Growing Patterns Lesson 1
Students explore growing patterns using the actual pattern and tables and determine a rule to tell what comes next.
(for remediation)- What's My Rule (From pg 33 in NCTM Addenda Series: Patterns (K-6)
What's My Rule?
Get ready. The purpose of this activity is to help students recognize number patterns and test their conjectures. Students solve number riddles by guessing the rules that apply to sets of three numbers. The focus is on describing a pattern both orally and in writing.
You can do this activity with or without a computer. To use the computer, you will need software that generates numerical patterns (such as the King's Rule [O'Brian 1985]) and, ideally, a display device for the overhead projector or a large-screen monitor. If these materials are not available, a similar activity can be conducted at the chalkboard.
Get going. Explain to the class that the goal of the activity is to guess a rule that you have made up. The rule generates number triples. Think of a rule and write on the chalkboard three numbers related by the rule. For example, if you rule is “the third number is the product of the first two,” write this sequence of numbers: 3 5 15. The students give three numbers that they think satisfy the rule. Respond yes or no and keep track of their guesses and your responses on the What’s My Rule Overhead. Ex:
Rule: Each number is 1 more than the previous number.
The class continues to name triples and to receive your feedback. When the students think they are ready to guess the rule, give them several triples. They must indicate whether or not the numbers satisfy the rule and describe the rule.
[Draw or project an illustration like the one below on the board.]
n+2 pattern over 4 consecutive days
To stimulate curiosity and increase student motivation, lead the class in a brief Visible Thinking routine called "I see, I think, I wonder". This type of experience sets the stage for inquiry-based learning. Encouraging students to be creative and thoughtful with their observations and interpretations, which helps promote discourse in the classroom. (MP 1, 7, 8) More research and graphic organizers for this routine can be found online.
Under the picture (above) post or state the following: "Complete each of the three sentences related to this picture."
Allow students 3-5 min to write their response prior to sharing in class. This can be done in a variety of ways. For example:
When sharing begins, encourage students to state all three of their responses at the same time, i.e. "I see… I think… I wonder…"
Scenario/Context to tell class
(related to n+2 pattern visual aid above) Each day, Mr. Green is adding trees to the border of his yard to help block some of the traffic noise around his house. If needed, explain how the illustration below fits the context (e.g., You can see that on the first day, he planted one tree. The second day, he planted 2 more trees…).
Discuss with students how the trees in the yard are changing each day, and have them model it with snap cubes or drawings.
Have students draw the next day (or two); this can be done with a partner if you choose. After students are done, engage a class discussion.
Be sure to highlight students' descriptions about how they are figuring out the number of trees for a given day. Remind students that this description is called a rule and ask students to articulate Mr. Green's rule (add two for his daily tree planting).
Discuss with students how using a table helps us organize the data. (MP7)
Have students create a table like the one below to check and see if their conjectures about day 5 and 6 are correct. Discuss how the use of a table to organize our thinking helps us look for (and recognize) patterns and relationships.
If you feel students need more visual representations use the Growing Patterns Activity Sheet.
Continue questioning students, guiding them to notice what is happening vertically and horizontally with the terms that they are labeling as "L numbers" (same number of dots in the vertical and horizontal segments of the letter).
Once this idea has been established have the students explore other letter patterns that follow this same "rule" (same number of dots in the vertical and horizontal segments of the letter).
Allow students to work with a partner (or independently) to figure out what the first four T letters would be. Have them model their findings with manipulatives and/or drawings as well as publishing their findings in a table (as we did for Mr. Green's trees).
Once done, have them figure out the first four S letters.
If time is an issue, you could split the class in half, having one group do the T letters and the other do the S letters.
For differentiation consider the following
Explore different growing patterns with toothpicks or other manipulatives. Invite students to create their own geometric design that follows a particular rule (e.g., create a growing pattern that has the rule "plus 5" to build the next design in the pattern).
Ask, if your creation follows the geometric growing patterns we have seen, what must be true (the design grows in the same way each time).
These other patterns can be used for differentiation (giving different groups different patterns to explore or having students choose the one they want to explore) and it can be done as a challenge for early finishers or students who need an additional challenge.
To close this activity, students should come together and share their findings with the class. Visit the Summarize/Synthesize section for what this would look like.
Have students share their findings about T and S number patterns. Be sure to have students explain how they illustrated the rules for each of the "letter" patterns.
Guiding questions during whole group discussion need to center around what students notice is happening in a growing pattern (the design grows in the same way each time) and what rule they use to describe how the pattern is growing (T numbers are growing by 3; S numbers are growing by 4).
What similarities and differences did you notice about the growing patterns we have discussed today? (Similarities- the design grows in the same way each time. Differences- Mr. Green's “trees” example was a pattern of growing by 2; T numbers are growing by 3; S numbers are growing by 4).
Share with the class that one of the most common misconceptions is thinking fractions with smaller denominators (1/2) are smaller than fractions with larger denominators (1/8). In this closing discussion the goal is to compare the size of 2 and 8 to the relative sizes of ½ and 1/8. This can be a whole group or small group discussion (SMP 7).
"If 2 is smaller than 8, why is ½ larger than 1/8? How do you know? What did we do in this lesson to help us better understand the difference between whole numbers and fractions?"
[This is discussed above: Ask, "As the denominator increases, what happens to the size of the pieces?" [The pieces get smaller as the denominator increases.]
Choose a letter and create the first 5 patterns; explain the pattern as well.
Leave your thoughts in the comments below.
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.
In this lesson, students use the idea of what comes next to determine the relationship between the pattern number and number of objects in the pattern (explicit rule).
Students explore a toothpick staircase problem to apply their skills of finding the rule to describe the relationship between corresponding terms.
Continue to explore relationships between terms by exploring a growing pattern that involves several rules.
Pose interesting, but more difficult-to-generalize growing patterns.
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS