Elementary Mathematics Benchmarks, Algebra Strand

Kindergarten

A.K.1 Identify, sort, and classify objects.

a. Sort and classify objects by attribute and identify objects that do not belong in a particular group.

• Recognize attributes that involve colors, shapes (e.g., triangles, squares, rectangles, and circles) and patterns (e.g., repeated pairs, bilateral symmetry).

  Example: Identify the common attribute of square in a square book, square table, and square window.

  Example: Distinguish different patterns in ABABABA, .

b. Recognize related addition and subtraction facts.

• Use objects to demonstrate "related facts" such as 7 - 4 = 3, 3 + 4 = 7, 7 - 3 = 4.

Grade 1

A.1.1 Recognize and extend simple patterns.

a. Skip count by 2s and 5s and count backward from 10.

b. Identify and explain simple repeating patterns.

• Find repeating patterns in the discrete number line, in the 12 x 12 addition table, and in the hundreds table (a 10 x 10 square with numbers arranged from 1 to 100).

  Note: Use examples based on linear growth (e.g., height, age).

• Create and observe numerical patterns on a calculator by repeatedly adding or subtracting the same number from some starting number.

c. Determine a plausible next term in a given sequence and give a reason.

Note: Without explicit rules, many answers to "next term" problems may be reasonable. So whenever possible, rules for determining the next term should be accurately described. Patterns drawn from number and geometry generally have clear rules; patterns observed in collected data generally do not.

A.1.2 Find unknowns in problems involving addition and subtraction.

a. Understand that addition can be done in any order but that subtraction cannot.

• Demonstrate using objects that the order in which things are added does not change the total, but that the order in which things are subtracted does matter.
• Use the fact that a + b = b + a to simplify addition problems.

  Examples: 2 + 13 = 13 + 2 = 15 (by adding on);
  7 + 8 + 3 = 7 + 3 + 8 = 10 + 8 = 18.

  Note: The relation a + b = b + a is known as the commutative property of addition. It reduces significantly the number of addition facts that need to be learned. However, the vocabulary is not needed until later grades.

• Demonstrate understanding of the basic formula a + b = c by using objects to illustrate all eight number sentences associated with any particular sum:

  Example: 8 + 6 = 14, 6 + 8 = 14; 14 = 8 + 6, 14 = 6 + 8;
  14 – 8 = 6, 6 = 14 – 8; 14 – 6 = 8, 8 = 14 – 6.

A.1.3 Understand how adding and subtracting are inverse operations.

a. Demonstrate using objects that subtraction undoes addition and vice versa.

• Subtracting a number undoes the effect of adding that number, thus restoring the original. Similarly, adding a number undoes the action of subtracting that number.

  Example: 2 + 3 = 5 implies 5 – 2 = 3, and 5 – 2 = 3 implies 2 + 3 = 5.

• Use the inverse relation between addition and subtraction to check arithmetic calculations.

  Note: Addition and subtraction are said to be inverse operations because subtraction undoes addition and addition undoes subtraction. However, this vocabulary is not needed until later grades.

  Caution: Subtraction is sometimes said to be equivalent to "adding the opposite," meaning that 5 – 3 is the same as 5 + –3. Here the "opposite" of a number is intended to mean the negative of a number. However, since negative numbers are not introduced until later grades, this formulation of the relation between addition and subtraction should be postponed.

Grade 2

A.2.1 Create, identify, describe, and extend patterns.

a. Fill in tables based on stated rules to reveal patterns.

• Find patterns in both arithmetic and geometric contexts.

b. Record and study patterns in lists of numbers created by repeated addition or subtraction.

• Create patterns mentally (by counting up and down), by hand (with paper and pencil), and by repeated action on a calculator.

  Examples: 3, 8, 13, 18, 23, . . .; 50, 46, 42, 38, 34, . . .

A.2.2 Find unknowns in simple arithmetic problems.

a. Solve equations and problems involving addition, subtraction, and multiplication with the unknown in any position.

Note: In the early grades, it is better to signify the unknown with a symbol, such as [ ], ?, or __, that carries the connotation of unknown rather than with an alphabetic letter such as x.

b. Understand and use the facts that addition and multiplication are commutative and associative.

• Use parentheses to clarify groupings and order of operation.
• Recognize terms such as commutative and associative.

