In chapter 7 of the math adventure novel Math and Magic in Wonderland, Lulu and Elizabeth collect and deliver a tribute of Tum Tums to the Bandersnatches. To accomplish this, they must calculate the height of a tree and build a word ladder with the correct number of rungs. They deliver the Tum Tums to seven bridges which, based on graph theory, look like the seven bridges of Königsberg. Let's play along...
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Seven Bridges of Königsberg problem
The Seven Bridges problem is a classic. It's based on a real-world problem presented to Euler in 1736 in which he was asked to find a walking route through the city of Königsberg that would cross each of its bridges exactly once. Euler realized that this was impossible and as he tried to prove why it was impossible, he started the field of graph theory. Here's a video explaining the answer to the Seven Bridges problem:
Your Turn to Play:
In Chapter 7, the girls had to leave half of their Tum Tums at each bridge. To calculate how many Tum Tums they would need in order for one to be left at the end, they had to work backwards and double the number of fruit for the eight bridge crossings. Lulu and Elizabeth soon learned how quickly a number grows as it is doubled. This is a power function (2 to the power of 8), also called an exponent. Here are some videos on exponents:
Lewis Carroll invented the game of "Word Ladders" (which he called "Doublets"). To play the game, start with a word and change one letter to form a new word. Keep going until you get to your desired ending word.
In the book, the girls calculated the height of the Tum Tum tree by using the proportionality of shadows. This is the method used by Thales to calculate the height of the Great Pyramid. You can read more about Thales's calculation here, or watch this video:
- Try this wonderful lesson on graph theory for kids.
- It is actually a sophisticated mental leap for children (and adults) to recognize that a problem doesn't have an answer (or not enough information is given). Use this week to practice some of these types of problems.
The power of exponents
In Chapter 7, the girls had to leave half of their Tum Tums at each bridge. To calculate how many Tum Tums they would need in order for one to be left at the end, they had to work backwards and double the number of fruit for the eight bridge crossings. Lulu and Elizabeth soon learned how quickly a number grows as it is doubled. This is a power function (2 to the power of 8), also called an exponent. Here are some videos on exponents:
Your Turn to Play:
- Copy the map of the Seven Bridges of Königsberg on your own piece of paper and use small objects (like crackers or cereal) to represent the Tum Tum fruit. Re-enact the problem by leaving half of your "stash" at each bridge crossing. If you want more than 1 piece of fruit left at the end, how many pieces would you have to carry with you?
- Make your own worksheet to practice exponents.
Word ladders
Lewis Carroll invented the game of "Word Ladders" (which he called "Doublets"). To play the game, start with a word and change one letter to form a new word. Keep going until you get to your desired ending word.
Your Turn to Play:
- Try some of these Word Letter Worksheets.
- The RobinWords online game is similar to Word Ladders and is great for developing vocabulary and spelling skills. How big of a word chain can you make?
Calculating the height of an object
In the book, the girls calculated the height of the Tum Tum tree by using the proportionality of shadows. This is the method used by Thales to calculate the height of the Great Pyramid. You can read more about Thales's calculation here, or watch this video:
Your Turn to Play:
The "Play Along" section at the end of Chapter 7 invited you to use this method to calculate the height of a tall object in your own neighborhood. Give it a try!
The "Play Along" section at the end of Chapter 7 invited you to use this method to calculate the height of a tall object in your own neighborhood. Give it a try!
Here is the Book Club / Math Circle schedule (you can join any time):
Week of August 1st:
- Book Club Kick-Off Party!
- Read Chapter 1: Mrs. Magpie's Manual
- Alliteration
- Memorizing digits of Pi
- Palindromes
- Calculating your age on other planets
- Read Chapter 2: Magic Square
- Making tangrams
- Acute, obtuse, and right angles
- Magic squares
- Adding consecutive numbers using Gauss's trick
- Read Chapter 3: Secret Codes
- Word permutations
- Cartesian coordinates
- Operations on odd and even numbers
- Read Chapter 4: Rabbit Trails
- Drawing a perfect circle
- Making a compass
- Finding the center of a circle
- Exploring Pi
- Famous mathematicians who followed rabbit trails
- Read Chapter 5: Two Worlds Join
- Mobius strips
- Fractals
- Tessellations
- Read Chapter 6: River Crossing
- River Crossing Problems
- Build a boat and explore buoyancy
- Read Chapter 7: Seven Bridges
- Seven Bridges of Königsberg problem
- The power of exponents
- Word ladders
- Thales's method for calculating the height of an object
- Read Chapter 8: Veracity
- Truth-tellers and liars
- Finding a fake coin using a balance scale
- Archimedes buoyancy principle
- Read Chapter 9: To Catch a Thief
- John Napier's Rooster
- Doubling pennies and calculating exponents
- Towers of Hannoi
- Read Chapter 10: The Vorpal Sword
- Square numbers
- Prime numbers
- Fibonacci Sequence
- Relativity and time dilation
- Acrostic poems
- Read Chapter 11: Two Great Powers
- Book Club Finishing Party with Prizes!
Thanks for joining us. I can't wait to read your comments!
We've been playing around with the seven bridges. This chapter is so fun.
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