What We TeachMathematical Structures, Ideas, and Practices that Promote

a Deep Understanding of Elementary Mathematics.

Dale OliverHumboldt State University

June, 2004

There are several different resources that mathematics faculty can use to decide which content areas to include in their courses for prospective teachers (see, for example, the MET document, the NCTM Principles and Standards for School Mathematics, or your state academic content standards.) Most sources imply or explicitly list a collection of understandings that are desirable for teachers. Not surprisingly, these lists include topics in Number, Operations, Algebra, Functions, Geometry, Measurement, Data Analysis, Statistics, and Probability. Most resources also imply or implicitly state the importance of the quality of these understandings by describing the cohesiveness, depth, and flexibility of mathematical knowledge that is required to teach well. It is this quality of understanding that I want to address here.

I like how Liping Ma, in Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States (1999), distinguishes the quality of the understandings of teachers about mathematics through discussion of the “basic” and the “fundamental” points of view. When elementary mathematics is viewed as “basic” mathematics, then the mathematics can be described as a collection of procedures. When elementary mathematics is viewed as “fundamental”, then the mathematics can be described as elementary (the beginning of the discipline), primary (containing the rudiments of advanced mathematics) and foundational (supporting future learning). Supporting the “fundamental” point of view, Ma argues that elementary teachers require a profound understanding of fundamental mathematics that has breadth, depth, and thoroughness.

How can mathematics faculty promote such an understanding, particularly one which has breadth, depth, and thoroughness? Recommendations 3 and 4 of the CBMS document on the Mathematical Preparation of Teachers give some general guidelines. One model for meeting the spirit of these guidelines requires math faculty who teach prospective elementary teachers to deliberately include the following features in their courses:

1. Organizing schemes (Mathematical structures and/or developmental sequences with connections to school mathematics) that promote breadth of understanding.

2. Study of key mathematical ideas and concepts from multiple points of view that promote depth of understanding.

3. Opportunities for prospective teachers to develop the habits of mind of a mathematical thinker that promote thoroughness of understanding.

What follows are examples of these three features in a given content area.

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Number and Operations (Number Sense)

A. Organizing Schemes

1. A diagram of nested subsets of the Real numbers. I use the diagram of nested subsets several times during our course on number sense and algebraic thinking. I use it for discussing types of numbers, closure of operations, identities, inverses, solutions to equations, and children’s development of number sense.

2. Definitions of “Number Sense” and “Numeracy”.

What is number Sense?

“Much of school mathematics depends on numbers, which are used to count, compute, measure, and estimate. Mathematical activity belongs here that centers primarily on the development of number concepts, on computation with numbers (addition, subtraction, multiplication, division, finding powers and roots, etc.), on numerations (systems for writing numbers, including base ten, fractions, negative numbers, rational numbers, percents, scientific notation, etc.), and on estimation. At higher levels this strand includes the study of prime and composite numbers, of irrational numbers and their approximation by rationals, of real numbers, and of complex numbers.”

Mathematics Framework for California Public Schools, K-12, 1999

“Like common sense, number sense produces good and useful results with the least amount of effort. It is not mindlessly mechanical, but flexible and synthetic in attitude.

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Counting Numbers

Whole Numbers

Integers

Rationals

Reals

It evolves from concrete experience and takes shape in oral, written, and symbolic expressions.”

National Research Council, 1989

A person who has Number Sense has some combination of A. an ability to represent number in a variety of ways,B. an awareness of the relationships between numbers,C. a knowledge of the effects of operations,D. an ability to interpret and use numbers in real world counting and measurement.

NCTM Addenda Series, 19943. Connections to State Standards.

Number sets through the grade levels CA Standards Operations through the grade levels

K: Counting Numbers through 30

1. Counting Numbers though 100

2. Counting numbers through 1,000Fractions – halves, thirds, sixths, eighths, twelfths.Decimals as related to money.

3. Counting numbers through 10,000Development of Fractions done in 2nd gradeDecimals to the hundredths.

4. Counting numbers through 1,000,000Decimals and fractions through hundredthsConcepts of negative integers (Number line, counting, temperature, owing)

5. Decimals number from thousandths to millionsWrite fractions as percents and decimalsDevelop positive and negative integers

6. Positive and negative fractions and decimalsMixed numbersPi

7. All of the aboveDifferentiate between rationals and irrationals

K: Addition and subtraction for 2 numbers each less than 10 (Concrete)

1. Addition facts to 20 (and corresponding differences)Addition and subtraction problems with a one-digit and a two-digit number.

