Elementary Mathematics Benchmarks, Grade 6

Number (N)

N.6.1 Understand and use negative numbers.

a. Know the definition of a negative number and how to locate negative numbers on the number line.

• If a is a positive number, –a is a number that satisfies a + (–a) = 0.
• On the number line, –a is the mirror image of a with respect to 0; it lies as far to the left of 0 as a lies to the right.

  Note: In elementary school, a negative number -a is sometimes called the "opposite" of a, but this terminology is not used in later grades.

• Negative numbers may be either whole numbers or fractions.
• The positive whole numbers together with their negative counterparts and zero are called integers.

  Note: The properties of negative numbers apply equally to integers and to fractions. Thus it is just as effective (and certainly easier) to limit to integers all examples that introduce the behavior of negative numbers.

  Note: The positive fractions together with the negative fractions and zero (which include all integers) are called rational numbers. In grade 6, these are all the numbers we have, so they are usually referred to just as "numbers." Later when irrational numbers are introduced, the distinction between rational and irrational will be important--but not now.

b. Understand why –(–a) = a for any number a, both when a is positive and when a is negative.

• Use parentheses, as in –(–a), to distinguish the subtraction operation (minus) from the negative symbol.

c. Use the number line to demonstrate how to subtract a larger number from a smaller one.

• If b > a, the point c on the number line that lies at distance b – a to the left of zero satisfies the relation a – b = c. Thus a – b = – (b – a).
• Subtracting a smaller from a larger number is the same as adding the negative of the smaller number to the larger. That is, if a > b then a - b = a + (-b).

  Note: Formally, a + (–b) = (a − b) + b + (-b) = (a – b) + 0 = a – b.

• Recognize that a + (-b) = a – b (even when b > a).

  Example: 3 – 8 = –5 because 5 + (3 – 8) = 5 + (3 + (–8)) = (5 + 3) + (–8) = 8 + (–8) = 0. Therefore, 3 – 8 satisfies the definition of –5 as being that number which, when added to 5, yields zero.

d. Recognize that all numbers, positive and negative, satisfy the same commutative, associative, and distributive laws.

Note: Demonstrations of these laws are part of Algebra, below. Here recognition and fluent use are the important issues.

N.6.2 Understand how to divide positive fractions and mixed numbers.

a. Understand that division of fractions has the same meaning as does division of whole numbers.

• Just as A/B = C means that A = C × B for whole numbers, so (a/b) / (c/d) = M/N means that (a/b) = (M/N) x (c/d).

b. Use and be able to explain the "invert and multiply" rule for division of fractions.

• Invert and multiply means: (a/b)/(c/d) = (a/b)x(d/c) = (ad)/(bc).

  Note: To verify that (a/b)/(c/d) = (ad)/(bc), we need to check that (ad)/(bc) satisfies the definition of (a/b)/(c/d), namely, that (a/b) = (ad)/(bc) x (c/d): (ad)/(bc) x (c/d) = (a/b)x(d/c) x (c/d) = (a/b) x 1 = a/b.

c. Find unknowns in division and multiplication problems using both whole and mixed numbers.

• Solve problems of the form a ÷ [ ] = b, a × [ ] = b, [ ] ÷ a = b.

  Examples: ÷ [ ] = 1; ÷ [] = ; = 1 × [ ]; 2 ÷ 1 = [ ]; 2 ÷ [ ] = 1 .

d. Create and solve contextual problems that lead naturally to division of fractions.

• Recognize that division by a unit fraction 1/n is the same as multiplying by its denominator n.

N.6.3 Understand and use ratios and percentages.

a. Understand ratio as a fraction used to compare two quantities by division.

• Recognize a:b and a/b as alternative notations for ratios.

  Note: A ratio is often thought of as a pair of numbers rather than as a single number. Two such pairs of numbers represent the same ratio if one is a non-zero multiple of the other or, equivalently, if when interpreted as fractions, they are equivalent.

  Example: 2:4 is the same ratio as 6:12, 8:16,or 1:2.

• Understand that quantities a and b can be compared using either subtraction (a - b) or division (a/b).
• Recognize that the terms numerator and denominator apply to ratios just as they do to fractions.

b. Understand that percentage is a standardized ratio with denominator 100.

• Recognize common percentages and ratios based on fractions whose denominators are 2, 3, 4, 5, or 10.

