1. Problems, understanding and solutions
Division
Problems, understanding and solutions

2. Why left to right?
Looking at efficiency of standard algorithms vs other ways

3. Addition Left to right Right to left Step 1 Step 2 Step 3
Right to left is better since there is no rewrite

4. Trouble coming? Subtraction Left to right Right to left
Similar to addition
No, since 4 > 2
Yes , since 3 < 6
8 – 5 = 3
4 – 2 – 1 = 1
13 – 6 = 7
Not sure whether to teach this to children learning subtraction for the 1st time
Surely share with adults and older children who need mathematically valid shortcuts

5. Multiplication 47  9
Left to right
Right to left
40  9 =
360
7  9 = 63
Step 1
7  9 = 63 +
40  9 = +
360
Step 2
423
360 3 42
Step 3
carry 6
Right to left is better since there is no rewrite

6. Division Left to right Right to left Step 1 Step 2 Step 3 Step 4
Left to right is better since there is no rewrite

7. Place “value”d?
Longer vs shorter algorithm

8. Long Short Very confusing!! Step 1 36 – 2????
Is that written correctly?
Step 2
But no need to rewrite the unit in the short

9. So…
Combining the best
Leaving out the rest

10. 86  3

11. 86  3
tens
units
2 0 8 2 8 20
units 3
8 0 6 3 86 – 60 –
6 0 26
2 0 6 – 24 –
2 0 4 2 2
tens 2

12. 38  5

13. 38  5
tens
units 7 3
tens 7 5
3 0 8 5 38 – 00 –
3 0 5 38 3 – 35 30
units 3

14. 538  4

15. 538  4 10 hundreds tens 1 units – 1 3 4 4 538 – 400 1 – 130 8 – 120
3 0 4 10
units
hundreds
tens 1
hundred
units 4
3 0 8 – 1 3 4 4
538
3 0 8 –
400 1
ten –
2 0
130 8 –
120
1 0 8 10
tens 18 –
1 0 6 – 16 2 2

16. 347  6

17. 347  6 hundreds tens units 5 7 – 30 40 3 6 347 – 000 340 7 – – 300 47
5 0 7
hundreds
tens
units 6
4 0 7 5 7 – 30
tens 40
units 3
hundreds 6
347
4 0 7 –
000
340 7 –
4 0 2 –
300 47 5 – 42 4
tens 5

18. 8643  7

19. 8643  7 thousands hundreds units tens 10 1 2 3 4 7 8643 1 – – 7000
3 0 4
thousands
hundreds
units
tens 1
1000 10 1 1 10
100 1 1 10 1 1 1
1000 10 1 1 10 10
hundreds
100 1 1 10 1 1 10 10 7
4 0 3 1 2 3 4
1000 1
1000 10 1 1 10
100 1 1 10 1 1 10 10 7
8643 1
thousand – 1
1000 10 1 1 10
100 1 1 10 1
100 10 10
100 –
7000 1
1000 10 1 1 10
100 1 1 1
100 10 10
100
4 0 3
1600 43
1000
100 20
tens
100 10 10
100 –
100
100 –
1400 30
units 2
hundreds
1000 3
tens
100
100 10 10 10
240 3 1 1 1 1 1
4 0 3
1000
100
100 –
210 –
1 0
1000
1000
1000
100
100
100 33
100
100
100 10 10 10 10 10 10 1 10 1 1 10 1 1 10 1
3 0 3 – 28 1 1 1 1 1 1
1000
100
100 10 5 10 –
2 0 8
1000
1000
1000 1 10 1
100
100
100 1 1
100
100
100 10 10 10 10 10 10 1 10 1 1 10 1 1 10 1 5 1 1 1 1 1 1

20. Estimation

21. Steps Round off divisor to the nearest multiple of 10
Estimate quotient (or quotient digit) at that step using the estimate
Calculate quotient digit × actual divisor
Check
if quotient digit × divisor > dividend: decrease quotient digit by 1 and repeat step 3
if not, proceed to check
If dividend – quotient digit × divisor > divisor: increase quotient digit by 1 and repeat step 3
If not, proceed to step 5
Complete division step with the (modified) quotient

22. 256  36

23. 36 rounded off to 40 256 ÷ 40 (or 25 ÷ 4) ≈ 6 6 × 36 = 216
Step 1:
36 rounded off to 40
256 ÷ 40 (or 25 ÷ 4) ≈ 6
Step 2:
6 × 36 = 216
Step 3:
Step 4: check
a. 216 < 256
b. 256 – 216 = 40 > 36
⇒ quotient = = 7
7 × 36 = 252
Step 3:
Step 5:
256 ÷ 36 = 7 remainder 4 = 256 – 252

24. 256  33

25. 33 rounded off to 30 256 ÷ 30 (or 25 ÷ 3) ≈ 8 8 × 33 = 264
Step 1:
33 rounded off to 30
256 ÷ 30 (or 25 ÷ 3) ≈ 8
Step 2:
8 × 33 = 264
Step 3:
Step 4: check
a. 264 > 256
⇒ quotient = 8 – 1 = 7
7 × 33 = 231
Step 3:
Step 5:
256 ÷ 33 = 7 remainder 25 = 256 – 231

26. Decimal dilemma!

27. Suggestion: Carry the decimal point throughout
2 strips (0.1s) converted to 20 small squares (0.01s)
Can’t divide 3 squares (1s)in 4
 0 squares (1s) distributed and 0 squares per group
 3 squares (1s) remain
20 small squares (0.01s) get divided
each group gets 5 small squares more i.e more per group
 0.75 total per group
3 squares (1s) converted to 30 strips (0.1s)
28 strips (0.1s) get divided
each group gets 7 strips i.e. 0.7 per group
 2 strips remain
Objection: More confusing points!!! No way!!!