Lesson 16 | Operations with Rational Numbers | 7th Grade Mathematics

Objective

Convert rational numbers to decimals using long division and equivalent fractions.

Common Core Standards

Core Standards

The core standards covered in this lesson

7.NS.A.2.D — 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.

The Number System

7.NS.A.2.D — 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.

Foundational Standards

The foundational standards covered in this lesson

5.NF.B.3

Number and Operations—Fractions

5.NF.B.3 — 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?
6.NS.B.2

The Number System

6.NS.B.2 — Fluently divide multi-digit numbers using the standard algorithm.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Define terminating decimals as numbers with a finite number of digits after the decimal point, and define repeating decimals as numbers with an infinite number of digits after the decimal point in which a digit or group of digits repeats indefinitely.
Convert fractions to decimals using long division.
Convert between decimals and fractions using place value understanding and equivalent fractions.
Recognize when it is more efficient to convert fractions to decimals using equivalent fractions with denominators of powers of 10 (MP.7).
Attend to precision in calculations and strategic use of place value understanding (MP.6).

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Lesson Materials

Calculators (1 per student)
Graph Paper (1 sheet per student)

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Using a calculator, find the decimal quotients below.

$${{1\over 2}, {1\over 3}, {1\over 4}, {1\over 5}, {1\over 6}, {1\over 7}, {1\over 8}, {1\over 9}, {1\over 10}, {1\over 11}}$$

Looking at your quotients, organize your answers into two categories. Describe your reasoning behind why you chose those two categories.

Guiding Questions

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References

EngageNY Mathematics Grade 7 Mathematics > Module 2 > Topic B > Lesson 14 — Examples 1 and 2

Grade 7 Mathematics > Module 2 > Topic B > Lesson 14 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

a. Write each decimal as a fraction.

$$0.35$$
$$1.64$$
$$2.09$$
$$-3.125$$

b. Write each rational number as a decimal.

$${-{5\over 6}}$$
$${9\over 16}$$
$${-7\over 9}$$
$${{25\over12}}$$

Guiding Questions

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Problem 3

Malia found a "short cut" to find the decimal representation of the fraction $${{117\over250}}$$. Rather than use long division, she noticed that because $${250 \times 4 = 1000}$$,

$${{{117\over250}}} ={ {117 \times 4}\over {250 \times 4}} = {468\over1000} = 0.468$$

a. For which of the following fractions does Malia's strategy work to find the decimal representation?

$${{1\over 3}, {3\over 4}, -{6\over 25}, {18\over 7}, {13\over 8}, -{113\over 40}}$$

For each one for which the strategy does work, use it to find the decimal representation.

b. In general, for which denominators can Malia's strategy work?

Guiding Questions

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References

Illustrative Mathematics Equivalent Fractions Approach to Non-Repeating Decimals

Equivalent Fractions Approach to Non-Repeating Decimals, accessed on Aug. 14, 2017, 4:14 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Fishtank Plus Content

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

What is the decimal value of $${{20\over7}}$$?

Problem 2

Write two examples of fractions that have a terminating decimal equivalent, and two examples of fractions that have a repeating decimal equivalent.

Problem 3

Compare each pair of rational numbers using an inequality sign.

a. $${{1{5\over 8}}}$$ ______ $${{1.6}}$$

b. $${{1{5\over 8}}}$$ ______ $$-{{1.6}}$$

c. $$-{{1{5\over 8}}}$$ ______ $${{1.6}}$$

d. $$-{{1{5\over 8}}}$$ _______ $$-{{1.6}}$$

Student Response

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Student Response

An example response to the Target Task at the level of detail expected of the students.

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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.

MARS Summative Assessment Tasks for Middle School Division
EngageNY Mathematics Grade 7 Mathematics > Module 2 > Topic B > Lesson 14 — Problem Set
EngageNY Mathematics Grade 7 Mathematics > Module 2 > Topic B > Lesson 13 — Exercises, Examples, and Problem Set
Illustrative Mathematics Decimal Expansion of Fractions
Open Middle Converting Fractions to Repeating Decimals — Challenge
Illustrative Mathematics Repeating Decimal as Approximation

Lesson 15

Lesson 17

Lesson Map

Topic A: Adding and Subtracting Rational Numbers

Represent rational numbers on the number line. Define opposites, absolute value, and rational numbers

7.NS.A.1

Compare and order rational numbers. Write and interpret inequalities to describe the order of rational numbers.

7.NS.A.1

Describe situations in which opposite quantities combine to make zero.

7.NS.A.1.A

Model the addition of integers using a number line.

7.NS.A.1.B 7.NS.A.1.D

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

7.NS.A.1.B 7.NS.A.1.D

Efficiently add and reason about sums of rational numbers.

7.NS.A.1.B 7.NS.A.1.D

Understand subtraction as addition of the opposite value (or additive inverse).

7.NS.A.1.C 7.NS.A.1.D

Find and represent the distance between two rational numbers as the absolute value of their difference.

7.NS.A.1.C

Subtract rational numbers with and without the number line.

7.NS.A.1.C 7.NS.A.1.D

Add and subtract rational numbers efficiently using properties of operations.

7.NS.A.1.D

Add and subtract rational numbers using a variety of strategies.

7.NS.A.1 7.NS.A.1.A 7.NS.A.1.B 7.NS.A.1.C 7.NS.A.1.D

Topic B: Multiplying and Dividing Rational Numbers

Determine the rules for multiplying signed numbers.

7.NS.A.2.A 7.NS.A.2.C

Multiply signed rational numbers and interpret products in real-world contexts.

7.NS.A.2.A 7.NS.A.2.C

Determine the rules for dividing signed numbers.

7.NS.A.2.B 7.NS.A.2.C

Divide signed rational numbers and interpret quotients in real-world contexts.

7.NS.A.2.B 7.NS.A.2.C

Convert rational numbers to decimals using long division and equivalent fractions.

7.NS.A.2.D

Multiply and divide with rational numbers using properties of operations.

7.NS.A.2 7.NS.A.2.C

Topic C: Using all Four Operations with Rational Numbers

Solve problems with rational numbers and all four operations.

7.NS.A.3

Operations with Rational Numbers

Objective

Common Core Standards

Core Standards

The Number System

Foundational Standards

Number and Operations—Fractions

The Number System

Criteria for Success

Tips for Teachers

Lesson Materials

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Problem Set

Target Task

Problem 1

Problem 2

Problem 3

Student Response

Student Response

Additional Practice

Lesson Map

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