Estimated Time
1 class period for collecting data and pondering it; then homework to generate the formula; then half a period for nailing down the formula and applying it to the floor.
Prerequisites
- Making a table of data
- Knowledge of prime, relatively prime, and composite numbers
Materials Needed
- A checkboard grid
- Rulers or thread
Ohio Standards Alignment
Topics
Representing visual patterns numerically, prime and relatively prime numbers, patterns in data
Overview
A mouse runs diagonally across a large black-and-white tiled floor. How many tiles does it touch? Since the floor is huge, data must be gathered from smaller cases, in the hope of generalizing to the larger case.
Learning Objectives:
- Representing a physical problem mathematically.
- Gathering and organizing data.
- Expressing a pattern in the data as a formula.
- Using the formula to answer the original question.
- Generalizing to three dimensions.
The "Hook"
City Hall has a splendid rectangular lobby with a floor of black and white tiles. The tiles are square, in a checkerboard pattern, lined up with the walls: 93 tiles in one direction and 231 in the other. Now, in this particular lobby there are two mouse holes, at diagonally opposite corners of the floor. One night a mouse comes shooting out of one of the mouse holes and runs fast, straight across the floor, and into the other mouse hole.
Critical Question
How many tiles does the mouse run across? How can we get a grip on this? That's a huge number of tiles!
(By the way, interior designers prefer that the tiles be lined up diagonally with the walls to make a more interesting visual pattern; evidently the city hall floor folks did not know this. Check the next time you see a checkerboard floor to see which way they go.)
The Investigation
The idea of using a gigantic checkerboard, or a picture of one, as a diagram of the floor is the first step (see sample grid).
Critical Question
No checkerboard is big enough to model the problem. What can we do?
Help students see that if they start with much smaller floors, they may be able to see a pattern that they can then use to answer the question.
Pass out the checkerboard worksheet, and either rulers or lengths of thread that can be stretched across the squares in various ways. Have students work in pairs, one with the ruler and the other writing down the number of squares crossed.
It will quickly become evident that there is a welter of data coming in--obviously some sort of a chart is needed. A blank chart is provided, but it might be better--if the students are up to it and you have time--to have them figure out how to make one. If you want to move more quickly, you could put the given chart onto a transparency and have students call out the information as they find it. (A completed chart is included.)
Once the chart is completed, then comes the serious scrutiny of the numbers. The pattern does not reveal itself easily! At this point you might leave the question hanging for overnight pondering by the students. There are many cases where the number of tiles crossed is almost the sum of the two dimensions, exactly one less than the sum of the two.
Critical Question
But what about the other cases? What kind of numbers don’t fit that easy pattern?
Once somebody has figured out what is going on, the formula can be applied to the large floor that is 93 by 231 tiles.
Teaching Tips
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Once you are finished, suggest an extension: What about a marble cake of alternating cubes of lemon and chocolate that is, say, 8 by 15 by 12 cubes? Suppose an ant burrows straight through from one corner to the corner that is diagonally opposite--what happens then? How could one model this to find out?
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Check yourself -- here is the
formula for the number of tiles the mouse touches.
Citation
From the teaching files of Rudd Crawford.