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8.8: Two-Step Patterns

Difficulty Level: At Grade Created by: CK-12

Two Step Patterns – Identify Patterns and Write Function Rules

Teacher Notes

Two Step Patterns require students to analyze the first 3 or 4 figures of a pattern, identify relationships between the pattern number and the number of items needed to draw the figure, describe the \begin{align*}10^{\text{th}}\end{align*} figure, and write the rule (in symbols) that relates the pattern number and the number of items. All rules require two steps: multiplication and either addition or subtraction. Encourage students to make a table to organize their data. The table will help them to observe the relationship and write the rule.

Solutions:

\begin{align*}1. \ \quad \text{Figure} \ 10 \ \text{has} \ 32 \ \text{tiles}; \ y = 3n + 2\!\\ {\;} \ \ \quad \text{Figure} \ 1 \ \text{has} \ 1 \ \text{row of} \ 3 \ \text{plus} \ 2; \ \text{Figure} \ 2 \ \text{has} \ 2 \ \text{rows of} \ 3 \ \text{plus} \ 2; \ \text{Figure} \ 3 \ \text{has} \ 3 \ \text{rows of} \ 3 \ \text{plus}\!\\ {\;} \ \ \quad 2; \ \text{Figure} \ 4 \ \text{has} \ 4 \ \text{rows of} \ 3 \ \text{plus} \ 2; \ \text{Figure} \ 10 \ \text{will have} \ 10 \ \text{rows of} \ 3 \ \text{plus} \ 2, \ \text{or} \ 32 \ \text{tiles.}\!\\ {\;} \ \ \quad \text{Figure} \ n \ \text{will have} \ n \ \text{rows of} \ 3 \ \text{plus} \ 2, \ \text{or} \ 3n + 2 \ \text{tiles.}\!\\ \\ 2. \ \quad \text{Figure} \ 10 \ \text{has} \ 31 \ \text{tiles}; \ y = 3n + 1\!\\ {\;} \ \ \quad \text{Figure} \ 1 \ \text{has} \ 1 \ \text{row of} \ 3 \ \text{plus} \ 1; \ \text{Figure} \ 2 \ \text{has} \ 2 \ \text{rows of} \ 3 \ \text{plus} \ 1; \ \text{Figure} \ 3 \ \text{has} \ 3 \ \text{rows of} \ 3 \ \text{plus}\!\\ {\;} \ \ \quad 1; \ \text{Figure} \ 4 \ \text{has} \ 4 \ \text{rows of} \ 3 \ \text{plus} \ 1; \ \text{Figure} \ 10 \ \text{will have} \ 10 \ \text{rows of} \ 3 \ \text{plus} \ 1, \ \text{or} \ 31 \ \text{tiles.}\!\\ {\;} \ \ \quad \text{Figure} \ n \ \text{will have} \ n \ \text{rows of} \ 3 \ \text{plus} \ 1, \ \text{or} \ 3n + 1 \ \text{tiles.}\!\\ \\ 3. \ \quad \text{Figure} \ 10 \ \text{has} \ 43 \ \text{tiles}; \ y = 4n + 3\!\\ {\;} \ \ \quad \text{Figure} \ 1 \ \text{has} \ 1 \ \text{row of} \ 4 \ \text{plus} \ 3; \ \text{Figure} \ 2 \ \text{has} \ 2 \ \text{rows of} \ 4 \ \text{plus} \ 3; \ \text{Figure} \ 3 \ \text{has} \ 3 \ \text{rows of} \ 4 \ \text{plus}\!\\ {\;} \ \ \quad 3; \ \text{Figure} \ 4 \ \text{has} \ 4 \ \text{rows of} \ 4 \ \text{plus} \ 3; \ \text{Figure} \ 10 \ \text{will have} \ 10 \ \text{rows of} \ 4 \ \text{plus} \ 3, \ \text{or} \ 43 \ \text{tiles. Figure}\!\\ {\;} \ \ \quad n \ \text{will have} \ n \ \text{rows of} \ 4 \ \text{plus} \ 3, \ \text{or} \ 4n + 3\ \text{tiles.}\end{align*}

