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Ten Million Problems

Lesson Plans > Mathematics > Problem Solving
 

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Bob’s heart beats 65 times per minute, and Bill’s heart beats 60 times per minute. How many seconds will pass before Bob’s heart has beat ten million times more than Bill’s heart?

If a quarter’s height is
1
16
inch, how tall (to the nearest yard) is a stack of quarters worth ten million dollars?

 

A perfect cube is a number which is the cube of an integer. What is the largest perfect cube less than ten million?

How many whole numbers less than ten million are multiples of either two or three?

If the positive integers were written out like this: 123456789101112…, what is the last integer which will be fully written out before the line reaches a length of ten million digits?

A group of people are playing ping pong, and each person wants to play exactly one game against each of the other people in the group. If there are at least ten million games played, what is the smallest possible number of people in the group?

What is the sum of the first ten million odd whole numbers?

If all the letters of the alphabet were written on a piece of paper, over and over again, which letter would be the ten millionth letter to be written?

What is the largest number less than ten million which is not a multiple of 2, 3, 4, 5, 6, 7, 8, or 9?

How many hours (to the nearest hour) would it take you to count to ten million if you counted one number every fifteen seconds?

What is the largest triangular number less than ten million?

How many positive integer factors does the number ten million have?

Lesson Plan/Article

Ten Million Problems

On the last day of my Algebra Two class, I informed the students that they were going to spend the class period solving ten million problems. After the look of shock, I explained that I meant they were doing problems that all had the number ten million in them. The students competed in teams to see who could solve the problems fastest or most accurately.

I explained that since all the problems included very large numbers, they would certainly want to use their calculators.

I also explained that sometimes when you are dealing with large numbers, it helps to make smaller versions of the problem to see if you can find patterns. "For example," I said, "If I asked you for the sum of the first ten million whole numbers, you wouldn't punch them into your calculator; you'd first find the sum of the first whole number, the first two whole numbers, then the first three whole numbers. Then you would study those results to see if you could find a pattern. If you can find a pattern, you'll use that pattern to solve the larger problem.

The rules of the game were as follows:

  1. Calculators are allowed.
  2. Problems will be given one at a time.
  3. Teams will have a maximum of five minutes per problem.
  4. As soon as any team has agreed on an answer, one team member shows the answer to the teacher - either in writing or on a calculator.
  5. If the answer is correct, the team will earn 25 points and that round is over.
  6. If the answer is incorrect, the other teams that are still in the round each earn 5 points, and can continue working.
  7. When time is called, all team who have not done so submit their answers.
  8. All correct answers earn 10 points.
  9. All answers which are "in the ballpark" earn 5 points.
  10. Between each round, teacher randomly selects a few students to swap teams. Points stay with the student rather than the team; even though students are always working as part of a team, the winners are individual students rather than teams.

The scoring system is interesting, because if you have four teams, and three teams submit incorrect answers, the fourth team automatically has 15 points, even if they don't have a solution in the ballpark. Thus, there is a bit of strategy involved; if you aren't certain of your answer, will you guess, knowing that you could be giving points to other teams? The odd scoring system also helps to keep the scores more balanced.

Scoring and rules can be modified as needed. The idea of swapping students around to different teams can be ditched, but the teacher should try to form teams that are reasonably even in skill levels.

The "ballpark" numbers in the solution key are my suggestions; these can be modified to make the game more or less challenging.

Make sure you verify that everyone on the team agrees with an answer before you even look at it - if even one person dissents, they're not ready to submit!

Variation: if you use the "team swapping" rule, you can also use the teacher to help make teams even. For example, if you have 23 students, but you want to have eight teams of three, the teacher can be the third person on one of the teams. The catch is this: the teacher cannot contribute to the problem solving process, but is allowed to reject (without penalty) up to three incorrect answers! Thus, the team has the disadvantage of solving the problem shorthanded, but has the advantage of getting to submit three wrong answers without getting booted from the round. Of course, the teacher randomly swaps teams as well.

This went over pretty well with the students; the students who usually complain a lot only complained a little, and most of the students seemed to enjoy the challenge. The odd scoring system upended everyone's expectations a bit; the students who are always at the top of the class were, in some cases, outdone by some of my diligent but less "flashy" students. All in all, I thought it worked very well.

We didn't do all the problems; I picked the ones I thought were best suited to my students' ability levels. We broke up the activity by doing a few rounds and then switching to doing a TedEd riddle video, and then doing some more problems. It was a great way to end the school year!

Lesson by Mr. Twitchell

Handouts/Worksheets

Ten Million Problems

  1. Bob’s heart beats 65 times per minute, and Bill’s heart beats 60 times per minute. How many seconds will pass before Bob’s heart has beat ten million times more than Bill’s heart?

     
  2. If a quarter’s height is
    1
    16
    inch, how tall (to the nearest yard) is a stack of quarters worth ten million dollars?

     
  3. How many positive integer factors does the number ten million have?

     
  4. A perfect cube is a number which is the cube of an integer. What is the largest perfect cube less than ten million?

     
  5. How many whole numbers less than ten million are multiples of either two or three?

     
  6. If the positive integers were written out like this: 123456789101112…, what is the last integer which will be fully written out before the line reaches a length of ten million digits?

     
  7. A group of people are playing ping pong, and each person wants to play exactly one game against each of the other people in the group. If there are at least ten million games played, what is the smallest possible number of people in the group?

     
  8. What is the sum of the first ten million odd whole numbers?

     
  9. What is the largest number less than ten million which is not a multiple of 2, 3, 4, 5, 6, 7, 8, or 9?

     
  10. How many hours (to the nearest hour) would it take you to count to ten million if you counted one number every fifteen seconds?


     
  11. What is the largest triangular number less than ten million?

     
  12. If all the letters of the alphabet were written on a piece of paper, over and over again, which letter would be the ten millionth letter to be written?

Ten Million Problems: Answer Key

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Ten Million Problems - Rules

  1. Calculators are allowed.
  2. Problems will be given one at a time.
  3. Teams will have a maximum of five minutes per problem.
  4. As soon as any team has agreed on an answer, write it down and the entire team stands to deliver your written answer to the teacher.
  5. If the answer is correct, the team will earn 25 points and that round is over.
  6. If the answer is incorrect, the other teams that are still in the round each earn 5 points, and can continue working.
  7. When time is called, all teams who have not done so submit their answers.
  8. All correct answers earn 10 points.
  9. All answers which are "in the ballpark" earn 5 points.
  10. Between each round, the teacher randomly selects a few students to swap teams. Points stay with the student rather than the team; even though students are always working as part of a team, the winners are individual students rather than teams.

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