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# 3.3: Circles and Arrows 1

Difficulty Level: At Grade Created by: CK-12

Can you figure out the value of each letter in the diagram below? In this concept, we will use problem solving steps to help us find the value of letters in circles and arrows diagrams.

### Guidance

In order to solve the problem above, use the problem solving steps.

• Start by describing what you see in the diagram.
• Next, figure out what your job is in the problem. In all of these problems your job will be to figure out the value of the two letters in the diagram.
• Then, make a plan for how you will solve. There is usually more than one way to solve the problem. You will want to figure out one letter first and then the next letter.
• Next, solve the problem.
• Finally, check to make sure that the values you found work for all four of the arrows.

#### Example A

Figure out the value of each letter.

Solution:

We can use the problem solving steps to help us solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{Two rows and two columns with numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ & \mathbf{My \ Job:} && \text{Figure out the values of letters} \ C \ \text{and} \ D.\\ & \mathbf{Plan:} && \text{Start with the first column. Solve for} \ C.\\ &&& \text{Then figure out} \ D.\\ & \mathbf{Solve:} && C+C=6, \ \text{so} \ C=3,\\ &&& \text{In the first row,} \ C+D=4\\ &&& \text{Replace} \ C \ \text{with} \ 3.\\ &&& \text{Then} \ 3+D=4. \ D=4-3, \ \text{or} \ 1.\\ & \mathbf{Check:} && \text{Replace all} \ C\text{s with} \ 3.\\ &&& \text{Replace all} \ D\text{s with} \ 1.\\ &&& \text{Add rows and columns. Check the sums.}\\ &&& \text{Row} \ 1: 3+1=4 \qquad \text{Column} \ 1: 3+3=6\\ &&& \text{Row} \ 2: 3+2=5 \qquad \text{Column} \ 2: 1+2=3\end{align*}

#### Example B

Figure out the value of each letter.

Solution:

We can use the problem solving steps to help us solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{Two rows and two columns with numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ & \mathbf{My \ Job:} && \text{Figure out the values of letters} \ E \ \text{and} \ F.\\ & \mathbf{Plan:} && \text{Start with the second row. Solve for} \ F.\\ &&& \text{Then figure out} \ E.\\ & \mathbf{Solve:} && F+F=4, \ \text{so} \ F=2,\\ &&& \text{In the first column,} \ E+F=7\\ &&& \text{Replace} \ F \ \text{with} \ 2.\\ &&& \text{Then} \ E+2=7. \ E=7-2, \ \text{or} \ 5.\\ & \mathbf{Check:} && \text{Replace all} \ E\text{s with} \ 5.\\ &&& \text{Replace all} \ F\text{s with} \ 2.\\ &&& \text{Add rows and columns. Check the sums.}\\ &&& \text{Row} \ 1: 5+1=6 \qquad \text{Column} \ 1: 5+2=7\\ &&& \text{Row} \ 2: 2+2=4 \qquad \text{Column} \ 2: 1+2=3\end{align*}

#### Example C

Figure out the value of each letter.

Solution:

We can use the problem solving steps to help us solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{Two rows and two columns with numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ & \mathbf{My \ Job:} && \text{Figure out the values of letters} \ G \ \text{and} \ H.\\ & \mathbf{Plan:} && \text{Start with the second column. Solve for} \ G.\\ &&& \text{Then figure out} \ H.\\ & \mathbf{Solve:} && G+G=8, \ \text{so} \ G=4,\\ &&& \text{In the first column,} \ 3+H=4\\ &&& \text{Then} \ H=4-3, \ \text{or} \ 1.\\ & \mathbf{Check:} && \text{Replace all} \ H\text{s with} \ 1.\\ &&& \text{Replace all} \ G\text{s with} \ 4.\\ &&& \text{Add rows and columns. Check the sums.}\\ &&& \text{Row} \ 1: 3+4=7 \qquad \text{Column} \ 1: 3+1=4\\ &&& \text{Row} \ 2: 1+4=5 \qquad \text{Column} \ 2: 4+4=8\end{align*}

#### Concept Problem Revisited

We can use the problem solving steps to help us solve this problem.

\begin{align*}& \mathbf{Describe:} && \text{Two rows and two columns with numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ & \mathbf{My \ Job:} && \text{Figure out the values of letters} \ A \ \text{and} \ B.\\ & \mathbf{Plan:} && \text{Start with the first row. Solve for} \ A.\\ &&& \text{Then figure out} \ B.\\ & \mathbf{Solve:} && A+A=4, \ \text{so} \ A=2,\\ &&& \text{In the first column,} \ A+B=5\\ &&& \text{Replace} \ A \ \text{with} \ 2.\\ &&& \text{Then} \ 2+B=5. \ B=5-2, \ \text{or} \ 3.\\ & \mathbf{Check:} && \text{Replace all} \ A\text{s with} \ 2.\\ &&& \text{Replace all} \ B\text{s with} \ 3.\\ &&& \text{Add rows and columns. Check the sums.}\\ &&& \text{Row} \ 1: 2+2=4 \qquad \text{Column} \ 1: 2+3=5\\ &&& \text{Row} \ 2: 3+6=9 \qquad \text{Column} \ 2: 2+6=8\end{align*}

### Vocabulary

In math, an unknown is a letter that stands for a number that we do not yet know the value of. In this concept, when you figured out the value of the letters in the circles and arrows diagrams you were solving for unknowns.

### Guided Practice

Figure out the value of each letter in each circles and arrows diagram.

1.

2.

3.

Answers:

1. \begin{align*}J = 1, K = 5\end{align*}
2. \begin{align*}L = 2, M = 3\end{align*}
3. \begin{align*}N = 5, P = 3\end{align*}

### Practice

Figure out the value of each letter in each circles and arrows diagram.

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Difficulty Level:
At Grade
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Date Created:
Jan 18, 2013
Last Modified:
Mar 23, 2016

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