Ray's New Practical Arithmetic · Arts. 235–237 · Unit 24: Involution and Evolution
110. Evolution: roots, and pointing off for square root
Goal: Your child knows square root and cube root, the perfect squares to 100, and points off a number in periods of two figures.
You'll need
- Paper and pencil, or a slate or small whiteboard
What's in the book
Page 300 defines Evolution and roots (Art. 235). Page 301 has the table of squares and cubes of 1 to 10. Page 302 shows why periods of two figures tell the number of figures in a square root (Art. 236). Page 303 teaches pointing off (Art. 237).
The lesson, step by step
Warm-up
Ask: What number times itself makes 49? Makes 81? What number used three times makes 8?
The idea
Evolution is the process of resolving a number into two or more equal factors.
The square root of a number is one of two equal factors.
The cube root of a number is one of three equal factors.
"Thus, 3 is the square root of 9; for 9 = 3 X 3."
A perfect power is one whose root can be obtained exactly.
An imperfect power is one whose root can be obtained only approximately.
Learn the table
Read the table of squares and cubes of 1 to 10 on page 301 together, and say the squares aloud until they come easily.
Why we point off
The square root of 100 is 10 and of 10000 is 100. So numbers of 3 or 4 figures have roots of 2 figures.
1st. If a number be pointed off into periods of two figures each, the number of periods will be the same as the number of figures in the square root.
How to do it:
Place a point over the order units, and then over every second order from units to the left and to the right.
Guided practice
Point off together:
Point off 368425.
Point off 6.843256.
With bars between periods: 36|84|25 (three periods, so the root has 3 figures) and 6.84|32|56.
Independent practice
Your child points off problem 3 and the numbers in problems 4–5 on page 303. These have no printed answers; see the answers box.
Answers to the book's problems
Art. 237 pointing off (the book prints no answers; a bar marks each break between periods): 1. 36|84|25. 2. 6.84|32|56. 3. 8|37|51.42|96|30 (a cipher is annexed to complete the last decimal period). 4. 86|43|26; 4.75|84|62; 75|84.37|69. 5. 9|72|85.46|13|80; 7|53|00; .04|68|27; .06|25; .62|50.
Worked out for this site from the scan. If a number in your copy differs, trust the book.