Practice · Solve shape-coded products in order · Sensim Math

Variant 1 easy

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 9 \qquad \blacktriangle \times \bullet = 18 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 9, triangle x circle = 18, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 9.
triangle x circle = 18.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 9, so circle is the number that times itself is 9. That is 3.

3 \times 3 = 9

Only 3 among one-digit numbers squares to 9, so circle = 3.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 18 and circle = 3, so triangle x 3 = 18, giving triangle = 6.

\blacktriangle \times 3 = 18 \Rightarrow \blacktriangle = 6

The missing factor of 18 with 3 is 6, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 16 since triangle = 6. So square x 2 = 16, giving square = 8. The shapes 3, 6, 8 are all different, so the sum is 3 + 6 + 8 = 17.

\blacksquare \times 2 = 16 \Rightarrow \blacksquare = 8,\quad 3+6+8 = 17

Doubling to reach 16 needs 8, and adding the three found values gives the total.

Answer: 17

4 · Reviewdoes it hold up?

Check all three: 3 x 3 = 9, 6 x 3 = 18, and 8 x 2 = 16 = '1 triangle' with triangle = 6; the shapes 3, 6, 8 are distinct one-digit numbers, so the sum 17 is correct.

Another way: You could list one-digit values: only circle = 3 fits the first equation, which then forces triangle = 6 and square = 8, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 9 and triangle from triangle x 3 = 18.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Variant 2 easy

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 16 \qquad \blacktriangle \times \bullet = 24 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 16, triangle x circle = 24, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 16.
triangle x circle = 24.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 16, so circle is the number that times itself is 16. That is 4.

4 \times 4 = 16

Only 4 among one-digit numbers squares to 16, so circle = 4.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 24 and circle = 4, so triangle x 4 = 24, giving triangle = 6.

\blacktriangle \times 4 = 24 \Rightarrow \blacktriangle = 6

The missing factor of 24 with 4 is 6, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 16 since triangle = 6. So square x 2 = 16, giving square = 8. The shapes 4, 6, 8 are all different, so the sum is 4 + 6 + 8 = 18.

\blacksquare \times 2 = 16 \Rightarrow \blacksquare = 8,\quad 4+6+8 = 18

Doubling to reach 16 needs 8, and adding the three found values gives the total.

Answer: 18

4 · Reviewdoes it hold up?

Check all three: 4 x 4 = 16, 6 x 4 = 24, and 8 x 2 = 16 = '1 triangle' with triangle = 6; the shapes 4, 6, 8 are distinct one-digit numbers, so the sum 18 is correct.

Another way: You could list one-digit values: only circle = 4 fits the first equation, which then forces triangle = 6 and square = 8, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 16 and triangle from triangle x 4 = 24.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Variant 3 easy

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 9 \qquad \blacktriangle \times \bullet = 24 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 9, triangle x circle = 24, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 9.
triangle x circle = 24.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 9, so circle is the number that times itself is 9. That is 3.

3 \times 3 = 9

Only 3 among one-digit numbers squares to 9, so circle = 3.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 24 and circle = 3, so triangle x 3 = 24, giving triangle = 8.

\blacktriangle \times 3 = 24 \Rightarrow \blacktriangle = 8

The missing factor of 24 with 3 is 8, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 18 since triangle = 8. So square x 2 = 18, giving square = 9. The shapes 3, 8, 9 are all different, so the sum is 3 + 8 + 9 = 20.

\blacksquare \times 2 = 18 \Rightarrow \blacksquare = 9,\quad 3+8+9 = 20

Doubling to reach 18 needs 9, and adding the three found values gives the total.

Answer: 20

4 · Reviewdoes it hold up?

Check all three: 3 x 3 = 9, 8 x 3 = 24, and 9 x 2 = 18 = '1 triangle' with triangle = 8; the shapes 3, 8, 9 are distinct one-digit numbers, so the sum 20 is correct.

Another way: You could list one-digit values: only circle = 3 fits the first equation, which then forces triangle = 8 and square = 9, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 9 and triangle from triangle x 3 = 24.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Variant 4 medium

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 25 \qquad \blacktriangle \times \bullet = 20 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 25, triangle x circle = 20, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 25.
triangle x circle = 20.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 25, so circle is the number that times itself is 25. That is 5.

5 \times 5 = 25

Only 5 among one-digit numbers squares to 25, so circle = 5.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 20 and circle = 5, so triangle x 5 = 20, giving triangle = 4.

\blacktriangle \times 5 = 20 \Rightarrow \blacktriangle = 4

The missing factor of 20 with 5 is 4, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 14 since triangle = 4. So square x 2 = 14, giving square = 7. The shapes 5, 4, 7 are all different, so the sum is 5 + 4 + 7 = 16.

\blacksquare \times 2 = 14 \Rightarrow \blacksquare = 7,\quad 5+4+7 = 16

Doubling to reach 14 needs 7, and adding the three found values gives the total.

Answer: 16

4 · Reviewdoes it hold up?

Check all three: 5 x 5 = 25, 4 x 5 = 20, and 7 x 2 = 14 = '1 triangle' with triangle = 4; the shapes 5, 4, 7 are distinct one-digit numbers, so the sum 16 is correct.

