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
The purpose of this activity is to give students a concrete experience of building the boundary of shapes and quantifying that length of the boundary, allowing them to conceptualize perimeter as a measurable geometric attribute. Students use
\(1\frac{1}{4}\)
-inch paper clips as the units for measuring the distances around four shapes. The reasoning here prepares them to reason about equal-size intervals that can be marked on the sides of a shape to measure its length (as students will see in the next activity).
Required Materials
Materials to Gather
Paper clips
Materials to Copy
What Does It Take to Build the Shapes?
Required Preparation
Each group of 4 needs 25-50 paper clips that are
\(1\frac{1}{4}\)
-inch long each.
If using 1-inch paper clips, use 80% scale when making copies of the blackline masters.
Launch
Groups of 4
Give each group a copy of the blackline master and 25–50 paper clips.
“Make a prediction: Which shape do you think will take the most paper clips to build?”
30 seconds: quiet think time
Poll the class on whether they think shape A, B, C, or D would take the most paper clips to build.
Activity
“Work with your group to find out which shape takes the most paper clips to build. You may need to take turns with the paper clips.”
5–7 minutes: small-group work time
Monitor for students who:
place paper clips one at a time and then count the total
place paper clips one at a time and then add up the numbers on all sides
use multiplication (
\(3 \times 3\)
for B,
\((2 \times 4) + (2 \times 2)\)
for C, or
\((2 \times 6) + 2\)
for D)
use estimation
Student Facing
Your teacher will give you four shapes on paper and some paper clips.
Work with your group to find out which shape takes the most paper clips to build. Explain or show how you know. Record your findings here. Draw sketches if they are helpful.
A
B
C
D
Student Response
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Activity Synthesis
Select previously identified students to share their responses and strategies for finding the number of paper clips. Record the different strategies for all to see.
“What we have done in this activity is to find the length of the boundary of shapes. The boundary of a flat shape is called its
perimeter
.”
“When we find the distance all the way around a shape, like we did with paper clips, we are measuring its perimeter.”
“To find the length of the perimeter of a shape, we can find the sum of its side lengths.”
“We can say that the perimeter of shape A is 13 paper clips long.”
“When we say ‘find the perimeter,’ we mean find the length of the perimeter.”


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