Objectives:
1. The teacher will understand how to create single unit models (models for 1)
that can be used to
model more than one fraction at a time. The teacher will do this as a
preliminary step to modeling
the addition or subtractions of fractional numbers .
2. The teacher will explore adding and subtracting fractions using wooden
blocks, Geoboards and
Cuisenaire Rods.
3. The teacher will use appropriate fraction termino logy and appropriate
manipulative terminology
when describing how to modeling fraction addition and subtraction with
manipulatives.
Terms and Ideas to Know
• Two fractions are LIKE FRACTIONS if they have the same denominator.
To add or subtract like
fractions we add or subtract the numerators .
Examples
i. [1/5] + [2/5] = [3/5] (3 = 1 + 2)
ii. [3/5] – [1/5] = [2/5] (2 = 3 – 1)
• Two fractions are UNLIKE FRACTIONS if they do not have the same
denominator. To add or
subtract unlike fractions we must first rewrite them with common denominators .
Examples
i. [1/3] + [1/4] = [4/(12)] + [3/(12)] = 7/(12)
ii. [1/3] - [1/4] = [4/(12)] - [3/(12)] = 1/(12)
• A COMMON DENOMINATOR for two unlike fractions is a new denominator
that both fractions
share. The common denominator for two fractions is also a common multiple of the
two
denominators . All unlike fractions can be rewritten with many different common
denominators.
• The easiest way to find one common denominator is to just multiply the
(unlike) denominators.
Examples:
i. One common denominator for 1/3 and 1/4 is 12 since [3 x 4 = 12].
ii. One common denominator for 1/3 and 1/6 is 18 since [3 x 6 = 18].
iii. One common denominator for 1/6 and 1/8 is 48 since [6 x 8 = 48].
• Although, we can always find a common denominator by multiplying the two
(unlike) denominators,
we prefer to find the Least Common Denominator (or LCD). For two unlike
fractions, the LCD is
the smallest of all common denominator. The LCD of two fractions is also the LCM
of the two
denominators . Unlike fractions have only least common denominator.
Examples:
i. One common denominator for 1/3 and 1/4 is 12 and, in fact, this is the Least
Common
Denominator for 1/3 and 1/4.
ii. One common denominator for 1/3 and 1/6 is 18. However, since [1/3] = [2/6],
6 is also a
common denominator, and, in fact, is the Least Common Denominator for 1/3 and
1/6.
iii. One common denominator for 1/6 and 1/8 is 48. However, since [1/6] =
[4/(24)] and [1/8] =
[3/(24)], 24 is also a common denominator, and, in fact, is the Least Common
Denominator for
1/6 and 1/8.
TOPIC: ADDITION AND SUBTRACTION OF FRACTIONS
•Materials: WOODEN CUBES
1.
a. Suppose one of your students comes to you and says: “Look at these cubes, I
think I have figured our
fractions. This shows that [1/2] + [1/3] = [2/5]!” (The arrows indicate which
blocks they are
pointing to.)
 |
 |
| This block is 1/2 |
This block is 1/3 |
Together these blocks are 2 of 5
blocks. Since you said fractions are
part of a whole.
[1/2] + [1/3] must be 2/5! |
As a group:
• Determine what your student did incorrectly
and
• Briefly discuss ideas for the correct solution path and jot notes here. Don't
model the solution path,
this will be explored in the rest of this lab.
b. Suppose your student now comes to you and says: “OK, now I get the idea of
“This is One” I think I
have figured our fractions now! This shows that [1/2] + [1/3] = [4/6]!” (The
arrows indicate which
blocks they are pointing to.)
These 6 cubes are 1
As a group:
• Determine what your student did incorrectly
and
• Briefly discuss ideas for the correct solution path and jot notes here. Don't
model the solution path,
this will be explored in the rest of this lab.
End note for #1:
• The idea behind this problem is to get you to re-think the importance of
setting the model for 1 and
to think about carefully using the model for 1.
• To help keep track of “What’s 1?” while modeling fraction operations it helps
to always “set aside”
(separate from the modeling of the fractions and the operation) the model for 1.
As a group, for each
problem in this lab, please use your “This is One” circle to “set aside” your
model for 1.
2. As a group, use wooden blocks and work through these steps to model the
addition of [1/3] + [1/4].
[1/3] + [1/4] = ?
a. We need to model 1/3 and we need to model 1/4, that is, we need to model
thirds and fourths first.
Neither of these models seems like it will work for BOTH thirds and fourths.
So this is NOT the correct
place to start. These two models are shown here to show that they DON'T WORK and
should not
usually be part of your modeling. Let’s try this:
b. Now model 1/3 and model 1/4 separately to make sure your model for 1 works
here. Set your model
up so that 1/3 looks like 1 of 3 parts and 1/4 looks like 1 of 4 parts
c. Now we will model 1/3 and we will model 1/4 together—and we will be
careful to keep the models
disjoint (non-overlapping). (Circle and label both 1/3 and 1/4 here.)
Looking at the
model for one we
see: The value of
each block is: _____
d. We can now count our blocks, look at our model for 1 (still in our "This
is One" circle) and see that
[1/3] + [1/4] = _______________.
3. As a group, use the wooden blocks to model and solve the following. Draw
clear , well-labeled
pictures of your work. (Don’t forget to use your “This is One” circle)
[1/6] + [3/8] = _____ ?
The value of each
block is: _____
4. Fraction Addition Guide (Wooden Cubes)
As a group write a brief summary of the steps that you need to show to clearly
model fraction
addition. Use your previous work as a guide and check to make sure you have all
of the steps that
a student would need to fol low your procedure . Use the terms addend and sum.
5. Practice Using Your Guide:
As a group, use the wooden blocks to model the following. Draw clear,
well-labeled pictures of your
work. (Don’t forget to use your “This is One” circle)
[2/3] + [1/6] = _____ ?
The value of each
block is: _____
6. As a group, explore how to use the wooden blocks to model [2/3] - [1/6] =
?
Draw clear, well-labeled pictures of your work here. (Don’t forget to use your
“This is One” circle)
The value of each
block is: _____
7. As a group write a brief summary of the steps that you need to show to
clearly model fraction
subtraction. Use your previous work as a guide and check to make sure you have
all of the steps that
a student would need to follow your procedure. Use the terms minuend, subtrahend
and difference .
Fraction Subtraction Guide (Wooden Cubes)
8. Practice Using Your Guide:
As a group, use the wooden blocks to model the following. Draw clear,
well-labeled pictures of your
work. (Don’t forget to use your “This is One” circle). To model 4/3 you will
need two copies of the
model for 1.
[4/3] - [1/6] = _____ ?
The value of each
block is: _____
