# MA.7.NSO.1.2

Rewrite rational numbers in different but equivalent forms including fractions, mixed numbers, repeating decimals and percentages to solve mathematical and real-world problems.

### Examples

Justin is solving a problem where he computes and his calculator gives him the answer 5.6666666667. Justin makes the statement that ; is he correct?
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Rational Number

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students rewrote positive rational numbers in different but equivalent forms as long as the decimal form is terminating. This expectation expands to all rational numbers in grade 7, including those with repeating decimals, as well as using this skill to solve mathematical and real-world problems. In grade 8, students will learn about irrational numbers as well as working to plot, order and compare rational and irrational numbers.
• When solving problems with numbers written in various forms, students must be able to convert between these forms to perform operations or make comparisons (MTR.2.1).
• Students should begin to develop charts, like the one below, that allow them to find patterns within the different forms of rational numbers. Students should have common fractions, decimals and percentages at their disposal in order to move to ones that are more difficult to determine.

• Students should have practice with and without the use of technology to rewrite rational numbers in different but equivalent forms.
• Students should work with simple problems to showcase how truncating repeated decimals may result in incorrect solutions.
• For example, using the fractional value of $\frac{\text{1}}{\text{3}}$ may provide a more precise answer than using the truncated decimal of 0.33.
• Students should use reasonableness to determine if it is appropriate to use a specific equivalent form over another one when problem solving (MTR.6.1).

### Common Misconceptions or Errors

• Students may not differentiate between terminating decimals, repeating decimals and rounded decimals, and they may not use them appropriately within the given contexts.
• Students may incorrectly truncate repeating decimals when problem solving.
• Students may incorrectly divide when the quotient is not a whole number.
• For example, students may use the remainder of a problem as a decimal representation.

### Strategies to Support Tiered Instruction

• Instruction includes the use of estimation to find the approximate decimal value of a fraction or mixed number before rewriting in decimal form to help with correct placement of the decimal point.
• Teacher provides opportunities for students to explore and discuss the differences between repeating and truncated decimals and the impact of truncating repeating decimals when solving problems.
• For example, provide students with the equation $y$= $\frac{\text{1}}{\text{3}}$$x$ and have them create a table of values comparing using $\frac{\text{1}}{\text{3}}$, 0.3, 0.333 and 0.33333 as the constant of proportionality. Students can discuss the differences in $y$-values and importance of using exact values in some cases and approximate values in others.

• Instruction includes co-creating a graphic organizer to highlight the differences between terminating decimals, repeating decimals, and rounded decimals.

Convert each of the following to an equivalent form in order to compare their values.
$\frac{\text{1}}{\text{5}}$ − 0.4  65%  − 2$\frac{\text{1}}{\text{3}}$ 5.75  $\frac{\text{9}}{\text{7}}$ 123%  2.3¯
• Part A. Graph the numbers on a number line to determine increasing order.

• Part B. Robin plotted her number line using all decimals, whereas Courtney plotted them using the original forms. Describe why both would be acceptable answers.

Instructional Task 2 (MTR.2.1, MTR.4.1, MTR.5.1)

Complete the table to identify equivalent forms of each number. Explain how you approached your solutions. Prompting questions: What patterns did you use? How did you start? Which values in the table are you most comfortable in starting with?

### Instructional Items

Instructional Item 1
All of the students in first period were given a glue stick to help build their interactive notebook. Benny said he has already used $\frac{\text{2}}{\text{3}}$ of his glue while Juniper has used 70% of hers. Which student has the most glue remaining for their notebooks?

Instructional Item 2
Ishana manages a corner store and wishes to give a discount to her customers for the holiday. If she subtracts 0.15 of the cost of any item in the store, what percent should her sale sign promote?

Instructional Item 3
Write three equivalent forms for 5$\frac{\text{7}}{\text{8}}$.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812020: Access M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.7.NSO.1.AP.2: Rewrite positive rational numbers in different but equivalent forms such as fractions, mixed numbers, repeating decimals and/or percentages to solve problems.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Educational Game

Fraction Quiz:

Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving fractions. You can even choose difficulty level, question types, and time limit.

