Operations with Rational Numbers

Lesson 5

Math

Unit 2

7th Grade

Lesson 5 of 18

Objective


Determine efficient ways to add rational numbers with and without the number line.

Common Core Standards


Core Standards

  • 7.NS.A.1.B — 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.
  • 7.NS.A.1.D — Apply properties of operations as strategies to add and subtract rational numbers.

Foundational Standards

  • 6.NS.C.6
  • 6.NS.C.7.C

Criteria for Success


  1. Determine an efficient way, through repeated reasoning, to add two rational numbers with the same sign: Add the absolute values of the numbers and use the common sign (MP.8).
  2. Determine an efficient way, through repeated reasoning, to add two rational numbers with opposite signs: Subtract the absolute values of the numbers and use the sign of the number with the greater absolute value (MP.8).
  3. Understand, using context, why efficient methods for adding rational numbers work. 
  4. Determine if a sum of two numbers is positive or negative without calculating the sum. 
  5. Find the sum of rational numbers with and without a number line.

Tips for Teachers


Though it is important for students to develop efficient methods to compute with rational numbers, it is equally important that they understand the reasoning behind these methods. The first two Anchor Problems aim to guide students toward understanding how to add rational numbers in general, by modeling travel along a number line (MP.4).

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Anchor Problems

25-30 minutes


Problem 1

Revisit Joshua’s road from Lesson 4 Anchor Problem #1 to answer the questions that follow.

a.    Joshua travels 5 miles east and then 2 miles east, represented by the equation $${5+2=7}$$.

  • Model this situation on the number line using arrows, and explain what each term in the equation represents in the context.
  • Explain, using the context of the situation, why adding two positive integers will always give you a greater positive integer. 

b.   Joshua travels 5 miles west and then 2 miles west, represented by the equation $${-5+(-2)=-7}$$.

  • Model this situation on the number line using arrows, and explain what each term in the equation represents in the context.
  • Explain, using the context of the situation, why adding two negative integers will always give you a lesser negative integer. 

Guiding Questions

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Student Response

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Problem 2

The number line below represents Joshua’s road. Use it to answer the questions below.

a.   Joshua takes two trips and represents them using the addition problem $${-5+2}$$.

  1. How far and in which direction did Joshua travel on his first trip?
  2. How far and in which direction did Joshua travel on his second trip? 
  3. In which direction did he travel farther, and by how much?
  4. How far and in which direction is he from home? Where is this on the number line?

b.   Joshua takes another two trips, starting from home, and represents them using the addition problem $${6+(-8)}$$. Answer the same questions (i.)–(iv.) from part A.

c.   Explain how you can find the sum of two numbers with opposite signs without using a number line.

Guiding Questions

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Student Response

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Problem 3

For problems A - H, determine if the sum will be positive, negative, or zero. You do not need to find the sum. For problems I - L, find the sum.

a.       $${4+(-1)}$$

b.      $${-14+(-10)}$$

c.     $${7+(-8)}$$

d.     $${12.25+13.7}$$

e.       $${-22+20}$$

f.     $${-5 {1\over2}+7}$$

g.       $${-4+4}$$

h.    $${-7+(-7)}$$

i.    Find the sum. $${42+(-10)}$$

j.   Find the sum: $${-15+23}$$

k.    Find the sum: $${-18 {1\over2}+\left ( -4 {3\over4} \right )}$$

l.     Find the sum: $${100+(-164)}$$

Guiding Questions

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Student Response

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Problem Set

15-20 minutes


Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

5-10 minutes


Problem 1

Which addition problems below will have a positive sum? Select all that apply.

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Student Response

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Problem 2

Write an addition problem that has a sum of $${-24}$$. Include one positive number and one negative number in your problem.

Student Response

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Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems where students explain or model the conclusions from Anchor Problems #1 and #2 in their own words.
  • Students can play a variation of Integer War using a standard deck of playing cards: Each student can draw two cards and find the sum—the higher sum wins, or each student can turn over a card, and whoever correctly determines the sum first can keep the two cards. 
  • Include problems where students are given a sum and determine two addends that add to that sum.
  • Challenge: The sum of three consecutive integers is -81. What are the three integers?

Next

Efficiently add and reason about sums of rational numbers.

Lesson 6
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Adding and Subtracting Rational Numbers

Topic B: Multiplying and Dividing Rational Numbers

Topic C: Using all Four Operations with Rational Numbers

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