# 2.3: Circles and Arrows

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

**Circles and Arrows – Solve for Unknowns**

In these problems, students are presented two-by-two grids containing circles and arrows. Circles contain numbers or letters that represent unknown quantities. Arrows point to sums of rows and columns. Each row and column represents an equation. For example, a row showing \begin{align*}A\end{align*}, \begin{align*}A\end{align*} and an arrow pointing to 4, can also be represented by the equation: \begin{align*}A + A = 4\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 and talk about alternative approaches. All sums in the first set of problems are 10 or less. In the Extras for Experts, sums are 18 or less.

**Solutions**

- \begin{align*}C = 3, D = 1\end{align*}
- \begin{align*}E = 5, F = 2\end{align*}
- \begin{align*}G = 4, H = 1\end{align*}
- \begin{align*}J = 1, K = 5\end{align*}
- \begin{align*}L = 2, M = 3\end{align*}
- \begin{align*}N = 5, P = 3\end{align*}
- \begin{align*}Q = 2, R = 4\end{align*}
- \begin{align*}S = 4, T = 2\end{align*}
- \begin{align*}U = 4, V = 6\end{align*}
- \begin{align*}W = 2, X = 3\end{align*}

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

\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 with 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*}

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

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

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

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

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

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

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

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

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

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

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