9. Demonstrate an understanding that the probability of either of two disjoint events occurring can be found by adding the probabilities for each and that the probability of one independent event following another can be found by multiplying the probabilities.

Technology 10. Use a variety of input and output devices such

As keyboards, scanners, cameras, microphones,.

Printers, projectors, CD-Roms.

11. Use spreadsheet applications.

12. Use electronic resources to practice skills and re-mediate deficits.

13. Present multimedia and/or online projects to audience inside and outside the classroom.

14. Demonstrate an understanding of terminology related to technology.

15. Access, print, save and retrieve resources using the network.

16. Use basic operating system features (e.g.: help menus and control panels).

Algebra 1

Number, Number Sense and Operations Standard

Number and 1. Identify and justify whether properties (closure,

Number Systems identity, inverse, commutative and associative)

hold for a given set and operations; e.g., even integers and multiplication.

2. Compare, order and determine equivalent forms for rational and irrational numbers.

Meaning of Operations 3. Explain the effects of operations such as

multiplication or division, and of computing

powers and roots on the magnitude of quantities.

Computation 4. Demonstrate fluency in computations using real

and Estimation numbers.

5. Estimate the solutions for problem situations involving square and cube roots.

Technology 6. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

7. Use word processing applications.

8. Use spreadsheet applications

9. Use database application

10. Use draw and paint applications

11. Integrate two or more applications

12. Use electronic resources to practice skills and re-mediate deficits

13. Make appropriate technology resource choices according to learning purposes and outcomes.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Convert rates within the same measurement

system; e.g., miles per hour to feet per second; kilometers per hour to meters per second.

Use Measurement 2. Use unit analysis to check computations

Techniques and Tools involving measurement.

3. Use the ratio of lengths in similar two-dimensional figures or three-dimensional objects to calculate the ratio of their areas or volumes respectively.

4. Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures.

5. Solve problems involving unit conversion for situations involving distances, areas, volumes and rates within the same measurement system.

Technology 6. Use the Internet and other electronic resources to research and digital media retrieval.

7. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

8. Evaluate and critique the quality and credibility of electronic information.

9. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

10. Use word processing applications

11. Use spreadsheet applications.

12. Use database applications

13. Use draw and paint applications

14. Integrate two or more applications.

15. Use electronic resources to practice skills and re-mediate deficits.

16. Make appropriate technology resource choices according to learning purposes and outcomes.

Geometry and Spatial Sense Standard

Characteristics and 1. Define the basic trigonometric ratios in right

Properties triangles: sine, cosine and tangent.

2. Apply proportions and right triangle trigonometric ratios to solve problems involving missing lengths and angle sizes in similar figures.

Visualization and 3. Analyze two-dimensional figures in a

Geometric Models coordinate plane; e.g., use slope and distance

formulas to show that a quadrilateral is a

parallelogram.

Technology 4. Use the Internet and other resources for research and digital media retrieval.

5. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

6. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

7. Use word processing applications.

8. Use spreadsheet applications.

9. Use database applications.

10. Use draw and paint applications.

11. Integrate two or more applications.

12. Use electronic resources to practice skills and re-mediate deficits.

13. Create multimedia and/or online projects.

14. Present multimedia and/or online projects to audience inside and outside the classroom.

15. Print, post, publish and/or distribute technology products.

16. Make appropriate technology resource choices according to learning purposes and outcomes.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Define function with ordered pairs in which

and Functions each domain element is assigned exactly one

range element.

2. Generalize patterns using functions or relationships (linear, quadratic and exponential), freely translating among tabular, graphical and symbolic representations.

3. Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations.

4. Demonstrate the relationship among zeros of a function, roots of equations, and solutions of equations graphically and in words.

5. Describe and compare characteristics of the following families of functions: linear, quadratic and exponential; e.g., general shape, number of roots, domain, range, rate of change and maximum or minimum.

Use Algebraic 6. Write and use equivalent forms of equations and

Representations inequalities in problem situations; e.g., changing

a linear equations to the slope-intercept form.

7. Use formulas to solve problems involving exponential growth and decay.

8. Find linear equations that represent lines that pass through a given set of ordered pairs, and find linear equations that represent lines parallel or perpendicular to a given line through a specific point.

9. Solve and interpret the meaning of 2 by 2 systems of linear equations graphically, by substitution and by elimination, with and without technology.

10. Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula

11. Add, subtract, multiply and divide monomials and polynomials (division of polynomials by monomials only).

12. Simplify rational expressions by eliminating common factors and applying properties of integer exponents.

Analyze Change 13. Model and solve problems involving direct and

inverse variation using proportional reasoning.

14. Describe the relationship between slope and the graph of a direct variation and inverse variation.

15. Describe how a change or a parameter in a linear or quadratic equation affects the related graphs.

Technology 16. Use the Internet and other electronic resources for research and digital media retrieval.

17. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

18. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

19. Use word processing applications.

20. Use spreadsheet applications.

21. Use database applications.

22. Use draw and paint applications.

23. Integrate two or more applications.

24. Use electronic resources to practice skills and re-mediate deficits.

25. Make appropriate technology resource choices according to learning purposes and outcomes.

Data Analysis and Probability Standard

Data Collection 1. Classify data as univariate (single variable) or

bivariate (two variables), and as quantitative

(measurement0 or qualitative (categorical) data.

2. Create a scatterplot for a set of bivariate data, sketch the “line of best fit,” and interpret the slope of the line of best fit.

Data Collection 3. Analyze and interpret frequency distributions

based on spread, symmetry, skewness, clusters

and outliers.

4. Describe and compare various types of studies (survey, observation, experiment), and identify possible misuses of statistical data.

5. Describe characteristics and limitations of sampling methods, and analyze the effects of random versus biased sampling; e.g., determine and justify whether the sample is likely to be representative of the population.

6. Make inferences about relationships in bivariate data, and recognize the difference between evidence of relationship (correlation) and causation.

Probability 7. Use counting techniques and the Fundamental

Counting Principle to determine the total

number of possible outcomes for mathematical

situations.

8. Describe, create and analyze a sample space and use it to calculate probability.

9. Identify situations involving independent and dependent events, and explain differences between and common misconceptions about probabilities associated with those events.

10. Use theoretical and experimental probability, including simulations or random numbers, to estimate probabilities and to solve problems dealing with uncertainty; e.g., compound events, independent events, simple dependent events.

Technology 11. Evaluate and critique the quality and credibility of electronic information.

12. Demonstrate an understanding of terminology related to technology.

13. Access, print, save and retrieve resources using the network.

14. Use basic operating system features (e.g.: example, using help menus and control panels).

15. Employ basic technology troubleshooting and maintenance techniques.

16. Understand and apply the basic working of the copyright law and appropriate usage of materials, including city resources.

17. Demonstrate appropriate behavior for technology use and shoe respect for technology equipment.

18. Apply and advocate the Westlake School District Acceptable Use Policy (AUP).

19. Understand the relationship that technology has to career opportunities, history and to today’s society and world.

Geometry

Number, Number Sense and Operations Standard

Number and 1. Connect physical, verbal and symbolic

Number Systems representations of irrational numbers; e.g.,

construct as a hypotenuse or on a number line.

Meaning of Operations 2. Explain the meaning of the nth root.

Computation 3. Use factorial notation and computations to

and Estimation represent and solve problem situations involving

arrangements.

4. Approximate the nth root of a given number greater than zero between consecutive integers when n is an integer; e.g., the 4^th root of 50 is between 2 and 3.

Technology 5. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

6. Use word processing applications.

7. Use spreadsheet applications.

8. Use database applications.

9. Use draw and paint applications.

10. Integrate two or more applications.

11. Use electronic resources to practice skills and re-mediate deficits.

12. Make appropriate technology resource choices according to learning purposes and outcomes.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Use Measurement 1. Explain how a small error in measurement may

Techniques and Tools lead to a large error in calculated results.

2. Calculate relative error.

3. Explain the difference between absolute error and relative error in measurement.

4. Give examples of how the same absolute error can be problematic in one situation but not in another; e.g., compare “accurate to the nearest foot” when measuring the height of a person versus when measuring the height of a mountain.

5. Determine the measures of central or inscribed angles and their associated major and minor arcs.

Technology 6. Use the Internet and other electronic resources for research and digital media retrieval.

7. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

8. Evaluate and critique the quality and credibility of electronic information.

9. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

10. Use word processing applications.

11. Use spreadsheet applications.

12. Use database applications.

13. Use draw and paint applications.

14. Integrate two or more applications.

15. Use electronic resources to practice skills and re-mediate deficits.

16. Make appropriate technology resource choices according to learning purposes and outcomes.

Geometry and Spatial Sense Standard

Characteristics and 1. Formally define and explain key aspects of

Properties geometric figures, including:

a) interior and exterior angles of polygons;

b) segments related to triangles (median, altitude, midsegment);

c) points of concurrency related to triangles (centroid, incenter, orthocenter, and

circumcenter);

d) circles (radius, diameter, chord, circumference, major arc, minor

arc, sector, segment, inscribed angle).

2. Recognize and explain the necessity for certain terms to remain undefined, such as point, line and plane.

3. Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including:

a) prove the Pythagorean Theorem;

b) prove theorems involving triangle similarity and congruence;

c) prove theorems involving properties of lines, angles, triangles and

quadrilaterals;

d) test a conjecture using basic constructions made with a compass

and straightedge or technology.

Spatial Relationships 4. Construct right triangles, equilateral triangles,

parallelograms, trapezoids, rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic geometry software.

5. Construct congruent or similar figures using tools, such as compass, straightedge, and protractor or dynamic geometry software.

Transformation and 6. Identify the reflection and rotation symmetries

Symmetry of two- and three-dimensional figures.

7. Perform reflections and rotations using compass and straightedge constructions and dynamic geometry software.

8. Derive coordinate rules for translations, reflections and rotations of geometric figures in the coordinate plane.

9. Show and describe the results of combinations of translations, reflections and rotations (compositions); e.g., perform compositions and specify the result of a composition as the outcome of a single motion, when applicable.

Visualization and 10. Solve problems involving chords, radii, and arcs

Geometric Models within the same circle.

Technology 11. Use the Internet and other electronic resources for research and digital media retrieval.

12. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

13. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

14. Use word processing applications.

15. Use spreadsheet applications.

16. Use database applications.

17. Use draw and paint applications.

18. Integrate two or more applications.

19. Use electronic resources to practice skills and re-mediate deficits.

20. Create multimedia and/or online projects.

21. Present multimedia and/or online projects to audience inside and outside the classroom.

22. Print, post, publish and/or distribute technology products.

23. Make appropriate technology resource choices according to learning purposes and outcomes.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Define function formally and with f(x) notation.

and Functions

2. Describe and compare characteristics of the following families of functions: square root, absolute value, cubic, basic trigonometric functions; e.g., general shape, possible number of roots, domain and range.

Use Algebraic 3. Solve equations and formulas for a specified

Representations variable; e.g., express the base of a triangle in

terms of the area and height.

4. Use algebraic representations and functions to describe and generalize geometric properties and relationships.

5. Solve simple linear and nonlinear equations and inequalities having square roots as coefficients and/or roots.

6. Solve equations and inequalities having rational expressions as coefficients and roots.

7. Solve systems of linear inequalities.

8. Graph the quadratic relationship that defines circles.

9. Recognize and explain that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals.

10. Solve everyday problems that can be modeled using linear, quadratic, exponential, or square root functions.

11. Solve everyday problems that can be modeled, using systems of linear equations and inequalities.

12. Describe the relationship between slope of a line through the origin and the tangent function of the angle created by the line and the positive x-axis.

Technology 13. Use the Internet and other electronic resources for research and digital media retrieval.

14. Use electronics to communicate and collaborate with others (e.g.: communicate with outside groups, classes and experts via e-mail and the Internet).

15. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

16. Use word processing applications.

17. Use spreadsheet applications.

18. Use database applications.

19. Use draw and paint applications.

20. Integrate two or more applications.

21. Use electronic resources to practice skills and re-mediate deficits.

22. Make appropriate technology resource choices according to learning purposes and outcomes.

Data Analysis and Probability Standard

Data Collection 1. Describe measure of center and the range

verbally, graphically and algebraically.

2. Represent and analyze bivariate data using appropriate graphical displays (scatterplots, parallel box-and-whisker plots, histograms with more than one set of data, tables, charts, spreadsheets) with and without technology.

3. Display bivariate data where at least one variable is categorical.

4. Identify outliers on a data display; e.g., use interquartile range to identify outliers on a box-and-whisker plot.

Statistical Methods 5. Provide examples and explain how a statistic

may or may not be an attribute of the entire

population; e.g., intentional and unintentional

bias may be present.

6. Interpret the relationship between two variables using multiple graphical displays and statistical measures; e.g., scatterplots, parallel box-and-whisker plots, and measures of center and spread.

Probability 7. Model problems dealing with uncertainty with

area models (geometric probability).

8. Differentiate and explain the relationship between the probability of an event and the odds of an event, and compute one given the other.

Technology 9. Evaluate and critique the quality and credibility of electronic information.

10. Demonstrate an understanding of terminology related to technology.

11. Access, print, save and retrieve resources using the network.

12. Use basic operating system features (e.g.: using help menus and control panels.).

13. Employ basic technology troubleshooting and maintenance techniques.

14. Understand and apply the basic workings of the copyright law and appropriate usage of materials, including city resources.

15. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

16. Apply and advocate the Westlake School District Acceptable Use Policy (AUP).

17. Understand the relationship that technology has to career opportunities, history and to today’s society and world.

Geometry C

Number, Number Sense and Operations Standard

Number and 1. Connect physical, verbal and symbolic

Number Systems representations of irrational numbers; e.g., construct as a hypotenuse or on a number line.

Technology 2. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

3. Use word processing applications.

4. Use spreadsheet applications.

5. Use database applications.

6. Integrate two or more applications.

7. Use electronic resources to practice skills and re-mediate deficits.

8. Make appropriate technology resource choices according to learning purposes and outcomes.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Use Measurement 1. Determine the measures of central or inscribed

Techniques and Tools angles and their associated major and minor

arcs.

2. Use the ratio of lengths in similar two-dimensional figures or three-dimensional objects to calculate the ratio of their areas or volumes respectively.

3. Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures.

Technology 4. Use the Internet and other electronic resources for research and digital media retrieval.

5. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

6. Evaluate and critique the quality and credibility of electronic information.

7. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

8. Use word processing applications.

9. Use spreadsheet applications.

10. Use database applications.

11. Use draw and paint applications.

12. Integrate two or more applications.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Make appropriate technology resource choices according to learning purposes and outcomes.

