Sneaky Sums—Unknowns in a Grid
Grades 4–8
Notes for the Teacher
Sneaky Sums is a game of logic that develops students’ early algebraic reasoning skills as
they deduce the numerical values assigned to four geometric shapes. A 4 × 4 grid is
randomly populated with circles, triangles, squares, hexagons, and blank spaces. Each
shape has been assigned a secret numerical value. Students can view the sum of the
shapes in any row or column. By using these sums as clues, students determine the
numerical values of each of the four shapes. Each new random assignment of shapes
and numerical values in the grid creates a fresh challenge to solve.
Objectives:
•
Students will use logical reasoning skills to determine the unknown numerical
values assigned to a collection of symbols.
•
Students will develop, refine, and share intuitive strategies for solving algebraic
equations without formal symbol manipulation.
Common Core Mathematical Practices: (1) Make sense of problems and
persevere in solving them; (2) Reason abstractly and quantitatively; (3) Construct viable
arguments and critique the reasoning of others; (5) Use appropriate tools strategically;
(7) Look for and make use of structure.
Common Core State Standards: 6.EE2, 4, 5, 6; 7.EE4; 8.EE8
Grade Range: Grades 4–8
Introduce:
Open Sneaky Sums--Unknowns in a Grid.gsp and distribute the worksheet. Use a
projector to show sketch page “Sneaky Sums.”
Press New Problem several times and ask students to observe what happens each time.
The pattern of shapes in the 4 × 4 grid changes each time the button is pressed. There
are four possible shapes that can appear in the grid: a circle, a triangle, a square, and a
hexagon. Not every space in the grid is necessarily filled with a shape; some spaces may
be blank.
Explain, “The computer has assigned secret numerical values from 1 to 10 to the circle,
triangle, square, and hexagon. It’s possible that two shapes might have the same value.
Your job is to figure out the value of each shape.”
Continue, “Sketchpad will give you clues. When you press a Show Sum button,
Sketchpad tells you the sum of the shapes in that row or column.”
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CC-BY-NC-SA 3.0
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Sneaky Sums—Unknowns in a Grid
Grades 4–8
Demonstrate by asking students to pick a row or column whose sum they’d like to see.
Press the Show Sum button corresponding to that row or column.
Explain that students can press as many Show Sums buttons as necessary to determine
the value of all shapes, but they should aim to view as few sums as necessary (a
minimum of four sums is always needed).
Do not solve a Sneaky Sums challenge together as a class just yet! Give students an
opportunity to develop their own strategies as they work with partners before sharing
their ideas with the entire class.
Explore:
Assign students to partners and send them in pairs to the computers. Have students
open Sneaky Sums--Unknowns in a Grid.gsp and go to page “Sneaky Sums.”
When playing Sneaky Sums for the first time, it’s best that students start with challenges
that aren’t too difficult. The initial layout of shapes on page “Sneaky Sums” is an
example of an introductory challenge: It contains five blank grid spaces, and the third
row contains only hexagons.
As students move on to new challenges, suggest that they press New Problem to create a
new challenge if they find a particular layout of shapes too difficult to solve. Ideally,
students will have the opportunity to play Sneaky Sums long enough that they can
develop the skill to solve more difficult problems.
Below are examples of three strategies that students will likely develop as they play
Sneaky Sums.
Strategy 1: Look for a row or column that contains just one type of shape.
In the example at left, the circled column has
three squares. Since the sum of the squares is 3,
each square has a value of 1.
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CC-BY-NC-SA 3.0
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Sneaky Sums—Unknowns in a Grid
Grades 4–8
Strategy 2: Look for pair of rows or columns (or one row and one column) that differ
by just one extra shape.
In the example at left, the circled row contains
two circles and a triangle. The circled column
is almost the same: It contains three circles
and a triangle. The value of the extra circle in
the column is the difference between the
column sum and the row sum:14 – 10, or 4.
Strategy 3: Look for rows or columns in which you know the value of all but
one shape.
By using Strategies 1 and 2 above, students
know that each square is equal to 1 and each
circle is equal to 4. Since square + triangle
+ circle = 7, then 1 + triangle + 4 = 7, or
triangle = 2.
From an algebraic perspective, students are learning about variables, equations,
simultaneous equations, and solving for unknowns as they apply these strategies. There
is no need, however, to formalize these concepts as students solve the Sneaky Sums
challenges. These algebraic ideas can develop organically as students use their own logic
to determine the numerical values of the shapes.
©2013 KCP Technologies, a McGraw-Hill Education Company
CC-BY-NC-SA 3.0
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Sneaky Sums—Unknowns in a Grid
Grades 4–8
Discuss:
Call students together to discuss and summarize what they have learned. Open Sneaky
Sums--Unknowns in a Grid.gsp and project page “Sneaky Sums.” As a class, solve
some new problems together. Below are two grids with a sample of the types of
explanations that students might give.
The sum of the triangles in the first row is 2. That means each triangle has a
value of 1. From the second row, I know that three circles plus one triangle
equals 31. Since the triangle equals 1, that means the three circles equal 30. Thus
each circle equals 10. From the third column, I know that a circle plus a square
equals 17. Since the circle equals 10, that means the square equals 7. Finally,
since circle + square + hexagon = 18 in the third row, I can substitute the values
of the circle and square to figure out that a hexagon equals 1.
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CC-BY-NC-SA 3.0
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Sneaky Sums—Unknowns in a Grid
Grades 4–8
The first column and the fourth column are almost identical—they each have a
square, a circle, and a hexagon. But the fourth column also has an extra square.
Since 18 – 12 = 6, that means the square equals 6. The first column and the third
row are nearly the same, too. They also share a square, a circle, and a hexagon.
But the third row also has an extra hexagon. Since 13 – 12 = 1, that means the
hexagon equals 1. Since square + circle + hexagon = 12 in the first column, I can
substitute the values of the square and hexagon to get 6 + circle + 1 = 12. So
circle = 5. Finally, I can use the second column to figure out that, since hexagon
+ hexagon + circle + triangle = 8 and hexagon = 1 and circle = 5, then
triangle = 1.
Explore More:
Students can create Sneaky Sums challenges for each other by using the “Make Your
Own” page of the sketch. Working in pairs, one student looks away while the other
student follows these directions:
1) Press New Symbols to create a new random pattern of shapes in the grid.
2) Enter numerical values for the circle, triangle, square, and hexagon. Once
the values are entered, press Hide Answers.
Students can take turns creating new challenges for each other.
Related Activities:
•
Mystery Sums, Part One—Unknown Addends
•
Mystery Sums, Part Two—Unknown Addends
•
Mystery Sums, Part Three—Dancing Addends
•
Balance—Solving for Unknowns, Part One
•
Balance—Solving for Unknowns, Part Two
•
Balance—Solving for Unknowns, Part Three
License (CC-BY-NC-SA 3.0)
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©2013 KCP Technologies, a McGraw-Hill Education Company
CC-BY-NC-SA 3.0
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Sneaky Sums—Unknowns in a Grid
Grades 4–8
Portions of this material are based upon work supported by the National Science Foundation under award
number DRL-0918733. Any opinions, findings, and conclusions or recommendations expressed in this
work are those of the author(s) and do not necessarily reflect the views of the National Science
Foundation.
©2013 KCP Technologies, a McGraw-Hill Education Company
CC-BY-NC-SA 3.0
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