Proper fractions The value of the numerator is less than the value of the denominator. Proper in...

Proper fractions

The value of the numerator is less than the value of the denominator.

Proper in this case does not mean correct or best.

Improper fractions

• The value of the numerator is greater than or equal to the value of the denominator.

Mixed numbers

• Meaning of

3

25

Writing mixed numbers as improper fractions

• The algorithm that is taught in schools obscures the meaning. This is true for many algorithms, which are “efficient” ways of carrying out operations.

Write mixed number as improper fraction and vice versa

312

4

23

Operations with fractions

• Addition

• Subtraction

• Multiplication

• Division

Adding and subtracting fractions

1/2 + 1/3

Multiplying fractions

• Repeated addition model

• Area model

Multiplication of fractions

• Fraction as operator

• The multiplication algorithm is best explained by the area model.

2/3 of 2 1/2

Mixed number times mixed number

Dividing fractions

• Division of fractions is most easily understood as repeated subtraction.

212

11 divided by 1 1/2

Multiplicative Inverses

• We know that division is the inverse of multiplication.

52

110

5210

Multiplicative inverses

• The multiplicative inverse of a is 1/a

• The multiplicative inverse of a/b is b/a

11

a

a

1a

b

b

a

Dividing fractions

• Because division is the inverse operation of multiplication, dividing a number by a fraction is equivalent to multiplying the number by the multiplicative inverse, called the reciprocal, of the fraction.

Exploration 5.12

• “Drawn to scale”

• Part 1 Use reasoning not algorithms to answer #1

• Part 2 Choose a model from the list that was not represented in the problems and make up a story problem using the fraction ¾. Are there any models that are not possible with fractions? Explain.

Operations with fractions

• Addition

• Subtraction

Operations with fractions

• Multiplication

Operations with fractions

• Division

Exploration 5.13

• Begin in class and finish for homework: 5.13 Part 1: #2-7 Part 2: Choose one of the models from the list that was not illustrated in the problems in Part 1 and write a story problem using the fraction ¾. Also, are there any models that are not possible with fractions? Explain.

• Homework problems from the textbook:pp. 303-305: 3b,d,e,f, 13, 21, 22, 25Note that in #3, you should not use algorithms to calculate the result; use reasoning to decide the answer to the question.

Extra Practice

• 1. You have from 10:00 - 11:30 to do a project. At 11, what fraction of time remains? At 11:20, what fraction of time remains?

• Use a diagram to explain how you know. Are there certain diagrams that are more effective? Discuss this with your group.

Extra Practice

• 2. Is 10/13 closer to 1/2 or 1?

• Use a diagram to explain how you know. Are there certain diagrams that are more effective? Discuss this with your group.

Extra Practice

• 3. If a/b = 3/4, will the value of (a + x)/(b + x) be less than, equal to, or greater than 3/4.

• Use a diagram to explain how you know. Are there certain diagrams that are more effective? Discuss this with your group.

Exploration 5.14

• Read the directions carefully and do #1

• Discuss with your partner

• Do #2

• Discuss with your partner

• Do # 3

Homework for Tuesday

• Exploration 5.15

• Read section 5.3 in your textbook

• Do problems pp. 305-307: 30, 31, 33, 36, 41, 44a,c,d,h,k, 45a, 48