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| 4.1 Concepts Tests for divisibility can be discovered and illustrated with base ten pieces (see section 4.1 for illustrations of divisibility tests for 2, 3, 5 and 9). Use base ten pieces to discover a test for divisibility by 13 or 17. Show diagrams to illustrate your test and explain how it works by using examples. (Hint: Form a three-digit number with base ten pieces and look for easy ways to take away groups of 13 (or 17) units at a time. 4.1 Concepts Tasks, such as the following, involving factors, multiples, prime numbers, and divisibility can afford opportunities for problem solving and reasoning: (1) Explain why the sum of the digits of any multiple of 3 is itself divisible by 3; and (2) A number of the form abcabc always has several prime-number factors. Which prime numbers are always factors of a number of this form? Why? NCTM Standards 2000, page 217Use sketches of base ten pieces to illustrate why the sum of the digits of any multiple of 3 is itself divisible by 3. Write a convincing argument using examples as to why numbers of the form abcabc always have several prime factors. In particular, what are the common factors of all of these numbers? 4.1 Teaching Doing mathematics involves discovery. Conjecture – that is, informed guessing – is a major pathway to discovery. Teachers and researchers agree that students can learn to make, refine, and test conjectures in elementary school. NCTM Standards 2000, page 57Sarah discovered that to test a number for divisibility by 8 multiply the digits to the left of the units digit by 2 and add the result to the units digit. If the resulting number is divisible by 8 then the original number is divisible by 8. For example, for 336, 2 x 33 + 6 = 72, so 336 is divisible by 8 because 72 is divisible by 8. Use diagrams of base ten number pieces to provide an explanation that would make sense to a middle school student as to why this method works. |