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FRACTIONS I
HANDBOOK
Desert Willow Family School
1
* Fractions slides can be obtained online at www.hand2mind.com under the name Picture Grids
Fractions I: Level 1 – Understanding Fractions on the Number Line
Display on Board
-∞
-1
0
Teacher’s Questions
∞ 1. How far does the number line go
1
Student’s Answer
1. To infinity.
to the right?
2. How far does the number line go 2. To negative infinity.
to the left?
0
1
½
with ½.)
½
Numerator = AMOUNT…..1
Denominator = NAME of the group...2
0
⅛
1
16
¼
4.The bottom number is the denominator, which shows how many parts
the whole is broken into. In our example, that would be two. The top
number is the numerator, which
shows how many of the parts we have.
5. Between 0 and 1.
1
5. Where are fractions located on
the number line? (Only consider
fractions that are not mixed numbers or improper fractions)
6. Yes.
½
1
6. Can you split the space between
zero and one in half? Can you split
that space in half? Again? Again?
½
1
7. How many fractions are there
between zero and one?
7. Infinity.
8. When can you stop dividing into
8. Never
0
0
3. Where are fractions? Since we 3. (Multiple possible answers;
are not dealing with negative num- If no response, ask: “Do
bers right now, we will only work
you know of any fractions?”
between 0 and 1.
Most students will answer
smaller pieces?
2
Fractions I: Level 1 – Understanding Fractions on the Number Line
Display on Board
0
Teacher’s Questions
½
0
¼
1
2
4
3
4
Student’s Answer
9. There is something important to
notice here…if we are talking about
4
4
fourths, the number line is equal
divided into 4 parts.
10. Notice that ½ has the same
0
½
(continued)
1
place as
2
4
10. An infinite number
. These are equivalent
fractions. If we break our number
0
1
2
3
4 5 6 7 8
8
8
8
8 8 8 8 8
¼
2
4
3
4
4
4
line into eight equal pieces, we get
eighths….how many ways are there
to say one half?
11. Look at the ones place…How
many ways are there to say one?
11. An infinite number
Same value….and infinite names….
there are an infinite ways to say ½:
2
4
½ = 4 = 8 Also, infinite number
of ways to say 1:
2
2
=
3
3
=
4
4
100
= 100 =
∞
∞
All these ways to write the number
1 are called “Names
3
of One.”
Fractions I: Level 2 – Adding/Subtracting Fractions with LIKE Denominators
Display on Board
Teacher’s Questions
Student’s Answer
1. What operations have we already 1. Adding, subtracting, multilearned to do with whole numbers? plying and dividing
2. Let’s start with adding… what do 2. Combining the numbers;
we know about adding?
putting together; getting a total
1/5 + 2/5
3. Can you get the fraction slides
out that represent these fractions
and add them using what we know
1
2
5
5
about adding?
+
3
4. What is the answer? (Lead the
5
=
1
5
1
5
1
5
4.
3/5
student to combine the slides to
display the answer by laying the
slides on top of each other.)
5. What do you notice about what
1
5
+
2
5
=
3
5
you do mathematically when you add
fractions?
5. You only add the numerators.
6. Why?
6. Because the denominator is
just the NAME of the whole,
the numerator is how many we
have of that whole... the AMOUNT.
4
1
+
apples
2
apples
=
3
apples
7. Yes, we don’t add the denominator
because it functions like a “NAME,” but
what is important to note is because they
have the SAME NAME, we can add the
numerators….
7. Add Apples to Apples!
(Note: Subtraction is performed in the same way as addition except that you
subtract the numerators, and the denominator acts as the NAME and does not
change.)
Fractions I: Level 3 – Adding/Subtracting Fractions with
Display on Board
Teacher’s Questions
Student’s Answer
1. Get out your fraction tiles for this
problem and see if you can solve it….
1
3
+
1
2. Didn’t we learn the first day of
fractions that there are many names
for each fraction? Could we write down
all the names and match them up, so we
can add them?
2. Yes
1
1
3
2
3. However…we do know how to do
something to a number to change it,
but not change its value….
