options: Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
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.


correct_option_index: 1