Lesson 18

The purpose of this warm-up is to elicit observations about partitions in number lines of different scales. In subsequent design activities, students will partition the sides of squares and other shapes into unit fractions. That process will be iterative, with the length being partitioned changing each time. The work here familiarizes students with the reasoning they will encounter later in the lesson.

• 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. 

• “How are all the number lines alike?” (They all go from 0 to 1. They have the labels 0 and 1.)
• “How are they different?” (They have different lengths.)
• “How would you partition the last number line to match the rest?” (Mark the halfway point between 0 and 1.)
• “Suppose we’d like all the number lines to be partitioned into fourths. How would you go about doing so?” (Find the halfway point between 0 and \(\frac{1}{2}\) and between \(\frac{1}{2}\) and 1.)

The purpose of this activity is for students to create a design using the fraction \(\frac{1}{2}\) as a constraint for length. Students partition each side of a given square into halves and mark a length of \(\frac{1}{2}\) on each side. They connect those midpoints to form another shape, partition the sides into halves again, and repeat the process to make increasingly smaller shapes. Students notice that the resulting shapes are also squares, and the squares in the pattern alternate between having vertical and horizontal sides and diagonal sides.

• Groups of 2
• “Let’s create a design using the fraction \(\frac{1}{2}\).”
• “Take a minute to read the activity statement. Then, turn and talk to your partner about what you are asked to do.”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Give each student a ruler or a straightedge.
• Provide access to extra paper, in case requested.

• “Work with your partner to complete the activity. Use a straightedge when you draw lines to connect points.”
• 10 minutes: partner work time
• Monitor for different strategies and tools students use to partition the sides of the squares, such as:
  • estimating or “eyeballing” the midpoint
  • folding opposite sides of each square in half
  • copying the side length of each square onto another paper, folding it in half, and using it to mark the midpoint of all four sides
  • using a ruler to measure

• Select previously identified students to share their strategies for partitioning the sides of each shape into halves. Ask them to demonstrate their methods as needed.
• “How did you know which endpoint to use as 0 or as a starting point to mark \(\frac{1}{2}\) of the length?” (It doesn’t matter. Starting from either end gives the same point.)
• Display one or more completed drawings (showing different numbers of iterations).
• “Why do you think we all ended up with the same design?” (Each time we marked the same set of points—the middle point of each side.)
• “In the next activity, we’ll creating a design with a different fraction.”

The purpose of this activity is for students to create a design using the fraction \(\frac{1}{4}\) as a constraint for length. The fraction \(\frac{1}{4}\) expands the number of possible designs that could be generated.

When the fractional length to be marked on the sides of a square was \(\frac{1}{2}\), students could use either end of a side as a starting point and would mark the same point. The shape that resulted from connecting the midpoints was always a square.

When the fractional length to be marked is \(\frac{1}{4}\), the location of the point changes depending on the starting point. Consequently, the shapes that result from connecting the points may be a square, another type of quadrilateral, or may vary each time. The shapes in turn determine how many iterations can be done. (For example, if the resulting shapes are narrow parallelograms, students may only be able to do 2 or 3 rounds before further partitioning becomes unfeasible.)

If time permits, encourage students color or decorate their drawings. Some students may also enjoy the challenge of creating another design using new constraints, such as:

• starting with a square of a different size or with another shape
• using another unit fraction or a non-unit fraction to mark the length of each side
• using a different unit fraction for each iteration

Students can observe regularity in repeated reasoning (MP8) in many different ways as the new shapes they make are often smaller versions of the previous shape, but this depends heavily on how they decide to mark off \(\frac{1}{4}\) of each side.

• Groups of 2–4
• “Let’s now create a design using the fraction \(\frac{1}{4}\).”
• “Take a minute to read the activity statement. Then, turn and talk to your partner about how you think the drawing process will be different this time.”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Give each student a ruler or a straightedge.
• Provide access to extra paper, in case requested.

• “Take a few minutes to work independently on the activity. Afterwards, share your drawing and your process with your group.”
• “Use a straightedge when you draw lines to connect points.”
• 8–10 minutes: independent work time
• 6–8 minutes: group discussion
• Monitor for different strategies and tools students use to partition the sides of the shapes.
• Also monitor for the starting point students use to identify a length of \(\frac{1}{4}\) on each side of the square. For example, they may:
  • always proceed in a clockwise or counterclockwise direction
  • start from the left endpoint for the horizontal sides, and start from the top endpoint for the vertical sides
  • start from either end of each side and in no particular order

• Select students to share their strategies for partitioning each side into fourths. Ask them to demonstrate their methods as needed.
• Invite students who created different designs to share the decisions they made along the way.
• “How did you decide which endpoint to use as a starting point for marking \(\frac{1}{4}\) of the side length?”
• “Did you mark the length the same way each time?”
• “Why do you think we ended up with different designs when the fraction is \(\frac{1}{4}\)?” (We didn’t mark off \(\frac{1}{4}\) from the same starting point or using the same order, so the locations of the points and the shapes from connecting the points were different.)