Fractions and Rational Numbers

• A rational number is a number whose value can be represented as the quotient or ratio of two integers a and b, where b is nonzero.

• A fraction is a number whose value can be expressed as the quotient or ratio of any two numbers a and b, where b is nonzero.

• How do rational numbers and fractions differ?

Terminology

• The number above the horizontal fraction line is called the numerator.

• The number below the horizontal fraction line is called the denominator.

Contexts for rational numbers

• There are 4 contexts or meanings for fractions.

• Let’s look at the different contexts in which this rational number has meaning.

4

3

Rational number as a measure

• Sylvia grew ¾ of an inch last year.

• We have some amount or object that has been divided into b equal amounts, and we are considering a of those amounts.

• Keep units.

• Diagram: length model

Rational number as a quotient

• 4 people want to share 3 candy bars equally. How much candy does each person get?

• We have an amount a that needs to be shared or divided equally into b groups.

• We should see a per at the end.• In this problem 3/4 candy bar per person tells us

how much each person gets.• Diagram: Area model

Rational number as operator

• ¾ of my shirts are blue. If I have 12 shirts, how many blue shirts do I own?

• a/b is a function machine that tells us the extent to which the given object or amount is stretched or shrunk.

• If we have 12, ¾ is telling us to take 3 out of every four.• Diagram: Discrete Model.

Rational number as a ratio

• At a college ¾ of the students are women. • A ratio is a relationship between two quantities.• Diagram:

W W W MW W W M

…• We can compare parts- women:men=3:1• We can compare parts to whole-

women:total=3:4

The unit and the whole

• Not always the same!• The whole is the given object or amount. The unit is the

amount to which we give a value of one.• Tom ate ¾ of a pizza.

– Whole is 1 pizza– Unit is 1 pizza

• Sylvia grew ¾ of an inch last year.– Whole is ¾– Unit is 1 in.

• 4 people want to share 3 candy bars equally. How much candy does each person get?– Whole is 3 candy bars.– Unit is 1 candy bar.

• ¾ of my shirts are blue. If I have 12 shirts, how many blue shirts do I own?– Whole is – Unit is

• At a college ¾ of the students are women.– Whole is – Unit is

Exploration 5.8

Part 1; #1-7

Part 2; #1-3

Part 3; #1-3

Part 4; #1,2

Contexts for fractions

• Measure or Part/Whole

• Ratio

• Operator

• Quotient

Figure 5.10

Equivalency

Equi - valent

equal value

522105

Equivalent Fractions

Factions are equivalent if they have equal value. In other words, they are equivalent if they represent the same quantity.

Figure 5.20

Figure 5.21

Comparing Fractions

Which one is bigger?

4

33

2

15

624

11

Exploration 5.8

Part 1; #1-7

Part 2; #1-3

Part 3; #1-3

Part 4; #1,2

Exploration 5.9

In #1, you are given three rectangles, same size and shape. Use three different ways to divide the rectangles into equal pieces.

Assignment:

Exploration 5.9 Part 1; #1-6 (skip 5c),

Part 2, and Part 3

Meanings for fraction: 2/5

• Part-whole: subdivide the whole into 5 equal parts, then consider 2 of the 5 parts.

0 1/5 2/5 3/5 4/5 1

discrete

area

Length (number line)

Meanings for fraction: 2/5• Ratio: a comparison two quantities. In this case, the first

quantity is the number of parts/pieces/things that have a certain quality, and the second quantity is the number of parts/pieces/things that do not share that quality.

2 blue : 3 non-blue2 : 3

Or2 blue : 5 total2 : 5

Meanings for fraction: 2/5

• Operator: instead of counting, the operator can be thought of as part of a set, or a stretch/shrink of a given amount.

• If the fraction is less than one, then the operator shrinks the given amount. If the fraction is greater than one, then the operator stretches the given amount.

• 2/5 of a set might mean take 2 out of every 5 elements of a set. It might also mean 2/5 as large as the original.

Meanings for fraction: 2/5

• Quotient: The result of a division.• There are two ways to think about this. • We can think of 2/5 as 2 ÷ 5: I have 2 candy

bars to share among 5 people; each person gets 2/5 of the candy bar. If students do not understand this model, then much of algebra will be less meaningful. What is π / 6? How do you solve x/3 = 12?

• We can also think of this as the result of a division: I have 16 candy bars to share among 40 people: 16/40 = 2/5.

Alexa

• We will watch Alexa in just a few moments. Look at her work on page 23 for the second problem. What do you think Alexa was doing in each diagram?

• Now pick one. Can this picture be used to mean Part-whole? Ratio? Operator? Quotient? If yes, say why. If no, say why not.

• Class Notes pp. 22-23

Now, we’ll watch Alexa

• Complete parts 1, 2, and Follow-up Question Part a in your groups.

• Now, try Part c (skip b for just a moment). Once you found the value for 5, explain it in words so that someone else will understand.

• Now, let’s try part b. Same thing--once you get an answer, try to write it up in words so that someone else will understand.

Part d

Extra Practice

• 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

• 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

• A teacher asks for examples of fractions that are equivalent to 3/4. One student replied:

• How would you respond to this answer?

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

11

2

2

Alexa’s problem

• Children’s thinking p. 22 b.

Which contexts can be explained using one of her diagrams?

Which ones cannot?

Why or why not?

CTA Yamalet Follow-up Questions

Expressing the fraction• As a ratio: 3 pieces of pizza have

mushrooms and 5 pieces of pizza do not have mushrooms

• As an operator: water splashed on the pizza and now only 3/8 of the pizza is edible. This does not have to be 3 slices but a portion of the pizza that has the same area as 3 pieces. The amount of pizza that can be eaten has shrunk.

• As a quotient: My family has 3 pizzas to share among 8 people. So, each person gets 3/8 of a pizza.

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3

Sean

• This one is long. Listen carefully to the language used by Sean and the language used by the teacher.

Sean’s work

• P. 29 not enough room for answers especially for part c. Use the back of the previous page.

• P. 33

Language in Mathematics

• Sometimes mathematics is called a language. It isn’t really but it has a lot of its own vocabulary, as well as symbols. Because these symbols are used universally across languages, it is sometimes called the “universal language”.

Language in Mathematics

• Words in math have a precise meaning.

• It is important for a teacher to use clear and correct language in talking about math—if the teacher is vague and imprecise, students have great difficulty learning concepts.

Watch David

Exploration 5.10

• #1 Compare the fractions in the table on Page 111. Indicate which is larger, smaller (or if they are equal) without using common denominators or converting to decimals (no calculators). Write an explanation of how you came up with your answer.

• #2&3 Discuss your reasoning with a partner and see if you can come up with some general rules for comparing fractions.

Homework for Wednesday

• Textbook problems: pp. 282-283: 5, 12b,d, 14, 22, 30

• Exploration 5.10: #1, 2, 3 Include justifications and discussions with partner(s)