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

Variant 1 answer: 22

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 2 answer: 18

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 3 answer: 17

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 4 answer: 19

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 5 answer: 16

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 6 answer: 20

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 7 answer: 21

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 8 answer: 23

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 9 answer: 17

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

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

Variant 10 answer: 20

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 $\blacktriangle$ .

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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.

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

Standards · min 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.

💡 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

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Standards · min grade 3

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Standards · min grade 3

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Standards · min grade 3

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Standards · min grade 3

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Standards · min grade 3

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Standards · min grade 3

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Standards · min grade 3

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Standards · min grade 3

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Standards · min grade 3

Understand

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Standards · min grade 3