  Examples: 20%, 25%, 33%, 40%, 50%, 66%, 90%, and 100% and their ratio, fraction, and decimal equivalents.

• Express the ratio between two quantities as a percent and a percent as a ratio or fraction.

c. Create and solve word problems involving ratio and percentage.

• Write number sentences and contextual problems involving ratio and percentage.

N.6.4 Understand and use exponents and scientific notation.

a. Calculate with integers using the law of exponents: bⁿ x b^m = b^n+m.

• Just as b × n (with b and n positive) can be understood as b added to itself n times, so bⁿ can be understood as b multiplied by itself n times.

  Note: The law of exponents for positive exponents is just a restatement of this definition, since both bⁿ x b^m and b^n+m mean b multiplied by itself n + m times.

  Note: If b > 0, b¹ = b, b⁰ = 1. The same is true of b < 0. If b = 0, 0¹ = 0, but 0⁰ is not defined.

• If n > 0, b^-n means 1/bⁿ (that is, 1 divided by b n times).

  Note: This definition of b^-n is designed to make the law of exponents work for all integers (positive or negative): bⁿ x b^-n means b multiplied by itself n times, then divided by b n times, yielding 1. Thus bⁿ x bⁿ = b^n+(-n) = b⁰ = 1.

  Example: 3^-2 = ()² = ; 3³ = 27; 3³ x 3^-2 = 3^(3-2) = 3¹ = 3 = 27 x = 27/9.

b. Understand scientific notation and use it to express numbers and to compute products and quotients.

• Recognize the importance of scientific notation to express very large and very small numbers.
• Locate very large and very small numbers on the number line.
• Understand the concept of significant digit and the role of scientific notatiion in expressing both magnitude and degree of accuracy.

c. Model exponential behavior with contextual illustrations based on population growth and compound interest.

N.6.5 Solve multi-step mathematical, contextual, and verbal problems using rational numbers.

a. Solve arithmetic problems involving more than one arithmetic operation using rational numbers.

• Calculate with and solve problems involving negative numbers, percentages, ratios, exponents, and scientific notation.

  Note: As usual, keep calculations simple in order to focus on the new concepts.

• Compare numbers expressed in different ways and locate them on the number line.
• Write number sentences involving negative numbers, percentages, ratios, exponents, and scientific notation.

b. Solve relevant contextual problems (e.g., sports, discounts, sales tax, simple and compound interest.

• Represent problems mathematically using diagrams, numbers, and symbolic expressions.
• Express answers clearly in verbal, numerical, symbolic, or graphical form.
• Use estimation to check answers for reasonableness and calculators to check for accuracy.
• Describe real situations that require understanding of and calculation with negative numbers, percentages, ratios, exponents, and scientific notation.

Probability and Statistics (PS)

PS.6.1 Understand the meaning of probability and how it is expressed.

a. The probability of an event is a number between zero and one that expresses the likelihood of an occurrence.

• The probibility of an occurrence is the ratio of the number of actual occurrences to the number of possible occurrences.
• Understand different ways of expressing probabilities—as percentages, decimals, or odds.

  Example: If the probability of rain is .6, the weather forecaster could say that there is a 60% chance of rain or that the odds of rain are 6:4 (or 3:2).

• If p is the probability that an event will occur, then 1 – p is the probability that it will not occur.

  Example: If the probability of rain is 60%, then the probability that it will not rain is 100% – 60% = 40%. (Equivalently, 1 – .60 = .40.)

Geometry (G)

G.6.1 Understand and use basic properties of triangles and quadrilaterals.

a. Understand and use the angle properties of triangles and quadrilaterals.

• The sum of angles in a right triangle is 180°, since two identical right triangles form a rectangle.

  

  Note: By definition, a rectangle has 4 right angles, so the sum of the angles of a rectangle is 4 x 90° = 360°. Each right triangle contains half 360°, or 180°.

  Note: Since one angle in a right triangle is 90°, the sum of the remaining two angles is also 90°.

• Since the sum of angles in a parallelogram is also 360°, the sum of angles in any triangle is also 360°/2 = 180°.

  Note: Following the line of argument used in grade 5 to find the area of a triangle, we note that (a) two identical copies of any triangle can be arranged to form a parallelogram, and (b) any parallelogram can be transformed into a rectangle with the same angle sum by moving a triangle from one side of the parallelogram to the other.

