In this activity, students generate a shape pattern given a rule and describe what they notice. Students number the shapes in the repeating pattern and describe what they notice about the numbers. Students use what they notice to make predictions about which shape will appear in a particular position of the pattern (MP2).

Although students may make predictions in a number of ways, during the Activity Synthesis, highlight reasoning that is based on the idea of multiples, especially student reasoning that shows making a connection between the size of the repeating unit (in this pattern, 3 shapes) and the patterns in the numbers that represent the position of the shapes.

• Invite 2–3 previously selected students to share their responses and reasoning to the last two problems.
• If all students used the numerical pattern that represents squares to help them find the 31st shape, ask:     
  • “Is it possible to predict the 31st shape with the numbers that represent the position of the triangles?” (Yes. The numbers would be 1, 4, 7, . . . We could keep adding 3 or multiples of 3 to one of these numbers to see if we get 31 at some point.)
  • “Is it possible to use the numbers that represent the position of the circles?” (Yes. The numbers would be 2, 5, 8, . . . We could keep adding 3 or multiples of 3 to one of these numbers.)
• “Why might you want to use the numerical pattern that represents the squares?” (The numbers are all multiples of 3, which are familiar numbers. We can reason using only multiplication, instead of adding repeatedly.)

In this activity, students generate more repeating shape patterns and use what they know about factors and multiples to analyze features of the patterns that are not explicit in their rules. For example, students may notice that some of the patterns repeat more than others in the first 12 terms and relate this to the size of the repeating unit and what they know about factors and multiples. Students may also notice how they can use what they know about multiplication and division to predict terms without generating the complete pattern.

• Invite previously selected students to share how they predicted the 25th shape.
• Consider asking:
  • “How did you use what you noticed to predict the 25th shape?”
• “Some of you answered the questions about Kiran’s shape pattern without drawing it. How did you predict the 12th shape? The 24th shape?” (I noticed there are 5 shapes that repeat, so I know it will only repeat 2 times before the 12th shape because 5 x 2 is 10. Then I looked to see what the second shape in the pattern is. I knew 24 is close to 25 and 5 x 5 = 25. I knew the 25th shape would be a square, then I went back one in Kiran’s pattern - triangle.)