2. Sums and differences of 3-digit numbersMental Arithmetic for sums and differences of 2-digit numbersModel and solve simple multiplication and division.

3. Multiplication tables through 10 x 10.Sums and differences of 4 digit numbers.Multiply and divide a multi-digit number by 1-digit number.Add, Subtract, Multiply, and Divide with money.Add and subtract simple fractions.

4. Estimate and Compute the sums or difference of whole numbers and positive decimals (2 places).

Multiply or Divide a multi-digit number by a 2-digit number.Add and subtract negative numbers.

5. Add, subtract, multiply, and divide decimals and negative numbers.

Proficient with multi-digit multiplication and division.Add, subtract, multiply, and divide simple fractions

6. Calculate (Add, subtract, multiply, and divide with all types of rational numbers.

7. All of the above. Use exponents and roots when working with rational numbers.

B. Key concepts from multiple viewpoints

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Here are two examples of concepts that I emphasize under “Number and Operations”. Whenever we are working on any of the “multiple approaches” for a concept, I conduct a classroom discussion connecting the activity or approach to that concept.

1. Place value

We study the concept of place value in various ways throughout the course, including Studying Egyptian and Mayan numeration systems, Doing arithmetic base 5, Analyzing algorithms for whole-number operations, Extending whole number operations to decimal forms of rational numbers, Developing Scientific Notation.

2. Binary Operations

We study the concept of binary operations in various ways throughout the course, including

Analyzing the 4-color game (see the Sunday afternoon presentation) to see binary operations as functions,

Studying different conceptual models and word problems for operations over the whole numbers, integers, and rationals,

Investigating compositions of functions, particularly of isometries.

C. Opportunities for developing habits of mind.

In the CBMS document, one of the items on the list of understandings for elementary teachers is, “understanding how any number represented by a finite or repeating decimal is rational, and conversely.” The activity started by Stuart Moskowitz in the laboratory class of the workshop is intended to get prospective teachers to thoroughly investigate a part of this understanding.

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Geometry and Measurement

A. Organizing Schemes

1. The van Hiele Levels of Geometric Thinking.1

Dina and Pierre van Hiele are two Dutch educators who were concerned about the difficulties that their students were having in geometry. This concern motivated their research aimed at understanding students’ levels of geometric thinking to determine the kinds of instruction that can best help students.

The five levels that are described below are not age-dependent, but, instead, are related more to the experiences students have had. The levels are sequential; that is, students must pass through the levels in order as their understanding increases. The descriptions of the levels are in terms of “students” – and remember that we are all students in some sense.

Level 0 – Visualization

Students recognize shapes by their global, holistic appearance.

Students at level 0 think about shapes in terms of what they resemble and are able to sort shapes into groups that “seem to be alike.” For example, a student at this level might describe a triangle as a “clown’s hat.” The student, however, might not recognize the same triangle if it is rotated so that it “stands on its point.”

Level 1 – Analysis

Students observe the component parts of figures (e.g., a parallelogram has opposite sides that are parallel) but are unable to explain the relationships between properties within a shape or among shapes.

Student at level 1 are able to understand that all shapes in a group such as parallelograms have the same properties, and they can describe those properties.

Level 2 – Informal deduction (relationships)

Students deduce properties of figures and express interrelationships both within and between figures.

Students at level 2 are able to notice relationships between properties and to understand informal deductive discussions about shapes and their properties.

1 Cathcart, et al. Learning Mathematics in Elementary and Middle Schools. p.282-3.

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Level 3 – Formal deduction

Students can create formal deductive proofs.

Students at level 3 think about relationships between properties of shapes and also understand relationships between axions, definitions, theorems, corollaries, and postulates. At this level, students are able to “work with abstract statements about geometric properties and make conclusions based more on logic than intuition” (Van de Walle).

Level 4 – Rigor

Students rigorously compare different axiomatic systems.

Students at this level think about deductive axiomatic systems of geometry. This is the level that college mathematics majors think about Geometry.

