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, with which they have worked in the last few lessons, students also see a different expression that represents the product. In the next activity, this expression will be combined with the distributive property to explain why, in all cases, multiplying a number by a fraction less than 1 results in a smaller number while multiplying by a fraction greater than 1 results in a greater number (MP8).

• Invite students to share their matches.
• “¿Cómo encontraron la recta numérica que correspondía a  ?” // “How did you find the matching number line for ?” (I saw that two of the number lines had on them and looked for the one that showed of . I knew which one it was because of is less than .)
• “¿Cómo encontraron la expresión que correspondía a ?” // “How did you find the matching expression for ?” (I looked for an expression with , and only one expression had another factor with the value .)
• “¿Cómo supieron si el valor de  era mayor que o menor que ?” // “How did you know whether the value of was greater than or less than ?” (I knew it was less because 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 1 increases the size of the number while multiplying by a fraction less than 1 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 as and then multiplying by gives: .

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

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