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# 8.7: Two-Step Patterns

Difficulty Level: At Grade Created by: CK-12

Look at the pictures below. Do you notice a pattern? Can you answer the questions? In this concept, we will practice writing rules to describe patterns.

### Guidance

In order to answer the questions about the pattern above, use the problem solving steps.

• Start by describing what you see in the figures.
• Next, figure out what your job is in this problem. In all of these problems your job will be to write a rule that works for the pattern and answer additional questions about the pattern.
• Then, make a plan for how you will solve. Describe the number of tiles in each figure in words. Look for a pattern in the shape of each figure. Finally, write the rule.
• Next, solve the problem.
• Finally, check to make sure that your rule works with all of the given figures.

#### Example A

Look at the figures below and answer the questions.

Solution:

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe:} && \text{Each figure is made of square tiles.}\\ &&& \text{Figure} \ 1 \ \text{has} \ 5 \ \text{tiles.}\\ &&& \text{Figure} \ 2 \ \text{has} \ 8 \ \text{tiles.}\\ &&& \text{Figure} \ 3 \ \text{has} \ 11 \ \text{tiles.}\\ &&& \text{Figure} \ 4 \ \text{has} \ 14 \ \text{tiles.}\\ \\ & \mathbf{My \ job:} && \text{Determine the number of tiles in Figure} \ 10.\\ &&& \text{Write the rule relating the Number of Tiles to the Figure Number.}\\ \\ & \mathbf{Plan:} && \text{Use the diagrams to figure out the relationship between the Figure Number and}\\ &&& \text{the Number of Tiles.}\\ \\ & \mathbf{Solve:} && \text{Figure} \ 1 \ \text{has} \ 1 \ \text{row of} \ 3 \ \text{tiles with two tiles on top. That is} \ 3 \times 1+2, \ \text{or} \ 5 \ \text{tiles.}\\ &&& \text{Figure} \ 2 \ \text{has} \ 2 \ \text{rows of} \ 3 \ \text{tiles with two tiles on top. That is} \ 3 \times 2+2, \ \text{or} \ 8 \ \text{tiles.}\\ &&& \text{Figure} \ 3 \ \text{has} \ 3 \ \text{rows of} \ 3 \ \text{tiles with two tiles on top. That is} \ 3 \times 3+2, \ \text{or} \ 11 \ \text{tiles.}\\ &&& \text{Figure} \ 4 \ \text{has} \ 4 \ \text{rows of} \ 3 \ \text{tiles with two tiles on top. That is} \ 3 \times 4+2, \ \text{or} \ 14 \ \text{tiles.}\\ &&& \text{Figure} \ 10 \ \text{will have} \ 10 \ \text{rows of} \ 3 \ \text{tiles with two on top. That is} \ 3 \times 10+2, \ \text{or} \ 32 \ \text{tiles.}\\ &&& \text{Figure} \ n \ \text{will have} \ n \ \text{rows of} \ 3 \ \text{tiles with two on top. That is} \ 3 \times n+2, \ \text{or} \ 3n+2 \ \text{tiles.}\\ &&& \text{The rule is} \ y=3n+2\\ \\ & \mathbf{Check:} && \text{Figure} \ 1: 3 \times 1+2=5\\ &&& \text{Figure} \ 2: 3 \times 2+2=8\\ &&& \text{Figure} \ 3: 3 \times 3+2=11\\ &&& \text{Figure} \ 4: 3 \times 4+2=14\end{align*}

#### Example B

Look at the figures below and answer the questions.

Solution:

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe:} && \text{Each figure is made of triangle tiles.}\\ &&& \text{Figure} \ 1 \ \text{has} \ 4 \ \text{tiles.}\\ &&& \text{Figure} \ 2 \ \text{has} \ 7 \ \text{tiles.}\\ &&& \text{Figure} \ 3 \ \text{has} \ 10 \ \text{tiles.}\\ &&& \text{Figure} \ 4 \ \text{has} \ 13 \ \text{tiles.}\\ \\ & \mathbf{My \ job:} && \text{Determine the number of tiles in Figure} \ 10.\\ &&& \text{Write the rule relating the Number of Tiles to the Figure Number.}\\ \\ & \mathbf{Plan:} && \text{Use the diagrams to figure out the relationship between the Figure Number and}\\ &&& \text{the Number of Tiles.}\\ \\ & \mathbf{Solve:} && \text{Figure} \ 1 \ \text{has} \ 1 \ \text{row of} \ 3 \ \text{tiles with one tile on top. That is} \ 3 \times 1+1, \ \text{or} \ 4 \ \text{tiles.}\\ &&& \text{Figure} \ 2 \ \text{has} \ 2 \ \text{rows of} \ 3 \ \text{tiles with one tile on top. That is} \ 3 \times 2+1, \ \text{or} \ 7 \ \text{tiles.}\\ &&& \text{Figure} \ 3 \ \text{has} \ 3 \ \text{rows of} \ 3 \ \text{tiles with one tile on top. That is} \ 3 \times 3+1, \ \text{or} \ 10 \ \text{tiles.}\\ &&& \text{Figure} \ 4 \ \text{has} \ 4 \ \text{rows of} \ 3 \ \text{tiles with one tile on top. That is} \ 3 \times 4+1, \ \text{or} \ 13 \ \text{tiles.}\\ &&& \text{Figure} \ 10 \ \text{will have} \ 10 \ \text{rows of} \ 3 \ \text{tiles with one on top. That is} \ 3 \times 10+1, \ \text{or} \ 31 \ \text{tiles.}\\ &&& \text{Figure} \ n \ \text{will have} \ n \ \text{rows of} \ 3 \ \text{tiles with one on top. That is} \ 3 \times n+1, \ \text{or} \ 3n+1 \ \text{tiles.}\\ &&& \text{The rule is} \ y=3n+1\\ \\ & \mathbf{Check:} && \text{Figure} \ 1: 3 \times 1+1=4\\ &&& \text{Figure} \ 2: 3 \times 2+1=7\\ &&& \text{Figure} \ 3: 3 \times 3+1=10\\ &&& \text{Figure} \ 4: 3 \times 4+1=13\end{align*}

#### Example C

Look at the figures below and answer the questions.

Solution:

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe:} && \text{Each figure is made of square tiles.}\\ &&& \text{Figure} \ 1 \ \text{has} \ 7 \ \text{tiles.}\\ &&& \text{Figure} \ 2 \ \text{has} \ 11 \ \text{tiles.}\\ &&& \text{Figure} \ 3 \ \text{has} \ 15 \ \text{tiles.}\\ &&& \text{Figure} \ 4 \ \text{has} \ 19 \ \text{tiles.}\\ \\ & \mathbf{My \ job:} && \text{Determine the number of tiles in Figure} \ 10.\\ &&& \text{Write the rule relating the Number of Tiles to the Figure Number.}\\ \\ & \mathbf{Plan:} && \text{Use the diagrams to figure out the relationship between the Figure Number and}\\ &&& \text{the Number of Tiles.}\\ \\ & \mathbf{Solve:} && \text{Figure} \ 1 \ \text{has} \ 1 \ \text{row of} \ 4 \ \text{tiles with three tiles on top. That is} \ 4 \times 1+3, \ \text{or} \ 7 \ \text{tiles.}\\ &&& \text{Figure} \ 2 \ \text{has} \ 2 \ \text{rows of} \ 4 \ \text{tiles with three tiles on top. That is} \ 4 \times 2+3, \ \text{or} \ 5 \ \text{tiles.}\\ &&& \text{Figure} \ 3 \ \text{has} \ 3 \ \text{rows of} \ 4 \ \text{tiles with three tiles on top. That is} \ 4 \times 3+3, \ \text{or} \ 7 \ \text{tiles.}\\ &&& \text{Figure} \ 4 \ \text{has} \ 4 \ \text{rows of} \ 4 \ \text{tiles with three tiles on top. That is} \ 4 \times 4+3, \ \text{or} \ 9 \ \text{tiles.}\\ &&& \text{Figure} \ 10 \ \text{will have} \ 10 \ \text{rows of} \ 4 \ \text{tiles with three on top. That is} \ 4 \times 10+3, \ \text{or} \ 21 \ \text{tiles.}\\ &&& \text{Figure} \ n \ \text{will have} \ n \ \text{rows of} \ 4 \ \text{tiles with three on top. That is} \ 4 \times n+3, \ \text{or} \ 4n+3 \ \text{tiles.}\\ &&& \text{The rule is} \ y=4n+3\\ \\ & \mathbf{Check:} && \text{Figure} \ 1: 4 \times 1+3=7\\ &&& \text{Figure} \ 2: 4 \times 2+3=11\\ &&& \text{Figure} \ 3: 4 \times 3+3=15\\ &&& \text{Figure} \ 4: 4 \times 4+3=19\end{align*}

