RIC-0563 Developing algebraic thinking

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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
Page 1 and 2: Developing algebraic thinking Writt
Page 3 and 4: DEVELOPING ALGEBRAIC THINKING Forew
Page 5 and 6: PROBLEM-SOLVING AND NUMBER TILES Te
Page 7 and 8: PROBLEM-SOLVING AND NUMBER TILES Te
Page 9 and 10: Place value picks R.I.C. Publicatio
Page 11 and 12: PLACE VALUE PICKS Teachers notes Pl
Page 13 and 14: PLACE VALUE PICKS Teacher’s notes
Page 15 and 16: PLACE VALUE PICKS 2 Using 9 number
Page 23 and 24: Same sums R.I.C. Publications ® ww
Page 25 and 26: SAME SUMS Teachers notes The follow
Page 27 and 28: SAME SUMS Same sums 1 Use the numbe
Page 33 and 34: SAME SUMS Consecutive Sums 11 Use t
Page 35 and 36: Primes and composites R.I.C. Public
Page 37 and 38: PRIMES AND COMPOSITES Teachers note
Page 39 and 40: LOTS OF PRIMES, LOTS OF COMPOSITES
Page 41 and 42: MORE PRIMES Use all 10 number tiles
Page 43 and 44: ALL PRIMES Use all 10 number tiles
Page 45: Divisibility rules R.I.C. Publicati
Page 49 and 50: DIVISIBILITY RULES Teachers notes D
Page 51 and 52: DIVISIBILITY RULES Divisibility rul
Page 59 and 60: Locker numbers R.I.C. Publications
Page 61 and 62: LOCKER NUMBERS Teachers notes Probl
Page 63 and 64: LOCKER NUMBERS Teachers notes Probl
Page 65 and 66: LOCKER NUMBERS 2 Use the number til
Page 67 and 68: Exciting exponents R.I.C. Publicati
Page 69 and 70: EXCITING EXPONENTS Teachers notes E
Page 71 and 72: EXCITING EXPONENTS 2 Place all 10 n
Page 73 and 74: EXCITING EXPONENTS 4 Place all 10 n
Page 75 and 76: Plus and times R.I.C. Publications
Page 77 and 78: PLUS AND TIMES Teachers notes Plus
Page 79 and 80: PLUS 5 Place any 5 of the digits 0
Page 81 and 82: TIMES Place any 5 of the digits 0 t
Page 83 and 84: Shapes R.I.C. Publications ® www.r
Page 85 and 86: SHAPES Teachers notes Triangle 10 T
Page 87 and 88: SHAPES Triangle 6 Place any 6 of th
Page 89 and 90: SHAPES Triangle 10 Place all 10 of
Page 91 and 92: SHAPES Pentagon Use all 10 number t
Page 93 and 94: Pentomino puzzles R.I.C. Publicatio
Page 95 and 96: PENTOMINO PUZZLES Teachers notes Gr
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PENTOMINO PUZZLES Use 5 number tile
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PENTOMINO PUZZLES Use 5 number tile
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Alphabet algebra a b e c d h f g R.
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ALPHABET ALGEBRA Teachers notes Alp
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ALPHABET ALGEBRA B Use all 10 numbe
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ALPHABET ALGEBRA D Use any 8 number
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ALPHABET ALGEBRA F Use all 10 numbe
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ALPHABET ALGEBRA H Use any 7 number
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ALPHABET ALGEBRA J Use any 8 number
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ALPHABET ALGEBRA L Use any 8 number
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ALPHABET ALGEBRA N Use any 9 number
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ALPHABET ALGEBRA P Use all 10 numbe
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ALPHABET ALGEBRA R Use all 10 numbe
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ALPHABET ALGEBRA T Use any 8 number
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ALPHABET ALGEBRA V Use any 7 number
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ALPHABET ALGEBRA X Use any 9 number
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ALPHABET ALGEBRA Z Use all 10 numbe
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RIC-0563 Developing algebraic thinking

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