Ray's New Higher Arithmetic · Art. 389 · Unit 26: Evolution
116. Similar figures and general exercises in roots
Goal: Your student uses the rules that areas of similar figures go as the squares, and volumes as the cubes, of their sides, and works mixed problems in roots.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
A remark on finding right triangles with whole-number sides, parallel lines and similar figures with five remarks, then general exercises in evolution (most answers printed), and the topical outline of Involution and Evolution.
The lesson, step by step
Warm-up
Ask: A 2-inch square and a 4-inch square: how many times bigger is the area of the second? (4.) Two cubes, 1 inch and 2 inches: how many times the volume? (8.)
Whole-number triangles
Read:
Take any two unequal numbers; the sum of their squares may represent a hypotenuse; the difference of their squares will then stand for one side, and double their product for the remaining side. Thus, from 3 and 2, form 13, 12, 5
Let your student make one from 4 and 1, and check it.
Similar figures
Read:
Similar Figures are figures having the same number of sides, and their like dimensions proportional.
The areas of similar figures are to each other as the squares of their like dimensions.
The solidities of similar solids are to each other as the cubes of their like dimensions.
Guided practice
One square is 12¼ times another: how many times does the side of the 1st contain the side of the 2d?
The lengths of two similar solids are 4 in. and 50 in.; the 1st contains 16 cu. in.: what does the 2d contain?
Independent practice
Your student picks five or six of the general exercises, for example:
The diagonals of two similar rectangles are as 5 to 12: how many times does the larger contain the smaller?
Review
Read the topical outline of Powers and Roots and have your student give one example for each line.
Answers to the book's problems
Ex. 2 (no printed answer in the scan): areas go as the squares of the diagonals, (12/5)² = 144/25, so the larger contains the smaller 5 19/25 times (5.76 times).
Worked out for this site from the scan. If a number in your copy differs, trust the book.