# 5.2: Circles and Arrows 3

**At Grade**Created by: CK-12

Look at the picture below. The arrows point to the sum of each row or column. Can you figure out the value of each letter? In this concept, we will practice solving for unknowns 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 6 of the arrows.

#### Example A

The arrows point to the sum of each row or column. Figure out the value of each letter.

**Solution:**

We can use the problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{I see rows and columns with numbers and letters}.\\ &&& \text{The arrows point to sums}.\\ & \mathbf{My \ Job} && \text{Figure out the value of each letter}.\\ & \mathbf{Plan} && \text{I will start with the first column to find the value of} \ C.\\ &&& \text{Then I will use the third column and find the value of} \ D.\\ & \mathbf{Solve} && \text{First Column:} \ C+C+C=9, \ \text{so} \ C=3.\\ &&& \text{Third Column:} \ 8+D+6=19. \ 19-8-6=5, \text{so} \ D=5.\\ & \mathbf{Check} && \text{First Row:} \ 3+5+8=16; \text{Second Row:} \ 3+2+5=10; \text{Third Row:} \ 3+3+6=12\\ &&& \text{First Column:} \ 3+3+3=9; \text{Second Column:} \ 5+2+3=10; \text{Third Column:} \ 8+5+6=19\end{align*}

#### Example B

The arrows point to the sum of each row or column. Figure out the value of each letter.

**Solution:**

We can use the problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{I see rows and columns with numbers and letters}.\\ &&& \text{The arrows point to sums}.\\ & \mathbf{My \ Job} && \text{Figure out the value of each letter}.\\ & \mathbf{Plan} && \text{I will start with the second row to find the value of } G.\\ &&& \text{Then I will use the first row and find the value of} \ F.\\ & \mathbf{Solve} && \text{Second Row:} \ G+G+7=23. \ 23-7=16, \text{so} \ G+G=16 \ \text{and} \ G=8.\\ &&& \text{First Row:} \ 9+F+F=21. \ 21-9=12, \text{so} \ F+F=12 \ \text{and} \ F=6.\\ & \mathbf{Check} && \text{First Row:} \ 9+6+6=21; \text{Second Row:} \ 8+8+7=23; \text{Third Row:} \ 8+4+5=17\\ &&& \text{First Column:} \ 9+8+8=25; \text{Second Column:} \ 6+8+4=18; \text{Third Column:} \ 6+7+5=18\end{align*}

#### Example C

The arrows point to the sum of each row or column. Figure out the value of each letter.

**Solution:**

We can use the problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{I see rows and columns with numbers and letters}.\\ &&& \text{The arrows point to sums}.\\ & \mathbf{My \ Job} && \text{Figure out the value of each letter}.\\ & \mathbf{Plan} && \text{In the second row, all letters are the same}.\\ &&& \text{I will start with that row}.\\ &&& \text{Then I will use the first row and find the value of} \ J.\\ & \mathbf{Solve} && \text{Second Row:} \ H+H+H=21, \text{so} \ H=7.\\ &&& \text{First Row:} \ J+J+4=22. \ 22-4=18, \text{so} \ J+J=18 \ \text{and} \ J=9.\\ & \mathbf{Check} && \text{First Row:} \ 9+9+4=22; \text{Second Row:} \ 7+7+7=21; \text{Third Row:} \ 3+8+7=18\\ &&& \text{First Column:} \ 9+7+3=19; \text{Second Column:} \ 9+7+8=24; \text{Third Column:} \ 4+7+7=18\end{align*}

#### Concept Problem Revisited

The arrows point to the sum of each row or column. Figure out the value of each letter.

We can use problem solving steps to help.

\begin{align*}& \mathbf{Describe} && \text{I see rows and columns with numbers and letters}.\\ &&& \text{The arrows point to sums}.\\ & \mathbf{My \ Job} && \text{Figure out the value of each letter}.\\ & \mathbf{Plan} && \text{In the second row, all letters are the same}.\\ &&& \text{I will start with that row}.\\ &&& \text{Then I will use the third row and find the value of} \ B.\\ & \mathbf{Solve} && \text{Second Row:} \ A + A + A = 6, \text{so} \ A = 2.\\ &&& \text{Third Row:} \ 7 + 5 = 12. \ 18 - 12 = 6, \text{so} \ B = 6.\\ & \mathbf{Check} && \text{First Row:} \ 3 + 2 + 6 = 11; \text{Second Row:} \ 2 + 2 + 2 = 6; \text{Third Row:} \ 7 + 5 + 6 = 18\\ &&& \text{First Column:} \ 3 + 2 + 7 = 12; \text{Second Column:} \ 2 + 2 + 5 = 9; \text{Third Column:} \ 6 + 2 + 6 = 14\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

The arrows point to the sum of each row or column. Figure out the value of each letter.

1.

2.

3.

**Answers:**

1. \begin{align*}K = 8; L = 5\end{align*}

2. \begin{align*}M = 9; N = 7\end{align*}

3. \begin{align*}P = 8; Q = 6\end{align*}

### Explore More

The arrows point to the sum of each row or column. Figure out the value of each letter.

### Image Attributions

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

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