Chapter 3: Whole-Number Computation
3.3 Algorithms for Whole-Number Multiplication and Division
3.3.1. Multiplication Algorithms: lots of ways to multiply
Standard algorithm
5
7
5 8
7 9
5 2 2
4 0 6 .
4 5 8 2
x
R  L; NO emphasis on place value (place value is not explicit in this method)
Partial product method
5
x 7
7
4 5
5 6
3 5 0
4 5 8
8
9
2
0
0
0
2
(9 x 8)
(9 x 50)
(70 x 8)
(70 x 50)
Information in RED is for notes ONLY – Do NOT include as part of algorithm.
R  L; Emphasis on place value (place value IS explicit in this method)
FOIL
58 x 79 = (50 + 8) x (70 + 9)
= (50 x 70) + (50 x 9) + (8 x 70) + (8 x 9)
= 3500 + 450 + 560 + 72
= 4582
Foil order (First – Outer – Inner – Last); Emphasis on place value (place value IS
explicit in this method); horizontal method (NOT done vertically); connects with
algebra; must do all steps as shown above for full credit
Lattice Multiplication
3
4
0
3
4
0
8
6
1
2
8
6
5
7
1
4
2
6
1
2
2
6
5
2
3
2
6
2
3
Step 1:
1
0
1
0
1
6
6
4
7
4 1
9
4
9
Step 2:
Fig. 2.3-2
Fig. 2.3-1
MUST
interpret this method, so final answer is: 12644
Order not important; NO emphasis on place value (place value is NOT explicit in this
method); Only need to know facts and be able to add
Look at that same problem done in partial product mode.
436
x 29
54
270
3600
120
600
8000
12644
Can you see how the place values and the diagonals match up?
Left to right multiplication
5
x 7
3 5 0
4 5
5 6
7
4 5 8
8
9
0
0
0
2
2
(50 x 70)
(50 x 9)
(8 x 70)
(8 x 9)
Information in RED is for notes ONLY – Do NOT include as part of algorithm.
L  R; Emphasis on place value (place value IS explicit in this method)
Russian peasant (Simple halving/doubling method)
78
156
312
624
1248
2496
3978
51
25
12
6
3
1
No Order; NO emphasis on place value (place value is NOT explicit in this method);
only need to be able to double (times 2) and add to use method
Why this method works:
78
156
312
624
1248
2496
3978
51
25
12
6
3
1
Halving
25 remainder 1
12 remainder 1
6 remainder 0
3 remainder 0
1 remainder 1
0 remainder 1
Base 2
1 x 20
1 x 21
0 x 22
0 x 23
1 x 24
1 x 25
Multiple of 78
78 x 1
78 x 2
78 x 0
78 x 0
78 x 16
78 x 32
Division
Standard Division Algorithm
357
72 25704
216
410
360
504
504
0
NO emphasis on place value (place value is not explicit in this method)
Repeated subtraction division
Here are two different looks at
72 25704
357
72 25704
72
25632
72
25560
72
25488
7200
18288
14400
3888
1440
2448
2160
288
288
0
done via repeated subtraction division.
357
72
72 25704
1
72
25632
1
72
25560
1
72
25488
100 7200
18288
200 14400
3888
20 1440
2448
30 2160
288
4
288
0
357
25704
72
1
25632
720
1
24912
720
1
24192
7200
100
16992
14400
200
2592
2160
20
432
432
30
0
4
72 25704
1
72
25632
10
720
24912
10
720
24192
100 7200
16992
200 14400
2592
30 2160
432
6
432
0
1
10
10
100
200
30
6
357
Emphasis on place value (place value IS explicit in this method); cannot “guess” every
value correctly or you end up back at the standard algorithm; helpful for students with
place value problems and trouble estimating
3.3.2. Now try this – try each of the following problems using each method of multiplication
discussed.
 23 x 96
 34 x 525
 68 x 62
3.3.3. Division algorithms
3.3.3.1. Standard division algorithm
3.3.3.2. Repeated subtraction – see example from class
3.3. Ongoing Assessment p. 153
3.3.1. Home work: Practice each of the methods, as appropriate, on the following
 87 x 290
 15 x 48
 62 x 51
 305 x 93



4524 x 522
14624  32
31562  43