Geometry and Spatial Sense Standard

Characteristics and 1. Formally define and explain key aspects of

Properties geometric figures, including:

a) interior and exterior angles of polygons;

b) segments related to triangles (median, altitude, midsegment);

c) circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle).

2. Recognize and explain the necessity for certain terms to remain undefined, such as point, line and plane.

a) prove the Pythagorean Theorem;

b) prove theorems involving triangle similarity and congruence;

Spatial Relationships 4. Construct right triangles, equilateral triangles,

parallelograms, trapezoids, rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic geometry software.

5. Construct congruent or similar figures using tools, such as compass, straightedge, and protractor or dynamic geometry software.

Transformation and 6. Identify the reflection and rotation symmetries

Symmetry of two- and three-dimensional figures.

7. Use coordinate rules for translations, reflections and rotations of geometric figures in the coordinate plane.

Visualization and 8. Solve problems involving chords, radii, and arcs

Geometric Models within the same circle.

9. Define the basic trigonometric ratios in right triangles: sine, cosine and tangent.

10. Use right triangle trigonometric ratios to solve problems involving missing lengths and angle sizes in similar figures.

Technology 11. Use the Internet and other electronic resources for research and digital media retrieval.

12. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

13. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

14. Use word processing applications.

15. Use spreadsheet applications.

16. Use database applications.

17. Use draw and paint applications.

18. Integrate two or more applications.

19. Use electronic resources to practice skills and re-mediate deficits.

20. Create multimedia and/or online projects.

21. Present multimedia and/or online projects to audience inside and outside the classroom.

22. Print, post, publish and/or distribute technology products.

23. Make appropriate technology resource choices according to learning purposes and outcomes.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Solve equations and formulas for a specified

and Functions variable; e.g., express the base of a triangle in terms of the area and height.

2. Use algebraic representations and functions to describe and generalize geometric properties and relationships.

3. Graph the quadratic relationship that defines circles.

4. Recognize and explain that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals.

5. Solve everyday problems that can be modeled using linear functions.

6. Solve everyday problems that can be modeled, using systems of linear equations and inequalities.

Analyze Change 7. Describe the relationship between slope of a line

through the origin and the tangent function of the angle created by the line and the positive x-axis.

Technology 8. Use the Internet and other electronic resources for research and digital media retrieval.

9. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

10. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

11. Use word processing applications.

12. Use spreadsheet applications.

13. Use database applications.

14. Use draw and paint applications.

15. Integrate two or more applications.

16. Use electronic resources to practice skills and re-mediate deficits.

17. Make appropriate technology resource choices according to learning purposes and outcomes.

Data Analysis and Probability Standard

Analyze Change 1. Describe measures of center and the range verbally, graphically and algebraically.

2. Represent and analyze bivariate data using appropriate graphical displays (scatter plots, parallel box-and-whisker plots, histograms with more than one set of data, tables, charts, spreadsheets) with and without technology.

3. Display bivariate data where at least one variable is categorical.

4. Identify outliers on a data display; e.g., use the interquartile range to identify outliers on a box-and-whisker plot.

Statistical Methods 5. Provide examples and explain how a statistic

may or may not be an attribute of the entire population; e.g., intentional or unintentional bias may be present.

6. Interpret the relationship between two variables using multiple graphical displays and statistical measures; e.g., scatterplots, parallel box-and-whisker plots, and measures of center and spread.

Probability 7. Differentiate and explain the relationships between the probability of an event and the odds of an event, and compute one given the other.

Technology 8. Evaluate and critique the quality and credibility of electronic information.

9. Demonstrate an understanding of terminology related to technology.

10. Access, print, save and retrieve resources using the network.

11. Use basic operating system features (e.g.: using help menus and control panels).

12. Employ basic technology troubleshooting and maintenance techniques.

13. Understand and apply the basic workings of the copyright law and appropriate usage of materials, including city resources.

14. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

15. Apply and advocate the Westlake School District Acceptable Use Policy (AUP).

16. Understand the relationship that technology has to career opportunities, history and to today’s society and world.

Advanced Algebra

Number, Number Sense and Operations Standard

Number and 1. Determine what properties hold for matrix

Number Systems addition and matrix multiplication; e.g., use

examples to show addition is commutative and

when multiplication is not commutative.

2. Determine what properties hold for vector addition and multiplication, and scalar multiplication.

3. Represent complex numbers on the complex plane.

Meaning of Operations 4. Use matrices to represent given information in a

problem situation.

5. Model (using the coordinate plane) vector addition and scalar multiplication.

Computation and 6. Compute sums, differences and products of

Estimation matrices using paper-and-pencil calculations for

simple cases, and technology for more

complicated cases.

7. Compute sums, differences, products and quotients of complex numbers.

8. Use fractional and negative exponents as optional ways of representing and finding solutions for problem situations; e.g., 27^2/3 = (27^1/3)²= 9.

