Sensim Math · Depth 한국어

4-1 · Finding Rules

Sum-and-difference rules in a number grid

4.OA.C.53.OA.D.9 · take · grade 4

Archetype: Generalize a Growing Pattern into a Rule · step in a 12-type progression

▶ Practice — 10 problems

On the calendar at the right, the 55 numbers inside the plus (➕) shape add up to 4545. When 55 numbers in the same shape are added and the sum is 6565, find the largest of those 55 numbers.

Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Show solution

Understand

On a monthly calendar (1st on Thursday, 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

Plan

#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.

Execute

#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(c-7)+(c-1)+c+(c+1)+(c+7) = 5c
The balanced neighbors above/below and left/right average out to the center.
#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÷5=945 \div 5 = 9
If 5 times the center is 45, the center must be 9.
#5 Look for a Pattern 4.OA.C.5
For a sum of 65, the center is 65 divided by 5.
65÷5=1365 \div 5 = 13
The same rule says the center is one fifth of the sum.
#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=2013 + 7 = 20
The number one row below the center is always the biggest in the plus.
Answer: 20

Review

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

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.

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