Content Session # 7 & 8

July 9, 2009

Which problem can we use our whole number multiplication knowledge to solve?• 1m of wire weighs 1.4 lb. How much will 6m

of the same wire weigh?

• 1m of wire weighs 6 grams. How much will 1.4 m of the sa,e wore weigh?

Double Number Line

?

6

1.40

0 1Length (in meters)

Weight (in grams)

Unitary Perspective of Numbers

• We have 6 groups of 14 0.1 grams.• 14 x 6 = 84; Altogether, we have 84 0.1 grams.• 84 0.1 grams 8.4 grams• 1.4 x 6 = 8.4

weight (in 0.1 grams)14

weight (in grams)

length (in meters)

?

6

1.4

10

0

Which problem can we use our whole number division knowledge to solve?• 4 m of iron pipe weighs 3.6 kg. How much will

1m of the same pipe weigh?

• 3.6 m of iron pipe weighs 9 kg. How much will 1m of the same pipe weigh?

Double Number Line

? 3.6

4

Weight (in kg)

Length (in meters)

10

0

Unitary Perspective of Numbers

• Divide 36 0.1-kg to make 4 equal groups.• 36 ÷ 4 = 9; Each group will have 9 0.1-kg.• 9 0.1-kg = 0.9 kg.• 3.6 ÷ 4 = 0.9

weight (in 0.1 grams)36

weight (in grams)

length (in meters)

3.6

4

?

10

0

What if we had 3.7 ÷ 4 mean?

• 37 0.1-kg: make 4 equal groups• BUT,

37 ÷ 4 = 9 rem. 1

• Each group will get 9 0.1-kg and there will be 1 0.1-kg left over.

• 3.7 ÷ 4 = 0.9 rem. 0.1

Dividing on: 3.7 ÷ 4

• Model 3.7 ÷ 4 using base-10 blocks – use a flat as 1.

• What will be left over?• Can we trade it? With what?

What is 8 ÷ 5?

Multiplying by Decimal Numbers

• 1m of wire weighs 6 grams. How much will 1.4 m of the same wire weigh?

weight (in grams)

length (in meters)

?

1.4

6

10

0

Unit Approach

0.114

weight (in grams)

length (in meters)

?

1.4

6

10

0

1.4 meters means 14 0.1 meters.So, we have 14 groups of weight of 0.1 meter.Weight of 0.1 meter = 6 ÷ 10 = 0.6Weight of 1.4 meters = 14 x 0.6

Make-It-Bigger ApproachWhat if we were looking for the weight of 14 meters of the wire?Weight of 14 meters = 14 x 6 = 84 grams14 meters are 10 times as long as 1.4 meters, so it must also weigh 10 times as much as the weight of 1.4 meters of the wire.Weight of 1.4 meters = 84 ÷ 10 = 8.4 grams

14

weight (in grams)

length (in meters)

?

1.4

6

10

0

Dividing by Decimal Numbers3.6 m of iron pipe weighs 9 kg. How much will 1m of the same pipe weigh?

weight (in grams)

length (in meters)

?

3.6

9

10

0

Unit Approach

0.1

weight (in grams)

length (in meters)

?

3.6

9

10

0

3.6 meters means 36 0.1 meters. So, if we divide 9 by 36, we get the weight of 0.1 meter of the wire. To get the weight f 1 meter of the wire, we have to multiply by 10.9 ÷ 36 = 0.25; 0.25 x 10 = 2.5 grams

Make-It-Bigger Approach

36

weight (in grams)

length (in meters)

?

3.6

9

10

0

36 meters of the same wire will weigh 10x9=90 grams.To find how much 1 meter will weigh, you just have to divide 90 by 36.

Area Model of Multiplication

• The two factors are represented by the two dimensions of a rectangle.

• The product is represented by the area of the rectangle.

• Represent 24 x 13 using base-10 blocks.

24 x 13 or 2.4 x 1.3

Multiplying Fractions by Whole #

You can paint 3/5 sq. feet with 1 pint of paint. How much can you paint with 2 pints?

210

Multiplying Fractions by Whole #

You can paint 3/5 sq. feet with 1 pint of paint. How much can you paint with 2 pints?

210

Dividing Fraction by Whole #You can paint 4/5 sq. feet with 3 pints of paint. How much can we paint with 1 pint?

3210

Dividing Fraction by Whole #You can paint 4/5 sq. feet with 3 pints of paint. How much can we paint with 1 pint?

3210

Multiplying by FractionsWith 1 pint of paint, you can paint 4/5 square feet. How much can we paint with 2/3 pints?

12

30

Multiplying by FractionsWith 1 pint of paint, you can paint 4/5 square feet. How much can we paint with 2/3 pints?

12

30

Dividing by FractionsWith 3/4 pints of paint we can paint 2/5 sq feet. How much can we paint with 1 pint?

13

40

Dividing by FractionsWith 3/4 pints of paint we can paint 2/5 sq feet. How much can we paint with 1 pint?

13

40

Area Model of Multiplication

3

4

Area Model of Multiplication

2

3

3

4

Area Model of Multiplication

2

3

3

4

Measurement Division

You need 2/3 cups of sugar to make 1 loaf of bread. If you have 8 cups of sugar, how many loaves of bread can you make?

Measurement Divisionwith Drawing

You need 2/3 cups of sugar to make 1 loaf of bread. If you have 8 cups of sugar, how many loaves of bread can you make?

Measurement Divisionwith Drawing

You need 2/3 cups of sugar to make 1 loaf of bread. If you have 8 cups of sugar, how many loaves of bread can you make?

Measurement Divisionwith Drawing

You need 2/3 cups of sugar to make 1 loaf of bread. If you have 8 cups of sugar, how many loaves of bread can you make?

Measurement Divisionwith Pattern Blocks

You need 2/3 cups of sugar to make 1 loaf of bread. If you have 8 cups of sugar, how many loaves of bread can you make?

Partitive Divisionwith Cubes and Drawing

3/4 pounds of coffee cost $6. How much does a pound of the same coffee cost?

$ 6

Partitive Divisionwith Cubes and Drawing

3/4 pounds of coffee cost $6. How much does a pound of the same coffee cost?

$ 2$ 2$ 2

Multiplication Makes Bigger?

1yd of cloth costs $3. What is the price of 11/3 yd of the same cloth? What is the price of 2/3 yd?

? ?

11

32

3

10

0

Division Makes Smaller?1 1/3m of thin wire weighs 12 g and 2/3m of thick wire weighs 12g. How many g does 1m of each weigh?

12

0

01

1

3

12?

?

2

310

0