  Note: It is not necessary for children at this grade to use or write these words, merely to recognize them orally and to know the properties to which they refer.

c. Recognize how multiplication and division are, like addition and subtraction, inverse operations.

A.2.3 Understand basic properties of odd and even numbers.

a. Explain why the sum of two even numbers is even and that the sum of two odd numbers is also even.

• Use diagrams to represent even and odd numbers and to explain their behavior.

  Example: The representation at the right shows that 14 is even and 13 is odd.

  

b. Answer similar questions about subtraction and multiplication of odd or even numbers.

Grade 3

A.3.1 Explore and understand arithmetic relationships among positive whole numbers .

a. Understand the inverse relationships between addition and subtraction and between multiplication and division, and the commutative laws of multiplication and addition.

• Show that subtraction and division are not commutative.

b. Find the unknown in simple equations that involve one or more of the four arithmetic operations.

Note: To emphasize the process of solving for an unknown, limit coefficients and solutions to small positive whole numbers.

Examples: 3 × ? = 3 + 6; ? ÷ 5 = 5 x 55; 36 = ? × ?.

c. Create, describe, explain and extend patterns based on numbers, operations, geometric objects and relationships.

• Explore both arithmetic (constant difference) and geometric (constant multiple) sequences.

  Examples: 100, 93, 86, 79, 72, . . .; 2, 4, 8, 16, . . .; 3, 9, 27, 81, . . . .

• Understand that patterns do not imply rules; rules imply patterns.

Grade 4

A.4.1 Use properties of arithmetic to solve simple problems.

a. Understand and use the commutative, associative, and distributive properties of numbers.

• Use these terms appropriately in oral descriptions of mathematical reasoning.
• Use parentheses to illustrate and clarify these properties.

b. Find the unknown in simple linear equations.

• Use a mixture of whole numbers, fractions, and mixed numbers as coefficients.

  Examples: 24 + n = n – 2; ¾ + p = 5/4 – p

  Note: "Simple" equations for grade 4 are those that require only addition or subtraction (e.g., 3/4 + [ ] = 7/4) or a single division whose answer is a whole number (e.g., 3 x [ ] = 12).

  Note: There is no need to use term linear since these are the only kinds of equations encountered in grade 4.

A.4.2 Evaluate simple expressions.

a. Find the value of expressions such as na + b, and na – b where a, b and n are whole numbers or fractions and where na ≥ b.

• Make tables and graphs to display the results of evaluating expressions for different values of n such as n = 1, 2, 3, . . . .

  Note: Evaluating an expression involves two distinct steps: substituting specific values for letter variables in the expression, and then carrying out the arithmetic operations implied by the expression. Working with expressions both introduces the processes of algebra and also reinforces skills in arithmetic.

  Note: Avoid negative numbers since systematic treatment of operations on negative numbers is not introduced until grade 6.

b. Evaluate expressions such as , where a, b, c, and n are whole numbers.

c. Evaluate expressions such as where a and b are single-digit whole numbers.

Example: The value of when a = 1, b = 2, c = 3, and n = 4 is .

Note: Addition of fractions is limited to cases included in the grade 4 expectations—namely, unit fractions with denominators under 10 and other fractions where one denominator is a multiple of the other.

Grade 5

A.5.1 Find the unknown in simple linear equations.

a. Equations that require only simple calculation should be solved mentally (that is, "by inspection"):

96 + 67 = b + 67

+ - = p

a + =

39 – k = 39 – 40

- + = d + -

+ = b +

78 + b = 57 + 79

53 + 76 = 51 +76 + d

A.5.2 Evaluate and represent simple expressions.

a. Translate between simple expressions, tables of data and graphs in the coordinate plane.

b. Understand and use the conventions for order of operations (including powers).

Example: ax² + bx = (a(x²)) + (bx), not (ax)² + bx.

c. Evaluate expressions such as

• nr where n is a whole number and r is a fraction.
• nab/(na-b) when n, a, and b are whole numbers and where na > b.
• + , where a, b, c, d are positive whole numbers.
• 1/ab where a and b are positive whole numbers.
• a/b where one of a or b is a positive whole number and one is a fraction.

  Note: Avoid expressions that introduce negative numbers since systematic treatment of operations on negative numbers is not introduced until grade 6.

  Note: Working with expressions both introduces the processes of algebra and also reinforces skills in arithmetic.

d. Understand the importance of not dividing by zero.