#### Concept Problem Revisited

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe:} && \text{Each figure is made of square tiles.}\\ &&& \text{Figure} \ 1 \ \text{has} \ 3 \ \text{tiles.}\\ &&& \text{Figure} \ 2 \ \text{has} \ 5 \ \text{tiles.}\\ &&& \text{Figure} \ 3 \ \text{has} \ 7 \ \text{tiles.}\\ &&& \text{Figure} \ 4 \ \text{has} \ 9 \ \text{tiles.}\\ \\ & \mathbf{My \ job:} && \text{Determine the number of tiles in Figure} \ 10.\\ &&& \text{Write the rule relating the Number of Tiles to the Figure Number.}\\ \\ & \mathbf{Plan:} && \text{Use the diagrams to figure out the relationship between the Figure Number and}\\ &&& \text{the Number of Tiles.}\\ \\ & \mathbf{Solve:} && \text{Figure} \ 1 \ \text{has} \ 1 \ \text{row of} \ 2 \ \text{tiles with one tile on top. That is} \ 2 \times 1+1, \ \text{or} \ 3 \ \text{tiles.}\\ &&& \text{Figure} \ 2 \ \text{has} \ 2 \ \text{rows of} \ 2 \ \text{tiles with one tile on top. That is} \ 2 \times 2+1, \ \text{or} \ 5 \ \text{tiles.}\\ &&& \text{Figure} \ 3 \ \text{has} \ 3 \ \text{rows of} \ 2 \ \text{tiles with one tile on top. That is} \ 2 \times 3+1, \ \text{or} \ 7 \ \text{tiles.}\\ &&& \text{Figure} \ 4 \ \text{has} \ 4 \ \text{rows of} \ 2 \ \text{tiles with one tile on top. That is} \ 2 \times 4+1, \ \text{or} \ 9 \ \text{tiles.}\\ &&& \text{Figure} \ 10 \ \text{will have} \ 10 \ \text{rows of} \ 2 \ \text{tiles with one on top. That is} \ 2 \times 10+1, \ \text{or} \ 21 \ \text{tiles.}\\ &&& \text{Figure} \ n \ \text{will have} \ n \ \text{rows of} \ 2 \ \text{tiles with one on top. That is} \ 2 \times n+1, \ \text{or} \ 2n+1 \ \text{tiles.}\\ &&& \text{The rule is} \ y=2n+1\\ \\ & \mathbf{Check:} && \text{Figure} \ 1: 2 \times 1+1=3\\ &&& \text{Figure} \ 2: 2 \times 2+1=5\\ &&& \text{Figure} \ 3: 2 \times 3+1=7\\ &&& \text{Figure} \ 4: 2 \times 4+1=9\end{align*}

### Vocabulary

One type of pattern is when the number of a certain object increases, decreases, or stays the same in a consistent way. In this concept, we saw patterns of tiles where the numbers of tiles were increasing. With any pattern you should be able to describe the pattern and how to get from one step of the pattern to the next. A rule is an equation that can describe a pattern. In this concept, we wrote rules for patterns that related the figure number to the number of tiles.

### Guided Practice

For each problem below, look at the figures and answer the questions.

1.

2.

3.

1. Figure 10 has 23 tiles. The rule is \begin{align*}y=2n+3\end{align*}.

2. Figure 10 has 24 tiles. The rule is \begin{align*}y=2n+4\end{align*}.

3. Figure 10 has 59 tiles. The rule is \begin{align*}y=5n+n-1\end{align*} or \begin{align*}y=6n-1\end{align*}.

### Practice

For each problem below, look at the figures and answer the questions.

### Notes/Highlights Having trouble? Report an issue.

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Date Created:
Jan 18, 2013