standard_description: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) ^( )Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
problem_activity: Optional activity
In this partner activity, students take turns matching a sequence to a definition. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
One sequence and one definition do not have matches. Students are tasked with writing the corresponding match.
Monitor for students using clear reasoning as they create a recursive definition for the sequence 18, 20, 22, 24 to share during the discussion.
Making a spreadsheet available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Tell students that for each sequence, one partner finds its matching recursive definition and explains why they think it matches. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next sequence, the students swap roles. If necessary, demonstrate this protocol before students start working.
Ensure that students notice that one sequence and one definition do not have matches, and they are tasked with writing the corresponding match for each.
Conversing: MLR8 Discussion Supports.
Use this routine to support small-group discussion as students should take turns finding a match. Display the following sentence frames for all to see: "Sequence _____ and definition _____ match because . . .” and “I noticed _____, so I matched . . .” Encourage students to challenge each other when they disagree, and to press for precise mathematical language. This will help students clarify their reasoning about each match.
Design Principle(s): Support sense-making; Maximize meta-awareness
Engagement: Provide Access by Recruiting Interest.
Leverage choice around perceived challenge. Invite groups of students to find matches for at least 4 of the sequences. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
Take turns with your partner to match a sequence with a recursive definition. It may help to first figure out if the sequence is arithmetic or geometric.
For each match that you find, explain to your partner how you know it’s a match.
For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
There is one sequence and one definition that do not have matches. Create their corresponding match.
Sequences:
3, 6, 12, 24
18, 36, 72, 144
3, 8, 13, 18
18, 13, 8, 3
18, 9, 4.5, 2.25
18, 20, 22, 24
Definitions:
\(G(1)=18, G(n)=\frac12 \boldcdot G(n-1), n\ge2\)
\(H(1)=3, H(n)=5 \boldcdot H(n-1), n\ge2\)
\(J(1)=3, J(n)=J(n-1)+5, n\ge2\)
\(K(1)=18, K(n)=K(n-1)-5, n\ge2\)
\(L(1)=18, L(n)=2 \boldcdot L(n-1), n\ge2\)
\(M(1)=3, M(n)=2 \boldcdot M(n-1), n\ge2\)
Student Response
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Anticipated Misconceptions
Some students may not be sure how to begin matching terms in a sequence to a definition. Encourage them to start by picking a definition and calculating the first few terms of the sequence it represents.
Activity Synthesis
Once all groups have completed the matching, ask “How did you decide which definitions to match to sequence 3, 6, 12, 24 and sequence 18, 36, 72, 144 when they both involve doubling?” (They are both geometric with a growth factor of 2, but since they have different first terms you can use those to match the sequences to
\(M\)
and
\(L\)
.)
Next, invite previously identified students to share the recursive definition they created for sequence 18, 20, 22, 24 and their strategy for writing it.
If time allows and students need extra practice graphing functions, assign students one function each from the task statement to sketch a graph for. After work time, select students to share their sketches, displaying them for all to see and compare.


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