Transcript Fractions, Part 1

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
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
Three-fourths 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 5.8
Part 1; #1-7
Part 2; #1-3
Part 3; #1-3
Part 4; #1,2
Use the manipulatives for all
parts.
Contexts for fractions
• Measure or Part/Whole
• Quotient
• Operator
• Ratio
Figure 5.10
Equivalency
Equi
equal
-
valent
value
5 10  2  52
Equivalent Fractions
Factions are equivalent if they have equal
value. In other words, they are equivalent
if they represent the same quantity.
Equivalent Fractions
Fraction strip
The strip is your whole.
Label fractional parts of this whole
What are the equivalent fractions?
Figure 5.20
Figure 5.21
Comparing Fractions
Which one is bigger?
3
4
2
3
6
15
11
24
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 (one of these is
not possible. Which one and why?)
Part 2, and Part 3
Test 3
• Thursday, April 15th
• Covers Number Theory and Fractions
Textbook: 4.1, 4.2, 4.3, 5.2, 5.3
Explorations: 4.2, 4.3, 4.5, 5.8, 5.9, 5.10,
5.12, 5.13, 5.14
Children’s Thinking Videos
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 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?
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.
3
8
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
• Textbook problems: pp. 282-283: 5,
12b,d, 14, 22, 30
• Exploration 5.10: #1, 2, 3 Include
justifications and discussions with
partner(s)