Pattern Building
The goal of today is to work on seeing and representing patterns.
To do this, we’re going to look at a progression of “steps” of pattern, and try to write down what we see, then find an expression that explains it.
Example: Consider the three steps to the right. How many blocks would be in Step 4? Step 10? Step n?
We’ll look at two different student’s approaches.
Student 1 notices that all three steps shown have a single dot on the far left and far right, so that’s 2 dots. There’s a top row and bottom row of dots, each of which is increasing by 1 each time.
So in step 1, we have 2 dots + 2 rows of 1 dot each: 2 + 2·1
We jot this down, and note the pattern, which we can then extend:
Step / What I See Here / Number of dots1 / 2 + 2 · 1 / 4
2 / 2 + 2 · 2 / 6
3 / 2 + 2 · 3 / 8
4 / 2 + 2 · 4 / 10
10 / 2 + 2 · 10 / 22
n / 2 + 2 · n / 2 + 2n
Student 2 notices that we start with 4 dots, and add 2 dots each time. So, in Step 2, we have 4 dots + 2 more. In step 3 we have 4 dots + 4 more, which is 2 more twice: 2·2, or 4 + 2·2
We jot this down, and note the pattern, which we can then extend:
Step / What I See Here / Number of dots1 / 4, or 4 + 2 · 0 / 4
2 / 4 + 2 · 1 / 6
3 / 4 + 2 · 2 / 8
4 / 4 + 2 · 3 / 10
10 / 4 + 2 · 9 / 22
n / 4 + 2 · (n – 1) / 4 + 2(n– 1)
Is one of the students wrong? Or are their answers the same?
We can check by simplifying Student 2’s answer:
4 + 2(n– 1)Distributing
4 + 2n – 2Combining like terms, 4 – 2 = 2
2 + 2n
The answers are the same, just written differently
Worksheet – PatternsName: ______
For each of the pattern sheets on the table, figure out What You See, and try to find out how many boxes will be needed for Step 4, Step 10, and Step n.
1)
Stage / What I See Here / Number of boxes1
2
3
4
10
n
2)
Stage / What I See Here / Number of boxes1
2
3
4
10
n
3)
Stage / What I See Here / Number of boxes1
2
3
4
10
n