Lesson 19

The purpose of this Estimation Exploration is for students to develop strategies for finding the product of a fraction and a mixed number. Since \(2 \frac{8}{9}\) is so close to 3, a good estimate is \(3 \times 28\) or 84. Students may refine this estimate using the distributive property 

\(\begin{array} 28 \times 2\frac{8}{9} &=& 28 \times \left(3 - \frac{1}{9}\right)\\ &=& 28 \times 3 - 28 \times \frac{1}{9} \end{array} \)

Since \(\frac{28}{9}\) is about 3, \(84 - 3\) or 81 is a very good estimate. Students will use these ideas in the lesson when they find products of fractions, whole numbers, and mixed numbers.

• Groups of 2
• Display the image.
• “What is an estimate that’s too high? Too low? About right?”

• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

• “How does \(28 \times 2\frac{8}{9}\) compare to \(28 \times 2\)? How do you know?” (It’s larger because \(2 \frac{8}{9}\) is greater than 2.)
• “Why is \(28 \times 3\) a good estimate?” (Because \(2\frac{8}{9}\) is really close to 3.)
• “Is \(28 \times 2\frac{8}{9}\) greater or less than \(28 \times 3\)? How do you know?” (Less because \(2\frac{8}{9}\) is less than 3.)

Optional: Reveal the actual value, \(80\frac{8}{9}\), and add it to the display.

The purpose of this activity is for students to apply what they have learned about multiplication and division of fractions to strategically write expressions with the greatest value. Students notice and explain patterns (MP7) such as:

• To make a product as large as possible, the two factors should be chosen as large as possible.
• To make a quotient or fraction as large as possible, the dividend should be as large as possible and the divisor as small as possible.

• Groups of 2

• 15 minutes: partner work time
• Monitor for students who reason about the size of the product or quotient based on the location of the digits.

• Invite previously selected students to share their responses.
• Display the second expression.
• Invite students to share their strategies for making the expression as large as possible.
• “Why is \(6 \div \frac{1}{5}\) a good choice for making this expression as large as possible?” (When I find the value I am multiplying 6 and 5. Those are the two biggest numbers so I know that will give me the biggest value.)
• “Is there another choice for filling in the blanks that gives the same value?” (Yes, I can also do \(5 \div \frac{1}{6}\).)

In the previous activity, students chose digits to create expressions whose value was as large as possible. The purpose of this activity is for students to create expressions with the smallest possible value. The expressions and digits that students use are the same so the patterns that they identified in the previous activity will apply here as well but they lead to a different choice of expressions.

• Groups of 2

• 7–8 minutes: independent work time
• 2–3 minutes: partner work time
• Monitor for students who reason about the size of the product or quotient based on the location of the digits.

• Display the last expression.
• “Why is \(\frac{1}{6}\div 5\) a good choice for making the expression as small as possible?” (The numerator is 1 so I want the denominator to be as large as possible. That’s why putting in the 6 and 5 is a good strategy.)
• “Is there another choice for filling in the blanks that gives the same value?” (Yes, I can also do \(\frac{1}{5} \div 6\).)
• “What is the value of \(\frac{1}{6} \div 5\) and \(\frac{1}{5} \div 6\) ? How do you know?” (\(\frac{1}{30}\) because I am either cutting \(\frac{1}{6}\) into 5 equal pieces or \(\frac {1}{5}\) into 6 equal pieces. Either way there are \(6 \times 5\) or 30 of those pieces in a whole.)