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Contents
Introduction
 | Pupil's should be taught effective calculating strategies. |
| Counting strategies impede pupils development of pupil's understanding of number. |
| Develop informal written methods at the same time. |
| Develop the ability to use alternate strategies whenever advantageous |
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1. Addition and Subtraction
| Continental two-step rather than English three-step methods. |
| All calculations should be horizontal. |
| X0 & Y rather than X tens and Y units (place value is of less importance). |
| Carefully graded exercises needed following: |
Type
T + T 50 + 50
T + U 50 + 3
TU + U without crossing the tens boundary 56 + 3
TU + U 56 + 7
TU + T 56 + 30
TU + TU without crossing the tens boundary 56 + 33
TU + TU 56 + 37
| Exercises within each of these stages of the type: |
23 + 4 = o
23 + o = 27
27 = 23 + o
o + 23 = 27
| Sums in which three numbers are to be added. |
| Inequalities such as: |
23 + o < 27
27 > 23 + o
23 + 2 o 27
| Analogous sums such as: |
5 + 4 3 + 6
50 + 40 13 + 6
73 + 6
| Related sums where one number is kept constant while the other is increased or
decreased: |
36 + 5 49 + 3
37 + 5 49 + 4
38 + 5 49 + 5
| Inverse operations |
68 + 5 73 - 5
| Splitting up one operation into smaller steps: |
58 + 5 = 58 + 2 + 3
or 37 + 26 = 37 + 20 + 6
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2. Multiplication
| Doubling ´ 2 (add the same number again) |
1 ´ 8 = 8
2 ´ 8 = 16
4 ´ 8 =
8 ´ 8 =
Investigate. Let the class double. Stop before 10 000. Discussion of infinity.
Insist on horizontal layout: 32 + 32 = 62 + 2 = 64
| Halving ¸ 2 (separate into 2 parts) |
10 ´ 8
5 ´ 8
….. see how far they can go …..5/16
´ 8 = 2½
| Bring back to: |
1 ´ 8
2 ´ 8
4 ´ 8
8 ´ 8
… and perhaps next lesson …
10 ´ 8
| Step 2: |
Other tables e.g. 9 times. Perhaps bring in fingers although the eventual aim is to
discourage.
| Step 3: |
Exercises such as Barking and Dagenham.
Working towards the Wigley table, reinforce daily with OHT.
| Step 4: |
Introduce the Rhind Papyrus w/s showing 13 ´ 15.
Lots of examples. (Alongside Gelosia ?).
As many different ways as possible of getting the answer.
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| Operation/Addition, the increase in one of the numbers is compensated for by the
decrease in the other: |
99 + 46 99 + 46
98 + 47 100 + 45
97 + 48 101 + 44
96 + 49 102 + 43
95 + 50 = 145
| Operation/Subtraction, both numbers increase or decrease together: |
134 - 98 569 - 103 1003 - 6
135 - 99 568 - 102 1002 - 5
136 - 100 = 36 567 - 101 1001 - 4
566 - 100 = 466 1000 - 3
999 - 2 = 997
| Operation/Multiplication, halve one number, double the other. |
24 ´ 25 6 ´ 4 8 ´ 113
12 ´ 50 3 ´ 8 4 ´ 226
6 ´ 100 = 600 12 ´ 2 2 ´ 452
24 ´ 1 = 24 1 ´ 904 = 904
Can extend into fractions etc.
| Operation/Division, divide by the same number or ´ by the
same factors on both sides. |
1. For each of these, what other additions give the same answer:
99 + 46 48 + 72
999 + 478 498 + 234
538 + 345
2. For each of these, what other subtractions give the same answer:
134 - 98 562 - 101
1000 - 3 1908 - 1902
3794 - 1749
3. For each of these, what other multiplications give the same answer:
4 ´ 84 3 ´ 48
2 ´ 33 8 ´ 113
256 ´ 121 35 ´ 41
4. For each of these, what other divisions give the same answer:
34 ¸ 4 250 ¸ 50
248 ¸ 14 345 ¸ 9
455 ¸ 35 783 ¸ 81
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2c. Factors
Start with factor trees
Generate primes
(Prime investigation ….Eratosthenes sieve … lots of cards …. history of
primes)
Key Words: factor, multiple, prime, product, (power)
18
2 9
3 3
18 = 2 ´ 3 ´
3
= 2 ´ 3² (notation)
Do factor trees of:
24, 32, 35, 37, 75, 125, 43, 111, 104, 139, 182, 198, 101, 207, 294, 300, 315, 450,
528, 588, 648, 693, 754, 668, 1111, 3432, 3528, 3773, 6032.
Investigate: Add consecutive odd numbers together
Extension: Powers, HCF, LCM, indices, dramatic difference (e.g. 7³ and 3 ´ 7).
2d Square Numbers
2e As Pupils Gain Confidence
Starters, spider diagrams, "what happens if you don't remember the answer to
…"
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Barbara Philip
March 1998.
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