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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?
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
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 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 = and have them create a table of values comparing using , 0.3, 0.333 and 0.33333 as the constant of proportionality. Students can discuss the differences in -values and importance of using exact values in some cases and approximate values in others.
- For example, provide students with the equation = and have them create a table of values comparing using , 0.3, 0.333 and 0.33333 as the constant of proportionality. Students can discuss the differences in -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.
Instructional Tasks
Instructional Task 1(MTR.2.1)Convert each of the following to an equivalent form in order to compare their values.
- 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 1All of the students in first period were given a glue stick to help build their interactive notebook. Benny said he has already used 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.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.
Related Courses
Related Access Points
Related Resources
Educational Game
Formative Assessments
Lesson Plans
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
Problem-Solving Tasks
STEM Lessons - Model Eliciting Activity
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.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.
MFAS Formative Assessments
Students are asked to use long division to convert four different fractions to equivalent decimals and to identify those that are rational.
Students are given a fraction to convert to a decimal and are asked to determine if the decimal repeats.
Students are given an integer division problem and asked to identify fractions which are equivalent to the division problem.
Student Resources
Educational Game
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
Problem-Solving Tasks
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
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.
Type: Problem-Solving Task
