Problem
Count how many odd numbers each calculation adds: 1st has 1 term, 2nd has 3, 3rd has 5, 4th has 7, 5th has 9. The term count goes up by 2 each time, so the 6th has 11 terms and the 7th has 13 terms.
Each step just tacks on the next two odd numbers, so the count of terms climbs by 2 every time.
4.OA.C.5Look For A PatternThe 7th calculation has 13 terms, so it runs through the first 13 odd numbers, ending at the 13th odd number, which is 2 x 13 - 1 = 25.
The odd numbers in order are 1, 3, 5, ...; the 13th of them is 25, so that is where the sum stops.
4.OA.C.5Look For A PatternLook at the results: 1 = 1x1, 9 = 3x3, 25 = 5x5, 49 = 7x7, 81 = 9x9. The result is always the number of terms multiplied by itself. The 7th calculation has 13 terms, so its result is 13 x 13.
Adding the first few odd numbers always builds a perfect square, so we can predict the total without adding all 13.
3.OA.D.9Solve An Easier Related ProblemThe seventh calculation's total equals the number of odd numbers it adds, 13, multiplied by itself, because adding consecutive odd numbers starting at 1 always builds a perfect square.
Show the running total as dots packed into a square, and each odd number you add becomes the L-shaped strip that grows the square to the next size up.
To grow a square to a side one dot longer, you lay down a new bottom row and a new side column of the same length and let them share the single corner dot, so the strip always holds an odd number of dots.
The larger square is exactly the smaller square plus that corner strip, with no dot left out and none counted twice, so the strip is just the extra dots needed to complete it.
Once the last odd strip is in place the dots fill a complete square, with as many rows as there are dots in each row, and that many rows of that many dots is that number times itself.
Counting equal rows of dots is the same as counting that many groups of that size, which is exactly what multiplying the two counts means.
Adding odd numbers starting at 1 always makes a square, so the 7th calculation (13 odd numbers) equals 13 x 13 = 169 with no long adding!