This Number Talk encourages students to look for structure in multiplication expressions and to rely on properties of operations to mentally solve problems. Reasoning about products of whole numbers helps to develop students’ fluency.
Launch
Display one expression.
“Give me a signal when you have an answer and can explain how you got it.”
1 minute: quiet think time
Activity
Record answers and strategies.
Keep expressions and work displayed.
Repeat with each expression.
Find the value of each expression mentally.
Student Response
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Advancing Student Thinking
Activity Synthesis
“How did the earlier expressions help you find the value of the last expression?”
Consider asking:
“Did anyone have the same strategy but would explain it differently?”
“Did anyone approach the problem in a different way?”
Activity 1
Standards Alignment
Building On
Addressing
3.NF.3.a
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
The purpose of this activity is for students to use diagrams to reason about equivalence and to reinforce their awareness of the relationship between fractions that are equivalent.
Students show that a shaded diagram can represent two fractions, such as and , by further partitioning given parts or by composing larger parts from the given parts. Unlike with the fraction strips, where different fractional parts are shown in rows and students could point out where and how they see equivalence, here students need to make additional marks or annotations to show equivalence.
In upcoming lessons, students will extend similar strategies to reason about equivalence on a number line—by partitioning the given intervals on a number line into smaller intervals or by composing larger intervals from the given intervals.
In the first problem, students construct a viable argument in order to convince Tyler that of the rectangle is shaded (MP3).
Action and Expression: Develop Expression and Communication. Synthesis: Identify connections between strategies that result in the same outcomes but use different approaches. Supports accessibility for: Memory, Visual-Spatial Processing
Launch
Groups of 2
Activity
“Work with your partner on the first problem. Discuss whether you agree with Jada and show your reasoning.”
3–4 minutes: partner work time
Pause for a brief discussion. Invite students to share their responses and their reasoning.
“Now, work independently on the rest of the activity.”
5 minutes: independent work time
Monitor for the different strategies students use to show equivalence, such as:
Drawing circles or brackets to show composing larger parts from the given parts.
Drawing lines to show new partitions.
Labeling parts of the fractions with two names.
Drawing a new diagram with different partitions but the same shaded amount.
Identify students, who use different strategies, to share during the Activity Synthesis.
The diagram represents 1.
What fraction does the shaded part of the diagram represent?
Jada says it represents . Tyler is not so sure.
Do you agree with Jada? If so, explain or show how you would convince Tyler that Jada is correct. If not, explain or show your reasoning.
Each diagram represents 1.
Show that the shaded part of this diagram represents both and .
Show that the shaded part of this diagram represents both and .
Show that the shaded part of this diagram represents both and .
Activity Synthesis
Select previously identified students to share their responses and their reasoning. Display their work for all to see.
As students explain, describe the strategies they use to show equivalence. Ask if others in the class showed equivalence the same way.
Activity 2
Standards Alignment
Building On
Addressing
3.NF.3.a
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
The purpose of this activity is for students to generate equivalent fractions, including for fractions greater than 1, given partially shaded diagrams. Students may use strategies from an earlier activity—partitioning a diagram into smaller equal parts, or making larger equal parts out of existing parts—or patterns they observed in the numerators and the denominators of equivalent fractions (MP7).
MLR8 Discussion Supports. Students should take turns naming the equivalent fractions they described and explaining their reasoning to their partner. Display the following sentence frame for all to see: “Observé _____, entonces pensé . . .” // “I noticed _____ , so I thought . . . .” Encourage students to challenge each other when they disagree. Advances: Speaking, Representing
Launch
Groups of 2
Display or draw a diagram with 2 fourths shaded:
“Notice there's a 1 below the diagram. This is another way to show which part of the diagram represents 1.”
“What fractions can the shaded parts of the diagram represent?” ()
Activity
“Now write two fractions that can represent the shaded parts of each diagram.”
3–5 minutes: independent work time
“Discuss with your partner the equivalent fractions you named. Be sure to share your reasoning for each fraction.”
2–3 minutes: partner discussion
Monitor for students who make statements such as:
The first diagram is because 4 of the 6 equal parts are shaded. It's also because every 2 sixths is 1 third, and 2 of the 3 thirds are shaded.
The second diagram is because 2 of the 8 equal parts are shaded. It's also because every 2 eighths is 1 fourth, and 1 of the 4 fourths is shaded.
Each diagram represents 1. Write 2 fractions represented by the shaded part of each diagram.
Here’s another diagram.
What fraction does the shaded part of the diagram represent?
Write another fraction that it represents.
Student Response
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Advancing Student Thinking
If students name a fraction based only on the given partitions, consider asking:
“Tell me how you named the fraction.”
“How could you use the diagram to find an equivalent fraction?”
Activity Synthesis
Select students to share their strategies for writing equivalent fractions for each diagram. Display the diagrams they marked or annotated.
“In what ways was the last diagram different from the first three?” (It shows 2 wholes. The shaded parts were greater than 1.)
“Was your strategy for finding equivalent fractions for this diagram different from the first three? Why or why not?” (No, it still involved making smaller equal parts. Yes, I partitioned the first 1 whole and the second 1 whole separately.)
If no students mention for the last diagram, ask, “Can you name an equivalent fraction other than and ?”
Lesson Synthesis
“Today we saw that the shaded parts of a diagram can be represented by multiple equivalent fractions.”
Display a diagram of labeled fraction strips from an earlier activity, and a couple of shaded diagrams that show equivalent fractions from this activity.
“How did we use the fraction strips to help us see and name equivalent fractions?” (We could see if some number of parts in one row is the same size as the parts in another row. The labels on the strips help us name the fractions that are equivalent.)
“How did the shaded diagrams in this activity help us see and name equivalent fractions?” (We could either partition the diagram into smaller equal parts, or put the parts together to make larger equal parts.)
Standards Alignment
Building On
Addressing
3.OA.5
Apply properties of operations as strategies to multiply and divide.
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