Another way: You could list one-digit values: only circle = 5 fits the first equation, which then forces triangle = 4 and square = 7, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 25 and triangle from triangle x 5 = 20.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Variant 5 medium

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 25 \qquad \blacktriangle \times \bullet = 30 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 25, triangle x circle = 30, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 25.
triangle x circle = 30.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 25, so circle is the number that times itself is 25. That is 5.

5 \times 5 = 25

Only 5 among one-digit numbers squares to 25, so circle = 5.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 30 and circle = 5, so triangle x 5 = 30, giving triangle = 6.

\blacktriangle \times 5 = 30 \Rightarrow \blacktriangle = 6

The missing factor of 30 with 5 is 6, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 16 since triangle = 6. So square x 2 = 16, giving square = 8. The shapes 5, 6, 8 are all different, so the sum is 5 + 6 + 8 = 19.

\blacksquare \times 2 = 16 \Rightarrow \blacksquare = 8,\quad 5+6+8 = 19

Doubling to reach 16 needs 8, and adding the three found values gives the total.

Answer: 19

4 · Reviewdoes it hold up?

Check all three: 5 x 5 = 25, 6 x 5 = 30, and 8 x 2 = 16 = '1 triangle' with triangle = 6; the shapes 5, 6, 8 are distinct one-digit numbers, so the sum 19 is correct.

Another way: You could list one-digit values: only circle = 5 fits the first equation, which then forces triangle = 6 and square = 8, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 25 and triangle from triangle x 5 = 30.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Variant 6 medium

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 16 \qquad \blacktriangle \times \bullet = 32 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 16, triangle x circle = 32, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 16.
triangle x circle = 32.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 16, so circle is the number that times itself is 16. That is 4.

4 \times 4 = 16

Only 4 among one-digit numbers squares to 16, so circle = 4.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 32 and circle = 4, so triangle x 4 = 32, giving triangle = 8.

\blacktriangle \times 4 = 32 \Rightarrow \blacktriangle = 8

The missing factor of 32 with 4 is 8, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 18 since triangle = 8. So square x 2 = 18, giving square = 9. The shapes 4, 8, 9 are all different, so the sum is 4 + 8 + 9 = 21.

\blacksquare \times 2 = 18 \Rightarrow \blacksquare = 9,\quad 4+8+9 = 21

Doubling to reach 18 needs 9, and adding the three found values gives the total.

Answer: 21

4 · Reviewdoes it hold up?

Check all three: 4 x 4 = 16, 8 x 4 = 32, and 9 x 2 = 18 = '1 triangle' with triangle = 8; the shapes 4, 8, 9 are distinct one-digit numbers, so the sum 21 is correct.

Another way: You could list one-digit values: only circle = 4 fits the first equation, which then forces triangle = 8 and square = 9, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 16 and triangle from triangle x 4 = 32.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Variant 7 medium

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 36 \qquad \blacktriangle \times \bullet = 24 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 36, triangle x circle = 24, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 36.
triangle x circle = 24.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 36, so circle is the number that times itself is 36. That is 6.

6 \times 6 = 36

Only 6 among one-digit numbers squares to 36, so circle = 6.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 24 and circle = 6, so triangle x 6 = 24, giving triangle = 4.

\blacktriangle \times 6 = 24 \Rightarrow \blacktriangle = 4

The missing factor of 24 with 6 is 4, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 14 since triangle = 4. So square x 2 = 14, giving square = 7. The shapes 6, 4, 7 are all different, so the sum is 6 + 4 + 7 = 17.

\blacksquare \times 2 = 14 \Rightarrow \blacksquare = 7,\quad 6+4+7 = 17

Doubling to reach 14 needs 7, and adding the three found values gives the total.

Answer: 17

4 · Reviewdoes it hold up?

Check all three: 6 x 6 = 36, 4 x 6 = 24, and 7 x 2 = 14 = '1 triangle' with triangle = 4; the shapes 6, 4, 7 are distinct one-digit numbers, so the sum 17 is correct.

Another way: You could list one-digit values: only circle = 6 fits the first equation, which then forces triangle = 4 and square = 7, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 36 and triangle from triangle x 6 = 24.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Variant 8 hard

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 25 \qquad \blacktriangle \times \bullet = 40 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 25, triangle x circle = 40, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 25.
triangle x circle = 40.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 25, so circle is the number that times itself is 25. That is 5.

5 \times 5 = 25

Only 5 among one-digit numbers squares to 25, so circle = 5.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 40 and circle = 5, so triangle x 5 = 40, giving triangle = 8.

\blacktriangle \times 5 = 40 \Rightarrow \blacktriangle = 8

The missing factor of 40 with 5 is 8, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 18 since triangle = 8. So square x 2 = 18, giving square = 9. The shapes 5, 8, 9 are all different, so the sum is 5 + 8 + 9 = 22.

\blacksquare \times 2 = 18 \Rightarrow \blacksquare = 9,\quad 5+8+9 = 22

Doubling to reach 18 needs 9, and adding the three found values gives the total.