9. Use vector addition and scalar multiplication to solve problems.

Technology 10. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

11. Use word processing applications.

12. Use spreadsheet applications.

13. Use database applications.

14. Use draw and paint applications.

15. Integrate two or more applications.

16. Use electronic resources to practice skills and re-mediate deficits.

17. Make appropriate technology resource choices according to learning purposes and outcomes.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measurement Units 1. Determine the number of significant digits in a

Measurement.

2. Use radian and degree angle measures to solve problems and perform conversions as needed.

Use Measurement 3. Derive a formula for the surface area of a cone

Techniques and Tools as a function of its slant height and the

circumference of its base.

4. Calculate distances, areas, surface areas and volumes of composite three-dimensional objects to a specified number of significant digits.

5. Solve real-world problems involving area, surface area, volume and density to a specified degree of precision.

Technology 6. Use the Internet and other electronic resources for research and digital media retrieval.

7. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

8. Evaluate and critique the quality and credibility of electronic information.

9. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

10. Use word processing applications.

11. Use spreadsheet applications.

12. Use database applications.

13. Use draw and paint applications.

14. Integrate two or more applications.

15. Use electronic resources to practice skills and re-mediate deficits.

16. Make appropriate technology resource choices according to learning purposes and outcomes.

Geometry and Spatial Sense Standard

Spatial Relationships 1. Use polar coordinates to specify locations on the

plane.

Transformations and 2. Represent translations using vectors.

Symmetry

3. Describe multiplication of a vector and a scalar graphically and algebraically and apply to problem situations.

4. Use trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines.

Visualization and 5. Identify, sketch and classify the cross sections

Geometric Models of three-dimensional objects.

Technology 6. Use the Internet and other electronic resources for research and digital media retrieval.

7. Use electronics to communicate and collaborate with others (e.g.: communicate with outside groups, classes and experts via e-mail and the Internet).

8. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

9. Use word processing applications.

10. Use spreadsheet applications.

11. Use database applications.

12. Use draw and paint applications.

13. Integrate two or more applications.

14. Use electronic resources to practice skills and re-mediate deficits.

15. Create multimedia and/or online projects.

16. Present multimedia and/or online projects to audience inside and outside the classroom.

17. Print, post, publish and/or distribute technology products.

18. Make appropriate technology resource choices according to learning purposes and outcomes.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Identify and describe problem situations

and Functions involving an iterative process that can be

represented as a recursive function; e.g.,

compound interest.

2. Translate a recursive function into a closed form expression or formula for the nth term to solve a problem situation involving an iterative process; e.g., find the value of an annuity after 7 years.

3. Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e.g., general shape, number of roots, domain and range, asymptotic behavior.

4. Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology.

5. Identify families of functions with graphs that have rotation symmetry or reflection symmetry about the y-axis, x-axis, or y = x.

Use Algebraic Representations 6. Represent the inverse function symbolically

and graphically as a reflection about y = x.

7. Model and solve problems with matrices and vectors.

8. Solve equations involving radical expressions and complex roots.

9. Solve 3 by 3 systems of linear equations by elimination and using technology, and interpret graphically what the solution means (a point, line, plane, or no solution).

10. Describe the characteristics of the graphs of conic sections.

Analyze Change 11. Describe how a change of a parameter in an

exponential, logarithmic or radical equation affects the graph of the equation.

Technology 12. Use the Internet and other electronic resources for research and digital media retrieval.

13. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

14. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

15. Use word processing applications.

16. Use spreadsheet applications.

17. Use database applications.

18. Use draw and paint applications.

19. Integrate two or more applications.

20. Use electronic resources to practice skills and re-mediate deficits.

21. Make appropriate technology resource choices according to learning purposes and outcomes.

Data Analysis and Probability Standard

Data Collection 1. Design a statistical experiment, survey or study

for a problem; collect data for the problem; and interpret the data with appropriate graphical displays, descriptive statistics, concepts of variability, causation, correlation and standard deviation.

2. Describe the role of randomization in a well-designed study, especially as compared to a convenience sample, and the generalization of results from each.

Statistical Methods 3. Describe how a linear transformation of

univariate data affects range, mean, mode, and

median.

4. Create a scatterplot of bivariate data, identify trends, and find a function to model the data.

5. Use technology to find the Least Squares Regression Line, the regression coefficient, and the correlation coefficient for bivariate data with a linear trend, and interpret each of these statistics in the context of the problem situation.

6. Use technology to compute the standard deviation for a set of data, and interpret standard deviation in relation to the context or problem situation.

7. Describe the standard normal curve and its general properties and answer questions dealing with data assumed to be normal.

8. Analyze and interpret univariate and bivariate data to identify patterns; note trends, draw conclusions and make predictions.

9. Evaluate validity of results of a study based on characteristics of the study design, including sampling method, summary statistics and data analysis techniques.

Probability 10. Understand and use the concept of random

variable; compute and interpret the expected value for a random variable in simple cases.

11. Examine statements and decisions involving risk; e.g., insurance rates and medical decisions.

Technology 12. Evaluate and critique the quality and credibility of electronic information.

