The Cookie Table That Did Not Match the Plan
Room 12 is having a small party on Thursday afternoon. The helper counts 20 cookies on the tray and sets out 8 chairs around the table. Then three things happen. First, three children from the reading group are still in the library, so their chairs stay empty. Second, one of the eight children, Dev, says he does not like cookies and will not take any. Third, Ms. Park says 4 cookies must stay on the tray for the custodian's birthday later that afternoon. Ms. Park wants the rest of the cookies split so that every child who is actually sitting at the table and eating cookies gets the same number, with nothing left over. She asks the helper to write one number sentence before handing out anything. The helper writes 20 divided by 8 equals 2 with 4 left over, and Ms. Park says, 'That sentence does not match the plan.' The helper has to find the sentence that does match before anyone touches the tray.
1. Write the number sentence that matches Ms. Park's plan, and give the number of cookies each child gets. Show where each number in your sentence comes from.
Read the worked answer
Start with who is eating. There are 8 chairs, but 3 children are in the library, so 8 minus 3 equals 5 children at the table. Dev takes none, so 5 minus 1 equals 4 children sharing cookies.
Now find how many cookies are shared: 20 on the tray minus 4 saved for the custodian equals 16.
Number sentence: 16 divided by 4 equals 4. Each of the four children gets 4 cookies.
Check: 4 children times 4 cookies equals 16 cookies, and 16 plus 4 saved equals 20, the original count.
The helper's sentence, 20 divided by 8, used the number of chairs and the number of cookies before saving. It answers a different question: 'If all 8 children shared all 20 cookies, how many each?' That is not the plan Ms. Park described.
2. Ms. Park says the helper's answer of '2 with 4 left over' is not just a different number, it is the wrong kind of answer for this plan. Explain what she means.
Read the worked answer
Ms. Park's plan has two conditions, and the helper's sentence ignores both.
The first is 'nothing left over.' 20 divided by 8 leaves a remainder of 4, so the helper's plan would put 4 more cookies back on the tray on top of the 4 already saved. That is not what she asked for, and it would also mean the shares are not equal.
The second is 'every child who is actually eating gets the same number.' The helper divided by chairs. An empty chair is not a person, and Dev's chair is not eating. Dividing by the wrong number can look right, because both 8 and 20 are numbers you can see on the table.
The habit to build is to write the sentence from the plan, not from the objects in front of you.
3. The custodian's birthday is cancelled, so no cookies need saving. Rewrite the number sentence and give the new number per child. Then say which of the two answers is larger and why that makes sense.
Read the worked answer
With nothing saved, all 20 cookies are shared by the same 4 eating children: 20 divided by 4 equals 5. Each child now gets 5 cookies instead of 4. Check: 4 times 5 equals 20.
The new answer is larger because the same four children are sharing a bigger pile. The 4 saved cookies have moved off the tray and into the shares, and since there are 4 eaters, each share goes up by exactly 1.
That gives a useful check on both answers. When the number of eaters stays the same, any change in the number of cookies available is divided equally among them, so the share changes by that amount divided by the number of eaters.