2. A connection to the State Standards

In general, most elementary school students are at levels 0 or 1; some middle school students are at level 2. The CA standards are written to begin the transition from levels 0 and 1 to level 2 as early as 5 th grade “Students identify, describe, draw and classify properties of, and relationships between, plane and solid geometric figures.” (5th grade, standard 2 under Geometry and Measurement) This emphasis on relationships is magnified in the 6th and 7th grade standards.

Interestingly, the sixth National Assessment of Educational Progress report (1997) states that “most of the students at all three grade levels (fourth, eight, and twelfth) appear to be performing at the ‘holistic’ level (level 0) of the van Heile levels of geometric thought.”

3. Definitions related to measurement

What is Measurement?

Measurement is the process of attaching numbers to certain qualities of objects and events.

Counting involves discrete objects Measuring involves continuous properties(How many?) (How much?)

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What is Measurable?

A measurable attribute of an object or event is a characteristic that can be quantified by comparing it to a unit.

Length Area Volume Capacity Weight MassTemperature Time Angles Position Distance SpeedAcceleration Pressure Pitch Energy Etc.

What is the Process of Measurement?

Choose an appropriate unit for the attribute being measured;

Compare the attribute of the object or event to the unit;

Obtain a measurement (always an approximation) — a number AND a unit.

What is a common instructional sequence?

1. Develop the meaning of the attribute being measured through activities involving perception and direct comparison. (Which is taller?)

2. Children begin to measure using arbitrary or nonstandard units. (How tall?)3. Children measure and estimate using standard units. (How tall in Feet and

Inches.)

Accuracy of Measurement

“To apply care to”When a person puts a lot of care into a measurement, the result is likely to be accurate.

Precision of Measurement

“to cut off in front”The smallest unit of measure used to express an approximate value (the exact value is “cut off” there).

B. Key concepts from multiple viewpoints

Here are two examples of concepts that I emphasize under “Geometry and Measurement”. Whenever we are working on any of the “multiple approaches” for a concept, I conduct a classroom discussion connecting the activity or approach to that concept.

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1. Similarity

We study the concept of similarity in various ways throughout the course, including

Investigating similar triangles in the context of parallel lines and circles, Conducting remote measurement activities (Heights of Trees), Exploring generalizations of the Pythagorean Theorem, Studying size transformations and scalings in a line, plane, or space.

2. The role of the Unit of measure

We study the concept of unit in various ways throughout the course, including

Use of non-standard and standard units in many different contexts. Conversion between one unit of measure and another unit of measure Dimensional analysis in solving problems. Extension of the concept to the meaning of the standard deviation

C. Opportunities for developing habits of mind.

I use a collection of investigations in Key Curriculum Press’ Exploring Geometry with the Geometer’s Sketchpad (1993). My favorites are related to the topic, “devising area formulas for triangles, parallelograms, and trapezoids; knowing the formula for the area of a circle (CBMS),” particularly one in which students test formulas for various quadrilaterals that involve either the length of a mid-segment or the length of the diagonals.

In addition, I have students work on selected investigations on the Geoboard. (See the next page.)

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Geoboard ActivitiesConsider the square area bounded by 4 adjacent pegs to be 1 square unit.

1. Find the area of this triangle. See if you can find more than one way to calculate the area.

2. Make shapes on your geoboard that can be made with only one geoband which does not cross itself anywhere. For each shape, calculate its area, record the shape and its area on your geoboard paper. Make a chart for your shapes, listing the number of pegs each shape touches (T), the number of pegs each totally enclosed in the interior of each shape (I) and the area of the shape (A). Look for a formula that describes the algebraic relationship among these three variables. (Hint -- find a systematic way to organize your data.)

T

I

A

3. How many triangular regions of different area can you create on your geoboard where all three vertices of the triangle are pegs of the board?

4. How many noncongruent triangles can you make on your geoboard where all three vertices of the triangle are pegs of the board?

5. How many distinct lengths of line segments can you make on your geoboard where both endpoints of the segment are pegs of the board?

6. How many different size squares can you make on your geoboard? Find the area of each.Extensions: Repeat questions 5 and 6 for an n by n geoboard.

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Algebra and Functions

A. Organizing Schemes

1. Algebra in School mathematics

What is “Algebra and Functions”, Pre-K through Grade 12?

