← Check multiple relation to find correspondence rule · Generalize a Growing Pattern into a Rule

Check multiple relation to find correspondence rule · 12 practice problems

4.OA.C.55.OA.B.35.OA.A.2

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 81

The table shows the correspondence between $△$ and $♡$ . When $△$ is $20$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$5$	$9$	$13$	$17$	$21$	$25$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 5, 9, 13, 17, 21, 25. We must find the heart that goes with a triangle of 20.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 5, 9, 13, 17, 21, 25.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 20.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 4 more than the one before it, all the way along.

5, 9, 13, 17, 21, 25

A constant jump of 4 suggests multiplying by 4.

2Compare the heart with 4 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 4 gives 4, 8, 12, 16, 20, 24. Line those up under the actual hearts: every one is off by the same 1.

4, 8, 12, 16, 20, 24

The multiple is close but not equal, so add 1.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 17.

4 \times 4 + 1 = 17

The rule survives a row it was not built from.

4Apply the rule to a triangle of 20

#5 Look for a Pattern 5.OA.A.2

Multiply 20 by 4, then add 1.

4 \times 20 + 1 = 81

The heart is 81.

Answer: 81

4 · Reviewdoes it hold up?

The heart for triangle 6 is 25, and 20 is 14 further along at 4 per step: 25 + 56 = 81.

Another way: Continue the table by adding 4 repeatedly from row 6 up to row 20; slower, but it lands on 81 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 4 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 20.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 2 easy answer: 74

The table shows the correspondence between $△$ and $♡$ . When $△$ is $25$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$2$	$5$	$8$	$11$	$14$	$17$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 2, 5, 8, 11, 14, 17. We must find the heart that goes with a triangle of 25.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 2, 5, 8, 11, 14, 17.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 25.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 3 more than the one before it, all the way along.

2, 5, 8, 11, 14, 17

A constant jump of 3 suggests multiplying by 3.

2Compare the heart with 3 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 3 gives 3, 6, 9, 12, 15, 18. Line those up under the actual hearts: every one is off by the same 1.

3, 6, 9, 12, 15, 18

The multiple is close but not equal, so subtract 1.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 11.

3 \times 4 - 1 = 11

The rule survives a row it was not built from.

4Apply the rule to a triangle of 25

#5 Look for a Pattern 5.OA.A.2

Multiply 25 by 3, then subtract 1.

3 \times 25 - 1 = 74

The heart is 74.

Answer: 74

4 · Reviewdoes it hold up?

The heart for triangle 6 is 17, and 25 is 19 further along at 3 per step: 17 + 57 = 74.

Another way: Continue the table by adding 3 repeatedly from row 6 up to row 25; slower, but it lands on 74 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 3 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 25.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 3 easy answer: 105

The table shows the correspondence between $△$ and $♡$ . When $△$ is $27$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$1$	$5$	$9$	$13$	$17$	$21$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 1, 5, 9, 13, 17, 21. We must find the heart that goes with a triangle of 27.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 1, 5, 9, 13, 17, 21.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 27.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 4 more than the one before it, all the way along.

1, 5, 9, 13, 17, 21

A constant jump of 4 suggests multiplying by 4.

2Compare the heart with 4 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 4 gives 4, 8, 12, 16, 20, 24. Line those up under the actual hearts: every one is off by the same 3.

4, 8, 12, 16, 20, 24

The multiple is close but not equal, so subtract 3.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 13.

4 \times 4 - 3 = 13

The rule survives a row it was not built from.

4Apply the rule to a triangle of 27

#5 Look for a Pattern 5.OA.A.2

Multiply 27 by 4, then subtract 3.

4 \times 27 - 3 = 105

The heart is 105.

Answer: 105

4 · Reviewdoes it hold up?

The heart for triangle 6 is 21, and 27 is 21 further along at 4 per step: 21 + 84 = 105.

Another way: Continue the table by adding 4 repeatedly from row 6 up to row 27; slower, but it lands on 105 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 4 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 27.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 4 easy answer: 88

The table shows the correspondence between $△$ and $♡$ . When $△$ is $18$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$3$	$8$	$13$	$18$	$23$	$28$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 3, 8, 13, 18, 23, 28. We must find the heart that goes with a triangle of 18.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 3, 8, 13, 18, 23, 28.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 18.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 5 more than the one before it, all the way along.

