Ray's New Practical Arithmetic · Arts. 242–244 · Unit 24: Involution and Evolution

113. Cube root: periods and the method

Goal: Your child points off a number in periods of three figures and follows the cube-root method, with the building-blocks picture behind it.

⏱ About 40 minutes · 6 steps

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You'll need

  • Paper and pencil, or a slate or small whiteboard
  • Small cubes or sugar cubes (optional)

What's in the book

Page 309 begins Cube Root (Art. 242), finished on page 310 with pointing off in threes (Art. 243). Page 311 gives Art. 244 with the worked example ∛13824 = 24. Pages 311–313 explain it with a cube and added slabs.

The lesson, step by step

  1. Warm-up

    Ask: 2³ = ? 3³ = ? 10³ = ? 20³ = ? (8, 27, 1000, 8000.)

  2. Pointing off

    1st. If a number be pointed off into periods of three figures each, the number of periods will be the same as the number of figures in the cube root.

    Place a point over the order units, and then over every third order from units to the left and to the right.

    Try: 876|453|921 and 37|683.562|400 (a bar marks each break between periods).

  3. Work the book's example

    Extract the cube root of 13824.

    Periods: 13|824. The largest cube in 13 is 8, root 2; 13 − 8 = 5; bring down 824: 5824.

    Square the root 2 and multiply it by 300; the result, 1200, is the trial divisor.

    1200 goes into 5824 about 4 times. Then 2 × 4 × 30 = 240 and 4 × 4 = 16. 1200 + 240 + 16 = 1456, and 1456 × 4 = 5824, remainder 0.

    Therefore, 13824 is a perfect cube, and its cube root is 24.

  4. The picture

    With blocks or a drawing: a 20-inch cube (8000). Add three slabs 20 × 20 × 4 (4800), three rods 20 × 4 × 4 (960) and a small cube 4 × 4 × 4 (64). 4800 + 960 + 64 = 5824.

    The 300, 30 and squared last figure in the rule come from these three kinds of pieces.

  5. Guided practice

    Point off together the numbers in problem 4 on page 310 and problem 5 on page 311. Then try ∛91125 together with the method.

  6. Check

    The pointing problems have no printed answers; see the answers box. ∛91125 = 45.

Answers to the book's problems

Art. 243 pointing off in threes (the book prints no answers; a bar marks each break between periods): 1. 876|453|921. 2. 7.356|849|227. 3. 37|683.562|400. 4. 138|975|462; 3.561|325|482; 684|536.256|403. 5. 2|756.568|430; 98|451.327|600; .856|375; .006|400. Also from the guided practice: the cube root of 91125 is 45.

Worked out for this site from the scan. If a number in your copy differs, trust the book.

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