Lesson 16: Solving Problems Involving Fractions

Solving Problems Involving Fractions

5 min

Teacher Prep

Setup

Students in groups of 2. Display problems for all to see. 1 minute of quiet think time, 1 minute of partner discussion, followed by a whole-class discussion.

Narrative

This Warm-up reinforces students’ understanding of what each of the four operations (addition, subtraction, multiplication, and division) does when performed on fractions. The same pair of fractions are used in each problem so that students can focus on the meaning of each operation. Because students are not to calculate exact values, to order the expressions they need to rely on what they know about the size of the fractions, as well as to look for and make use of structure (MP1).

Launch

Arrange students in groups of 2. Display problems for all to see. Ask students to put the expressions in order based on their value, from least to greatest, but without calculating the exact values. Instead, they should estimate the value of each expression by reasoning about the operation and the fractions. Ask students to give a signal as soon as they have determined an order and can support it with an explanation.

Give students 1–2 minutes of quiet think time and another minute to discuss their reasoning with a partner and come to an agreement.

Student Task

Without calculating, order the expressions according to their values from least to greatest.
Be prepared to explain your reasoning.

$\frac34 + \frac23$

$\frac34 - \frac23$

$\frac34 \boldcdot \frac23$

$\frac34 \div \frac23$

Sample Response

The order from smallest to largest is $\frac34-\frac23$ , $\frac34\boldcdot \frac23$ , $\frac34 \div \frac23$ , $\frac34+\frac23$ . Sample reasoning:

$\frac34+\frac23$ and $\frac34 \div \frac23$ are both greater than 1.
$\frac34-\frac23$ and $\frac34\boldcdot \frac23$ are both less than 1.
$\frac34$ is greater than $\frac12$ and $\frac23$ is also greater than $\frac12$ , so $\frac34+\frac23$ is greater than 1.
$\frac34$ is 0.75 and $\frac23$ is about 0.67, so their sum is a little less than 1.5.
$\frac34 \div \frac23$ can be viewed as “How many $\frac 23$ s are in $\frac34$ ?”. Since $\frac34$ is just a little over $\frac23$ , the quotient is a little more than 1. If we were to draw a tape diagram, we can see it is just a little bit more than 1.

$\frac34\boldcdot \frac23$ can be viewed as $\frac34$ of $\frac23$ , so the product is less than $\frac23$ but more than $\frac13$ .
$\frac34$ is $\frac14$ more than $\frac12$ , and $\frac23$ is greater than $\frac12$ by an even smaller amount, so $\frac34-\frac23$ is less than $\frac14$ .

Activity Synthesis (Teacher Notes)

Invite 1–2 groups to share how they ordered their expressions from least to greatest. Record it for all to see.

To involve more students in the conversation, consider asking:

“Does anyone want to add on to $\underline{\hspace{.5in}}$ ’s strategy?”
“Did anyone compare the expressions in a different way?”
“Do you agree or disagree? Why?”

If there are disagreements, ask students with opposing views to explain their reasoning, and discuss it to reach an agreement on a correct order.

Anticipated Misconceptions

Some students may assign the division expression to be the one with the lowest value because they still assume that the quotient will always be less than the dividend. Prompt them to test their assumption with a counterexample, such as $2 \div \frac{1}{3}$ or $\frac{1}{2} \div \frac{1}{4}$ . If the assumption is common, consider addressing it during a whole-class discussion.

Standards

Addressing

6.NS.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
6.NS.A.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for $(2/3) \div (3/4)$ and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that $(2/3) \div (3/4) = 8/9$ because $3/4$ of $8/9$ is $2/3$. (In general, $(a/b) \div (c/d) = ad/bc$.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

25 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Solving Problems Involving Fractions

5 min

Teacher Prep

Setup

Students in groups of 2. Display problems for all to see. 1 minute of quiet think time, 1 minute of partner discussion, followed by a whole-class discussion.

Narrative

Launch

Give students 1–2 minutes of quiet think time and another minute to discuss their reasoning with a partner and come to an agreement.

Student Task

Without calculating, order the expressions according to their values from least to greatest.
Be prepared to explain your reasoning.

$\frac34 + \frac23$

$\frac34 - \frac23$

$\frac34 \boldcdot \frac23$

$\frac34 \div \frac23$

Sample Response

The order from smallest to largest is $\frac34-\frac23$ , $\frac34\boldcdot \frac23$ , $\frac34 \div \frac23$ , $\frac34+\frac23$ . Sample reasoning:

$\frac34+\frac23$ and $\frac34 \div \frac23$ are both greater than 1.
$\frac34-\frac23$ and $\frac34\boldcdot \frac23$ are both less than 1.
$\frac34$ is greater than $\frac12$ and $\frac23$ is also greater than $\frac12$ , so $\frac34+\frac23$ is greater than 1.
$\frac34$ is 0.75 and $\frac23$ is about 0.67, so their sum is a little less than 1.5.
$\frac34 \div \frac23$ can be viewed as “How many $\frac 23$ s are in $\frac34$ ?”. Since $\frac34$ is just a little over $\frac23$ , the quotient is a little more than 1. If we were to draw a tape diagram, we can see it is just a little bit more than 1.

$\frac34\boldcdot \frac23$ can be viewed as $\frac34$ of $\frac23$ , so the product is less than $\frac23$ but more than $\frac13$ .
$\frac34$ is $\frac14$ more than $\frac12$ , and $\frac23$ is greater than $\frac12$ by an even smaller amount, so $\frac34-\frac23$ is less than $\frac14$ .

Activity Synthesis (Teacher Notes)

Invite 1–2 groups to share how they ordered their expressions from least to greatest. Record it for all to see.

To involve more students in the conversation, consider asking:

“Does anyone want to add on to $\underline{\hspace{.5in}}$ ’s strategy?”
“Did anyone compare the expressions in a different way?”
“Do you agree or disagree? Why?”

If there are disagreements, ask students with opposing views to explain their reasoning, and discuss it to reach an agreement on a correct order.

Anticipated Misconceptions

Standards

Addressing

6.NS.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
6.NS.A.1·Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for $(2/3) \div (3/4)$ and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that $(2/3) \div (3/4) = 8/9$ because $3/4$ of $8/9$ is $2/3$. (In general, $(a/b) \div (c/d) = ad/bc$.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

Solving Problems Involving Fractions

Warm-upOperations with Fractions5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityPairs of Problems25 minKCs

Knowledge Components

Solving Problems Involving Fractions

Warm-upOperations with Fractions5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityPairs of Problems25 minKCs

5 min

25 min

5 min

25 min