# 6.6: Circle Sums

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

**Circle Sums - Solve for Values of Unknowns**

**Teacher Notes**

Circle Sums show three-by-three grids of circles that contain either letters that represent unknown quantities or numbers. Arrows point to sums of rows and columns. For example, in Row 1, \begin{align*}A + C + B =14\end{align*}, and in Column 1, \begin{align*}A + A + B = 17\end{align*}. Students use grid relationships to figure out the values of the letters. Each grid contains three different letters and one number. Note: There is often more than one solution path.

**Solutions**

- \begin{align*}A = 4, \ B = 3, \ C = 5\end{align*}
- \begin{align*}A = 6, \ B = 4, \ C = 8\end{align*}
- \begin{align*}A = 5, \ B = 6, \ C = 3\end{align*}
- \begin{align*}A = 7, \ B = 8, \ C = 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{Rows and columns contain numbers or letters.}\\ &&& \text{Arrows point to sums.}\\ &&& \text{There are three different letters.}\\ &&& \text{No rows or columns contain only one type of letter.}\\ & \mathbf{My \ job} && \text{Figure out the value of each letter,} \ A, \ B, \ \text{and} \ C.\\ & \mathbf{Plan} && \text{Row} \ 3: \ B + A + 4 = 16. \ \text{Figure out the value of} \ A + B.\\ &&& \text{Column} \ 1: A + A + B = 17.\\ &&& \text{Replace} \ A + B \ \text{with its sum to figure out the value of} \ A.\\ &&& \text{Then replace all A’s with that number and solve for other letters.}\\ & \mathbf{Solve} && \text{Row} \ 3: \ B + A + 4 = 16. \ \text{So} \ B + A = 16 - 4, \ \text{or} \ 12.\\ &&& \text{Column} \ 1: A + A + B = 17. \ \text{Replace} \ A + B \ \text{with} \ 12.\\ &&& \text{Then} \ 12 + A = 17 \ \text{and} \ A = 17 - 12, \ \text{or} \ 5.\\ &&& \text{Column} \ 2: \ \text{Replace}\ A \ \text{with} \ 5. \ \text{Then} \ C + C + 5 = 9, 2C = 9 - 5, \ \text{or} \ 4 \ \text{and} \ C = 2.\\ &&& \text{Column} \ 3: \ \text{Replace}\ C \ \text{with} \ 2. \ \text{Then} \ B + 2 + 4 = 13, \text{and} \ B = 13 - 6, \ \text{or} \ 7.\\ &&& A = 5, B = 7, C = 2.\\ & \mathbf{Check} && \text{Replace letters with their values. Check with the sums.}\\ &&& \text{Rows top to bottom:} \ 5 + 2 + 7 = 14 \ \text{Columns left to right:} \ 5 + 5 + 7 = 17\\ &&& \qquad \qquad \qquad \qquad \quad \ 5 + 2 + 2 = 9 \qquad \qquad \qquad \qquad \qquad \quad 2 + 2 + 5 = 9\\ &&& \qquad \qquad \qquad \qquad \quad \ 7 + 5 + 4 = 16 \qquad \qquad \qquad \qquad \qquad \ 7 + 2 + 4 = 13\end{align*}

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

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

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

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

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