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Letter Grids – Solve for Unknowns

Teacher Notes

Letter grids show three rows and four columns of circles, each containing a number or letter. Letters represent unknown values. Same letters have same values. Arrows point to sums of rows and columns. Students use relationships among contents of row and columns to figure ou the values of the five unknowns. For each grid, one of the columns is a subset of one of the rows. For example, in the teaching example, Column 4 contains A, B, and C. Row 3 also contains A, B and C, as well as an additional C. The ABC in Row 3 can be replaced with its sum (18), and the value of the remaining C can be determined.

Encourage students to record the value of a variable everywhere it appears in the grid, once it has been determined. This will facilitate the solution process by helping students see rows and columns where only one variable remains.

Note: For some problems, there are different solution paths.

Solutions:

\mathbf{Letter \ Grid \ 1 :}  \ M = 13, \ P = 10, \ Q = 20, \ T = 6, \ W=2\!\\\\\\\mathbf{Letter \ Grid \ 2 :} \ C = 2, \ D = 4, \ E = 7, \ F = 5, \ G=6\!\\\\\\\mathbf{Letter \ Grid \ 3 :} \ A = 3, \ B = 1, \ X = 11, \ Y = 12, \ Z=4\!\\\\\\\mathbf{Letter \ Grid \ 4 :} \ E = 9, \ G = 7, \ L = 10, \ R = 5, \ S=4\!\\\\\\\mathbf{Letter \ Grid \ 5 :} \ D = 3, \ E = 8, \ F = 5, \ G = 15, \ H=20\!\\\\\\\mathbf{Letter \ Grid \ 6 :} \ S = 1, \ T = 20, \ U = 5, \ X = 12, \ W=9

Letter Grid

The arrows point to the sum of each row or column. Same letters stand for same numbers. Figure out the value of each letter

&\mathbf{Describe} && \text{There are three rows and four columns.}\\&&& \text{Circles have letters or numbers.}\\&&& \text{Sums of rows and columns are in the squares.}\\&&& \text{Arrows point to sums.}\\\\& \mathbf{My \ Job} && \text{Figure out values of letters,} \ A, B, C, D, \ \text{and} \ E.\\\\&\mathbf{Plan} && \text{Look for a column that contains all letters shown a row. Replace the group of}\\&&& \text{same letters with its value and continue.}\\\\&\mathbf{Solve} && \text{All of Column} \ 4 \ \text{is contained in Row} \ 3. \ \text{Replace} \ ABC \ \text{in Row} \ 3 \ \text{with} \ 18. \ \text{Then} \ 18 + C = 22,\\&&& \text{and} \ C = 4. \ \text{Replace all} \ C' \text{s in the grid with} \ 4.\\&&& \text{Column} \ 3: D + D + 4 = 22. \ \text{So}, \ 2D = 18 \ \text{and} \ D = 9. \ \text{Replace all} \ D' \text{s with} \ 9.\\&&& \text{Row} \ 1: 2 + A + 9 + A = 27. \ \text{So}, \ 2A + 11 = 27 \ \text{and} \ 2A = 16 \ \text{and} \ A = 8. \ \text{Replace} \ A' \text{s with} \ 8.\\&&& \text{Column} \ 4: 8 + B + 4 = 18. \ \text{So}, \ B + 12 = 18 \ \text{and} \ B = 6. \ \text{Replace} \ B'\text{s with} \ 6.\\&&& \text{Column} \ 1: 2 + E + 8 = 11. \ \text{So}, \ E + 10 = 11, \ \text{and} \ E = 1. \ \text{Replace} \ E \ \text{with} \ 1.\\\\&\mathbf{Check} && \text{Check rows or columns}\\&&& \text{Row} \ 1: 2 + 8 + 9 + 8 = 27\\&&& \text{Row} \ 2: 1 + 8 + 9 + 6 = 24\\&&& \text{Row} \ 3: 8 + 6 + 4 + 4 = 22

Letter Grid 1

Letters stand for numbers.

Same letters are same numbers.

Numbers in squares are row and column sums.

Arrows point to the sums.

What are the values? \mathbf{M} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{P} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{Q} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{T} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{W} = \underline{\;\;\;\;\;\;\;\;}

Letter Grid 2

Letters stand for numbers.

Same letters are same numbers.

Numbers in squares are row and column sums.

Arrows point to the sums.

What are the values? \mathbf{C} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{D} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{E} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{F} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{G} = \underline{\;\;\;\;\;\;\;\;}

Letter Grid 3

Letters stand for numbers.

Same letters are same numbers.

Numbers in squares are row and column sums.

Arrows point to the sums.

What are the values? \mathbf{A} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{B} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{X} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{Y} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{Z}= \underline{\;\;\;\;\;\;\;\;}

Letter Grid 4

Letters stand for numbers.

Same letters are same numbers.

Numbers in squares are row and column sums.

Arrows point to the sums.

What are the values? \mathbf{E} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{G} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{L} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{R} = \underline{\;\;\;\;\;\;\;\;} \ \mathbf{S} = \underline{\;\;\;\;\;\;\;\;}

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