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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 performed operations with integers, multiplied and divided positive multi-digit numbers with decimals to the thousandths and computed products and quotients of positive fractions by positive fractions, including mixed numbers with procedural fluency. In grade 7, students perform all four operations with positive and negative rational numbers with procedural fluency. In grade 8, they will expand to operations with rational numbers including exponents and radicals, and will perform operations with rational numbers expressed in scientific notation.- This benchmark is the completion of arithmetic operations with rational numbers (MTR.3.1).
- Instruction includes the possibility that the division of two fractions can be written as a complex fraction. This connection will be important when students work with algebraic expressions in later grades.
- Students should develop fluency with and without the use of a calculator when performing operations with rational numbers.
Common Misconceptions or Errors
- Students may think the product of a fraction and another fraction is greater than either factor. Use manipulatives or models referenced in previous grade levels to support conceptual understanding (MTR.2.1).
- Students may incorrectly believe that dividing by is the same as dividing by 2.
- Students may incorrectly solve complex fractions by multiplying the two fractions.
Strategies to Support Tiered Instruction
- Instruction includes the use of fraction tiles to represent operations with positive fractions while simultaneously recording the equivalent numerical expressions.
- Instruction includes the use of base ten blocks to represent operations with positive decimals while simultaneously recording the equivalent numerical expressions.
- Instruction includes the use of two-color counters to represent operations with positive and negative whole numbers while simultaneously recording the equivalent numerical expressions.
- Teacher co-creates a graphic organizer with students to review operations with positive fractions and operations with integers to assist when applying operations with rational numbers.
- Instruction includes using manipulatives or models referenced in previous grade levels to support conceptual understanding.
Instructional Tasks
Instructional Task 1 (MTR.7.1)Daliah purchases eggs by the dozen for her two children. Each day, Zane eats carton and Amare eats carton. A carton of 12 eggs costs $1.65.
- Part A. How much does Daliah spend on eggs for her two children in 30 days?
- Part B. During one of her shopping trips, Daliah finds that her grocery store has started to sell cartons of 18 eggs for $2.25. If she begins to purchase these cartons, how much does Daliah spend on eggs for her two children in 30 days? After how many days will Daliah spend more than $50? Explain your reasoning.
Instructional Task 2 (MTR.3.1)
Given = −2 and = , calculate the following:
- +
- −
- ·
Instructional Items
Instructional Item 1Determine the product of and −1.2.
Instructional Item 2
What is the value of the expression 7.24 − 5.01 − 78.4?
Instructional Item 3
What is the value of the expression
*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
Educational Software / Tool
Formative Assessments
Lesson Plans
Perspectives Video: Experts
Problem-Solving Task
Tutorial
Video/Audio/Animation
STEM Lessons - Model Eliciting Activity
In this Model Eliciting Activity, MEA, students will use the order of operations to develop and apply a scoring system to evaluate different lawn tractors for a company. Students will justify their rankings using their analysis, calculations, and scoring system.
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. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
In this Model Eliciting Activity, MEA, students will analyze and interpret data to recommend the best vehicle purchases for a school district. Students will work collaboratively to perform calculations that can be used to make comparisons and create composite scores for each vehicle.
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. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
In this Model Eliciting Activity, MEA, students will research a list of companies to invest in through purchasing stocks. Students will calculate the amount invested and readjust their investment choices.
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.
In this Model Eliciting Activity, MEA, students will use unit rates and scoring systems to analyze and interpret data to recommend the best store from which a family restaurant should purchase its weekly non-frozen food items.
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. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
This is a NASA-themed, MEA (Model Eliciting Activity) lesson that challenges students to solve a real world open ended problem, while promoting collaboration through teamwork. This lesson asks each group of students to choose five positions and assign salaries to the positions with a given budget of $500,000. The students' original decision (and "twist") will be based on information from the client's letter(s) and data set(s). Groups are to write a detailed letter to the client of the procedure used.
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. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx
In this Model Eliciting Activity, MEA, students will apply their knowledge of rational numbers and order of operations to analyze and compare data to provide recommendations on the best camera to use in an introductory photography class.
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. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://www.cpalms.org/cpalms/mea.aspx
In this Model Eliciting Activity, MEA, "Sticks and Stones May Break My Bones", teams of students work as forensic anthropologists and use equations to determine the height and gender of persons to whom a series of newly discovered bones may belong.
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.
The Whirl Wind Corporation would like to install Wind Turbines in the Mojave Desert. The company produces various models of these turbines and is looking for help in selecting the best one for the job.
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 add, subtract, multiply, and divide positive and negative fractions.
Students are asked to evaluate expressions involving multiplication of rational numbers and use the properties of operations to simplify calculations.
Students are asked to rewrite complex fractions as simple fractions in lowest terms.
Students are asked to describe a real-world context for a given expression involving the product of two rational numbers.
Students are asked to combine rational numbers, including fractions and decimals, and use the properties of operations to simplify calculations.
Students are asked to explain why the product of a positive and a negative rational number is negative.
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
Educational Software / Tool
In this activity, students solve arithmetic problems involving whole numbers, integers, addition, subtraction, multiplication, and division. This activity allows students to track their progress in learning how to perform arithmetic on whole numbers and integers. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Educational Software / Tool
Problem-Solving Task
The purpose of this task is to help solidify students' understanding of signed numbers as points on a number line and to understand the geometric interpretation of adding and subtracting signed numbers. There is a subtle distinction between a fraction and a rational number. Fractions are always positive, and when thinking of the symbol ab as a fraction, it is possible to interpret it as a equal-sized pieces where b pieces make one whole.
Type: Problem-Solving Task
Tutorial
In this example, we will work with three numbers in different formats: a percent, a decimal, and a mixed number.
Type: Tutorial
Parent Resources
Problem-Solving Task
The purpose of this task is to help solidify students' understanding of signed numbers as points on a number line and to understand the geometric interpretation of adding and subtracting signed numbers. There is a subtle distinction between a fraction and a rational number. Fractions are always positive, and when thinking of the symbol ab as a fraction, it is possible to interpret it as a equal-sized pieces where b pieces make one whole.
Type: Problem-Solving Task