13. Demonstrate an understanding of terminology related to technology.

14. Access, print, save and retrieve resources using the network.

15. Use basic operating system features (e.g.: using help menus and control panels).

16. Employ basic technology troubleshooting and maintenance techniques.

17. Understand and apply the basic workings of the copyright law and appropriate usage of materials, including city resources.

18. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

19. Apply and advocate the Westlake School District Acceptable Use Policy (AUP)

20. Understand the relationship that technology has to career opportunities, history and to today’s society and world.

PreCalculus

Number, Number Sense and Operations Standard

Number and 1. Determine what properties (closure, identity,

Number Systems inverse, commutative and associative) hold for

operations with complex numbers.

Computation and 2. Apply combinations as a method to create

Estimation coefficients for the Binomial Theorem, and

make connections to everyday and workplace problem situations.

3. Apply informal concepts of successive approximation, upper and lower bounds, and limits in measurement situations; e.g., measurement of some quantities, such as volume of a cone, can be determined by sequences of increasingly accurate approximations.

Technology 4. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

5. Use word processing applications.

6. Use spreadsheet applications.

7. Use database applications.

8. Use draw and paint applications.

9. Integrate two or more applications.

10. Use electronic resources to practice skills and re-mediate deficits.

11. Make appropriate technology resource choices according to learning purposes and outcomes.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Use Measurement 1. Solve problems involving derived

Techniques and Tools measurements; e.g., acceleration and pressure.

2. Use radian measure in the solution of problems involving angular velocity and acceleration.

Technology 4. Use the Internet and other electronic resources for research and digital media retrieval.

5. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

6. Evaluate and critique the quality and credibility of electronic information.

7. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

8. Use word processing applications.

9. Use spreadsheet applications.

10. Use database applications.

11. Use draw and paint applications.

12. Integrate two or more applications.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Make appropriate technology resource choices according to learning purposes and outcomes.

Geometry and Spatial Sense Standard

Transformations and 1. Use matrices to represent translations,

Symmetry reflections, rotations, dilations and their

compositions.

2. Derive and apply the basic trigonometric identities; i.e., angle addition, angle subtraction, and double angle.

Visualization and 3. Relate graphical and algebraic representations

Geometric Models of lines, simple curves and conic sections.

4. Recognize and compare specific shapes and properties in multiple geometries; e.g., plane, spherical and hyperbolic.

Technology 5. Use the Internet and other electronic resources

for research and digital media retrieval.

6. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

7. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

8. Use word processing applications.

9. Use spreadsheet applications.

10. Use database applications.

11. Use draw and paint applications.

12. Integrate two or more applications.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Create multimedia and/or online projects.

15. Present multimedia and/or online projects to audience inside and outside the classroom.

16. Print, post, publish and/or distribute technology products.

17. Make appropriate technology resource choices according to learning purposes and outcomes.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Analyze the behavior of arithmetic and

and Functions geometric sequences and series as the number of

terms increases.

2. Translate between the numeric and symbolic form of a sequence or series.

3. Describe and compare the characteristics of transcendental and periodic functions; e.g., general shape, number of roots, domain and range, asymptotic behavior, extrema, local and global behavior.

4. Represent the inverse of a function symbolically (transcendental functions).

Use Algebraic 5. Set up and solve systems of equations, using

Representations matrices and graphing with and without

Technology.

6. Make arguments about mathematical properties using mathematical induction.

7. Make mathematical arguments using the concepts of limit.

8. Compare estimates of the area under a curve over a bounded interval by partitioning the region with rectangles; e.g., make successive estimates using progressively smaller rectangles.

9. Translate freely between polar and Cartesian coordinate systems.

Analyze Change 10. Use the concept of limit to find instantaneous

rate of change for a point on a graph as the slope

of a tangent at a point.

Technology 11. Use the Internet and other electronic resources for research and digital media retrieval.

12. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

13. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

14. Use word processing applications.

15. Use spreadsheet applications.

16. Use database applications.

17. Use draw and paint applications.

18. Integrate two or more applications.

19. Use electronic resources to practice skills and re-mediate deficits.

20. Make appropriate technology resource choices according to learning purposes and outcomes.

Data Analysis and Probability Standard

Data Collection 1. Identify and use various sampling methods

(voluntary response, convenience sample, random sample, stratified random sample, and census) in a study.

Statistical Methods 2. Transform bivariate data so it can be modeled

by a function; e.g., use logarithms to allow nonlinear relationship to be modeled by linear function.

3. Describe the shape and find all summary statistics for a set of univariate data and describe how a linear transformation affects shape, center and spread.

4. Apply the concept of a random variable to generate and interpret probability distributions, including binomial, normal and uniform.

5. Use sampling distributions as the basis for informal inference.

6. Use theoretical or experimental probability, including simulations, to determine probabilities in real-world problem situations involving uncertainty, such as mutually exclusive events, complementary events and conditional probability.

Technology 7. Evaluate and critique the quality and credibility of electronic information.

