What patterns do you see on a zebra?
Maya counted an odd number of stripes on a zebra’s leg. If the zebra’s leg has an even number of black stripes, how many white stripes might it have?
- Choose any odd number for the total. What if there were 19 stripes in all? What if there were 23?
- Maya counted an even number of stripes on a different zebra leg. Could there be an odd number of black stripes? Could there be an even number? Explain.
- Write your own story problem about the stripes on a zebra. Show your problem to a friend and ask them to solve it.
This task asks students to find a possible number of white stripes on a zebra’s leg given an odd total number of stripes and an even number of black stripes. Solving this problem requires students to make connections among repeating patterns and sums of even and odd numbers.
There are multiple strategies that students might use to solve this problem. They may directly model an AB pattern with an odd total number of elements, count how many objects they used to represent each stripe, and then assign the color black to the even number of objects and white to the other number of objects. Some students may begin by choosing an even number for the black stripes, then determine how many white stripes are needed to make an odd total number of stripes. Another strategy may include showing a Doubles fact (e.g., 5 + 5 = 10 or 10 + 10 = 20) and then adding or removing 1 to make a combination with an odd total. Many students may need to make adjustments as they solve. For example, students may first get an answer of 11 black stripes and 10 white stripes, then revise their work to get an even number of black stripes (10 black and 11 white). As needed, some students may find it helpful to group objects into pairs or make equal groups to help identify whether quantities are even or odd. Instead of giving a specific number for the answer, some students might make a generalization about the number of white stripes, such as: There has to be an odd number of white stripes because the sum of an even number and an odd number is always odd, but if you added 2 odd numbers then you’d always get an even number.
Various apps may be used to represent the context of this problem.
- In the Number Frames app, students may alternate different-colored markers, using one color to represent black stripes and another to represent white stripes (as shown here and here).
- The Pattern Blocks app can be used to directly model the AB pattern of black and white stripes. Students may try several combinations until they end up with an even number of black stripes and an odd number of total stripes (as shown here). Some students may find it helpful to then rearrange the pattern blocks into pairs to help identify the even and odd numbers (as shown here).
- The Number Rack app can be used to show a combination with an odd total, with the black stripes assigned to one or more rows and the white stripes assigned to one or more rows (as shown here). Alternatively, students may assign the colors black and white to the red beads and white beads, use groups of 5 beads to make a Double fact, and remove or add 1 bead to make an odd total (as shown here).
No matter which the app students choose to use, their response will be based on an assumption about the stripes on the zebra’s leg. Students are likely to assume that the zebra’s leg has clearly defined stripes in an AB pattern. However, the stripes on a real zebra may form more complex patterns, with stripes that merge with and split from each other. If your student makes an assumption that the stripes are not clearly defined, they may make a combination of an even number of black stripes and an odd number of even stripes with two nonconsecutive numbers, such as 8 black stripes and 11 white stripes. If your student can justify their thinking, other combinations that fit the description may also be acceptable.
Extend students’ thinking by encouraging them to make generalizations about the sums of even and odd numbers. Ask questions such as:
- Think about the sums of even and odd numbers. What do you notice?
- Will the number of white stripes be even or odd? How do you know?