Answer: 22

4 · Reviewdoes it hold up?

Check all three: 5 x 5 = 25, 8 x 5 = 40, and 9 x 2 = 18 = '1 triangle' with triangle = 8; the shapes 5, 8, 9 are distinct one-digit numbers, so the sum 22 is correct.

Another way: You could list one-digit values: only circle = 5 fits the first equation, which then forces triangle = 8 and square = 9, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 25 and triangle from triangle x 5 = 40.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Variant 9 hard

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 36 \qquad \blacktriangle \times \bullet = 48 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 36, triangle x circle = 48, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 36.
triangle x circle = 48.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 36, so circle is the number that times itself is 36. That is 6.

6 \times 6 = 36

Only 6 among one-digit numbers squares to 36, so circle = 6.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 48 and circle = 6, so triangle x 6 = 48, giving triangle = 8.

\blacktriangle \times 6 = 48 \Rightarrow \blacktriangle = 8

The missing factor of 48 with 6 is 8, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 18 since triangle = 8. So square x 2 = 18, giving square = 9. The shapes 6, 8, 9 are all different, so the sum is 6 + 8 + 9 = 23.

\blacksquare \times 2 = 18 \Rightarrow \blacksquare = 9,\quad 6+8+9 = 23

Doubling to reach 18 needs 9, and adding the three found values gives the total.

Answer: 23

4 · Reviewdoes it hold up?

Check all three: 6 x 6 = 36, 8 x 6 = 48, and 9 x 2 = 18 = '1 triangle' with triangle = 8; the shapes 6, 8, 9 are distinct one-digit numbers, so the sum 23 is correct.

Another way: You could list one-digit values: only circle = 6 fits the first equation, which then forces triangle = 8 and square = 9, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 36 and triangle from triangle x 6 = 48.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Variant 10 hard

Each shape stands for the same number wherever it appears. The shapes $\bullet$ , $\blacktriangle$ , and $\blacksquare$ are three different one-digit numbers. Using the equations below, find the value of $\bullet + \blacktriangle + \blacksquare$ .

$\bullet \times \bullet = 81 \qquad \blacktriangle \times \bullet = 36 \qquad \blacksquare \times 2 = 1\,\blacktriangle$

Here $1\,\blacktriangle$ is the two-digit number whose tens digit is $1$ and whose ones digit is $\blacktriangle$ .

Your answer

Show solution

1 · Understandwhat's really being asked

Three shapes stand for three different one-digit numbers. From the equations circle x circle = 81, triangle x circle = 36, and square x 2 = the two-digit number '1 triangle', find the sum circle + triangle + square.

Givens

circle x circle = 81.
triangle x circle = 36.
square x 2 = 1(triangle), the two-digit number with tens digit 1 and ones digit triangle.
The three shapes are different one-digit numbers.

Unknowns

The values of circle, triangle, and square.
Their sum circle + triangle + square.

Constraints

Each shape is a single-digit whole number.
Different shapes are different numbers.

2 · Planchoose the strategy

#3 Eliminate Possibilities · also uses: #6 Guess and Check

Solve in order from the equation that can be determined alone: the square of circle pins circle, then triangle follows, then square. Each step has only one valid one-digit value.

3 · Execute3 carry out the plan

1Find the circle

#3 Eliminate Possibilities 3.OA.A.4

circle x circle = 81, so circle is the number that times itself is 81. That is 9.

9 \times 9 = 81

Only 9 among one-digit numbers squares to 81, so circle = 9.

2Find the triangle

#3 Eliminate Possibilities 3.OA.A.4

triangle x circle = 36 and circle = 9, so triangle x 9 = 36, giving triangle = 4.

\blacktriangle \times 9 = 36 \Rightarrow \blacktriangle = 4

The missing factor of 36 with 9 is 4, a basic division fact.

3Find the square and sum

#6 Guess and Check 3.OA.D.8

The two-digit number 1(triangle) is 14 since triangle = 4. So square x 2 = 14, giving square = 7. The shapes 9, 4, 7 are all different, so the sum is 9 + 4 + 7 = 20.

\blacksquare \times 2 = 14 \Rightarrow \blacksquare = 7,\quad 9+4+7 = 20

Doubling to reach 14 needs 7, and adding the three found values gives the total.

Answer: 20

4 · Reviewdoes it hold up?

Check all three: 9 x 9 = 81, 4 x 9 = 36, and 7 x 2 = 14 = '1 triangle' with triangle = 4; the shapes 9, 4, 7 are distinct one-digit numbers, so the sum 20 is correct.

Another way: You could list one-digit values: only circle = 9 fits the first equation, which then forces triangle = 4 and square = 7, the same result.

Standardsmin grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding circle from circle x circle = 81 and triangle from triangle x 9 = 36.
3.OA.D.8 Solve two-step word problems using four operations within 100 — Using the two-digit clue to find square and summing the three shape values.

💡Takeaway. Start with the shape you can pin down, then unlock the rest in order -- just Grade 3 multiplication facts!

Solve shape-coded products in order · 10 practice problems

Generated variants — 10

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?