Download document

Exploration 5.8
Part 1; #1-7
Part 2; #1-4
Part 3; Choose one of the
manipulatives to do #1,3
Part 4; Choose one of the
manipulatives to do #1,2
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.
3
4
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
• Another example:
– 7/20 of my paycheck goes to my car payment.
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:
WWWM
WWWM
…
• We can compare parts- women:men=3:1
• We can compare parts to wholewomen: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 5.9
In #1, you are given three rectangles, same
size and shape. Use three different ways
to divide the rectangles into equal pieces.
Work on this in class and finish for Monday.
• If the given diagram has a value of 2 ½,
show 1.
• If the given diagram has a value of ¼,
show 1/3.
Contexts for fractions
• Measure or Part/Whole
• Ratio
• Operator
• Quotient
Figure 5.10
Happy
Halloween!
Equivalency
Equi
equal
-
valent
value
5 10  2  52
Equivalent Fractions
• Fractions are equivalent if they have equal
value.
Equivalent Fractions
Comparing Fractions
• Which one is bigger?
3
4
2
3
6
15
11
24
Meanings for fraction: 2/5
• Part-whole: subdivide the whole into 5
equal parts, then consider 2 of the 5 parts.
discrete
area
Length (number line)
0 1/5 2/5 3/5 4/5 1
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-blue
2:3
Or
2 blue : 5 total
2:5
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
1
you respond to
1
2
this answer?
2
• Use a diagram to explain how you know. Are
there certain diagrams that are more effective?
Discuss this with your group.
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?
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.