# 4.4: Circles and Arrows

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

**Circles and Arrows – Solve for Unknowns**

Circles and Arrows show grids with two rows and three columns. In each grid, circles contain numbers or letters. Letters represent unknown quantities. Arrows point to sums of rows and columns. All rows and columns represent equations. For example, a row that shows \begin{align*}A, A\end{align*}, 4 and a sum of 10, can be represented as \begin{align*}A + A + 4 = 10\end{align*}. Students use the grid relationships to figure out the values of the letters. There are often several ways to begin to solve each problem. Encourage students to consider alternative solution approaches and to talk about them.

**Solutions**

- \begin{align*}C = 5, D = 4\end{align*}
- \begin{align*}E = 3, D = 8\end{align*}
- \begin{align*}G = 2, H = 5\end{align*}
- \begin{align*}J = 6, K = 4\end{align*}
- \begin{align*}K = 8, M = 4\end{align*}
- \begin{align*}N = 3, P = 7\end{align*}
- \begin{align*}Q = 4, R = 2\end{align*}
- \begin{align*}S = 6, T = 5\end{align*}
- \begin{align*}U = 3, V = 8\end{align*}
- \begin{align*}W = 9, X = 4\end{align*}

**Figure out the value of each letter.**

\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 with 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*}

**Extras for Experts: Circles and Arrows – Solve for Unknowns**

**Figure out the value of each letter.**

**Solutions**

- \begin{align*}A = 8; B = 4\end{align*}
- \begin{align*}c = 5; D = 9\end{align*}
- \begin{align*}E = 1; F = 7\end{align*}
- \begin{align*}G = 6; H = 2\end{align*}
- \begin{align*}J = 4; K = 3\end{align*}
- \begin{align*}L = 2; M = 8\end{align*}
- \begin{align*}N = 7; P = 5\end{align*}
- \begin{align*}Q = 6; R = 9\end{align*}