← Sum-and-difference rules in a number grid · Generalize a Growing Pattern into a Rule

Sum-and-difference rules in a number grid · 10 practice problems

4.OA.C.53.OA.D.9

Generated variants — 10

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

Variant 1 easy answer: 20

On the calendar at the right, the $5$ numbers inside the plus shape add up to $45$ . When $5$ numbers in the same shape are added and the sum is $65$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Thu, dates 1 to 30), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 45. For a plus whose five numbers sum to 65, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 45
The target plus sums to 65

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 45, so the center is 45 divided by 5 = 9. This matches a plus centered on the 9.

45 \div 5 = 9

If 5 times the center is 45, the center must be 9.

3Find the target center

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

For a sum of 65, the center is 65 divided by 5.

65 \div 5 = 13

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 6 (above), 12 (left), 13 (center), 14 (right), 20 (below). The largest is the number directly below the center, which is 13 + 7 = 20.

13 + 7 = 20

The number one row below the center is always the biggest in the plus.

Answer: 20

4 · Reviewdoes it hold up?

Adding the five numbers 6 + 12 + 13 + 14 + 20 = 65, which matches the given sum, and 20 (still within 1-30) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 9; a center of 13 gives below = 20 and total 65, confirming the largest is 20.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!

Variant 2 easy answer: 23

On the calendar at the right, the $5$ numbers inside the plus shape add up to $45$ . When $5$ numbers in the same shape are added and the sum is $80$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Sun, dates 1 to 31), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 45. For a plus whose five numbers sum to 80, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 45
The target plus sums to 80

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 45, so the center is 45 divided by 5 = 9. This matches a plus centered on the 9.

45 \div 5 = 9

If 5 times the center is 45, the center must be 9.

3Find the target center

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

For a sum of 80, the center is 80 divided by 5.

80 \div 5 = 16

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 9 (above), 15 (left), 16 (center), 17 (right), 23 (below). The largest is the number directly below the center, which is 16 + 7 = 23.

16 + 7 = 23

The number one row below the center is always the biggest in the plus.

Answer: 23

4 · Reviewdoes it hold up?

Adding the five numbers 9 + 15 + 16 + 17 + 23 = 80, which matches the given sum, and 23 (still within 1-31) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 9; a center of 16 gives below = 23 and total 80, confirming the largest is 23.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!

Variant 3 easy answer: 24

On the calendar at the right, the $5$ numbers inside the plus shape add up to $45$ . When $5$ numbers in the same shape are added and the sum is $85$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Wed, dates 1 to 31), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 45. For a plus whose five numbers sum to 85, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 45
The target plus sums to 85

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 45, so the center is 45 divided by 5 = 9. This matches a plus centered on the 9.

45 \div 5 = 9

If 5 times the center is 45, the center must be 9.

3Find the target center

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

For a sum of 85, the center is 85 divided by 5.

85 \div 5 = 17

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 10 (above), 16 (left), 17 (center), 18 (right), 24 (below). The largest is the number directly below the center, which is 17 + 7 = 24.

17 + 7 = 24

The number one row below the center is always the biggest in the plus.

Answer: 24

4 · Reviewdoes it hold up?

Adding the five numbers 10 + 16 + 17 + 18 + 24 = 85, which matches the given sum, and 24 (still within 1-31) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 9; a center of 17 gives below = 24 and total 85, confirming the largest is 24.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!

Variant 4 medium answer: 24

On the calendar at the right, the $5$ numbers inside the plus shape add up to $50$ . When $5$ numbers in the same shape are added and the sum is $85$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Tue, dates 1 to 30), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 50. For a plus whose five numbers sum to 85, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 50
The target plus sums to 85

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 50, so the center is 50 divided by 5 = 10. This matches a plus centered on the 10.

50 \div 5 = 10

If 5 times the center is 50, the center must be 10.

3Find the target center

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

For a sum of 85, the center is 85 divided by 5.

85 \div 5 = 17

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 10 (above), 16 (left), 17 (center), 18 (right), 24 (below). The largest is the number directly below the center, which is 17 + 7 = 24.

17 + 7 = 24

The number one row below the center is always the biggest in the plus.

