Order of operations

Find the value of 36 divided by 4 plus 7 times 5. Try it first. Pause if you like. The plus in the middle joins two amounts. 36 divided by 4 is one amount. 7 times 5 is the other one. The divide and the times sit inside those amounts, so both amounts are finished before the plus joins anything. 36 divided by 4 shares 36 counters into 4 equal rows. Every row holds the same number of counters. One row is 9, so 36 divided by 4 is 9. Those 9 counters are the first amount, and the 7 times 5 is written again exactly as it was. 7 times 5 is 7 columns with 5 counters in each. Each column holds five counters. One. Two. Three. Four. Five. Six. Seven. All 7 columns together are one block, and that block is 35. Those 35 counters are there before the plus does anything. Now the plus joins them: 9 counters in one row and 35 in the block. That is 44. So 36 divided by 4 plus 7 times 5 is 44. Put brackets round the front of the same line, and it is a different line. The brackets make 36 divided by 4 plus 7 into one amount, so that amount is made first. Inside the brackets, 36 divided by 4 is 9. And 9 plus 7 is 16. Then 16 times 5 is 80. Same 4 numbers and the same 3 signs, and the value is 80 instead of 44. Reading straight along our line gives 80 as well, because it treats the 7 as if those brackets were there. The plus joins two amounts, and each amount is finished first. 9 in the row, and 35 in the block. A bracket is a way of making a plus into an amount as well. With no brackets, we divide and multiply first, then add and subtract.