Another discovery we had was there
are infinite names for 1….Let’s use
as our “Name of One”.
x
3
1
3
1. I can’t solve it because they
don’t have the same denominator;
they have different names; they
are different types of fruit….
=
2
+
1
UNCOMMON Denominators
2
2
=
3. Multiply a number by 1….
2
2
2
6
4. What do we know about multiplication?
4. It is an area with 2 dimensions…an over and and up.
5
area =
1
1
3
x1
1
5. We don’t want to change the value
3
of the fraction, just the name…so
2
as shown by the fractions slide
2
same area as
1
3
x
2
2
=
2
1
3
2
6
has the
…same value, same name,
…..EQUAL VALUE….
6
5. Equivalent fractions!
Fractions I: Level 3 – Adding/Subtracting Fractions with UNCOMMON Denominators
(CONTINUED)
Display on Board
Teacher’s Questions
Student’s Answer
(Focus should be on the conceptual understanding of changing the
name of the fraction, during teaching of multiplication of
fractions, students will be shown to multiply across.)
1
+
3
1
=
2
6. So let’s return to our problem….
What denominator do you want both
of these fractions to have?
6. (Let student choose a number)
Let’s try 5 ……(a common guess
because five is two plus three)
7. What “Name of One” do you want 7. (Let student choose a name of
to multiply 3 by to get a 5 in the
one. It usually takes about 2
denominator?
errors to see what will work.)
1
?
3
1
3
+
(?) +
1
2
1
2
?
8
8
(?) =
=
Oh, five won’t work because I
don’t get the same denominator…
(Student needs to discover that it
needs to be a multiple of BOTH
numbers and doesn’t need to be
lowest at this time.)
?
5
Let’s try six….because I can
multiply both fractions by a name
6
(
of one
2
2
and
3
) to get a six
3
in the denominator….
1
3
2
1 3
?
2
2
6
( )+
( ) =
3
(Now show using slides.)
+
2
3
+
6
6
=
8. Now can add the fractions?
8. Yes, the answer is
5
6
(After further practice, proceed with subtraction, using the same process as addition.)
Fractions I: Level 4 – Finding a Lowest Common Denominator (LCD)
Display on Board
Method 1: List the Multiples
Teacher’s Questions
Student’s Answer
1. Let’s start with the largest denominator….
Let’s list the multiples for the largest denominator until you get one that the other
denominator will go into….
2
5
1
+
3
=
5: 5, 10, 15
Stop…3 goes into 15
Fifteen will be your common denominator
Method 2: Prime Factorization (used with larger denominators)
1
84
+
1
96
84
2
42
7
6
=
1. We begin by prime factoring each denominator
and boxing in all primes.
96
3 32
4 8
7
2
3
2 · 7 ·2 · 3
2 2 24
2 2
3 · 2 · 2 · 2 · 2 · 2 (Note: multiplying ALL primes produces the denominator)
Cross out any prime factors that are shared…..and bring down
only once. (By crossing out the shared prime factors, you
reduce the math because crossing it out takes out that
common factor which gives a lower common denominator.)
84
96
2
42
3 32
2. So we have two 2’s in common and one 3……
7
6
4 8
cross out and bring down ONLY ONCE….
2 3
2 2 2 4
2 2
2x 2x 3
Fractions I: Level 4 – Finding a Lowest Common Denominator (LCD)
Display on Board
Teacher’s Questions
(continued)
Student’s Answer
3. Now all the rest of the primes
drop down….
84
2
7
42
2
6
3
96
3 32
4 8
2 2 24
2 2
2 x 2 x 3 x 7x 2 x 2 x 2
4. You multiply all these prime factors together to get your
lowest common denominator (LCD).
5. What do you get when you multiply
these together?
..…that is your LCD.
….672
6. We can use this box of primes to help find
the “Name of One” to multiply each fraction by….
8
2 x 2 x 3 x 7x 2 x 2 x 2
7. Notice that if you cross out the primes that multiply
to get 84, which are 2, 7, 2, and 3, the rest of the
numbers multiply out to the “Name of One” to multiply
1
1
8
84
8
+
84
1
?