Type: Educational Game

## Formative Assessments

Find Decimal Using Long Division:

Students are asked to use long division to convert four different fractions to equivalent decimals and to identify those that are rational.

Type: Formative Assessment

Quotients of Integers:

Students are given an integer division problem and asked to identify fractions which are equivalent to the division problem.

Type: Formative Assessment

Fraction to Decimal Conversion:

Students are given a fraction to convert to a decimal and are asked to determine if the decimal repeats.

Type: Formative Assessment

## Lesson Plans

The Watergate Effect part 3:

Students will create a circle graph to display categorical data of the public presidential approval rates after the Supreme Court Case United States v. Nixon. Students will graph results independently and compare them to the circle graphs created during the Watergate Effect Part 1 Lesson (Resource ID#: 208926) and the Watergate Effect Part 2 Lesson (Resource ID#: 210122) to discuss the trend of the data over the entirety of the Supreme Court case.

Type: Lesson Plan

The Watergate Effect part 2:

Students will create a circle graph to display categorical data of the public presidential approval rates during the Supreme Court Case United States v. Nixon. Students will graph results in pairs/groups and compare them to the circle graph created during the Watergate Effect Part 1 Lesson (Resource ID#: 208926).

Type: Lesson Plan

Generating Equivalent Forms of Numbers Using the Legislative Branch of the Government:

Students will rewrite fractions, decimals, and percentages in equivalent forms to compare the number of seats that each state has in the U.S. House of Representatives to the total number of seats in the House of Representatives, in this integrated lesson plan.

Type: Lesson Plan

The Watergate Effect Part 1:

Students will create a circle graph to display categorical data of the public presidential approval rates of Richard Nixon before the Supreme Court Case United States v. Nixon. Students will calculate percentages and central angle degrees to graph results in pairs/groups and analyze the results in this integrated lesson plan.

Type: Lesson Plan

Fractions and Percentages of the Legislative Branch:

Students will use fractions, decimals, and percentages to compare the number of seats that Florida (as well as other states) has in the U.S. House of Representatives to the total amount of seats in the House of Representatives in this integrated lesson plan.

Type: Lesson Plan

Rewriting Rational Numbers to Analyze International Organizations (Part 1: United Nations):

Students will analyze regional membership of the United Nations to represent the part to a whole relationship as a fraction. Students will rewrite rational numbers in equivalent forms while examining the purpose of the United Nations and the United States’ role as a member in this integrated lesson plan.

Type: Lesson Plan

Independent Compound Probability:

During this lesson, students will use Punnett Squares to determine the probability of an offspring's characteristics.

Type: Lesson Plan

Fast Food Frenzy:

In this activity, students will engage critically with nutritional information and macronutrient content of several fast food meals. This is an MEA that requires students to build on prior knowledge of nutrition and working with percentages.

Type: Lesson Plan

Students will learn how to calculate markup, markdown, percent increase, and percent decrease. Using sales "ad" inserts from the internet, newspapers, and store flyers, students will understand how these concepts apply to real-world situations.

Type: Lesson Plan

Who's Being Irrational?:

In this lesson, students will learn how irrational numbers differ from rational numbers. The students will complete a graphic organizer that categorizes rational and irrational numbers. Students will also be able to identify irrational numbers found in the real world.

Type: Lesson Plan

Savvy Shopping:

This is the second part of the CPalms lesson titled Markup and Make Money. In Savvy Shopping students will shop at their peers' store and buy items. If it is discounted, they will have to calculate the revised price. They will then find the total price including the tax.

Type: Lesson Plan

Have you ever heard students ask the question, "Why do I have to learn this?" This lesson answers that question because it requires the students to apply their knowledge in real world scenarios but does not teach a basic conceptual understanding of percentages. The teacher may use the whole lesson or select specific problems.

Type: Lesson Plan

Rational vs Irrational:

Students will organize the set of real numbers and be able to identify when a number is rational or irrational. They will also learn the process of how to change a repeating decimal to its equivalent fraction.

Type: Lesson Plan

Predicting the decimal equivalent for a fraction - terminating or repeating?:

This lesson encourages students to make an important discovery. Will a given fraction yield a terminating or repeating decimal? Discussion includes why knowing this is important. The lesson is structured to allow exploration, discovery, and summarization.

