Ray's New Practical Arithmetic · Arts. 239–241 · Unit 24: Involution and Evolution
112. Square root by factoring; right triangles; side of a square
Goal: Your child finds square roots by prime factors, uses the rule of the right-angled triangle to find a missing side, and finds the side of a square from its area.
You'll need
- Paper and pencil, or a slate or small whiteboard
- Grid paper and a ruler
What's in the book
Page 306 begins square root by factoring (Art. 239), with problems on page 307. Page 307 starts right-angled triangles (Art. 240), and page 308 shows the 3-4-5 triangle and gives two rules with problems. Page 309 has more triangle problems and the side of a square (Art. 241). Answers are printed.
The lesson, step by step
Warm-up
Ask: What is 3² + 4²? What is 5²? Notice anything?
By factoring
Ray's example: 441 = 3 × 3 × 7 × 7, so √441 = 3 × 7 = 21.
Resolve the number into its prime factors, and find the product of one of each two equal factors.
Right-angled triangles
The side opposite the right-angle is called the hypotenuse; the other two sides, the base and the perpendicular.
The key fact:
The square described on the hypotenuse of a right-angled triangle is equal to the sum of the squares described on the other two sides.
Draw Ray's triangle with sides 3 in. and 4 in. on grid paper, with a square on each side. Count: 9 + 16 = 25.
The two rules
1st. To the square of the base add the square of the perpendicular, and extract the square root of the sum.
2d. From the square of the hypotenuse subtract the square of the other given side, and extract the square root of the difference.
Guided practice
Do these together:
A castle 45 yd. high is surrounded by a ditch 60 yd. wide: what length of rope will reach from the outside of the ditch to the top of the castle?
The area of a square field is 6241 sq. rd.: what is the length of one side?
Independent practice and check
Your child works the factoring problems on page 307 and the other triangle and square-side problems on pages 308–309. 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.