Grade 6

A.6.1 Understand that the system of negative and positive numbers obeys and extends the laws governing positive numbers.

a. The sum and product of two numbers, whether positive or negative, integer or fraction, satisfy the commutative, associative and distributive laws.

• For any numbers a, b, c, (whether positive, negative or zero),

  a + b = b + a, a x b = b x a (commutative);
  a + (b + c) = (a + b) + c; a × (b × c) = (a × b) × c (associative);
  a × (b + c) = (a × b) + (a × c) (distributive).

  Example: (–3) × 5 = –(3 × 5), because ((–3) × 5) + (3 × 5) = ((–3) + 3) × 5) = 0 x 5 = 0. Hence the sum of (–3) × 5 and 3 × 5 is 0, so (–3) × 5 = –(3 x 5).

  Note: This example can usefully be demonstrated on the number line in a way that avoids the formality of parentheses required above.

b. Understand why the product of two negative numbers must be positive.

• Since a negative number, -a, is defined by the equation -a + a = 0, the distributive law forces the product of two negative numbers to be positive.

  Example: To show that (–3) × (–5) = 3 × 5, we demonstrate that the sum of the left side [(–3) x (–5)] with the negative of the right [3 × 5] is zero:
  ((–3) × (–5)) + (–(3 × 5)) = ((–3) × (–5)) + ((–3) × 5) = (–3) × ((–5) + 5) = (–3) × 0 = 0.

  The key middle step uses the distributive law.

c. Understand why the quotient of two negative numbers must be positive.

• Division is the same as multiplication by a reciprocal. If a number b is negative, so is its reciprocal 1/b. So if a and b are both negative, a/b is positive since it equals the product of two negative numbers: a x (1/b).

  Example: If p = –12/-3, then –12 = p × (–3). Since 4 x –3 = –12, this yields p = 4, hence –12/–3 = 12/3 = 4.

A.6.2 Represent and use algebraic relationships in a variety of ways.

a. Recognize and observe notational conventions in algebraic expressions.

• Understand and use letters to stand for numbers.
• Recognize the use of juxtaposition (e.g., 3x, ab) to stand for multiplication and the convention in these cases of writing numbers before letters.
• Recognize the tradition of using certain letters in particular contexts.

  Note: Most common: k for constant, n for whole number, t for time, early letters (a, b, c) for parameters, late letters (u, v, x, y, z) for unknowns.

• Recognize different conventions used in calculator and computer spreadsheets (e.g., * for multiplication, ^∧ for power).
• Understand and use conventions concerning order of operations and use parentheses to specify order when necessary.

  Note: By convention, powers are calculated before multiplication (or division) and multiplication is done before addition (or subtraction).

  Example: 3x² means 3(x · x), not (3x) · (3x); 3x² – 7x means (3x²) – (7x), not (3x² - 7) · x.

• Evaluate expressions involving all five arithmetic operations (addition, subtraction, multiplication, division, and power).

  Note: For the most part this is review. What is new is the introduction of powers and the shift to use of letters (rather than boxes, question marks, and other placeholders) to represent generic or unknown numbers.

  Examples: 3x² – 7x + 2, when x = 3 or 1/3; 2x^-3 + x, when x = 2 or 1/2 or 0 or -3; 3x² – 2xy, when x = 2 and y = 6.

b. Solve problems involving translation among and between verbal, tabular, numerical, algebraic, and graphical expressions.

• Write an equation that generates a given table of values.

  Note: Limit examples to linear relationships with integer domains.

• Graph ordered pairs of integers on a coordinate grid.

  Example: Prepare scatter plots of related data such as students' height vs. arm length in inches.

• Generate data and graph relationships concerning measurement of length, area, volume, weight, time, temperature, money, and information.
• Understand why formulas or words can represent relationships, whereas tables and graphs can generally only suggest relationships.

  Note: Unless the rule for a table (or graph) is specified (e.g., in a tax table), the numbers themselves cannot determine a relationship that extends to numbers not in the table.

  Note: The issue here is about the lack of precision inherent in a graph, not about the possibility, which is present in some graphs, of ambiguity concerning which of two very different points on the graph are associated with a given input value. The common example of ambiguity concerns the graph of the equation for a circle, but such graphs are not among those studied in grades K – 8.

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