Sensim Math · Depth 한국어

3-2 · Multiplication

Consecutive integers around a middle value

3.OA.D.93.OA.D.8 · take · grade 3

Archetype: Sum of Evenly Spaced Numbers via the Middle · step in a 5-type progression

▶ Practice — 12 problems

Whole numbers listed in a row, such as 1, 2, 31,\ 2,\ 3 or 99, 100, 10199,\ 100,\ 101, are called consecutive whole numbers. When the sum of three consecutive whole numbers is 2727, find the product of these three numbers.

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Understand

Three whole numbers in a row (each one bigger than the last) add up to 27. Find the product of those three numbers.

Givens
  • The three numbers are consecutive whole numbers (like 8, 9, 10).
  • Their sum is 27.
Unknowns
  • The three consecutive numbers, and their product.
Constraints
  • Consecutive whole numbers differ by exactly 1.

Plan

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

Three consecutive numbers are (middle - 1), (middle), (middle + 1). Their sum is 3 times the middle, so the middle equals the sum divided by 3. Then check by listing the three numbers.

Execute

#5 Look for a Pattern 3.OA.D.9
Write the three numbers around the middle one: middle - 1, middle, middle + 1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.
(m−1)+m+(m+1)=3×m=27(m-1) + m + (m+1) = 3 \times m = 27
Evenly spaced numbers balance around their center, so their sum is the center times how many there are.
#6 Guess and Check 3.OA.C.7
Since 3 times the middle is 27, the middle number is 27 divided by 3.
m=27÷3=9m = 27 \div 3 = 9
Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.
#5 Look for a Pattern 3.OA.C.7
The numbers are 8, 9, 10 (check: 8 + 9 + 10 = 27). Multiply them together.
8×9×10=72×10=7208 \times 9 \times 10 = 72 \times 10 = 720
Multiply two at a time; times 10 just appends a zero.
Answer: 720

Review

8 + 9 + 10 = 27 matches the given sum. The product 720 is close to 9 cubed (729), which makes sense for three numbers near 9.

Guess and check: try 7,8,9 (sum 24, too small), then 8,9,10 (sum 27). It works, so multiply to get 720.

Standards · min grade 3

  • 3.OA.D.9 Identify arithmetic patterns and explain using properties of operations — Seeing that three consecutive numbers sum to 3 times the middle one.
  • 3.OA.C.7 Fluently multiply and divide within 100 — Dividing 27 by 3 and multiplying the three numbers.
💡 Numbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!