RIC-0563 Developing algebraic thinking
DIVISIBILITY RULES Teachers notes Introduction Looking at the algebra We often need to know quickly whether one number will divide evenly into another number. Knowing simple divisibility rules can make later factoring tasks in algebra much easier. The activities in this section provide opportunities to use the most common divisibility tests to create numbers satisfying the rules. Starting with tasks involving a limited number of tiles and divisibility rules, the pages become increasingly more difficult until all 10 number tiles are used. A number is said to be divisible by another number if and only if the first number divides evenly into the second number, leaving a remainder of 0. Divisibility tests are helpful in algebra when students began factoring various polynomials. Below is a list of the most common divisibility rules. A number is divisible by: • 2, if it ends with an even number in the ones place. That is, the ones digit is 0, 2, 4, 6, or 8. Examples: 54, 806, 3798. • 3, if the sum of the digits in the number is divisible by 3. Examples: 57, 321, 5412. • 4, if the last two-digit number is divisible by 4. Examples: 124, 6380, 999912. • 5, if the ones digit is 0 or 5. Examples: 75, 200, 4070, 99995 • 6, if the number is divisible by 2 and by 3. Examples: 48, 234, 7236. • 8, if the last three-digit number is divisible by 8. Examples: 1240, 3568, 765032. • 9, if the sum of the digits in the number is divisible by 9. Example: 468, 5247, 71559. • 10, if the number ends in 0. Example: 50, 700, 4080. Tests for 7, 11, and other numbers do exist, but they are sometimes very awkward to use. Long division or using a calculator are often quicker. 46 DEVELOPING ALGEBRAIC THINKING www.ricgroup.com.au R.I.C. Publications ® ISBN 978-1-74126-088-5
DIVISIBILITY RULES Teachers notes Divisibility rules For Divisibility Rules 1, 2, 3 and 4 only 6 number tiles are used for 1, 2, 3 and 4 each task and only two divisibility rules are involved. Many solutions exist for each task. In creating these numbers, students must consider place value ideas, along with divisibility tests. The least number would have the 1-digit in the hundreds place, most likely followed by 0 in the tens place. The greatest number would have a 9 in the hundreds place, possibly followed by 8 in the tens place. Below are the least and greatest numbers satisfying the divisibility rules. Least Greatest Divisibility Rules 1 (2 and 3) 102 984 Divisibility Rules 2 (3 and 5) 105 975 Divisibility Rules 3 (3 and 4) 108 984 Divisibility Rules 4 (4 and 9) 108 972 Divisibility rules In Divisibility rules 5 and Divisibility rules 6, nine number tiles are 5 and 6 used to create three 3-digit numbers satisfying the listed divisibility tests. The question: ‘Where should I start?’ brings problem-solving into play. Divisibility by 2 requires an even digit in the ones place and divisibility by 4 requires the last 2-digit number to be divisible by 4. In each task, these conditions provide what appears to be the logical ‘starting place.’ Having met those conditions, students have many possibilities for creating numbers divisible by 3. Divisibility rules Divisibility rules 7, 8, 9, 10, 11 and 12 each use all 10 numbers tiles to 7, 8, 9, 10, create two 5-digit numbers satisfying two divisibility rules. Again, the 11 and 12 nature of a particular test essentially dictates a starting place, unless a strategy of ‘Guess and check’ is used. Divisibility rules Divisibility rules 13, 14 and 15 are the most difficult activities in this 13, 14 and 15 section. Each requires students to use all 10 number tiles to create several multiplace numbers satisfying a variety of divisibility rules. Certain digits are strategically placed to provide students with limited choices for particular digits. For all three tasks, students cannot merely start on the first number and work through the numbers in order. R.I.C. Publications ® www.ricgroup.com.au DEVELOPING ALGEBRAIC THINKING 47 ISBN 978-1-74126-088-5
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DIVISIBILITY RULES<br />
Teachers notes<br />
Introduction<br />
Looking at<br />
the algebra<br />
We often need to know quickly whether one number will divide evenly<br />
into another number. Knowing simple divisibility rules can make later<br />
factoring tasks in algebra much easier. The activities in this section<br />
provide opportunities to use the most common divisibility tests to create<br />
numbers satisfying the rules. Starting with tasks involving a limited<br />
number of tiles and divisibility rules, the pages become increasingly<br />
more difficult until all 10 number tiles are used.<br />
A number is said to be divisible by another number if and only if the<br />
first number divides evenly into the second number, leaving a remainder<br />
of 0. Divisibility tests are helpful in algebra when students began<br />
factoring various polynomials. Below is a list of the most common<br />
divisibility rules.<br />
A number is divisible by:<br />
• 2, if it ends with an even number in the ones place. That is, the ones<br />
digit is 0, 2, 4, 6, or 8.<br />
Examples: 54, 806, 3798.<br />
• 3, if the sum of the digits in the number is divisible by 3.<br />
Examples: 57, 321, 5412.<br />
• 4, if the last two-digit number is divisible by 4.<br />
Examples: 124, 6380, 999912.<br />
• 5, if the ones digit is 0 or 5.<br />
Examples: 75, 200, 4070, 99995<br />
• 6, if the number is divisible by 2 and by 3.<br />
Examples: 48, 234, 7236.<br />
• 8, if the last three-digit number is divisible by 8.<br />
Examples: 1240, 3568, 765032.<br />
• 9, if the sum of the digits in the number is divisible by 9.<br />
Example: 468, 5247, 71559.<br />
• 10, if the number ends in 0.<br />
Example: 50, 700, 4080.<br />
Tests for 7, 11, and other numbers do exist, but they are sometimes very<br />
awkward to use. Long division or using a calculator are often quicker.<br />
46 DEVELOPING ALGEBRAIC THINKING www.ricgroup.com.au R.I.C. Publications ®<br />
ISBN 978-1-74126-088-5
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