“This strand involves two closely related subjects. Functions are rules that assign to each element in an initial set an element in a second set….they are encountered informally in the elementary grades and grow in prominence and importance with the student’s increasing grasp of algebra in the higher grades. …

Algebra…starts informally. It appears initially in its proper form in the third grade as “generalized arithmetic.” In later grades algebra is the vital tool needed for solving equations and inequalities and using them as mathematical models of real situations.”

Mathematics Framework for California Public Schools, K-12, 1999

“Instructional programs from prekindergarten through grade 12 should enable all students to

understand patterns, relations, and functions; analyze change in various contexts. represent and analyze mathematical situations and structures using algebraic

symbols use mathematical models to represent and understand quantitative relationships”

Principles and Standards for School Mathematics, NCTM, 2000

Sample tasks that are algebraic

The ideas for these samples came from the NCTM Principles and Standards for School Mathematics.

Primary Grades – Front and Back Cards

A primary teacher had prepared two groups of cards for her students. In the first group, the number on the front and back of each card differed by 1. In the second group, these numbers differed by 2.

The teacher showed the students a card with 12 written on it and explained, "On the back of this card, I've written another number." She turned the card over to show the number 13. Then she showed the students a second card with 15 on the front and 16 on the back. As she continued showing the students the cards, each time she asked the students, "What do you think will be on the

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back?" Soon the students figured out that she was adding 1 to the number on the front to get the number on the back of the card.

Then the teacher brought out a second set of cards. These were also numbered front and back, but the numbers differed by 2, for example, 33 and 35, 46 and 48, 22 and 24. Again, the teacher showed the students a sample card and continued with other cards, encouraging them to predict what number was on the back of each card. Soon the students figured out that them numbers on the backs of the cards were 2 more than the numbers on the fronts.

When the set of cards was exhausted, the students wanted to play again. "But," said the teacher, "we can't do that until I make another set of cards." One student spoke up, "You don't have to do that, we can just flip the cards over. The cards will all be minus 2."

As a follow-up to the discussion, this teacher could have described what was on each group of cards in a more algebraic manner. The numbers on the backs of the cards in the first group could be named as "front number plus 1" and the second as "front number plus 2." Following the student's suggestion, if the cards in the second group were flipped over, the numbers on the backs could then be described as "front number minus 2." Such activities, together with the discussions and analysis that follow them, build a foundation for understanding the inverse relationship.

Elementary Grades – The Growing Squares Problem

An elementary teacher asked her students to describe patterns they see in the "growing squares" display and express the patterns in mathematical sentences. Students were encouraged to explain these patterns verbally and to make predictions about what will happen if the sequence is continued.

Middle Grades – The Super Chocolates Problem

Super Chocolates are arranged in boxes so that a caramel is placed in the center of each array of four chocolates, as shown below. The dimensions of the box tell you have many columns and how many rows of chocolates come in the box. Develop a method to find the number of caramels in any box if you know its dimensions. Explain and justify your methods using words, diagrams , or expressions.

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High School – Edging a Garden.

Suppose that you have 24 feet of garden edging with which to mark out a rectangular garden. Describe the relationship between the width of the garden that you create and it’s area. Analyze this relationship to find the width of the garden that corresponds to the largest area.

2. A progression of definitions

Variable

A variable is simply a name we introduce into a mathematical setting or discussion with the intention of assigning it some value.

Perhaps the fundamental difficulty for many students making the transition from arithmetic to algebra is their failure to recognize that the symbol x stands for a number. For example, the equation 3(2x-5)+4(xs-2)=12 simply means that a certain number x has the property that when the arithmetic operations 3(2x-5)+4(x-2) are performed on it as indicated, the result is 12. The problem is to find that number (solution).

Mathematics Framework for California Public Schools, K-12, 1999, Page 158

It is important to realize that developing students' understanding of variable is complicated, precisely because the concept of variable itself is multifaceted. Consider the following equations and their variables:

30 = 5x A = LW y = 3x.

Unknown In the first equation, x stands for a specific, non varying number that can be determined by solving the equation.

Measured Quantity The second equation is usually considered a formula with A, L, and W standing for the quantities area, length, and width. These quantities actually feel more like knowns than unknowns.

Independent and Dependent Variables The third equation shows x independently taking on a variety of values, and the corresponding y varying depending on the value of x.

