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.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”

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

• “¿En qué se parecen y en qué se diferencian \(28 \times 2\frac{8}{9}\) y \(28 \times 2\)? ¿Cómo lo saben?” // “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.)
• “¿Por qué \(28 \times 3\) es una buena estimación?” // “Why is \(28 \times 3\) a good estimate?” (Because \(2\frac{8}{9}\) is really close to 3.)
• “¿\(28 \times 2\frac{8}{9}\) es mayor o menor que \(28 \times 3\)? ¿Cómo lo saben?” // “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.)

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.
• “¿Por qué \(6 \div \frac{1}{5}\) es una buena opción para lograr que esta expresión sea lo más grande posible?” // “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.)
• “¿Hay alguna otra opción para escribir en los espacios en blanco con la que se obtenga el mismo valor?” // “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.
• “¿Por qué \(\frac{1}{6}\div 5\) es una buena opción para lograr que la expresión sea lo más pequeña posible?” // “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.)
• “¿Hay alguna otra opción para escribir en los espacios en blanco con la que se obtenga el mismo valor?” // “Is there another choice for filling in the blanks that gives the same value?” (Yes, I can also do \(\frac{1}{5} \div 6\).)
• “¿Cuál es el valor de \(\frac{1}{6} \div 5\) y de \(\frac{1}{5} \div 6\)? ¿Cómo lo saben?” // “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.)