### Clarifications

*Clarification 1:*Instruction focuses on making connections between visual models, the relationship between multiplication and division, reciprocals and algorithms.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**6

**Strand:**Number Sense and Operations

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Area Model
- Commutative Property
- Dividend
- Divisor
- Expression

### Vertical Alignment

Previous Benchmarks

http://flbt5.floridaearlylearning.com/standards.html

Next Benchmarks

### Purpose and Instructional Strategies

In grade 5, students multiplied fractions by fractions with procedural reliability and explored how to divide a unit fraction by a whole number and a whole number by a unit fraction. In grade 6, students become procedurally fluent with multiplication and division of positive fractions. The expectation is to utilize skills from the procedural reliability stage to become fluent with an efficient and accurate procedure, including a standard algorithm. In grade 7, students will become fluent in all operations with positive and negative rational numbers.- Instruction includes making connections to the distributive property when multiplying fractions.
- For example, when multiplying 1$\frac{\text{1}}{\text{2}}$ by $\frac{\text{3}}{\text{4}}$, it can be written as (1+ $\frac{\text{1}}{\text{2}}$) $\frac{\text{3}}{\text{4}}$ to determine $\frac{\text{9}}{\text{8}}$ as the product.

- Instruction includes making connections to inverse operations when multiplying or dividing fractions.
- For example, when determining $\frac{\text{3}}{\text{4}}$ ÷ $\frac{\text{5}}{\text{8}}$, students can write the equation $\mathrm{x(}$$\frac{\text{5}}{\text{8}}$) = $\frac{\text{3}}{\text{4}}$ and then solve for x$$.

- Instruction focuses on appropriate academic vocabulary, such as reciprocal. Avoid focusing on tricks such as “keep-change-flip.” Using academic language and procedures allow for students to connect to future mathematics
*(MTR.5.1).*- For example, $\frac{\text{3}}{\text{4}}$ ÷ $\frac{\text{5}}{\text{8}}$ can be read as “How many five-eighths are in three-fourths?”

- Instruction includes using concrete and pictorial models, writing a numerical sentence that relates to the model and discovering the pattern or rules for multiplying and dividing fractions by fractions
*(MTR.2.1, MTR.3.1, MTR.5.1).*- Area Model
- Linear Model
- Bar Model

- Area Model

- Instruction includes providing opportunities for students to analyze their own and others’ calculation methods and discuss multiple strategies or ways of understanding with others
*(MTR.4.1).* - Students should develop fluency with and without the use of a calculator when performing operations with positive fractions.

### Common Misconceptions or Errors

- Students may forget that common denominators are not necessary for multiplying or dividing fractions.
- Students may have incorrectly assumed that multiplication results in a product that is larger than the two factors. Instruction continues with students assessing the reasonableness of their answers by determining if the product will be greater or less than the factors within the given context.
- Students may have incorrectly assumed that division results in a quotient that is smaller than the dividend. Instruction continues with students assessing the reasonableness of their answers by determining if the quotient will be greater or less than the dividend within the given context.

### Strategies to Support Tiered Instruction

- Teacher encourages and allows for students who have a firm understanding of multiplying and dividing decimals to convert the provided fractional values to their equivalent decimal form before performing the desired operation and converting the solution back to fractional form.
- Instruction includes the use of fraction tiles, fraction towers, or similar manipulatives to make connections between physical representations and algebraic methods.
- Instruction includes the co-creation of a graphic organizer utilizing the mnemonic device Same, Inverse Operation, Reciprocal (S.I.R.) for dividing fractions, which encourages the use of correct mathematical terminology, and including examples of applying the mnemonic device when dividing fractions, whole numbers, and mixed numbers.
- Teacher provides students with flash cards to practice and reinforce academic vocabulary.
- Instead of multiplying by the reciprocal to divide fractions, an alternative method could include rewriting the fractions with a common denominator and then dividing the numerators and the denominators.
- For example, $\frac{\text{5}}{\text{6}}$÷ $\frac{\text{3}}{\text{2}}$ is equivalent to $\frac{\text{5}}{\text{6}}$÷ $\frac{\text{9}}{\text{6}}$ which is equivalent to $\frac{\text{5/9}}{\text{1}}$ which is equivalent to $\frac{\text{5}}{\text{9}}$.

- Instruction provides opportunities to assess the reasonableness of answers by determining if the product will be greater or less than the factors within the given context.
- Instruction provides opportunities to assess the reasonableness of answers by determining if the quotient will be greater or less than the dividend within the given context.

### Instructional Tasks

*Instructional Task 1*

Jasmine wants to build a 2$\frac{\text{5}}{\text{6}}$ meters long garden path paved with square stones that measure $\frac{\text{1}}{\text{4}}$ meter on each side. There will be no spaces between the stones.

**(MTR.2.1, MTR.4.1)**- Part A. Create a model that could be used to answer the following question: How many stones are needed for the path?
- Part B. How many stones are needed for the path?

*A container at a juicing plant holds 6$\frac{\text{2}}{\text{3}}$ tons of oranges. The plant can juice 1$\frac{\text{1}}{\text{2}}$ tons of oranges per day. At this rate, how long will it take to empty the container?*

Instructional Task 2

Instructional Task 2

**(MTR.3.1, MTR.6.1)***Explain using visual models why $\frac{\text{4}}{\text{5}}$× $\frac{\text{2}}{\text{3}}$= $\frac{\text{8}}{\text{15}}$ .$\frac{\text{}}{}$*

Instructional Task 3

Instructional Task 3

**(MTR.2.1)**

### Instructional Items

*Instructional Item 1*

What is the value of the expression $\frac{\text{3}}{\text{5}}$ ÷$\frac{\text{5}}{\text{8}}$ ?

*What is the value of the expression 8$\frac{\text{1}}{\text{10}}$÷$\frac{\text{5}}{\text{8}}$ ?*

Instructional Item 2

Instructional Item 2

**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

## Perspectives Video: Experts

## Professional Development

## Teaching Ideas

## Tutorials

## Video/Audio/Animation

## MFAS Formative Assessments

Students are asked to complete two fraction division problems – one with fractions and one with mixed numbers.

Students are asked to write and evaluate a numerical expression involving division of fractions and mixed numbers to model and solve a word problem.

Students are asked to explain the relationship between a fraction division word problem and either a visual model or an equation.

## 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

## Tutorials

In this tutorial, you will see how mixed numbers can be divided.

Type: Tutorial

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial

## Video/Audio/Animation

When working with fractions, divisions can be converted to multiplication by the divisor's reciprocal. This chapter explains why.

Type: Video/Audio/Animation

## Parent Resources

## Tutorials

This five-minute video answers the question "Must one always invert and multiply?" when dividing fractions. An alternative algorithm is presented which works well in certain cases. The video focuses on sense-making in using either method, and on judging the reasonableness of answers.

Type: Tutorial

The video describes how to multiply fractions and state the answer in lowest terms.

Type: Tutorial