Two Step Patterns – Identify Patterns and Write Function Rule

\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*}

Extra for Experts: Two Step Patterns – Identify Patterns and Write Function Rule

Solutions:

\begin{align*}1. \ \quad \text{Figure} \ 10 \ \text{has} \ 23 \ \text{tiles}; \ y = 2n + 3\!\\ {\;} \ \ \quad \text{Figure} \ 1 \ \text{has} \ 1 \ \text{column of} \ 2 \ \text{plus} \ 3; \ \text{Figure} \ 2 \ \text{has} \ 2 \ \text{columns of} \ 2 \ \text{plus} \ 3; \ \text{Figure} \ 3 \ \text{has} \ 3\!\\ {\;} \ \ \quad \text{columns of} \ 2 \ \text{plus} \ 3; \ \text{Figure} \ 4 \ \text{has} \ 4 \ \text{columns of} \ 2 \ \text{plus} \ 3; \ \text{Figure} \ 10 \ \text{will have} \ 10 \ \text{columns of} \ 2\!\\ {\;} \ \ \quad \text{plus} \ 3, \ \text{or} \ 23 \ \text{tiles.}\!\\ {\;} \ \ \quad \text{Figure} \ n \ \text{will have} \ n \ \text{columns of} \ 2 \ \text{plus} \ 3, \ \text{or} \ 2n + 3 \ \ \text{tiles.}\!\\ \\ 2. \ \quad \text{Figure} \ 10 \ \text{has} \ 24 \ \text{tiles}; \ y = 2n + 4\!\\ {\;} \ \ \quad \text{Figure} \ 1 \ \text{has} \ 1 \ \text{row of} \ 2 \ \text{plus} \ 4; \ \text{Figure} \ 2 \ \text{has} \ 2 \ \text{rows of} \ 2 \ \text{plus} \ 4; \ \text{Figure} \ 3 \ \text{has} \ 3 \ \text{rows of} \ 2 \ \text{plus}\!\\ {\;} \ \ \quad 4; \ \text{Figure} \ 4 \ \text{has} \ 4 \ \text{rows of} \ 2 \ \text{plus} \ 4; \ \text{Figure} \ 10 \ \text{will have} \ 10 \ \text{rows of} \ 2 \ \text{plus} \ 4, \ \text{or} \ 24 \ \text{tiles.}\!\\ {\;} \ \ \quad \text{Figure} \ n \ \text{will have} \ n \ \text{rows of} \ 2 \ \text{plus} \ 4, \ \text{or} \ 2n + 4 \ \text{tiles.}\!\\ \\ 3. \ \quad \text{Figure} \ 10 \ \text{has} \ 59 \ \text{tiles}; \ y = 5n + n - 1, \ \text{or} \ 6n - 1\!\\ {\;} \ \ \quad \text{Figure} \ 1 \ \text{has} \ 1 \ \text{house of} \ 5 \ \text{tiles; Figure} \ 2 \ \text{has} \ 2 \ \text{houses of} \ 5 \ \text{tiles plus} \ 1 \ \text{more tile; Figure} \ 3 \ \text{has}\!\\ {\;} \ \ \quad 3 \ \text{houses of} \ 5 \ \text{tiles plus} \ 2 \ \text{more tiles; Figure} \ 4 \ \text{has} \ 4 \ \text{houses of} \ 5 \ \text{tiles plus} \ 3 \ \text{more tiles; Figure}\!\\ {\;} \ \ \quad 10 \ \text{will have} \ 10 \ \text{houses of} \ 5 \ \text{tiles plus} \ 9 \ \text{more tiles, or} \ 59 \ \text{tiles.}\!\\ {\;} \ \ \quad \text{Figure} \ n \ \text{will have} \ n \ \text{houses of} \ 5 \ \text{tiles plus} \ n - 1 \ \text{more tiles, or} \ 5n + n - 1 \ \text{tiles.}\end{align*}

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CK.MAT.ENG.SE.1.Algebra-Explorations-K-7.8.8