Lesson 17

The purpose of this Estimation Exploration is to estimate the product of a fraction and a large whole number. Students know how to find the exact answer but it would require many calculations. Making an estimate will help develop the intuition that because \(\frac{5}{3}\) is greater than 1, the product has to be greater than the other factor. Students can make a better estimate by replacing the whole number 9,625 with a friendlier number that they can find \(\frac{1}{3}\) of mentally. Throughout this lesson, students will continue to compare the size of products to the size of one of the factors. 

• Groups of 2
• Display the expression.
• “¿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.

• “¿Cómo sabemos que el producto va a ser mayor que 9,625?” // “How do we know the product is going to be greater than 9,625?” (\(\frac{5}{3}\) is more than 1 so the product is greater than the other factor, 9,625.)
• “¿Cómo sabemos que el producto va a ser mayor que 15,000?” // “How do we know the product is going to be greater than 15,000?” (\(\frac{1}{3}\times 9,\!000=3,\!000\) so \(\frac{5}{3}\times 9,\!000=15,\!000\) and we are trying to figure out what \(\frac{5}{3}\) of more than 9,000 is.)

The goal of this activity is for students to match expressions and diagrams and then compare the value of each expression with one of the factors. To match the expressions with the diagrams students will likely use the meaning of multiplication. For example, \(\frac{2}{7} \times 3\) means 2 of 7 equal parts of 3 wholes. The area diagram shows the 7 parts with 2 shaded whereas the number line only shows the relative locations of \(\frac{2}{7} \times 3\) and 3, requiring students to understand the relationship between \(\frac{2}{7} \times 3\) and 3 in order to pick the right match. Once they have made the matches, the diagrams help to visualize that \(\frac{2}{7} \times 3\) is less than 3 and the activity synthesis highlights this. When students match diagrams and expressions they look for and identify structure in the number line and area diagrams (MP7).

• Groups of 2

• 1–2 minutes: quiet think time
• 8–10 minutes: partner work time
• Monitor for students who compare the numbers by:
  • finding the value of the expressions
  • using the number lines which show how the value of each expression compares to 3 or 5
  • using the area diagrams which also show how the value of each expression compares to 3 or 5

• Display the expression: \(\frac{2}{7} \times 3\)
• “¿Cómo decidieron cuáles diagramas le corresponden a la expresión?” // “How did you decide which diagrams match the expression?” (For the area diagram I took the rectangle with length 3 and width less than 1. For the number line, I picked the one with 3 and a point that was less than 3.)
• Invite students to share how they compared \(\frac{2}{7} \times 3\) with 3, highlighting these strategies:
  • reasoning about the size of \(\frac{2}{7}\)
  • using the number line
  • using the area diagram

The purpose of this activity is for students to compare a product to an unknown factor based on the size of the other factor. In this case, students cannot calculate the values of the products to compare but instead rely on their understanding of fractions and the meaning of multiplication. Students also use a number line to help them visualize the different distances after listing them in order. For this part of the activity the expectation is that they will use what they already know about the order of the distances to determine which point corresponds to each student. They might, however, also reason about the quantities. For example twice Priya’s distance can be found by marking off Priya’s position on the number line a second time (MP2).

• Groups of 2
• 1–2 minutes: quiet think time

• 6–8 minutes: partner work time

• Display:
  \(\frac{3}{5} \times 2\)
  \(\frac{6}{7} \times 2\)
  \(\frac{14}{8} \times 2\)
  \(2 \times 2\)
• “¿Qué observan acerca de estas expresiones?” // “What do you notice about these expressions?” (They represent the amount that each person ran, if Priya’s distance is 2. They are all a number multiplied by 2. They are listed in increasing order.)
• “Si Priya hubiera corrido 4 millas, ¿qué expresiones de multiplicación podríamos escribir para representar cuántas millas corrieron los demás estudiantes?” // “What if Priya ran 4 miles? What multiplication expressions can we write to represent how many miles each of the other students ran?” (It would be just like the expressions above except that the 2 would be replaced with a 4.)
• Record expressions for all to see:
  \(\frac{3}{5} \times 4\)
  \(\frac{5}{7} \times 4\)
  \(\frac{14}{8} \times 4\)
  \(2 \times 4\)
• “Si cambia la distancia de Priya, ¿cambia el orden de las distancias? ¿Por qué sí o por qué no?” // “Does the order of the distances change when Priya’s distance changes? Why or why not?” (No, the order of the products is the same as the order of the other factor, the multiple of Priya’s distance.)