8. Demonstrate an understanding of terminology related to technology.

9. Access, print, save and retrieve resources using the network.

10. Use basic operating system features (e.g.: using help menus and control panels).

11. Employ basic technology troubleshooting and maintenance techniques.

12. Understand and apply the basic workings of the copyright law and appropriate usage of materials, including city resources.

13. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

14. Apply and advocate the Westlake School District Acceptable Use Policy (AUP).

15. Understand the relationship that technology has to career opportunities, history, and to today’s society and world.

College Prep Math

Number, Number Sense and Operations Standard

Number and 1. Compare, order, and determine equivalent forms

Number Systems for rational and irrational numbers.

2. Determine properties of matrix addition and multiplication.

3. Represent complex numbers in the complex plane.

Meaning of Operations 4. Explain the effects of operations such as multiplication, division or exponents. Estimate the solutions for problem situations involving square and cube roots.

5. Explain the meaning of nth root.

6. Use matrices to represent given information in a problem situation.

Computation and 7. Estimate the solutions for problem situations

Estimation involving square and cube roots.

8. Approximate the nth root of a given number.

9. Compute sums, differences and products of matrices.

10. Compute sums, differences, products and quotients of complex numbers.

11. Use fractional and negative exponents.

Technology 12. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

13. Use word processing applications.

14. Use spreadsheet applications.

15. Use database applications.

16. Use draw and paint applications.

17. Integrate two or more applications.

18. Use electronic resources to practice skills and re-mediate deficits.

19. Make appropriate technology resource choices according to learning purposes and outcomes.

Measurement Standard

Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.

Measuring Units 1. Convert rates within the same measurement

System.

2. Use radian and degree angle measures to solve problems and perform conversions as needed.

Use Measurement 3. Use scale drawings and right triangle

Techniques and Tools trigonometry to solve problems that include

Unknown distances and angle measures.

Technology 4. Use the Internet and other electronic resources for research and digital media retrieval.

5. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

6. Evaluate and critique the quality and credibility of electronic information.

7. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

8. Use word processing applications.

9. Use spreadsheet applications.

10. Use database applications.

11. Use draw and paint applications.

12. Integrate two or more applications.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Make appropriate technology resource choices according to learning purposes and outcomes.

Geometry and Spatial Sense Standard

Characteristics and 1. Define the basic trigonometric ratios in right

Properties triangles.

2. Apply proportions and right triangle ratios to solve problems.

3. Formally define and explain key aspects of geometric figures.

Visualization and 4. Analyze two dimensional figures in a coordinate

Geometric Models plane; e.g. use slope to show a quadrilaterial is a

parallelogram.

Technology 5. Use the Internet and other electronic resources for research and digital media retrieval.

6. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

7. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

8. Use word processing applications.

9. Use spreadsheet applications.

10. Use database applications.

11. Use draw and paint applications.

12. Integrate two or more applications.

13. Use electronic resources to practice skills and re-mediate deficits.

14. Create multimedia and/or online projects.

15. Present multimedia and/or online projects to audience inside and outside the classroom.

16. Print, post, publish and/or distribute technology products.

17. Make appropriate technology resource choices according to learning purposes and outcomes.

Patterns, Functions, and Algebra Standard

Use Patterns, Relations 1. Relate the various representations of a

and Functions relationship.

2. Generalize patterns and sequences by describing the nth term.

3. Identify functions as linear or non-linear based on information given in a table, graph or equation.

4. Define functions with ordered pairs in which each domain element is assigned to exactly one range element.

5. Generalize patterns using functions or relationships (linear, quadratic and exponential) and freely translate among tabular, graphical and symbolic representations.)

6. Demonstrate the relationships among the zeros of a function, roots of an equation, and solutions of an equation graphically and in words.

7. Describe and compare the characteristics of the following families of functions; linear, quadratic and exponential.

8. Define function formally and with f(x) notation.

9. Describe and compare characteristics of the following families of functions: square root, cubic, absolute value and basic trigonometric functions.

10. Describe and compare the characteristics of the following family of functions: quadratics with complex roots, polynomials of any degree and rational functions.

11. Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology.

12. Identify families of functions with graphs that have rotation symmetry or reflection symmetry and the y-axis or the line y=x.

Use Algebraic Representations 13. Describe the relationship between the graph of a

line and its equation, including being able to explain the meaning of slope as a constant rate of change and y-intercept in real world problems.

14. Use symbolic algebra, graphs and tables to represent situations and solve problems.

15. Write, simplify and evaluate algebraic expressions to generalize situations and solve problems.

16. Solve linear equations and inequalities graphically, symbolically and using technology.

17. Solve and interpret the meaning of a 2 by 2 system of linear equations graphically, by substitution and by elimination, with and without technology.

18. Interpret the meaning of the solution of a 2 by 2 system of equations.

19. Compute and interpret slope, midpoint and distance given a set of ordered pairs.

20. Write and use equivalent forms of equations and inequalities in problem situations.

21. Use formulas to solve problems involving exponential growth and decay.

22. Find linear equations that represent lines that pass through a given set of ordered pairs, and find linear equations that represent lines parallel or perpendicular to a given line through a given point.

