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: Narrative
In this warm-up, students practice estimating a reasonable answer using known information, rounding, and multiplicative reasoning strategies. For example, students may create a diagram and arrive at
\(7 \times 7 \times 7 \times 7\)
as an estimate of the number of fish. Then, they may approximate
\((7 \times 7) \times (7 \times 7)\)
with
\(50 \times 50\)
, or estimate
\((7 \times 7 \times 7) \times 7\)
with
\((50 \times 7) \times 7\)
or
\(350 \times 7\)
. Some students may notice that the question is vague and asks, “How many are going to the park?” rather than “How many people are going to the park?” and account only for the number of children and teachers in their estimation. If students ask for clarification, ask them to make their own assumptions and explain why they made those assumptions when they share their estimates.
Launch
Groups of 2
Display the description.
Activity
“What is an estimate that’s too high? Too low? About right?”
2 minutes: quiet think time
1 minute: partner discussion
Record responses.
Student Facing
Seven teachers are going to the park.
Each teacher is taking 7 students.
Each student is bringing 7 fishbowls.
Each fishbowl has 7 fish.
[IMG]
How many are going to the park?
Record an estimate that is:
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "text-align:center"}, "_value": "too low"}, {"_attributes": {"scope": "col", "style": "text-align:center"}, "_value": "about right"}, {"_attributes": {"scope": "col", "style": "text-align:center"}, "_value": "too high"}]}]}, {"tr": [{"td": [{"span": [{"_attributes": {"class": ["math"]}, "_value": "\\(\\phantom{\\hspace{2.5cm} \\\\ \\hspace{2.5cm}}\\)"}]}, {"span": [{"_attributes": {"class": ["math"]}, "_value": "\\(\\phantom{\\hspace{2.5cm} \\\\ \\hspace{2.5cm}}\\)"}]}, {"span": [{"_attributes": {"class": ["math"]}, "_value": "\\(\\phantom{\\hspace{2.5cm} \\\\ \\hspace{2.5cm}}\\)"}]}]}]}]}]}
Student Response
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Activity Synthesis
Consider asking:
“Is anyone’s estimate less than 500? Greater than 5,000?”
“Did anyone include the number of teachers and students in their estimate?”
Invite students to share their estimation strategies. After each explanation, ask if others reasoned the same way.
Consider revealing the actual value of 2,457 teachers, students, and fish.


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