96
by…..
which is 2 x
=
2x 2=?
8. Now what is the “Name of One” you will use for
1
2 x 2 x 3 x 7x 2 x 2 x 2
1
8
84
8
+
8
7
96
7
7
+
672
1
672
7. Eight
1
96
8. We do the same, cross out
the primes that multiply to get
96….which are 3,2,2,2,2…
7
=
so our “Name of One” is
= ____
9. Now we can add the fractions…..
9. So the answer is
Fractions I: Level 5 – Multiplying Fractions
Display on Board
1
x
3
1
Teacher’s Questions
=
2
?
7
15
672
Student’s Answer
1. What do we know about multiplication?
1. It is an area problem. You need
an over and an up….
2. Can you get out the slides that
represent this problem?
(Notice that one slide is rotated 90° )
x
1
1
3
2
3. Lay them on top of each other to
show the intersecting area representing this problem…
1
3
9
1
1
6
2
4. What fraction does the intersection
of the two areas represent?
1
3
x
1
1
=
2
4.
1
6
…..the answer!
5. Notice you are just multiplying
6
across…but the answer is smaller.
Why? Let’s look back at the slides…
1
3
1
6
1
2
6. With whole numbers 2 x 3, which is 2 “groups of” 3
Do we have ½ groups of 3 ?
We have:
½ “part
of””
1
3
½ of a third….it is a “PART
or
½ of
6. No
1
3
of a PART”
(As a demonstration, give the student half piece of paper and ask for 1/3
of it…..the paper gets smaller!)
Fractions I: Level 6.1 – Multiplication Word Problems
Display on Board
1
x
3
1
2
Teacher’s Questions
1
=
3
½
1
x
3
1
2
Student’s Answer
of the school is first graders.
of the first graders are boys.
Q: What fraction of the whole school
is first-grade boys?
1
A.
3
10
1
6
1
2
1
6
Fractions I: Level 6.2 – Simplifying Fractions
Display on Board
Teacher’s Questions
Student’s Answer
(Avoid the word “reduce” because it sounds like we are changing the amount,
but the amount is never changed.)
3
8
3
15
4
x
x
=
15
4
8
1. We can simplify fractions before
we multiply….because of the commutative
property of multiplication where order
doesn’t matter…so we can rewrite our
equation as…
=
1
5
1
x
2
so each fraction can now be SIMPLIFIED.
1
1
3
8
x
4
=
15
2
1
x
2
1
5
5
However, we don’t even have to
rearrange the fraction just cross-simplify.
Fractions I: Level 7 – Division of Fractions with Conceptual Understanding
Display on Board
Teacher’s Questions
Student’s Answer
(Division of fractions are the first time students will see that a
bigger number can go into (be divided by) a smaller number.)
1
6
÷
2
3
=
1. Using your fraction slides, can you
tell me which fraction is larger?
“divide by”
1. 2/3
2. This is important to understand about
dividing fractions is that you can divide
a bigger fraction into a smaller, so we
need to understand how to read this equation.
11
1
6
÷
2
3
=
3. We read this equation as 2/3
“into”
goes into 1/6 because we can’t make
sense of “divide by” so we use “into.”
We also see equations written like this…
2
3
“into”
“divide by”
1
We will need to know how to read
division of fractions both ways.
6
4. Now let’s use our fraction slides to
help us understand this better…..
1
6
1/6 into 2/3
“into”
A smaller fraction into a larger…
2
so we ask “HOW MANY” fit?
3
Think of a smaller envelope fitting into
a larger envelope…how many will fit in?
4. Four!
4
4
Did you notice the answer is a whole number?
When you divide a small fraction INTO a larger
fraction, the answer is always GREATER THAN ONE.
The answer can be a whole number OR a whole number and a fraction.
Fractions I: Level 7 – Division of Fractions with Conceptual Understanding
Display on Board
Teacher’s Questions
Student’s Answer
We can also divide a larger fraction
into a smaller fraction….
2
3
÷
1
6
=
“into”
1. Using your fraction slides, can you
tell me which fraction is smaller?
