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How Many Hot Dogs?

47
skips
53
questions

Act One

• Robert Kaplinsky

December 13, 2012

How many hot dogs were on the grill before Kobayashi ordered the first hot dog?

• 1.

How many hot dogs were on the grill before Kobayashi ordered the first hot dog?

• 2.

What is a guess that is too low?

• 3.

What is a guess that is too high?

• 4.

• 5.

How many hot dogs did each person order each round?

• 6.

What pattern do you see?

• 7.

How can we record this information?

• 8.

How many total hot dogs did each person eat in the first four rounds?

Act Two

• Teacher note
Kobayashi, speaks in Japanese and it may not be clear at first that his one finger is asking for one hot dog. Once, Sonya says that she wants two hot dogs, the meaning behind him holding up three fingers should be clear.
I define each â€œroundâ€ as each of them getting to order hot dog(s). So in Round 1, Kobayashi eats one hot dog and Sonya eats two hot dogs. In Round 2, Kobayashi eats three additional hot dog and Sonya eats four additional hot dogs. This continues until Kobayashi tries to order nine hot dogs to begin Round 5 and finds that the store has run out.
The purpose of the first question is to give students a question that they can sink their teeth into. There are many ways for them to determine that together the pair ate 36 hot dogs in the first four rounds (1 + 2 + â€¦ + 8). If you stop the video at 0:12, you can sort of count the hot dogs. I can clearly see four columns with at least eight hot dogs per column. For the sake of this problem, I am going to pretend that they each have exactly nine hot dogs in each of the four columns.
Depending on the challenge questions you ask, this problem will require only linear models or also quadratic models. I picked Round 20 and Round n, but feel free to modify this and add an earlier round like Round 10.

Act Three

• Teacher note
Something important to note is that there are many ways to get the answers. For example, with question 7, one student might notice that Kobayashi eats one less hot dog each round and so after n rounds, he has to eat n hot dogs to catch up. Another students might reason that after n rounds, Kobayashi has eaten n^2 hot dogs and Sonya has eaten n^2+n hot dogs. Subtraction the n^2 from n^2+n leaves just n, which is how many hot dogs behind he is. Those are two different approaches that give the same answer, and it would make for a great discussion as to how that happened.

Sequel

• 9.

How many hot dogs would each person eat in Round 20?

• 10.

How many hot dogs would each person eat in Round n?

• 11.

How many total hot dogs would each person have eaten in Round 20?

• 12.

How many total hot dogs would each person have eaten in Round n?

• 13.

How many hot dogs would Kobayashi have to eat to catch up to Sonya after Round 20 is over?

• 14.

How many hot dogs would Kobayashi have to eat to catch up to Sonya after Round n is over?

• Teacher note
In case the last two questions are unclear, what I mean is that Kobayashi starts to fall behind Sonya at the end of the first round because she orders one more than him. This continues and he falls farther behind her. So, how many hot dogs would he have to eat to catch up to her?