Universal variables The fourth equation generalizes an arithmetic pattern where n can take all real numbers (but not zero).

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Expression

An expression is a description of how to calculate something.

Expressions are sometimes called rules or formulas. When you evaluate an expression you actually follow the description and carry out the prescribed sequence of steps in the calculation, and the resulting object is called the value of the expression.

Expressions are made up of numbers, variables, operations, and known functions. Parenthesis and the accepted order of operations govern the interpretation of the description of calculation. For example,

is an expression whose value is 3.

If we evaluate the expression

when x assumes the value 2, then the value of the expression is .

The expressions

and

have the same value when x = 2 (there are other ways to reorganize parenthesis and operations to produce the same value). These two expressions are equivalent, in that they will produce the same value for any given value of the variable x. One common task in algebra is to simplify expressions. A “simplified expression” is a description of how to calculate something “efficiently”. However, what two different people think of as being efficient can vary greatly – so the performing a “simplification” may produce more than one acceptable result.

Relation

A relation is a mathematical way to describe a relationship between elements of two different sets of objects or elements within the same set of objects.

The more common relations that we use in elementary mathematics are, “is equal to”, “is less than”, “is greater than”, “is not equal to”, “is a multiple of”, and “is a factor of”. We use ordered pairs to represent these relations.

Statement

A mathematical statement is a sentence to which a truth-value (True or False) may be assigned.

The most common statements in school mathematics are equations and inequalities (corresponding to the relations of equality and inequality, respectively).

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3. A State Standards view of the development of the concept of functions

A function is a device which has a fixed internal mechanism which accepts some objects as input and produces some object as output.

Functions are special kinds of relations. Mathematically speaking, functions are relations with the characteristic that every acceptable first coordinate (input) is paired with exactly one second coordinate (output). Functions are dependable devices, in that for a given input, the function always produces the same output.

Here is how the CA Framework suggests organizing the development of functions:

Grades K – 7 An informal introduction to functions – usually in the context of linear equations or in recognizing or writing rules or expressions for observed patterns or applied contexts.

Algebra I … “introduce this concept of functions in the language of ordered pairs. Introducing this concept needs to be done carefully because students at this stage of their mathematical development may not be ready for this level of abstraction. However, during a first-year algebra course is the stage at which students should see and use the fuctional notation f(x) for the first time.” (Page 161)

Geometry The special class of function called isometries – distance preserving transformations of the plan – are concrete examples of functions. Rotations, reflections, and translations should be studied as functions (input – preimage, output – image) and composition of functions should be explored for the first time.

Algebra II “A first high point of Algebra II is the study of quadratic (polynomial) functions. … A second high point of Algebra II is the introduction of two of the basic functions in all of mathematics: ex and log x. … A third high point of Algebra II is the study of arithmetic and geometric series [also functions]” (pages 169-171)

Trigonometry “The trigonometric functions studied are defined geometrically rather than in terms of algebraic equations, but one of the goals of this course is to acquaint students with a more algebraic viewpoint toward these functions. Students should have a clear understanding that the definition of the trigonometric functions is made possible by the notion of similarity between triangles.

Mathematical analysis: (Precalculus) The functional viewpoint is emphasized in this course.

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B. Key concepts from multiple viewpoints

Here are two examples of concepts that I emphasize under “Algebra and Functions”. Whenever we are working on any of the “multiple approaches” for a concept, I conduct a classroom discussion connecting the activity or approach to that concept.

1. Functions

We study the concept of function in various ways throughout the course, including Identifying and describing patterns in number sequences, Modeling patterns in number sequences with continuous functions Conceptualizing operations as functions, Studying functional relationships and their graphs through activities from the

Shell Red book Viewing transformations of the plane as functions on the points of the plane.

2. Equality

We study the concept of equality in various ways throughout the course, including the role of the concept in

Computation Equivalence Simplifying Expressions Solving Equations Modeling relationships

C. Opportunities for developing habits of mind.

One of the goals for teacher understanding under Algebra and Functions in the CBMS document is “being able to read and create graphs of functions.” To develop flexibility in thinking about functional relationships, students work in small groups to discuss selected activities from The Language of Functions and Graphs (ISBN # 0901628433), Shell Centre for Mathematical Education.