Answer: 24

4 · Reviewdoes it hold up?

Adding the five numbers 10 + 16 + 17 + 18 + 24 = 85, which matches the given sum, and 24 (still within 1-30) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 10; a center of 17 gives below = 24 and total 85, confirming the largest is 24.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!

Variant 5 medium answer: 25

On the calendar at the right, the $5$ numbers inside the plus shape add up to $55$ . When $5$ numbers in the same shape are added and the sum is $90$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Fri, dates 1 to 28), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 55. For a plus whose five numbers sum to 90, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 55
The target plus sums to 90

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 55, so the center is 55 divided by 5 = 11. This matches a plus centered on the 11.

55 \div 5 = 11

If 5 times the center is 55, the center must be 11.

3Find the target center

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

For a sum of 90, the center is 90 divided by 5.

90 \div 5 = 18

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 11 (above), 17 (left), 18 (center), 19 (right), 25 (below). The largest is the number directly below the center, which is 18 + 7 = 25.

18 + 7 = 25

The number one row below the center is always the biggest in the plus.

Answer: 25

4 · Reviewdoes it hold up?

Adding the five numbers 11 + 17 + 18 + 19 + 25 = 90, which matches the given sum, and 25 (still within 1-28) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 11; a center of 18 gives below = 25 and total 90, confirming the largest is 25.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!

Variant 6 medium answer: 25

On the calendar at the right, the $5$ numbers inside the plus shape add up to $55$ . When $5$ numbers in the same shape are added and the sum is $90$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Sat, dates 1 to 30), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 55. For a plus whose five numbers sum to 90, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 55
The target plus sums to 90

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 55, so the center is 55 divided by 5 = 11. This matches a plus centered on the 11.

55 \div 5 = 11

If 5 times the center is 55, the center must be 11.

3Find the target center

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

For a sum of 90, the center is 90 divided by 5.

90 \div 5 = 18

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 11 (above), 17 (left), 18 (center), 19 (right), 25 (below). The largest is the number directly below the center, which is 18 + 7 = 25.

18 + 7 = 25

The number one row below the center is always the biggest in the plus.

Answer: 25

4 · Reviewdoes it hold up?

Adding the five numbers 11 + 17 + 18 + 19 + 25 = 90, which matches the given sum, and 25 (still within 1-30) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 11; a center of 18 gives below = 25 and total 90, confirming the largest is 25.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!

Variant 7 medium answer: 26

On the calendar at the right, the $5$ numbers inside the plus shape add up to $60$ . When $5$ numbers in the same shape are added and the sum is $95$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Mon, dates 1 to 31), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 60. For a plus whose five numbers sum to 95, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 60
The target plus sums to 95

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 60, so the center is 60 divided by 5 = 12. This matches a plus centered on the 12.

60 \div 5 = 12

If 5 times the center is 60, the center must be 12.

3Find the target center

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

For a sum of 95, the center is 95 divided by 5.

95 \div 5 = 19

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 12 (above), 18 (left), 19 (center), 20 (right), 26 (below). The largest is the number directly below the center, which is 19 + 7 = 26.

19 + 7 = 26

The number one row below the center is always the biggest in the plus.

Answer: 26

4 · Reviewdoes it hold up?

Adding the five numbers 12 + 18 + 19 + 20 + 26 = 95, which matches the given sum, and 26 (still within 1-31) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 12; a center of 19 gives below = 26 and total 95, confirming the largest is 26.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!

Variant 8 hard answer: 27

On the calendar at the right, the $5$ numbers inside the plus shape add up to $40$ . When $5$ numbers in the same shape are added and the sum is $100$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Thu, dates 1 to 30), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 40. For a plus whose five numbers sum to 100, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 40
The target plus sums to 100

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 40, so the center is 40 divided by 5 = 8. This matches a plus centered on the 8.

40 \div 5 = 8

If 5 times the center is 40, the center must be 8.

3Find the target center

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

For a sum of 100, the center is 100 divided by 5.

100 \div 5 = 20

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 13 (above), 19 (left), 20 (center), 21 (right), 27 (below). The largest is the number directly below the center, which is 20 + 7 = 27.

20 + 7 = 27

The number one row below the center is always the biggest in the plus.

