# 4.3: Circles and Arrows 2

**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 this 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 5 of the arrows.

#### Example A

Figure out the value of each letter.

**Solution:**

We can use the problem solving steps to help us.

\begin{align*}& \mathbf{Describe:} && \text{There are two rows and three columns of numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ & \mathbf{My \ Job:} && \text{Figure out the value of the letters} \ C \ \text{and} \ D.\\ & \mathbf{Plan:} && \text{Start with column 2. Solve for}\ C.\\ &&& \text{Replace}\ C \ \text{with its value in row 1. Solve for}\ D.\\ & \mathbf{Solve:} && \text{Column} \ 2: C + C = 10. \ \text{Therefore}, \ C = 5.\\ &&& \text{Row} \ 1: 6 + C + D = 15, \ \text{so}, \ 6 + 5 + D = 15. \ \text{So}, \ D = 15 - 11, \ \text{or}\ 4.\\ & \mathbf{Check:} && \text{Replace all}\ C\text{s with 5. Replace all}\ D\text{s with 4.}\\ &&& \text{Add rows and columns. Check the sums.}\\ &&& \text{Row} \ 1: 6 + 5 + 4 = 15 \quad \text{Column} \ 1: 6 + 4 = 10\\ &&& \text{Row} \ 2: 4 + 5 + 3 = 12 \quad \text{Column} \ 2: 5 + 5 = 10\\ &&& \qquad \qquad \qquad \qquad \qquad \quad \text{Column} \ 3: 4 + 3 = 7\end{align*}

#### Example B

Figure out the value of each letter.

**Solution:**

We can use the problem solving steps to help us.

\begin{align*}& \mathbf{Describe:} && \text{There are two rows and three columns of numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ & \mathbf{My \ Job:} && \text{Figure out the value of the letters} \ E \ \text{and} \ F.\\ & \mathbf{Plan:} && \text{Start with column 1. Solve for}\ E.\\ &&& \text{Replace}\ E \ \text{with its value in row 1. Solve for}\ F.\\ & \mathbf{Solve:} && \text{Column} \ 1: E + E = 6. \ \text{So}, \ E = 3.\\ &&& \text{Row} \ 1: E + F + 2 = 13, \ \text{so}, \ 3 + F + 2 = 13. \ \text{Therefore}, \ F = 13 - 5, \ \text{or}\ 8.\\ & \mathbf{Check:} && \text{Replace all}\ E\text{s with 3. Replace all}\ F\text{s with 8.}\\ &&& \text{Add rows and columns. Check the sums.}\\ &&& \text{Row} \ 1: 3 + 8 + 2 = 13 \quad \text{Column} \ 1: 3 + 3 = 6\\ &&& \text{Row} \ 2: 3 + 7 + 8 = 18 \quad \text{Column} \ 2: 8 + 7 = 15\\ &&& \qquad \qquad \qquad \qquad \qquad \quad \text{Column} \ 3: 2 + 8 = 10\end{align*}

#### Example C

Figure out the value of each letter.

**Solution:**

We can use the problem solving steps to help us.

\begin{align*}& \mathbf{Describe:} && \text{There are two rows and three columns of numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ & \mathbf{My \ Job:} && \text{Figure out the value of the letters} \ G \ \text{and} \ H.\\ & \mathbf{Plan:} && \text{Start with row 2. Solve for}\ H.\\ &&& \text{Replace}\ H \ \text{with its value in Column 3. Solve for}\ H.\\ & \mathbf{Solve:} && \text{Row} \ 2: 3 + H + H = 13, \ \text{so} \ H + H = 10. \ \text{Therefore}, \ H = 5.\\ &&& \text{Column} \ 3: G + H = 7, \ \text{so} \ G + 5 = 7. \ \text{Therefore}, \ G = 7 - 5, \ \text{or}\ 2.\\ & \mathbf{Check:} && \text{Replace all}\ H\text{s with 5. Replace all}\ G\text{s with 2.}\\ &&& \text{Add rows and columns. Check the sums.}\\ &&& \text{Row} \ 1: 2 + 9 + 2 = 13 \quad \text{Column} \ 1: 2 + 3 = 5\\ &&& \text{Row} \ 2: 3 + 5 + 5 = 13 \quad \text{Column} \ 2: 9 + 5 = 14\\ &&& \qquad \qquad \qquad \qquad \qquad \quad \text{Column} \ 3: 5 + 2 = 7\end{align*}

#### Concept Problem Revisited

**Solution:**

We can use the problem solving steps to help us.

\begin{align*}& \mathbf{Describe:} && \text{There are two rows and three columns of numbers and letters.}\\ &&& \text{Arrows point to sums.}\\ & \mathbf{My \ Job:} && \text{Figure out the value of the letters} \ A \ \text{and} \ B.\\ & \mathbf{Plan:} && \text{Start with column 3. Solve for}\ B.\\ &&& \text{Replace}\ B \ \text{with its value in Column 2. Solve for}\ A.\\ & \mathbf{Solve:} && \text{Column} \ 3: B + B = 12. \ \text{So}, \ B = 6.\\ &&& \text{Column} \ 2: A + 6 = 9. \ \text{So}, \ A = 9 - 6, \ \text{or}\ 3.\\ & \mathbf{Check:} && \text{Replace all}\ A\text{s with 3. Replace all}\ B\text{s with 6.}\\ &&& \text{Add rows and columns. Check the sums.}\\ &&& \text{Row} \ 1: 3 + 3 + 6 = 12 \quad \text{Column} \ 1: 3 + 5 = 8\\ &&& \text{Row} \ 2: 5 + 6 + 6 = 17 \quad \text{Column} \ 2: 3 + 6 = 9\\ &&& \qquad \qquad \qquad \qquad \qquad \quad \text{Column} \ 3: 6 + 6 = 12\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

For each problem, figure out the value of each letter.

1.

2.

3.

4.

**Answers:**

1. \begin{align*}J = 6, K = 4\end{align*}

2. \begin{align*}L = 8, M = 4\end{align*}

3. \begin{align*}N = 3, P = 7\end{align*}

4. \begin{align*}Q = 4, R = 2\end{align*}

### Practice

For each problem, figure out the value of each letter.

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### Image Attributions

Students find the value of letters using the relationships between the letters presented in a two-by-three grid. Students use problem solving steps to help them to solve the problems.