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

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.

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

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

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.

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

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.

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.

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.

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

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
problem_activity: Narrative
In this activity, students use rectangular diagrams and similar reasoning as in earlier activities to represent the multiplication of 2 two-digit numbers. They analyze a progression of diagrams, starting with those that represent multiplication of two-digit and one-digit numbers (18 and 6), a two-digit number and a ten (18 and 10), and then 2 two-digit numbers (18 and 16).
Students may decompose factors in different ways. For example, those who are familiar with multiples of 25 may find it intuitive to decompose
\(25 \times 46\)
as
\(25 \times 40\)
and
\(25 \times 6\)
, rather than decomposing 25 into
\(20 + 5\)
.
MLR8 Discussion Supports.
Pair gestures with verbal directions to clarify the meaning of any unfamiliar terms such as partial product.
Advances: Listening, Representing
Action and Expression: Develop Expression and Communication.
Give students access to base-ten blocks. If students use the blocks for the last question, ask them to also draw a diagram that represents their work with the blocks.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Launch
Display the three diagrams in the activity.
“What do you notice? What do you wonder?”
1 minute: quiet think time
1 minute: partner discussion
Activity
“Take a few quiet minutes to work on the first problem. Then, share your thinking with your partner.”
3 minutes: independent work time
3 minutes: partner discussion
Invite students to share their responses: “What multiplication expression can be represented by each diagram?”
“Complete the rest of the activity.”
3 minutes: independent or partner work time
Monitor for students who decompose both factors into tens and ones and those who choose to keep one of the factors intact.
Student Facing
For each diagram, write a multiplication expression that the diagram can represent. Then, find the value of the expression. Use equations to show or explain your reasoning.
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How are the diagrams alike? How are they different? Discuss with your partner.
Use a diagram to find each product.
\(13 \times 21\)
\(25 \times 46\)
Student Response
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Activity Synthesis
Select two students to share their diagrams, solutions, and reasoning for the last problem.
“How did you decompose the factors and the diagram?” (I decomposed the 13 into 10 and 3 and the 21 into 20 and 1.)
“What expression could we write to show the partial product represented by each part of the diagram?”
“How do these partial products help us find the value of
\(13 \times 21\)
?” (Adding them gives us the value of
\(13 \times 21\)
:
\(200 + 10 + 3 + 60 = 273\)
)


correct_option_index: 9