# 5.4: Circles and Arrows

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

**Circles and Arrows - Solve for Unknowns**

Circles and Arrows show grids in which circles contain numbers and letters that represent unknown quantities. Arrows point to sums of rows and columns. For example, in the top row, \begin{align*}3 + A + B = 11\end{align*}. In the first column, \begin{align*}3 + A + 7 = 12\end{align*}. Students use the grid relationships to figure out the values of the letters. Although only one approach is shown, there are often several ways to begin to solve each problem. Encourage students to consider alternative approaches for each problem.

**Solutions**

- \begin{align*}C = 3; D = 5\end{align*}
- \begin{align*}F =6; G = 8\end{align*}
- \begin{align*}H = 7; J = 9\end{align*}
- \begin{align*}K = 8; L = 5\end{align*}
- \begin{align*}M = 9; N = 7\end{align*}
- \begin{align*}P = 8; Q = 6\end{align*}

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

\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'll start with that row}.\\ &&& \text{Then I'll do the bottom row and find the value of} \ B.\\ & \mathbf{Solve} && \text{Middle Row:} \ A + A + A = 6, \text{so} \ A = 2.\\ &&& \text{Bottom Row:} \ 7 + 5 = 12. \ 18 - 12 = 6, \text{so} \ B = 6.\\ & \mathbf{Check} && \text{Top Row:} \ 3 + 2 + 6 = 11; \text{Middle Row:} \ 2 + 2 + 2 = 6; \text{Bottom Row:} \ 7 + 5 + 6 = 18\\ &&& \text{Left Column:} \ 3 + 2 + 7 = 12; \text{Middle Column:} \ 2 + 2 + 5 = 9; \text{Right Column:} \ 6 + 2 + 6 = 14\end{align*}