Unit 2: Operations with Rational Numbers
Students extend the operations of addition, subtraction, multiplication, and division to include positive and negative rational numbers, and build fluency with evaluating numerical expressions.
In Unit 2, seventh-grade students extend the operations of addition, subtraction, multiplication, and division to include positive and negative rational numbers. Standards 7.NS.1 and 7.NS.2 represent a culmination in the extension of the four operations to all rational numbers. In this unit, students model addition and subtraction on the number line, and through repeated reasoning and application of properties of operations, they determine efficient rules for computing with rational numbers (MP.8). Students gain the ability to model a greater scope of real-world contexts to include situations involving elevation, temperature changes, debts and credits, and proportional relationships with negative rates of change (MP.4). They also develop greater fluency with evaluating numerical expressions, using the properties of operations to increase their flexibility in approach.
Starting in first grade, students learn about the commutative and associative properties of addition, and the relationship between addition and subtraction. In third grade, students extend their understanding of the properties of operations to include multiplication and the distributive property. Throughout the years, students have applied these properties and relationships between the operations to whole numbers, fractions, and decimals. In seventh grade, all of these skills and concepts come together as students now operate with all rational numbers, including negative numbers.
In several upcoming units, seventh-grade students will rely on their increased number sense and ability to compute with rational numbers, in particular in Unit 3, Numerical and Algebraic Expressions, and in Unit 4, Equations and Inequalities. By the time students enter eighth grade, students should have a strong grasp on operating with rational numbers, which will be an underlying skill in many algebraic concepts. In eighth grade, students are introduced to irrational numbers, rounding out their understanding of the real number system before learning about complex numbers in high school.
Included in the materials for this unit are some activities that aim to support and build students’ fluency with integer computations, especially mental math. See our Guide to Procedural Skill and Fluency for additional information and strategy and activity suggestions.
Pacing: 22 instructional days (18 lessons, 3 flex days, 1 assessment day)
For guidance on adjusting the pacing for the 2021-2022 school year, see our 7th Grade Scope and Sequence Recommended Adjustments.
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The central mathematical concepts that students will come to understand in this unit
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 7th Grade Course Material Overview.
Terms and notation that students learn or use in the unit
To see all the vocabulary for Unit 2, view our 7th Grade Vocabulary Glossary.
Topic A: Adding and Subtracting Rational Numbers
Represent rational numbers on the number line. Define opposites and absolute value.
Compare and order rational numbers. Write and interpret inequalities to describe the order of rational numbers.
Describe situations in which opposite quantities combine to make zero.
Model the addition of integers using a number line.
Determine efficient ways to add rational numbers with and without the number line.
Efficiently add and reason about sums of rational numbers.
Understand subtraction as addition of the opposite value (or additive inverse).
Find and represent the distance between two rational numbers as the absolute value of their difference.
Subtract rational numbers with and without the number line.
Add and subtract rational numbers efficiently using properties of operations.
Add and subtract rational numbers using a variety of strategies.
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Topic B: Multiplying and Dividing Rational Numbers
Determine the rules for multiplying signed numbers.
Multiply signed rational numbers and interpret products in real-world contexts.
Determine the rules for dividing signed numbers.
Divide signed rational numbers and interpret quotients in real-world contexts.
Convert rational numbers to decimals using long division and equivalent fractions.
Multiply and divide with rational numbers using properties of operations.
Topic C: Using all Four Operations with Rational Numbers
Solve problems with rational numbers and all four operations.
The content standards covered in this unit
— Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
— Describe situations in which opposite quantities combine to make 0.
For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
— Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
— Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
— Apply properties of operations as strategies to add and subtract rational numbers.
— Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
— Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
— Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
— Apply properties of operations as strategies to multiply and divide rational numbers.
— Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
— Solve real-world and mathematical problems involving the four operations with rational numbers.
Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
Standards covered in previous units or grades that are important background for the current unit
— Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
— Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
— Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
— Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.
For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
— Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
— Apply properties of operations as strategies to add and subtract.
Students need not use formal terms for these properties.
To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.)
— Apply properties of operations as strategies to multiply and divide.
Students need not use formal terms for these properties.
Example: Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
Example: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)
— Understand division as an unknown-factor problem.
For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
— Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
— Fluently divide multi-digit numbers using the standard algorithm.
— Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
— Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
— Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
— Understand ordering and absolute value of rational numbers.
— Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars.
— Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Standards in future grades or units that connect to the content in this unit
— Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
For example, we define 5<sup>1/3</sup> to be the cube root of 5 because we want (5<sup>1/3</sup>)³ = 5(<sup>1/3</sup>)³ to hold, so (5<sup>1/3</sup>)³ must equal 5.
— Rewrite expressions involving radicals and rational exponents using the properties of exponents.
— Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
— Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
— Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²).
For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Numerical and Algebraic Expressions
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