Something to Think About

What does a student need to know before they can understand how to multiply two decimals?

e.g. What does 0.2 x 0.3 mean?

Secondary Numeracy Project

Multiplication,Division, and

Algebra

Making Chocolate Bars Cadbury is making a variety of rectangular

chocolate bars. They have different numbers of pieces in them, with different shapes.

For example, their 6-piece chocolate bar can have two shapes:

A 6 by 1 bar

and a 3 by 2 bar

Task One Work out and draw the different

shapes Cadbury can make for chocolate bars with

a) 8 piecesb) 12 piecesc) 18 pieces

18 Piece Candy Bars

Task Two Write down all the different shapes Cadbury

can make for chocolate bars with:

a) 24 pieces b) 16 pieces

c) 30 pieces d) 36 pieces

e) 40 pieces f) 48 pieces

Task Three Copy and complete this table for the number of pieces in

chocolate bars with these lengths and widths:

87654321 9

1

2

3

4

5

6

7

8

9

Length of chocolate bar

Widthof

chocolate bar

Times Tables

000000000 0

16141210864 18

242118129

16 322824

15

20

40353025 45

484236

5649

64

27

36

54

63

72

81

876543210x 109

0

1

2

3

4

5

6

7

8

9

10

87654321 9

2

3

4

5

6

7

8

9

0

0

0

0

0

0

0

0

0

6

128

201510

30241812

4235282114

564840322416

72635445362718

0

100

10

20

30

40

50

60

70

80

90

80706050403020 90

Times Tables - Hot Spots

000000000 00

16141210864 18

27

16 36

15

20

40353025 45

54

63

242118129

322824

484236

5649

64 72

81

2

3

4

5

6

7

8

9

10

0

0

0

0

0

0

0

0

0

0

876543210x 109

0

1

2

3

4

5

6

7

8

9

10

87654321 109

20

30

40

50

60

70

80

90

6

128

201510

30241812

4235282114

564840322416

72635445362718

80706050403020 90

Algebra in Multiplication

Computing using the distributive law Recognising structure in numbers Exploring algebraic expressions

7 x 98

What’s the answer?

How did you do it?

7 x 98 = 7 x 100 - 7 x 2

597 x 5

What’s the answer?

How did you do it?

597 x 5 = 600 x 5 - 3 x 5

407 x 8

What’s the answer?

How did you do it?

407 x 8 = 400 x 8 + 7 x 8

True or False?

3 x (400 - 8) = 3 x 400 – 3 x 8

597 x 11 = 600 x 11 - 3 x 7

915 x 8 = 900 x 8 + 15 x 8

7 x 98 = 7 x 100 - 7 x 2

True or False?

12 x 40 – 8 x 12 = 8 x (40 - 12)

79 x 11 = 70 x 11 + 11 x 9

80 x 7 - 5 x 7 = 85 x 7

8 x 300 - 7 x 8 = 8 x 293

What’s missing?

5 x (70 - 2) = 5 x 70 – 5 x 2

6 x 600 + 7 x 6 = 6 x 607

58 x 13 = 50 x 13 + 8 x 13

80 x 6 - 6 x 6 = 74 x 6

Fill in the Gaps

5(y - 2) = 5y - 10

6(m + 7) = 6m + 42

7( m + 9 ) = 7m + 63

8c - 48 = 8(c - 6)

Expand

5( y + 6) = 5y + 30

7x - 70 = 7( x – 10 )

4( 2m + 3 ) = 8m + 12

18r - 54 = 6( 3r – 9 )

Factorise

3 ( n + 2 ) = 3n + 6

8( x – 4 ) = 8x - 32

20p + 15 = 5 ( 4p + 3 )

7( 6 - 5y ) = 42 - 35y

a(b ± c) = a b ± ac

Recognising Structure

Algebra in Long Multiplication

Using arrays to illustrate multiplication

Exploring polynomial expansions Touching factoring

5 x 21

What’s the answer?

5

20

1

14 x 21

What’s the answer?

10 4

20

1

23 x 21

What’s the answer?

20

20

3

1

23 x 21

Abstraction #1

20

20

3

1

23 x 21

Abstraction #1

20

20

3

1

400 60

20 3

400 + 80 + 3 = 483

23 x 21

Abstraction #2

20

20

3

1

400 60

20 3

400 + 80 + 3 = 483

Link to the Algorithm

20

20

3

1

400 60

20 3

23 x 21 3 20 60+400 483

(x+3)(x+2)Expanding Expressions

x

x

+3

+2

x2 3x

2x 6

x2 +5x +6

(x-3)(x-2)

Expanding Expressions

x

x

-3

-2

x2 -3x

-2x 6

x2 -5x +6

x-3x x-2 6

-2x -3xx2 x2-5x+6

Algorithm?

x

x

-3

-2

x2 -3x

-2x 6

Notice that both diagonals have the same product:

-2x x -3x = 6x2

x2 x 6 = 6x2

Leading to Factorising

x

x

-3

-2

x2 -3x

-2x 6

X2-5x+6

Factorising Expressions

x2 -5x

6

Sum:

Product:

6x2

-3x -2x-3x

-2x

X2-5x+6

Factorising Expressions

x

x

-3

-2

x2 -3x

-2x 6

Now factor each pair

X2-5x+6 = (x-2)(x-3)

Multiplying Decimals

5

6

0.2

0.3

30 1.2

1.5 ?

When do we introduce students to 5.2 x 6.3?Not until they know how to multiply fractions

Tips for Using T&T Books Planning Guide (p 10 on) Division (p 122 on) Learning Intention Linked knowledge/Numeracy books Materials needed Colour coded questions (p 3)

Now YOU!

Choose Multiplication/Division Algebra (year 9 or 10)

Plan a unit for your students Use HIBS Planning Notes, T & T

Books, purple books