23. Add, subtract, multiply and divide monomials and polynomials.

24. Simplify rational expressions by eliminating common factors and applying properties of integer exponents.

25. Solve equations and formulas for a specified variable.

26. Solve simple linear and non-linear equations and inequalities having square roots as coefficients and solutions.

27. Solve equations and inequalities having rational expressions as coefficients and solutions.

28. Solve systems of linear inequalities.

29. Graph the quadratic relationship that describes circles.

30. Recognize and explain that the slopes of parallel lines are equal and the slopes of perpendicular lines are opposite reciprocals.

31. Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions.

32. Represent the inverse of a function symbolically and graphically as a reflection about y = x.

33. Solve equations involving radical expressions and complex roots.

34. Model and solve problems with matrices and vectors.

Analyze Change 35. Differentiate and explain types of changes in mathematical relationships.

36. Describe and compare how changes in an equation affect the related graphs.

37. Use graphing calculators or computers to analyze change.

38. Model and solve problems involving direct and inverse variation.

39. Describe the relationship between slope and the graph of a direct variation and inverse variation.

40. Describe the relationship between the slope of a line through the origin and the tangent function of the angle created by the line and the positive x-axis.

Technology 41. Use the Internet and other electronic resources for research and digital media retrieval.

42. Use electronics to communicate and collaborate with others. For example, communicate with outside groups, classes and experts via e-mail and the Internet.

43. Use a variety of input and output devices such as keyboards, scanners, cameras, microphones, printers, projectors, CD-ROMs.

44. Use word processing applications.

45. Use spreadsheet applications.

46. Use database applications.

47. Use draw and paint applications.

48. Integrate two or more applications.

49. Use electronic resources to practice skills and re-mediate deficits.

50. Make appropriate technology resource choices according to learning purposes and outcomes.

Data Analysis and Probability Standard

Data Collection 1. Differentiate between discrete and continuous data and appropriate ways to represent each.

2. Classify data as univariate or bivariate and as quantitative or qualitative.

3. Describe measures of center and the range verbally, graphically and algebraically.

4. Represent and analyze bivariate data using appropriate graphical displays with and without technology.

Statistical Methods 5. Compare two sets of data using measures of center (mean, median, mode) and measures of spread.

6. Explain the mean’s sensitivity to extremes and its use in comparison with the median and mode.

7. Make conjectures about possible relationships in a scatterplot and approximate line of best fit.

8. Construct convincing arguments based on analysis of data and interpretation of graphs.

9. Create a scatterplot for a set of bivariate data, sketch the line of best fit and interpret the slope of the line of best fit.

10. Create a scatterplot of bivariate data, identify trends and find a function to model the data.

11. Use technology to find the Least Squares Regression Line, the regression coefficient and the correlation coefficient for bivariate data with a linear trend and interpret each of these statistics in the context of the problem situation.

Technology 12. Evaluate and critique the quality and credibility of electronic information.

13. Demonstrate an understanding of terminology related to technology.

14. Access, print, save and retrieve resources using the network.

15. Use basic operating system features (e.g., using help menus and control panels).

16. Employ basic technology troubleshooting and maintenance techniques.

17. Understand and apply the basic workings of the copyright law and appropriate usage of materials, including city resources.

18. Demonstrate appropriate behavior for technology use and show respect for technology equipment.

19. Apply and advocate the Westlake School District Acceptable Use Policy (AUP).

20. Understand the relationship that technology has to career opportunities, history and to today’s society and world.

MATHEMATICS COURSE OF STUDY

Committee Members

DISTRICT BELIEFS, VISION AND MISSION

BELIEFS

MISSION STATEMENT

WE EDUCATE FOR EXCELLENCE…

Mathematics Program Philosophy

NCTM Principles & Standards, page 79

NCTM Principles & Standards, page 35

MEASUREMENT: K-2

Introduction

The Research Base Adding it Up, pages 282-283

GEOMETRY: K-2

ALGEBRA: K-2 Benchmarks, page 217

DATA ANALYSIS AND PROBABILITY: K-2 SMART Mathematics

Course of Study, page 52

The Research Base Benchmarks, page 353

MEASUREMENT: 6-8

GEOMETRY: 6-8

ALGEBRA: 6-8

DATA ANALYSIS AND PROBABILITY: 6-8

MEASUREMENT: 9-12

GEOMETRY: 9-12

ALGEBRA: 9-12

DATA ANALYSIS AND PROBABILITY: 9-12

Use Measurement 5. Make conversions within the same measurement

Computation and 6. Simplify numerical expressions involving

Characteristics and 1. Use proportional reasoning to describe and

figures to solve problems using proportional

Probability 7. Compute probabilities of compound events; e.g.,

Characteristics and 1. Use proportional reasoning to describe and

Visualization and 9. Draw representations of three-dimensional

BENCHMARKS:

Spatial Relationships 4. Represent and analyze shapes using coordinate

Honors Differentiation