Another of the understandings is “solving word problems via algebraic manipulation.” The “fair game” activity is one of my favorite examples.

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Two-color Chip Game

Decide which partner will be player A and which will be player B.

Game 1

Begin with two chips of one color and one of the other color in the bag.

Shake the bag. Each player pick one chip from the bag (without

looking). Person A wins if the chips are the same color; person B

wins if the chips are different colors

Is this a fair game?

Game 2:

Replay game 1 with three of one color chip and one of the other color.

Is this a fair game?

For what values of n and m (where there are n chips of one color and m chips of the other color in the bag) will this game be fair?

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Summary of number and operations content, CBMS

Understanding models and interpretations of operations with whole numbers (i.e., the set of non-negative integers):

having a large repertoire of interpretations of addition, subtraction, multiplication and division, and of ways they can be applied.

understanding relationships among operations.

Developing a strong sense of place value in the base-10 number system: understanding how place value permits efficient representation of number. recognizing the value of each place as ten times larger than the value of the next

place to the right and the implications of this for ordering numbers and for estimation and approximation.

seeing how the operations of addition, multiplication, and exponentiation are used in representing numbers.

recognizing the relative magnitude of numbers.

Understanding multidigit calculations, including standard algorithms, "mental math,'' and non-standard methods commonly created by students:

recognizing how the base-10 structure of number is used in multidigit computations.

recognizing how decimal notation allows for approximation by "round numbers'' (multiples of powers of 10).

recognizing the properties of commutativity, associativity, and distributivity as useful tools for organizing thinking about computation.

developing flexibility in mental computation and estimation.

Developing the concepts of integer and rational number and extending the operations to these larger domains:

understanding what integers are and the meaning of sign and magnitude. understanding what rational numbers are, understanding fractions and decimals as

representations of rationals, and developing a sense of their relative size. knowing interpretations and applications for the arithmetic operations in the

extended domains. understanding the relationship between fractions and the operations of

multiplication and division. understanding how whole number arithmetic extends to integers and rational

numbers. understanding how any number represented by a finite or repeating decimal is

rational, and conversely. understanding how and why whole number decimal arithmetic extends to finite

decimals and, in particular, how place value extends to decimal fractions.

Summary of geometry and measurement content, CBMS

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Developing visualization skills: becoming familiar with projections, cross-sections, and decompositions of common two-

and three-dimensional figures. representing three-dimensional shapes in two dimensions and constructing three-

dimensional objects from two-dimensional representations.

Developing familiarity with basic shapes and their properties: knowing fundamental objects of geometry. developing an understanding of angles and how they are measured. becoming familiar with plane isometries---reflections (flips), rotations (turns), and

translations (slides)---and symmetries. understanding congruence and similarity. learning technical vocabulary and understanding the importance of definition.

Understanding the process of measurement: recognizing different aspects of size. understanding the idea of unit and the need to select a unit appropriate to the attribute

being measured. knowing the standard (English and metric) systems of units. comparing units. understanding that measurements are approximate and that different units affect

precision.

Understanding length, area, and volume: knowing what is meant by one, two, and three dimensions. seeing rectangles as arrays of squares, rectangular solids as arrays of cubes. recognizing the behavior of measure (length, area, and volume) under uniform dilations. devising area formulas for triangles, parallelograms, and trapezoids; knowing the formula

for the area of a circle; becoming familiar with formulas for prisms, cylinders, and other three-dimensional objects understanding the independence of perimeter and area; surface area and volume.

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Summary of Algebra and Functions content, CBMS

Generalizing arithmetic and quantitative reasoning: learning to use a variety of representations, including conventional algebraic notation, to

articulate and justify generalizations. understanding algebraic expressions as shorthand for describing calculation;

understanding algebraic identities as statements of equivalence of expressions. understanding different forms of argument and learning to devise deductive arguments. solving word problems via algebraic manipulation.

Discovering how the field axioms govern arithmetic: recognizing commutativity, associativity, distributivity, identities, and inverses as

properties of operations on a given set. seeing computation algorithms as applications of particular axioms.

Understanding functions: becoming familiar with the notion of function. being able to read and create graphs of functions, formulas (closed and recursive), and

tables. studying the characteristics of particular classes of functions on integers, especially

linear, quadratic, and exponential functions.

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