Ray's New Higher Arithmetic · Arts. 371–372 · Unit 25: Involution
109. Squaring and cubing a sum
Goal: Your student squares and cubes two-part numbers by the tens-and-units pattern, and sees it in a drawing.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
- Paper, ruler and scissors
What's in the book
Parallel operations: the square of a sum (with tens and units), a drawn square divided into two squares and two rectangles, then the cube of a sum, with examples (answers printed).
The lesson, step by step
Warm-up
Ask: What is 20²? 4²? 20 × 4?
The idea
Read:
The square of the sum of two numbers is equal to the square of the first, plus twice the product of the first by the second, plus the square of the second.
Show it with 64 = 60 + 4: 3600 + 480 + 16 = 4096.
Draw it
Read:
Draw a square. From points in the sides, at equal distances from one of the corners, draw two straight lines across the figure.
Have your student draw a 6 by 6 square and cut it at 4. Name the four pieces: a 4 by 4 square, a 2 by 2 square, and two 4 by 2 rectangles.
Practice the square
Read the instruction:
Square the following numbers, considering each as the sum of two quantities, and applying the principle announced in Art. 371.
Work 19, 29, 59 and 125 this way.
The cube
Read the book's principle for the cube of a sum. Work its example together: 25³ = (22 + 3)³ = 10648 + 4356 + 594 + 27 = 15625.
Wrap-up
Ask: Why will these two patterns matter next? (They are the reason the square root and cube root methods work.)