# Corresponding Relationships in Growing Patterns - Two Step Rules

Lesson 6 of 6

5th grade

60-120 minutes

**Description**

Pose interesting, but more difficult-to-generalize growing patterns.

**Materials**

### Introduce

Revisit an adapted version of Mr. Green's tree growing problem from Day 1 of this ARC:

Students access prior knowledge by engaging in a brief discussion about the "what comes next" rule (add two). Engage in a Think/Pair/Share activity where you ask students to think about if there is another relationship going on in this pattern (between Day and Number of Trees). Give students a minute or two to work on this independently, then have them discuss what they are thinking with a partner. After a few minutes share their ideas in whole group discussion. Once the relationship between Day and Number of Trees has been established have students generate rules they think match this growing pattern (SMP4). After a few minutes have them justify the rule they came up with to a partner (SMP8).
Reconvene the whole group and ask what they think the rule is for Mr. Green's tree planting. Students may have generated the rule "double it and add 1", but if they haven't, then they should create a table, graph, or visual model.

### Explore

Two growing patterns (Happy Counting and Hexagons) are provided for this part of the lesson. These tasks can be sequenced (Smileys then Hexagons), or students can choose one, or you can give half of the class one, and half of the class the other, and they can later pair-share their rule for the relationship.

Distribute the Happy Counting and/or Hexagon Trails Activity Sheet to students. For Happy Counting, also distribute counters; For Hexagon Trails, distribute hexagon shapes from pattern blocks. Have students work individually or in partners on their task.

As students are working, circulate and make a note of the different counting strategies that students are using. During the discussion later you will use this evidence of student thinking as examples. (SMP8)

- “What is the
*rule *for ‘what comes next’?”
- “What is changing with each new step?”
- “What is the same with each new step?”
- “How can you test to see if your
*rule* is correct?”

After students have completed their Activity Page, regroup students so that they find someone who did the SAME task, but wrote the rule differently. Tell students to each share their rule and to work together to decide if both rules are correct or not, and to be ready to tell how the rule connects to the pattern. Give students enough time for most to familiarize themselves with someone else's strategy.

If everyone did the same task, repeat this process for the next task. If you had some students doing one, and some doing the other, have students find a partner who solved a DIFFERENT task. On each student's turn, have them explain to their partner how they know their rule works for the pattern they explored.

### Synthesize

As a whole class, discuss the *rules* for the patterns explored and help students to write the numeric or algebraic expression for the *rules*.

For example, if Susan says "First we figured out that each of the hexagons on the end had 5 sides to count. All of the ones in the middle have only 4 sides. See! We have 6 hexagons in the middle and each one has 4 sides!"

Teacher writes 5 + 4 + 4 + 4 + 4 + 5

Ask students if there is a shorter way to write this expression. Listen for evidence that students are thinking about using multiplication to simplify the addition [5 + (4 x 4) + 5 or 4 x 4 + 10].

- As an extension, translate these numeric expressions into algebraic expressions. In this example, that would be 4(n - 2) + 10.
- Have students describe how these patterns (Mr. Green’s Trees, Happy Counting, and/or Hexagon Trails) are alike and different from the patterns they have done previously (they still grow each time by + 2 or some amount; these three patterns had a constant amount or starting value). This can be done informally by pairing with a partner and discussing, having students write in their math journal, or as an exit slip/ticket out the door activity.

### Assessment

#### Optional

Revisit any of the* pattern*s explored in Lessons 1-3, and discover the *formula *for your chosen growing pattern.

### Teacher Reflection

- How could I modify this lesson (or previous ones) to better help struggling students?
- What representations were students most comfortable using (
*tables*, graphs, or visual models)? How could I encourage them to explore other representations?
- What additional guiding questions could I ask to help students move from the representations to using
*formulas*?
- What were some misconceptions I noticed students making? How could I eliminate these?
- How did I encourage students to explain and defend their thinking?

Leave your thoughts in the comments below.