2. Now we are dividing a smaller fraction
into a larger fraction…how do we read
12
(continued)
1. 1/6
this equation?
A bigger fraction into a smaller….
2.
1/6 into 2/3
4. Now let’s use our fraction slides to
help us understand this better…..
2
3
2/3 into 1/6
“into”
1
A larger fraction into a smaller…
so we ask “HOW MUCH” fits?
6
Think of a larger envelope fitting into
smaller envelope…how MUCH will fit in?
4.
1/4
(Notice that the answers are inverse 4 and ¼)
jkjk
1
4
Did you notice the answer is a fraction?
When you divide a larger fraction INTO a smaller
fraction, the answer is always LESS THAN ONE,
a fraction….only a piece of the larger fraction
can fit INTO the smaller fraction.
(Note: To help with the visual representation of dividing fraction INTO each other, pick “friendly” fractions.
Friendly fractions contain denominators that are multiples of each other. For example, 1/5 ÷ 1/3 aren’t friendly
fractions because 5 and 3 are not multiples of each other, and the slides don’t allow you to see the division
conceptually. However, 1/5 ÷ 1/10 contain friendly fractions because 5 and 10 are multiples, which allow us to
conceptually see the answer as 2.)
…Now that we conceptually understand
what is happening when we divide fractions,
let’s see how to divide fractions mathematically….
Fractions I: Level 7 – Division of Fractions with Conceptual Understanding
Display on Board
Teacher’s Questions
6. What is another way to write our
division problem?
2
3
÷
1
6
=
….so let’s rewrite it like this….
13
(continued)
Student’s Answer
6. As a fraction.
2
3
=
….this is a very ugly fraction….it is complex!
1
7. We could solve this problem, but it is not
6
the most efficient method. Is there anything we could do to eliminate the divisor
(denominator)?
What number would the bottom fraction
need to be equal to in order for us to be
allowed to get rid of it?
7. One?
8. How can I get the denominator (divisor)
to equal one?
8. Multiply it by its reciprocal.
2
3
1
6
x
6
1
= 1
9. Now we have changed the problem, so
what can we do so we aren’t changing the
problem?
2
3
1
6
2
3
X
x
x
6
9. Multiply the top by
6
1
10. Isn’t this the same as multiplying by a NAME of ONE?
1
6
…So what happens when we multiply out this complex fraction?
1
10. 1/6 x 6/1 is one!
6
11. What do you notice about solving a
1
=
12
3
= 4
division of fractions problem?
11.
2
3
÷
1
6
is the same thing
1
as
2
X
6
3
1
(Note: Students should be able to demonstrate why you multiply by the inverse before using the division “short-cut” of
“just flip and multiply to get your answer.”)
Fractions I: Level 8 – Division Word Problems
Display on Board
Teacher’s Questions
There are three different types of
division word problems:
1. CONTAINED IN …. where we are
14
Student’s Answer
asking “How many are contained in this…?
“How many can I make out of this….?
“How many fit into this…?
Example:
9
I have
of a yard of fabric. I
10
am making an art project that requires
1
of a yard of fabric. How many art
15
projects can I make?
2. AREA PROBLEM….knowing that division
is the inverse of multiplication, turn the
multiplication problem of L x W = AREA
into a division problem…AREA ÷ L = W
or AREA ÷ W = L
Example:
I own a rectangular field that measures
square mile. If one border measures
9
10
of a
1
of a mile, what is the dimension of
10
the other side?
2. MISSING FACTOR PROBLEM….similar
to the area problem, in that you know the
answer and one factor to the multiplication
problem and you are trying to find the
other factor…. (the fraction slides are useful in this type of problem)
Example:
1
3
1
Red
and
Spck.
2
So the multiplication problem would be…. 1/3 of the fish in the
aquarium are red. If ½ of the red fish are speckled, what fraction
of the fish in the aquarium are red and speckled.
1
3
1
6
x
÷
1
2
=
1
3
1
6
Now start with the answer…. 1/6 of all the fish in the aquarium are red
and speckled. If 1/3 of all the fish are red, what fraction of the fish
=
1
2
in the aquarium are speckled?
15