3, 8, 13, 18, 23, 28

A constant jump of 5 suggests multiplying by 5.

2Compare the heart with 5 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 5 gives 5, 10, 15, 20, 25, 30. Line those up under the actual hearts: every one is off by the same 2.

5, 10, 15, 20, 25, 30

The multiple is close but not equal, so subtract 2.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 18.

5 \times 4 - 2 = 18

The rule survives a row it was not built from.

4Apply the rule to a triangle of 18

#5 Look for a Pattern 5.OA.A.2

Multiply 18 by 5, then subtract 2.

5 \times 18 - 2 = 88

The heart is 88.

Answer: 88

4 · Reviewdoes it hold up?

The heart for triangle 6 is 28, and 18 is 12 further along at 5 per step: 28 + 60 = 88.

Another way: Continue the table by adding 5 repeatedly from row 6 up to row 18; slower, but it lands on 88 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 5 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 18.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 5 medium answer: 86

The table shows the correspondence between $△$ and $♡$ . When $△$ is $15$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$2$	$8$	$14$	$20$	$26$	$32$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 2, 8, 14, 20, 26, 32. We must find the heart that goes with a triangle of 15.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 2, 8, 14, 20, 26, 32.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 15.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 6 more than the one before it, all the way along.

2, 8, 14, 20, 26, 32

A constant jump of 6 suggests multiplying by 6.

2Compare the heart with 6 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 6 gives 6, 12, 18, 24, 30, 36. Line those up under the actual hearts: every one is off by the same 4.

6, 12, 18, 24, 30, 36

The multiple is close but not equal, so subtract 4.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 20.

6 \times 4 - 4 = 20

The rule survives a row it was not built from.

4Apply the rule to a triangle of 15

#5 Look for a Pattern 5.OA.A.2

Multiply 15 by 6, then subtract 4.

6 \times 15 - 4 = 86

The heart is 86.

Answer: 86

4 · Reviewdoes it hold up?

The heart for triangle 6 is 32, and 15 is 9 further along at 6 per step: 32 + 54 = 86.

Another way: Continue the table by adding 6 repeatedly from row 6 up to row 15; slower, but it lands on 86 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 6 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 15.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 6 medium answer: 104

The table shows the correspondence between $△$ and $♡$ . When $△$ is $33$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$8$	$11$	$14$	$17$	$20$	$23$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 8, 11, 14, 17, 20, 23. We must find the heart that goes with a triangle of 33.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 8, 11, 14, 17, 20, 23.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 33.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 3 more than the one before it, all the way along.

8, 11, 14, 17, 20, 23

A constant jump of 3 suggests multiplying by 3.

2Compare the heart with 3 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 3 gives 3, 6, 9, 12, 15, 18. Line those up under the actual hearts: every one is off by the same 5.

3, 6, 9, 12, 15, 18

The multiple is close but not equal, so add 5.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 17.

3 \times 4 + 5 = 17

The rule survives a row it was not built from.

4Apply the rule to a triangle of 33

#5 Look for a Pattern 5.OA.A.2

Multiply 33 by 3, then add 5.

3 \times 33 + 5 = 104

The heart is 104.

Answer: 104

4 · Reviewdoes it hold up?

The heart for triangle 6 is 23, and 33 is 27 further along at 3 per step: 23 + 81 = 104.

Another way: Continue the table by adding 3 repeatedly from row 6 up to row 33; slower, but it lands on 104 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 3 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 33.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 7 medium answer: 114

The table shows the correspondence between $△$ and $♡$ . When $△$ is $22$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$9$	$14$	$19$	$24$	$29$	$34$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 9, 14, 19, 24, 29, 34. We must find the heart that goes with a triangle of 22.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 9, 14, 19, 24, 29, 34.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 22.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 5 more than the one before it, all the way along.

9, 14, 19, 24, 29, 34

A constant jump of 5 suggests multiplying by 5.

2Compare the heart with 5 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 5 gives 5, 10, 15, 20, 25, 30. Line those up under the actual hearts: every one is off by the same 4.

