Problems, understanding and solutions Division Problems, understanding and solutions
Why left to right? Looking at efficiency of standard algorithms vs other ways
Addition Left to right Right to left Step 1 Step 2 Step 3 Right to left is better since there is no rewrite
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
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
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
Place “value”d? Longer vs shorter algorithm
Long Short Very confusing!! Step 1 36 – 2???? Is that written correctly? Step 2 But no need to rewrite the unit in the short
So… Combining the best Leaving out the rest
86 3
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
38 5
38 5 tens units 7 3 tens 7 5 3 0 8 5 38 – 00 – 3 0 5 38 3 – 35 30 units 3
538 4
538 4 10 hundreds tens 1 units – 1 3 4 4 538 – 400 1 – 130 8 – 120 1 0 0 3 0 4 10 units hundreds tens 1 hundred units 4 5 0 0 3 0 8 4 0 0 – 1 3 4 4 538 1 0 0 3 0 8 – 400 1 ten – 1 0 0 2 0 130 8 – 120 1 0 8 10 tens 18 – 1 0 6 – 16 2 2
347 6
347 6 hundreds tens units 5 7 – 30 40 3 6 347 – 000 340 7 – – 300 47 5 0 7 hundreds tens units 6 3 0 0 4 0 7 5 7 – 3 0 0 30 tens 40 units 3 hundreds 6 347 4 0 7 – 000 340 7 – 4 0 2 – 300 47 5 – 42 4 tens 5
8643 7
8643 7 thousands hundreds units tens 10 1 2 3 4 7 8643 1 – – 7000 1 0 0 0 2 0 0 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 8 0 0 0 6 0 0 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 7 0 0 0 – 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 1 0 0 0 6 0 0 4 0 3 1600 43 1000 100 20 tens 100 10 10 100 – 1 0 0 0 4 0 0 100 100 – 1400 30 units 2 hundreds 1000 3 tens 100 100 10 10 10 240 3 1 1 1 1 1 2 0 0 4 0 3 1000 100 100 – 210 – 2 0 0 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
Estimation
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
256 36
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 = 6 + 1 = 7 7 × 36 = 252 Step 3: Step 5: 256 ÷ 36 = 7 remainder 4 = 256 – 252
256 33
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
Decimal dilemma!
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. 0.05 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!!!