Depth · one concept from first principles · step 2 of 4
Whole numbers listed in a row, such as or , are called consecutive whole numbers. When the sum of three consecutive whole numbers is , find the product of these three numbers.
Write the numbers around the middle: middle-1, middle, middle+1. The -1 and +1 cancel, so the sum is exactly 3 times the middle number.
Evenly spaced numbers balance around their center, so their sum is the center times how many there are.
3.OA.D.9Look For A PatternThe sum of three consecutive whole numbers is exactly 3 times the middle number.
The smallest number is 1 below the middle and the largest is 1 above, so sliding that single unit from the largest down to the smallest makes all three equal to the middle while the total stays the same.
Moving 1 unit from one number to another only rearranges the same total among the three numbers, so the sum does not change.
However you split a fixed total into parts, those parts always add back to that same total.
Once all three are the middle number, the sum is middle plus middle plus middle, which is three equal groups of the middle.
Adding the same amount three times is the same as taking 3 groups of it, which is 3 times that amount.
Since 3 times the middle is 27, divide to get the middle number: 27 ÷ 3 = 9.
Dividing by 3 undoes the 'times 3', a basic Grade 3 fact.
3.OA.C.7Guess And CheckThe numbers are 8, 9, 10 (check: 8 + 9 + 10 = 27). Multiply them together.
Multiply two at a time; times 10 just appends a zero.
3.OA.C.7Look For A PatternNumbers in a row balance around the middle, so the sum is just the middle times how many -- pure Grade 3 sense!
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Lattice / Grid Pattern from Small Cases, Pair-Sum Invariant, Small-Domain Constrained Search. Why the shapes repeat →