Ray's New Practical Arithmetic · Arts. 233–234 · Unit 24: Involution and Evolution
109. Involution: squares, cubes and powers
Goal: Your child knows square, cube, power and exponent, and raises numbers (and decimals) to any power.
You'll need
- Paper and pencil, or a slate or small whiteboard
- Grid paper and a handful of small cubes or blocks (optional)
What's in the book
Page 298 defines Involution and powers (Art. 233). Page 299 explains exponents and gives Art. 234 with a worked example (the cube of 75), the rule and problems 2–17. Answers are printed.
The lesson, step by step
Warm-up
Draw a square 3 squares on a side on grid paper. How many small squares? Then picture a cube of blocks 3 on each edge. How many blocks?
The idea
Involution is the multiplication of a number into itself one or more times.
A power is the product obtained by involution.
The second power, or square, is the product obtained by taking the number twice as a factor.
The third power, or cube, is the product obtained by taking the number three times as a factor.
The book explains the names: a square's area and a cube's solid contents are products of equal factors.
Exponents
The exponent is a number denoting the power to which the given number is to be raised.
So 3² = 9 and 5³ = 125.
Work the book's example
Find the cube of 75.
75 × 75 = 5625 (the square); 5625 × 75 = 421875 (the cube).
Obtain a product in which the number is taken as a factor as many times as there are units in the exponent of the power.
Guided practice
Do these together:
Find the square of 65.
Find the fourth power of .25.
Independent practice and check
Your child works problems 3–17 on page 299 (fractions included: square the numerator and the denominator). The book prints the answer at the end of each problem line. Have your child cover the answers with a strip of paper, work the problem, then slide the strip down to check.
Mental-math wrap-up
Squares 1–12 and cubes 1–5, aloud, as fast as possible.