5, 10, 15, 20, 25, 30

The multiple is close but not equal, so add 4.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 24.

5 \times 4 + 4 = 24

The rule survives a row it was not built from.

4Apply the rule to a triangle of 22

#5 Look for a Pattern 5.OA.A.2

Multiply 22 by 5, then add 4.

5 \times 22 + 4 = 114

The heart is 114.

Answer: 114

4 · Reviewdoes it hold up?

The heart for triangle 6 is 34, and 22 is 16 further along at 5 per step: 34 + 80 = 114.

Another way: Continue the table by adding 5 repeatedly from row 6 up to row 22; slower, but it lands on 114 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 5 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 22.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 8 medium answer: 83

The table shows the correspondence between $△$ and $♡$ . When $△$ is $40$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$5$	$7$	$9$	$11$	$13$	$15$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 5, 7, 9, 11, 13, 15. We must find the heart that goes with a triangle of 40.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 5, 7, 9, 11, 13, 15.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 40.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 2 more than the one before it, all the way along.

5, 7, 9, 11, 13, 15

A constant jump of 2 suggests multiplying by 2.

2Compare the heart with 2 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 2 gives 2, 4, 6, 8, 10, 12. Line those up under the actual hearts: every one is off by the same 3.

2, 4, 6, 8, 10, 12

The multiple is close but not equal, so add 3.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 11.

2 \times 4 + 3 = 11

The rule survives a row it was not built from.

4Apply the rule to a triangle of 40

#5 Look for a Pattern 5.OA.A.2

Multiply 40 by 2, then add 3.

2 \times 40 + 3 = 83

The heart is 83.

Answer: 83

4 · Reviewdoes it hold up?

The heart for triangle 6 is 15, and 40 is 34 further along at 2 per step: 15 + 68 = 83.

Another way: Continue the table by adding 2 repeatedly from row 6 up to row 40; slower, but it lands on 83 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 2 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 40.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 9 hard answer: 187

The table shows the correspondence between $△$ and $♡$ . When $△$ is $30$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$13$	$19$	$25$	$31$	$37$	$43$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 13, 19, 25, 31, 37, 43. We must find the heart that goes with a triangle of 30.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 13, 19, 25, 31, 37, 43.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 30.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 6 more than the one before it, all the way along.

13, 19, 25, 31, 37, 43

A constant jump of 6 suggests multiplying by 6.

2Compare the heart with 6 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 6 gives 6, 12, 18, 24, 30, 36. Line those up under the actual hearts: every one is off by the same 7.

6, 12, 18, 24, 30, 36

The multiple is close but not equal, so add 7.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 31.

6 \times 4 + 7 = 31

The rule survives a row it was not built from.

4Apply the rule to a triangle of 30

#5 Look for a Pattern 5.OA.A.2

Multiply 30 by 6, then add 7.

6 \times 30 + 7 = 187

The heart is 187.

Answer: 187

4 · Reviewdoes it hold up?

The heart for triangle 6 is 43, and 30 is 24 further along at 6 per step: 43 + 144 = 187.

Another way: Continue the table by adding 6 repeatedly from row 6 up to row 30; slower, but it lands on 187 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 6 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 30.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 10 hard answer: 86

The table shows the correspondence between $△$ and $♡$ . When $△$ is $12$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$9$	$16$	$23$	$30$	$37$	$44$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 9, 16, 23, 30, 37, 44. We must find the heart that goes with a triangle of 12.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 9, 16, 23, 30, 37, 44.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 12.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 7 more than the one before it, all the way along.

9, 16, 23, 30, 37, 44

A constant jump of 7 suggests multiplying by 7.

2Compare the heart with 7 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 7 gives 7, 14, 21, 28, 35, 42. Line those up under the actual hearts: every one is off by the same 2.

7, 14, 21, 28, 35, 42

The multiple is close but not equal, so add 2.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 30.

7 \times 4 + 2 = 30

The rule survives a row it was not built from.

4Apply the rule to a triangle of 12

#5 Look for a Pattern 5.OA.A.2

Multiply 12 by 7, then add 2.

7 \times 12 + 2 = 86

The heart is 86.

Answer: 86

4 · Reviewdoes it hold up?