Answer: 27

4 · Reviewdoes it hold up?

Adding the five numbers 13 + 19 + 20 + 21 + 27 = 100, which matches the given sum, and 27 (still within 1-30) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 8; a center of 20 gives below = 27 and total 100, confirming the largest is 27.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!

Variant 9 hard answer: 29

On the calendar at the right, the $5$ numbers inside the plus shape add up to $40$ . When $5$ numbers in the same shape are added and the sum is $110$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Wed, dates 1 to 31), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 40. For a plus whose five numbers sum to 110, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 40
The target plus sums to 110

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 40, so the center is 40 divided by 5 = 8. This matches a plus centered on the 8.

40 \div 5 = 8

If 5 times the center is 40, the center must be 8.

3Find the target center

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

For a sum of 110, the center is 110 divided by 5.

110 \div 5 = 22

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 15 (above), 21 (left), 22 (center), 23 (right), 29 (below). The largest is the number directly below the center, which is 22 + 7 = 29.

22 + 7 = 29

The number one row below the center is always the biggest in the plus.

Answer: 29

4 · Reviewdoes it hold up?

Adding the five numbers 15 + 21 + 22 + 23 + 29 = 110, which matches the given sum, and 29 (still within 1-31) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 8; a center of 22 gives below = 29 and total 110, confirming the largest is 29.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!

Variant 10 hard answer: 30

On the calendar at the right, the $5$ numbers inside the plus shape add up to $50$ . When $5$ numbers in the same shape are added and the sum is $115$ , find the largest of those $5$ numbers.

Show solution

1 · Understandwhat's really being asked

On a monthly calendar (1st on Sun, dates 1 to 31), a plus shape covers a center number and its four neighbors above, below, left, and right. One example plus sums to 50. For a plus whose five numbers sum to 115, find the largest of the five numbers.

Givens

The plus shape holds a center number plus the numbers directly above (center - 7), below (center + 7), left (center - 1), and right (center + 1)
An example plus sums to 50
The target plus sums to 115

Unknowns

The center number of the target plus
The largest of the five numbers in that plus

Constraints

On a weekly calendar, the number above is 7 less and the number below is 7 more than the center
Left/right neighbors are 1 less / 1 more than the center

2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #6 Guess and Check

The four neighbors are symmetric around the center (-7, -1, +1, +7), which cancel in pairs, so the sum equals 5 times the center. Finding the center then gives the largest number.

3 · Execute4 carry out the plan

1See why the sum is 5 times the center

#5 Look for a Pattern 3.OA.D.9

The neighbors are center-7, center-1, center+1, center+7. Adding all five numbers, the +7 and -7 cancel and the +1 and -1 cancel, leaving five copies of the center.

(c-7)+(c-1)+c+(c+1)+(c+7) = 5c

The balanced neighbors above/below and left/right average out to the center.

2Check with the example

#6 Guess and Check 4.OA.C.5

The example sum is 50, so the center is 50 divided by 5 = 10. This matches a plus centered on the 10.

50 \div 5 = 10

If 5 times the center is 50, the center must be 10.

3Find the target center

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

For a sum of 115, the center is 115 divided by 5.

115 \div 5 = 23

The same rule says the center is one fifth of the sum.

4Find the largest number

#5 Look for a Pattern 3.OA.D.9

The five numbers are 16 (above), 22 (left), 23 (center), 24 (right), 30 (below). The largest is the number directly below the center, which is 23 + 7 = 30.

23 + 7 = 30

The number one row below the center is always the biggest in the plus.

Answer: 30

4 · Reviewdoes it hold up?

Adding the five numbers 16 + 22 + 23 + 24 + 30 = 115, which matches the given sum, and 30 (still within 1-31) is indeed the largest.

Another way: Guess and check (tool 6): try centers near the example's 10; a center of 23 gives below = 30 and total 115, confirming the largest is 30.

Standardsmin grade 4

4.OA.C.5 Generate a number or shape pattern following a given rule — Using the calendar's +7/-7 and +1/-1 structure to relate the sum to the center
3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Recognizing that the symmetric neighbors cancel so the sum is five times the center

💡Takeaway. This only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!