standard_description: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
problem_activity: Narrative
The purpose of this activity is for students to match different diagrams and expressions representing the same product. In the previous several lessons, students have studied different ways to find products of a whole number and a fraction using arithmetic properties such as the distributive property. Earlier in the unit they learned about the connection between fractions and division. They combine these skills as the expressions and diagrams they work with incorporate both the distributive property and the interpretation of a fraction as division (MP7). Students may match diagrams to expressions differently than the ways that are listed in the sample responses. Encourage students to match the expressions and diagrams in a way that makes sense to them, as long as they can accurately explain how the expression is represented in the diagram they chose.
MLR7 Compare and Connect.
Invite students to prepare a visual display that shows their thinking about their favorite diagram and expression. Encourage students to include details that will help others understand what they see, such as using different colors, arrows, labels, or notes. If time allows, invite students time to investigate each others’ work.
Advances: Representing
Engagement: Develop Effort and Persistence.
Chunk this task into more manageable parts. Give students a subset of options to start with and introduce the remaining numbers once students have completed their initial set of matches.
Supports accessibility for: Organization, Social-Emotional Functioning
Launch
Groups of 2
“You are going to match expressions and diagrams that show different ways to find the value of the product
\(4 \times 5 \frac{2}{3}\)
.”
Activity
5–7 minutes: independent work time
2–3 minutes: partner discussion
Student Facing
Each diagram represents a way to calculate
\(4 \times 5 \frac{2}{3}\)
. Each expression is equivalent to
\(4 \times 5 \frac{2}{3}\)
. Match the diagrams and expressions. Show or explain your reasoning.
A
[IMG]
B
[IMG]
1.
\((4 \times 5) +  \left(4 \times \frac{2}{3}\right)\)
C
[IMG]
2.
\((4 \times 6) - \left(4 \times \frac{1}{3}\right)\)
\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)
3.
\(4 \times \frac{17}{3}\)
D
[IMG]
\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm} \\ \hspace{2.5cm}}\)
4.
\((4 \times 17) \div 3\)
Choose your favorite diagram and expression to find the value of
\(4 \times 5 \frac{2}{3}\)
. Explain why it is your favorite.
Student Response
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Activity Synthesis
Display the diagrams B and D.
“How are the diagrams the same? How are they different?” (They both have the same shaded area and the same divisions inside. The top one shows the length of the shaded rectangle. The bottom one does not though it has enough information to find the shaded area.)
Invite students to share their favorite ways to find the product.


aligns: no