  Note: Alternatively, following the secondary argument offered in grade 5, one can drop a perpendicular to divide any triangle into two right triangles. The sum of the interior angles of each of these right triangles is 180°, but when put together they include two superfluous right angles. Subtracting these yields 180° + 180° – (90° + 90°) = 180° as the sum of the interior angles of any triangle.

  

• Since any quadrilateral can be divided into two triangles the sum of the angles in a quadrilateral is also 2 × 180° = 360°.

  

b. Use a protractor, ruler, square, and compass to draw triangles and quadrilaterals from data given in either numerical or geometric form.

• Draw a variety of triangles (right, isosceles, acute, obtuse) and quadrilaterals (squares, rectangles, parallelograms, trapezoids) of different dimensions.
• Verify basic properties of triangles and quadrilaterals by direct measurement.

  Note: Verification by measurement requires many examples, especially some with relatively extreme or uncommon dimensions.

  Examples: In parallelograms, opposite sides and opposite angles are equal; in rectangles, diagonals are equal.

  Example: Cut any triangle out of paper and tear it into three parts so that each part contains one of the triangle’s vertices. Notice that when the angles are placed together, the edge is straight (180°).

  

• Explore properties of triangles and quadrilaterals with dynamic geometry software.

  Note: Is a parallelogram a trapezoid? It depends on the definition of trapezoid. If a trapezoid is defined as a quadrilateral with at least one pair of parallel edges, parallelograms become special cases of trapezoids. However, dictionaries usually define a trapezoid as a quadrilateral with exactly one pair of parallel edges, thereby distinguishing between parallelograms and trapezoids. Mathematicians generally prefer nested definitions as conditions become more or less restrictive. For example, all positive whole numbers are integers, all integers are rational numbers, and all rational numbers are real numbers. So in the world of mathematics, squares are rectangles, rectangles are parallelograms, and parallelograms are trapezoids.

G.6.2 Understand and use the concepts of translation, rotation, reflection, and congruence in the plane.

a. Recognize that every rigid motion of a polygon in the plane can be created by some combination of translation, rotation, and reflection.

• Translation, rotation, and reflection move a polygonal figure in the plane from one position to another without changing its length or angle measurements.
• Explore the meaning of rotation, translation, and reflection through drawings and hands-on experiments.

  Example: The figure below illustrates how a rigid motion can be decomposed into a series of three steps: translation, reflection, and rotation.

  

b. Understand several different characterizations and examples of congruence.

• Two figures in the plane are called congruent if they have the same size and same shape.
• Two shapes are congruent if they can be made to coincide when superimposed by means of a rigid motion.
• Two polygons are congruent if they have the same number of sides and if their corresponding sides and angles are equal.

  Note: Historically—beginning with Euclid—congruence applied only to polygons and used this as the definition. Indeed, the important properties of congruence are typically only about polygonal figures.

  

• Congruent figures in the plane are those that can be laid on top of one another by rotations, reflections, and translations.

  Note: Using rotations, reflections, and translations to define congruence gives precise meaning to the intuitive idea of congruence as "same size and same shape," thus permitting a precise definition of congruence for shapes other than polygons.

  Note: Technological aids (transparencies, dynamic geometry programs) help greatly in studying rigid motions.

c. Identify congruent polygonal figures.

• Understand why the two triangles formed by drawing a diagnonal of a parallelogram are congruent.
• Understand why the two triangles formed by bisecting the vertex angle of an isosceles triangle are congruent.

G.6.3 Understand and use different kinds of symmetry in the plane.

a. Symmetries in the plane are actions that leave figures unchanged.

• Explore and explain the symmetry of geometric figures from the standpoint of rotations, reflections and translations.

b. Identify and utilize bilateral and rotational symmetry in regular polygons.

• A regular polygon is a polygon whose sides and angles are all equal.
• Bilateral symmetry means there is a reflection that leaves everything unchanged.
• Regular polygons have rotational symmetries.

c. Identify and utilize translational symmetry in tessellations of the plane.

• Most tessellations have translation symmetry.

d. Use reflections to study isosceles triangles and isosceles trapezoids.

• Understand why the base angles of an isosceles triangle are equal.
• Understand why the bisector of the angle opposite the base is the perpendicular bisector of the base.

  Note: Draw an angle bisector on an isosceles triangle. Fold the drawing along the angle bisector (that is, reflect across the angle bisector). Then the base vertices collapse on each other: Both angles are equal, thus the angle bisector also bisects the base.

  

Algebra (A)

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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