Lesson 19

The purpose of this What Do You Know About _____ is for students to share what they know about and how they can represent the product \(\frac{15}{14}\times\frac{23}{30}\). The numbers were intentionally chosen to make finding the exact value of the product challenging.

• Display the expression.
• “¿Qué saben sobre \(\frac{15}{14}\times\frac{23}{30}\)?” // “What do you know about \(\frac{15}{14}\times\frac{23}{30}\)?”
• 1 minute: quiet think time

• Record responses.
• “¿Cómo podríamos encontrar el valor del producto \(\frac{15}{14}\times\frac{23}{30}\)?” // “How could we find the value of the product \(\frac{15}{14}\times\frac{23}{30}\)?” (Find the product of the numerators and the product of the denominators.)

• “¿\(\frac{15}{14}\times\frac{23}{30}\) es menor que, igual a o mayor que \(\frac{23}{30}\)? ¿Por qué?” // “Is \(\frac{15}{14}\times\frac{23}{30}\) less than, equal to, or greater than \(\frac{23}{30}\)? Why?” (It is greater since \(\frac{15}{14}\) is greater than 1.)

The goal of this activity is to continue to compare the size of a product of fractions to the size of the second factor. In addition to the number line representation which students have worked with in the last few lessons, they also see a different expression that represents the product. In the next activity, this expression will be combined with the distributive property to explain in all cases why multiplying a number by a fraction less than one results in a smaller number while multiplying by a fraction greater than one results in a larger number (MP8).

• Groups of 2

• 1–2 minutes: quiet think time
• 6–8 minutes: partner work time

• Invite students to share their matches.
• “¿Cómo encontraron la recta numérica que correspondía a \(\frac{3}{4} \times \frac{5}{2}\)?” // “How did you find the matching number line for \(\frac{3}{4} \times \frac{5}{2}\)?” (I saw that two of the number lines have \(\frac{5}{2}\) on them and looked for the one that showed \(\frac{3}{4}\) of \(\frac{5}{2}\). I knew which one it was because \(\frac{3}{4}\) of \(\frac{5}{2}\) is less than \(\frac{5}{2}\).)
• “¿Cómo encontraron la expresión que correspondía a \(\frac{3}{4} \times \frac{5}{2}\)?” // “How did you find the matching expression for \(\frac{3}{4} \times \frac{5}{2}\)?” (I looked for an expression with \(\frac{5}{2}\) and only one of them had another factor with the value \(\frac{3}{4}\).)
• “¿Cómo supieron si el valor de \(\frac{3}{4} \times \frac{5}{2}\) era mayor que o menor que \(\frac{5}{2}\)?” // “How did you know whether the value of \(\frac{3}{4} \times \frac{5}{2}\) was greater than or less than \(\frac{5}{2}\)?” (I knew it was less because \(\frac{3}{4}\) is less than 1. That was what helped me find the right number line.)

The goal of this activity is to use the distributive property to explain why multiplying a number by a fraction greater than one increases the size of the number while multiplying by a fraction less than one decreases the size of the number. Expressions are particularly useful here because they show explicitly how the size of the number relates to the product. For example writing \(\frac{3}{5}\) as \(1 - \frac{2}{5}\) and then multiplying by \(\frac{4}{7}\) gives: \(\displaystyle \left(1 - \frac{2}{5}\right) \times \frac{4}{7} = \frac{4}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\)

The revealing part of this calculation is that the structure of the right hand side shows that it is less than \(\frac{4}{7}\) without calculating the exact value (MP7). It must be less than \(\frac{4}{7}\) because it is \(\frac{4}{7}\) minus some other number.

• Groups of 2

• 1–2 minutes: quiet think time
• 8–10 minutes: partner work time
• Monitor for students who use the expressions in the first problem to make the comparisons and then generalize about what happens when you multiply a number by any fraction greater than 1 or less than 1.

• Invite students to share their expressions for the products in the first problem.
• Display the equation: \(\left(1 - \frac{2}{5}\right)\times \frac{4}{7} = \frac{4}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\)
• “¿Cómo pueden darse cuenta de que el valor de la expresión es menor que \(\frac{4}{7}\)?” // “How can you see that the value of the expression is less than \(\frac{4}{7}\)?” (It’s \(\frac{4}{7}\) minus something.)
• “¿Este razonamiento también funciona para \(\left(1 - \frac{3}{8}\right)\times \frac{4}{7}\)?” // “Does this reasoning also work for \(\left(1 - \frac{3}{8}\right)\times \frac{4}{7}\)?” (Yes, it’s again \(\frac{4}{7}\) minus some other number.)
• “¿Este razonamiento también va a funcionar si se multiplica un número menor que 1 por \(\frac{4}{7}\)?” // “Will this reasoning work whenever you multiply a number less than 1 by \(\frac{4}{7}\)?” (Yes, I’ll always get \(\frac{4}{7}\) minus an amount so that’s less than \(\frac{4}{7}\).)