The heart for triangle 6 is 44, and 12 is 6 further along at 7 per step: 44 + 42 = 86.

Another way: Continue the table by adding 7 repeatedly from row 6 up to row 12; slower, but it lands on 86 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 7 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 12.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 11 hard answer: 85

The table shows the correspondence between $△$ and $♡$ . When $△$ is $11$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$5$	$13$	$21$	$29$	$37$	$45$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 5, 13, 21, 29, 37, 45. We must find the heart that goes with a triangle of 11.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 5, 13, 21, 29, 37, 45.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 11.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 8 more than the one before it, all the way along.

5, 13, 21, 29, 37, 45

A constant jump of 8 suggests multiplying by 8.

2Compare the heart with 8 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 8 gives 8, 16, 24, 32, 40, 48. Line those up under the actual hearts: every one is off by the same 3.

8, 16, 24, 32, 40, 48

The multiple is close but not equal, so subtract 3.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 29.

8 \times 4 - 3 = 29

The rule survives a row it was not built from.

4Apply the rule to a triangle of 11

#5 Look for a Pattern 5.OA.A.2

Multiply 11 by 8, then subtract 3.

8 \times 11 - 3 = 85

The heart is 85.

Answer: 85

4 · Reviewdoes it hold up?

The heart for triangle 6 is 45, and 11 is 5 further along at 8 per step: 45 + 40 = 85.

Another way: Continue the table by adding 8 repeatedly from row 6 up to row 11; slower, but it lands on 85 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 8 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 11.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.

Variant 12 hard answer: 127

The table shows the correspondence between $△$ and $♡$ . When $△$ is $14$ , find the value of $♡$ .

$△$	$1$	$2$	$3$	$4$	$5$	$6$	$\cdots$
$♡$	$10$	$19$	$28$	$37$	$46$	$55$	$\cdots$

Show solution

1 · Understandwhat's really being asked

A table pairs each triangle value 1 through 6 with a heart value. The heart values are 10, 19, 28, 37, 46, 55. We must find the heart that goes with a triangle of 14.

Givens

The triangle runs 1, 2, 3, 4, 5, 6 and the heart runs 10, 19, 28, 37, 46, 55.
The same rule holds for every row, and beyond the table.

Unknowns

The heart value when the triangle is 14.

Constraints

The rule must fit every row shown, not just one.

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#6 Guess and Check

The heart column climbs evenly, which points at multiplying by the size of the jump. That alone will not match the table, so compare the two columns row by row and see what constant is missing.

3 · Execute4 carry out the plan

1Notice the steady jump

#5 Look for a Pattern 4.OA.C.5

Each heart is 9 more than the one before it, all the way along.

10, 19, 28, 37, 46, 55

A constant jump of 9 suggests multiplying by 9.

2Compare the heart with 9 times the triangle

#9 Solve an Easier Related Problem 5.OA.B.3

Multiplying each triangle by 9 gives 9, 18, 27, 36, 45, 54. Line those up under the actual hearts: every one is off by the same 1.

9, 18, 27, 36, 45, 54

The multiple is close but not equal, so add 1.

3Check the rule on a known pair

#6 Guess and Check 5.OA.A.2

Test the rule against a row we can see: triangle 4 should give heart 37.

9 \times 4 + 1 = 37

The rule survives a row it was not built from.

4Apply the rule to a triangle of 14

#5 Look for a Pattern 5.OA.A.2

Multiply 14 by 9, then add 1.

9 \times 14 + 1 = 127

The heart is 127.

Answer: 127

4 · Reviewdoes it hold up?

The heart for triangle 6 is 55, and 14 is 8 further along at 9 per step: 55 + 72 = 127.

Another way: Continue the table by adding 9 repeatedly from row 6 up to row 14; slower, but it lands on 127 too, which is what makes the rule trustworthy.

Standardsmin grade 5

4.OA.C.5 Generate a number or shape pattern following a given rule — Spotting the constant jump of 9 down the heart column.
5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Comparing the two columns to find the constant the multiple misses by.
5.OA.A.2 Write simple expressions that record calculations with numbers — Writing the rule and evaluating it at 14.

💡Takeaway. A steady jump tells you what to multiply by, but not the whole rule -- check the multiple against the table to find what is left over.