Problem
The neighbors are center-7, center-1, center+1, center+7 — adding all five, the 7s and 1s cancel, leaving five copies of the center.
The balanced neighbors above/below and left/right average out to the center.
3.OA.D.9Look For A PatternThe five numbers in the plus always add up to five times the center number.
The four neighbors are the center shifted by matched amounts — 7 up and 7 down, 1 left and 1 right — so when all five are added the shifts pair off and only five plain copies of the center are left.
The five numbers can be added in any order and grouped any way, so the '7 less' can be placed beside the '7 more' and the '1 less' beside the '1 more'.
Changing the order of the numbers being added does not change their total.
Changing which numbers you group and add first does not change their total.
Inside each pair the two shifts wipe each other out — adding 7 cancels taking 7, and adding 1 cancels taking 1 — so each pair puts nothing extra onto the center.
Adding an amount and then taking the same amount away brings you back to where you started.
Once the two shifts have cancelled, what is added to the center is zero, and adding zero leaves the center exactly the same.
With every shift gone, the center is the only number left in each of the five spots, and adding one number to itself five times is five times that number.
The example sum is 45, so the center is 45 divided by 5 = 9.
If 5 times the center is 45, the center must be 9.
4.OA.C.5Guess And CheckFor a sum of 65, the center is 65 divided by 5 = 13.
The same rule says the center is one fifth of the sum.
4.OA.C.5Look For A PatternThe five numbers are 6, 12, 13 (center), 14, 20 — the largest sits directly below the center: 13 + 7 = 20.
The number one row below the center is always the biggest in the plus.
3.OA.D.9Look For A PatternThis only needs Grade 4 pattern sense: a balanced plus always sums to 5 times its middle number!