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

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

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

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.

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.

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

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.

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

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.
problem_activity: Narrative
In this activity, students are given expressions that represent strategies for finding the area of rectangles. The strategies are based on the distributive property and the associative property of multiplication. Students interpret the expressions by marking or shading area diagrams and connect each expression to the product of two factors (MP2). For instance, they see that to find the value of
\(2 \times (2 \times 6)\)
is to find the value of
\(4 \times 6\)
or
\(6 \times 4\)
.
MLR8 Discussion Supports:
Synthesis: Create a visual display of the diagrams. As students share their strategies, annotate the display to illustrate connections. For example, trace the area showing 5 columns of 3, and write
\(5 \times 3\)
.
Advances: Speaking, Representing
Engagement: Provide Access by Recruiting Interest.
Leverage choice around perceived challenge. Invite students to select at least 2 of the 3 problems to complete.
Supports accessibility for: Organization, Attention, Social-emotional skills
Required Materials
Materials to Gather
Colored pencils, crayons, or markers
Launch
Groups of 2
“Take a minute to read the directions of the activity. Then, talk to your partner about what you are asked to do.”
1 minute: quiet think time
1 minute: partner discussion
Answer any clarifying questions from students.
Give students access to colored pencils, crayons, or markers.
Activity
“Mark or shade each diagram to represent how each student found the area.”
3–5 minutes: independent work time
“Share with your partner how you used the rectangles to show each expression.”
3–5 minutes: partner discussion
Student Facing
Here are some rectangles and expressions that show how three students saw the area of the rectangles.
Noah
[IMG]
\((5\times 3)+(2 \times 3)\)
Priya
[IMG]
\(2 \times (2 \times 6)\)
Tyler
[IMG]
\((5 \times 8) + (3 \times 8)\)
For each rectangle:
Name the two factors that can be multiplied to find its area.
Mark or shade each rectangle to show how each student saw the area. Be prepared to explain your reasoning.
Student Response
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Activity Synthesis
“What are the two factors you can multiply to find the area of Noah’s rectangle?” (7 and 3)
“How are those numbers related to the expression that he wrote:
\((5\times 3)+(2 \times 3)\)
?”  (
\(7 \times 3\)
is 21. Finding
\(5\times3\)
, which is 15, then adding
\(2\times3\)
, which is 6, also gives 21.)
“Where do you see the two factors in his expression?” (The 7 is 5 and 2 combined. The 3 is in the
\(5 \times 3\)
and
\(2 \times 3\)
.)
Repeat the line of questioning with Priya’s rectangle and expression.


correct_option_index: 3

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