All work on extending 1 dimensional similarity (distances) to 2 or 3 dimensions (area and volume). This is very counter-intuitive for students, and I believe they need multiple experiences to retrain their intuition. Of these, (1) is probably the toughest because students jump to linear relationship for area and volume. (3) is the best for countering that ill assumption, although (4) can help also.
1. Big Trouble.
Finn Mac Cumhail, (pronounced Finn McCool; no, really) leader of the ancient Fianna warriors, and gifted with "magic, insight and the power of words" when he was the first to eat of the Salmon of Knowledge, and ended up a giant. (Only in Ireland do magic powers come with the gift of gab.) One of his rival giants, Benandonner, lived across the sea in Scotland. Benandonner wasn't able to swim across the sea to Ireland for a proper gigantic challenge so Finn tore pieces of volcanic rock into columns to make the causeway to Scotland.
Benandonner came across to Ireland and Finn's house, where Finn was dressed up as a baby. Yes, a baby over 15 feet long! The "baby" bit the Scottish giant's hand off and the Scot took off for Scotland, terrified at how big Finn himself must be if his baby was so big.
Draw a picture for each of these questions. Label edges with dimensions.
a) If Finn was really a 15 foot long baby, how tall would the father be? (State any assumptions clearly.)
b) Say a typical 6-foot tall Celtic Warrior weighs 9 stone. (Ancient weight measure.) How much might the 15 foot tall Finn weigh? (Weight, density being equal, corresponds roughly with volume.)
c) If it takes three square yards of wolf pelt to make a fierce looking warrior garb for your typical 6 foot warrior, how many much material would Finn need to make a costume? If that takes two wolves for 3 square yards, how many wolves for Finn?
d) Give the measurements (dimensions, area, volume, weight, etc.) of a giant sized something you might find in Finn’s house. (Iron cooking skillets feature heavily in the Benandonner story, but don’t feel limited by that.)
(Tomie DePaola did a version of this story, but he mixes up Finn, Benandonner an Cuchalain - pronounced 'Kuh-kullen' - another Irish hero of myth.)
2. Tangram
Requires multiple tangram sets or copies of paper tangrams. Can eliminate step (1) for time.
1) Use all the Tangram pieces of one set to make a square.2) Since all squares are similar (and why is that?) this large square is similar to the small square in the set. What is the scale factor?
3) If the small square has area = 1, what is the area of the large square?
4) Use the tangram pieces to make a figure and two other figures that are similar to the first. (Bigger and even bigger, or bigger and smaller, or...)
5) Prove the similarity of your figures in (4) by using ratios.
See also, the teacher.net Grandfather Tang lesson.
3. 3-D Similarity
Requires: 100 cubes or so
1) Build the building with mat plan (also called a base plan):
3) Prove your building is similar with ratios of corresponding sides.
4) Build or design a building three times larger than the original. Explain how you know what is needed.
5) Find the volume and surface area of each building. What relationship do the enlarged surface areas and volumes have with the original? Why is it like that?
6) Can you design a building which has a buildable enlargement of 125%? Find their surface and volumes. What scale factor relationship do the buildings' area and volume have? How does that compare to (5)?
4. Dilation
Requires: computer access
Open the Hexagon Dilation geogebra sketch or webpage.
In this sketch, the blue hexagon is dilated from the red point by a scale factor of S. The sketch allows you to change S, and move the dilation point or any of the blue vertices. It also measures the area and perimeter of the hexagon and the dilation.
The check box lets you show a square with area equal to 1 square unit for comparison, and its dilation by a scale factor S also.
1) Try varying the scale factor S. What do you notice? What questions do you wonder about?
2) Collect data on the areas and perimeters for a fixed blue hexagon and its dilation as you vary S.
3) Can you find a pattern in your data? Can you find a formula for the purple area and perimeter in terms of the original measurement and S?
4) Use your formula to make a prediction for a scale factor and original area of your choice. Use the sketch to check. Does your formula work for a scale factor that is a decimal? Does it work for a scale factor less than 1?
5) Compare the edges of the original and the edges of the image. What do you notice as you vary S? As you move the center of dilation?
6) Can you predict the coordinates of the image of a vertex if the center of dilation is at the origin? If it is not at the origin?
Extension: Open the sketch gigantotron.ggb (or webpage) and investigate 3-D similarity. What questions would you ask to investigate?
(Now also on GeoGebraTube and a mobile applet.)
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