# 6.4: Circles and Arrows

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

**Circles and Arrows - Solve for Unknowns**

**Teacher Notes**

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*}6 + A + B =13\end{align*}. In the first column, \begin{align*}6 + B + B = 16\end{align*}. Students use the grid relationships to figure out the values of the letters. Each grid contains three different letters and one number.

**Solutions**

- \begin{align*}D = 4, \ E = 1, \ F = 10\end{align*}
- \begin{align*}G = 7, \ H = 3, \ J = 6\end{align*}
- \begin{align*}K = 8, \ M = 4, \ N = 3\end{align*}
- \begin{align*}P = 6, \ R = 2, \ S = 8\end{align*}
- \begin{align*}Q = 5, \ T = 7, \ U =4\end{align*}
- \begin{align*}V = 3, \ W = 9, \ Z = 5\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{There are rows and columns with numbers and letters.}\!\\ &&& \text{Arrows point to sums}.\!\\ &&& \text{There are three different letters}.\!\\ & \mathbf{Job} && \text{Figure out the value of each letter,} \ a, b, \ \text{and} \ c.\!\\ & \mathbf{Plan} && \text{In the first column, there are} \ 2 \ \text{Bs and} \ 6.\!\\ &&& \text{Start with that column.}\!\\ & \mathbf{Solve} && \text{Column} \ 1: \ 6 + B + B = 16, \ \text{so} \ B + B = 10 \ \text{and} \ B = 5.\!\\ &&& \text{Row} \ 1 \ \text{has} \ 6, B \ \text{and} \ A. \ \text{Replace} \ B \ \text{with} \ 5 \ \text{and solve for} \ A, 6 + A + 5 = 13, \ \text{so} \ A + 11 =\!\\ &&& 13 \ \text{and} \ A = 2.\!\\ &&& \text{Column} \ 2: \ \text{replace} \ A \ \text{and} \ B \ \text{with their values, and find} \ C. \ 2 + 5 + C = 15, \ \text{so} \ 7 + C\!\\ &&& =15, \ \text{and} \ C = 8.\!\\ & \mathbf{Check} && \text{Rows top to bottom}: \ \ 6 + 2 + 5 = 13 \ \text{Columns left to right}: \ 6 + 5 + 5 = 16\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 5 + 5 + 8 = 18 \qquad \qquad \qquad \qquad \qquad \ \ 2 + 5 + 8 = 15\!\\ &&& \qquad \qquad \qquad \qquad \qquad \ 5 + 8 + 2 = 15 \qquad \qquad \qquad \qquad \qquad \ \ 5 + 8 + 2 = 15\end{align*}

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

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

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

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