Type: Lesson Plan

This activity will allow students to explore the concept of simple probability using a random selection of multi-colored beads.

Type: Lesson Plan

Markup and Make Money:

In this lesson students will create their own imaginary store with at least 15 items to sell. They will begin with a discussion and then learn about markup. They will use their knowledge to calculate prices and create a display for their store. This is the first of 2 lessons (next lesson is Savvy Shopping, Resource ID 48879), which allows students to shop in their peer's store to calculate discount and tax.

Type: Lesson Plan

Really! I'm Rational!:

In this lesson students will gain an understanding of how repeating decimals are converted into a ratio in the form of by completing an exploration worksheet. They will conclude that any number which can be written in this form is called a rational number.

Type: Lesson Plan

Equivalent fractions approach to non-repeating decimals:

The purpose of the task is to get students to reflect on the definition of decimals as fractions (or sums of fractions), at a time when they are seeing them primarily as an extension of the base-ten number system and may have lost contact with the basic fraction meaning. Students also have their understanding of equivalent fractions and factors reinforced.

Repeating Decimal as Approximation:

The student is asked to complete a long division which results in a repeating decimal, and then use multiplication to "check" their answer. The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation.

Estimating Square Roots:

By definition, the square root of a number n is the number you square to get n. The purpose of this task is to have students use the meaning of a square root to find a decimal approximation of a square root of a non-square integer. Students may need guidance in thinking about how to approach the task.

Converting Decimal Representations of Rational Numbers to Fraction Representations:

Requires students to "convert a decimal expansion which repeats eventually into a rational number." Despite this choice of wording, the numbers in this task are rational numbers regardless of the choice of representation. For example, 0.333¯ and 1/3 are two different ways of representing the same number.

Calculating and Rounding Numbers:

In this task, students explore some important mathematical implications of using a calculator. Specifically, they experiment with the approximation of common irrational numbers such as pi (p) and the square root of 2 (v2) and discover how to properly use the calculator for best accuracy. Other related activities involve converting fractions to decimal form and a concrete example where rounding and then multiplying does not yield the same answer as multiplying and then rounding.

## STEM Lessons - Model Eliciting Activity

Fast Food Frenzy:

In this activity, students will engage critically with nutritional information and macronutrient content of several fast food meals. This is an MEA that requires students to build on prior knowledge of nutrition and working with percentages.

## MFAS Formative Assessments

Find Decimal Using Long Division:

Students are asked to use long division to convert four different fractions to equivalent decimals and to identify those that are rational.

Fraction to Decimal Conversion:

Students are given a fraction to convert to a decimal and are asked to determine if the decimal repeats.

Quotients of Integers:

Students are given an integer division problem and asked to identify fractions which are equivalent to the division problem.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Educational Game

Fraction Quiz:

Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving fractions. You can even choose difficulty level, question types, and time limit.

Type: Educational Game

Repeating Decimal as Approximation:

The student is asked to complete a long division which results in a repeating decimal, and then use multiplication to "check" their answer. The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation.

Estimating Square Roots:

By definition, the square root of a number n is the number you square to get n. The purpose of this task is to have students use the meaning of a square root to find a decimal approximation of a square root of a non-square integer. Students may need guidance in thinking about how to approach the task.

Converting Decimal Representations of Rational Numbers to Fraction Representations:

Requires students to "convert a decimal expansion which repeats eventually into a rational number." Despite this choice of wording, the numbers in this task are rational numbers regardless of the choice of representation. For example, 0.333¯ and 1/3 are two different ways of representing the same number.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Repeating Decimal as Approximation:

The student is asked to complete a long division which results in a repeating decimal, and then use multiplication to "check" their answer. The purpose of the task is to have students reflect on the meaning of repeating decimal representation through approximation.

Estimating Square Roots:

By definition, the square root of a number n is the number you square to get n. The purpose of this task is to have students use the meaning of a square root to find a decimal approximation of a square root of a non-square integer. Students may need guidance in thinking about how to approach the task.