Analyze patterns and find rules: This Comes Next
Lesson 2 of 6
5th grade
60 minutes
Description
Students continue to explore growing patterns and rules to determine what comes next. They analyze, describe, and justify their rules for naming patterns. Since students are likely to see growing patterns in a different way compared to their classmates, this is an opportunity to engage them in communicating about mathematics. This lesson requires students to explain correspondences among their verbal descriptions of the patterns, tables, and graphs that will help them eventually build an equation to solve the problem.
Materials
Introduce
To continue the context from Lesson 1, explain that Mr. Green is putting sticks in a repeating triangular pattern in his garden to plant different things in each triangle. Engage in class discussion about how many toothpicks Mr. Green needs to create one triangle, two triangles, three triangles, etc.)
If students need a sheet to help organize their thinking for this activity, you can use the Growing Patterns Activity Sheet.
Have students write a rule that tells Mr. Green how many new toothpicks he will need each time he adds a new triangle. Students then test their rule to determine how many toothpicks will be needed to create 5 triangles.
Have students model it with the toothpicks to verify findings, and compare their answers with other students in the class. (SMP 4)
Prompt students to think about and discuss with a partner how we kept track of our findings previously (when Mr. Green was planting trees in lesson one in this ARC). Ask them to share ideas; when the use of a table is offered, encourage explicit explanation that recording findings in a table demonstrates use of structure in their math thinking and helps them look for patterns and relationships. (SMP 7) Have students create the table below, add three more figures in the pattern, and expand the table to include these new figures.
Number of Triangles | Number of Toothpicks |
1 | 3 |
2 | 5 |
3 | 7 |
4 | |
5 | |
Students share results with a partner, discussing how to express the number pattern that describes what is happening in the table and how it helps them determine what comes next. [Add 2 toothpicks for each additional triangle.] Allow for whole class sharing.
Discuss with students the idea of different representations (visual model and table), and encourage conversation about other ways they could represent this information. If the answer of a graph is not shared, explain that they are going to expand and connect their thinking to include how the information they have in the table could be represented in a graph (next activity).
Explore
Continue with the same context, providing a new prompt,
“What do you think would change if Mr. Green used a repeating square pattern instead of a repeating triangular pattern in his garden?”
Repeated Square Toothpick Pick Problem
Instruct students to write a rule that tells Mr. Green how many new toothpicks he will need each time he adds a new square. Have the students explain how they got their answers by using both a visual model and a table.
Have students recall the name for this pattern (a growing pattern), then have them consider if the toothpicks were removed, could they use their rule to determine how many toothpicks are needed to create Figure 10? (You start with 1 edge and add three toothpicks for each figure, or three times the figure number plus 1.)
Figure Number | Number of Toothpicks |
1 | 4 |
2 | 7 |
3 | 10 |
4 | 13 |
10 | ? |
(The important part of the lesson is not to have students generate the same verbal descriptions. Many will have different ways to describe the rule. The important idea is to allow students to verbalize the rule that they see and give them time to see how their rule and another student's can result in the same output. You start with 1 edge and add three for each figure, or three times the figure number plus 1.)
The remainder of this lesson will address correspondences among verbal descriptions of patterns, tables, and graphs.
Remind students that earlier when asked, "Is there another way we could represent the information we have in our table?" we said we could use a graph.
- How could we use the numbers from our table to generate points on a graph? If necessary, remind students that an ordered pair consists of two numbers.
- What do you think is one of the ordered pairs we would place on a graph to represent Mr Green's square gardens?
It might be a helpful visual aide for students to shrink the table so it looks like this:
Figure Number | Square Pattern |
1 | 4 |
2 | 7 |
3 | 10 |
4 | 13 |
5 | 16 |
6 | 19 |
Once students recognize and articulate the ordered pairs correctly, have them plot these on graph paper. Allow time for them to conduct a buddy check before leading a closing discussion.
For differentiation, consider the following:
Describe the rule using letters and numbers. As students say the rule for what comes next, they can also write it using expressions. For add three this can be n + 3. Or, you could write equations emphasizing the language of next. Next = Now + 3. This is a great connection to the recursive rules that students will generate in algebra.
Synthesize
Reflect
The following questions are focused on assessing whether students are learning to generalize and describe what comes next across representations (picture, table, graph). These can be delivered as whole class discussion or used as an Exit Slip or a Ticket Out the Door for assessment purposes.
What will come next in this pattern (first growing pattern)? How do you know? [There will be 16 toothpicks for Figure 5. I used the counting numbers.]
How is each figure changing? [Each figure has 3 more toothpicks.]
How is the table changing each time you enter a new figure? [You are adding one more figure and 3 more toothpicks.]
How is the graph changing each time you enter a new figure? [You are going up three each time for each new figure.]
What is the rule that tells you what comes next? [Can you say/write the rule using words? (Add 3.).]
Assessment
Use the exit slip to assess student mastery.
Teacher Reflection
- If students were able to analyze and describe growing patterns, what extension activities are appropriate?
- What aspects of this lesson were the hardest for students to grasp? How could I modify this lesson to differentiate?
- How did students demonstrate fluidity from moving in between visual models, tables, and graphs?
- What other examples of growing patterns could I use in this lesson for continued practice?
- How did I encourage students to explain and defend their thinking?
Leave your thoughts in the comments below.