Lesson 6

The purpose of this warm-up is for students to visualize the idea of perimeter and elicit observations about distances around a shape. It also familiarizes students with the context and materials they will be working with in the next activity, where they will use paper clips to form the boundary of shapes and compare or quantify their lengths.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “Can you predict how many paper clips it might take to go around the whole shape?”
• Consider asking: “The paper clips in the picture are standard paper clips. If we use jumbo paper clips, which are larger, would we need more or fewer paper clips to build the shape?”

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).

• 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.

• 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.

• “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

• 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.”

In this activity, students find the perimeter of shapes—first on dot paper, and then using the tick marks on the sides of the shapes. Students may need a reminder that when we measure length, we count the number of length-units, not the number of endpoints. While students may count the tick marks on all sides and add them, they may also observe that some side lengths are the same, especially on shapes A and B, and use this structure and multiplication by 2 to find the perimeter efficiently (MP8).

• Groups of 2
• “Earlier, we used paper clips to measure the distance around shapes. What are some other units we could use to measure distances or lengths?” (The side length of a square on grid paper, the distance between points on dot paper, centimeter, inch, foot)
• “Let’s find the length of the perimeter of some shapes on dot paper and some shapes whose side lengths are shown with tick marks.”

• “Work independently to find the perimeter of each shape. Afterwards, share your responses with your partner.”
• 5 minutes: independent work time
• 3–5 minutes: partner discussion
• Monitor for the different strategies students use to find the perimeter of each shape, such as counting one unit at a time, adding the number of length units of all sides, and using multiplication.

If students lose track of the side lengths they are counting, consider asking:

• “How are you finding the distance around the shape?”
• “How could you keep track of the side lengths as you work?”

• Select students to share their responses and strategies. Start with those who counted individual units and move toward those who use operations.
• Record the strategies for all to see.
• “Which shape, C, D, or E, has the greatest perimeter?” (C. Its perimeter is 33 units. The other two are 30 and 32 units.)
• Consider asking: “Which shape has a greater perimeter: the rectangle in the first problem or the triangle?” (The triangle has 33 units while the rectangle has 22 units, but the units aren’t the same length, so we don’t know which perimeter is greater.)