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U n t er r i ch t spl a n
F rac t io n R id d l e s Like
De no minat o rs
Altersgruppe: 4 t h Gr ade , 5 t h Gr ade
Texas - TEKS: G3 .7 .GM .D, G3 .7 .GM .E , G4 .8.GM .C
Riverside USD Scope and Sequence: 4 .N F .3 d [4 .6]
Oklahoma Academic Standards Mathematics: 4 .N .2.4
Virginia - Mathematics Standards of Learning (2009): 3 .7 , 4 .5 b,
4 .5 d, 5 .6
Common Core: 4 .N F .B .3 d
Mathematics Florida Standards (MAFS): 4 .N F .2.3 d
Alaska: 3 .M D.2, 3 .M D.A , 4 .M D.2
Minnesota: 5 .3 .2.2, 5 .3 .2.3
Fairfax County Public Schools Program of Studies: 3 .7 .a.5 , 4 .5 .b.1,
4 .5 .b.2, 4 .5 .b.3 , 4 .5 .b.4 , 4 .5 .d.1, 5 .6.a.1, 5 .6.a.2, 5 .6.a.4
Nebraska Mathematics Standards: M A .4 .1.1.j , M A .4 .1.1.l
South Carolina: 3 .M DA .2, 4 .M DA .2
Indiana: 2.M .4 , 3 .M .1, 4 .M .3
Georgia Standards of Excellence: M GS E 4 .N F .3 d
Virginia - Mathematics Standards of Learning (2016): 4 .5 .b, 4 .5 .c ,
5 .6.a
Online-Ressourcen: P o ur Y o ur se l f I nt o i t
Opening
T eacher
present s
St udent s
play
Class
discussion
20
12
12
5
min
min
min
min
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ZIE L E :
Experience addition and subtraction of fractions and mixed
numbers with liquid volume as the medium
Practice arithmetic involving whole numbers, fractions, and
mixed numbers
Learn to connect addition and subtraction of fractions with like
denominators to addition and subtraction of whole numbers
Develop strategies for arriving at a specific value using
addition and subtraction but that do not depend on guessing
Ope ni ng | 20 min
Present the following scenario to your class: A chocolate bar is
made up of 12 separable pieces (shown below).
It may be helpful to use a manipulative that you can split into
pieces.
Ask the class: What fraction of the chocolate bar am I eating if I eat
1 piece?
Ideally, the class will come to the conclusion fairly quickly that
you are eating 1/12 of the chocolate bar.
If some students are still struggling with the basic idea of
fractions, try to use this as a teaching point, saying: I am eating
1 piece out of a possible 12 pieces in the whole, which means 1
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is the numerator and 12 is the denominator.
Ask the class: What fraction of the chocolate bar am I eating if I eat
5 pieces?
Again, the class should arrive at 5/12 fairly quickly.
Ask the class: What fraction of the chocolate bar am I eating if I eat
6 pieces?
The immediate response, based on the previous two answers, may
be 6/12.
By this point, the idea of equivalent fractions should be
somewhat familiar to your students, so you can ask them if there
is another way to think about this fraction.
If your students appear to have any difficulties, consider
showing a vertical split down the middle of the bar and laying
one half on top of the other (or just pointing out that the two
groupings are the same size).
Ask the class: What fraction of the chocolate bar am I eating if I eat
12 pieces?
Some students may say: You are eating the whole thing!
This type of student response is actually perfect, despite the fact
that it does not answer your question with a fraction.
Respond with: Yes, we did eat the whole chocolate bar. What
fraction does that whole represent?
Here, you are trying to connect the idea of a whole (1) being
represented by the form , where is a whole number for the
purposes of this lesson.
You need not jump to a generalization, but part of the aim of
this lesson is to use this fact.
Specifically, conclude that 12/12 = 1.
Move to a scenario where you are splitting the chocolate bar into
two “chunks.”
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For example, say: You want to share this chocolate with a friend. What are
some ways you could split up the chocolate bar?
After any suggestions, relate the response to fractions.
A common response is to “split the bar in half.”
Follow up this answer with the question of how much each
person receives.
Answers may vary, depending on whether or not the fractions are
simplified, but try to start with 6/12 and 6/12.
Connect this to the sum: 6/12 + 6/12 = 12/12 = 1.
You can also consider this as a difference: 12/12 - 6/12 = 6/12.
The difference can be seen as “giving away” 6/12 from the
original 12/12, thus be left with 6/12.
Encourage less egalitarian splits, perhaps by adding a comment
such as: What does it look like if your friend only wants 2 pieces?
Again, connect the number or pieces with the fraction of the
chocolate bar that each person gets (2/12 and 10/12 here).
Follow this with either the sum 2/12 + 10/12 or the difference
12/12 - 2/12, relating the number sentence to the actual scenario
as appropriate.
Finally, consider scenarios wherein more than one chocolate bar
(whole) is used.
For example, say that you went trick-or-treating and collected 4
chocolate bars.
Ask your students: How can we think about 4 in terms of fractions?
The success of this hinges on the strength of your students’
understanding of 1 whole chocolate bar being 12/12.
Start with the idea that 4 = 1 + 1 + 1 + 1. Ask your class: How did we
write 1 as a fraction before?
This should lead to the idea that 4 = 12/12 + 12/12 + 12/12 +
12/12.
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The challenge of this is to avoid a conclusion like 4 = 48/48.
While many students can comfortably add and subtract fractions
with numerator smaller than the denominators, sums of this type
of cause problems. Try to reiterate that the denominator stays
the same throughout.
Thus, the conclusion should be that 4 = 48/12, or that 1 piece is
1/12, so from 4 bars we have 48 pieces, or 48/12.
Essentially, you can point out that you are referring to the same
general whole within the denominator!
Once your class feels some comfort with this idea, present a
modification, such as: You ate 8 pieces. How much chocolate is
left?
The phrasing is purposefully a little vague here. Ideally, your
students will be able to answer this in more than one way.
One response could be that you had 48/12 and ate 8/12, so this is
the same as having 48/12 - 8/12 = 40/12 left.
Many students may be better served by backing up from 4 = 48/12
for a moment and simply looking at how much of one bar is left if
8 pieces are eaten.
This process starts with 12/12 - 8/12 = 4/12. Then, your
students should remember that you still have 3 full chocolate
bars. So, the quantity can be represented by 3 + 4/12, which can
be written as a mixed number, another perfectly acceptable
answer with a slightly different interpretation within the
scenario (whole bars instead of pieces).
Regardless of the route taken, develop a connection between the
mixed number and the fraction representations.
Continue to a scenario wherein you have eaten more than one
chocolate bar, or where you wish to share more than one chocolate
bar with a friend.
This will involve more addition and subtraction, but now both
values will be mixed numbers (which, again, can be thought of as
fractions).
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The concepts will be identical, but you will have appropriately
laid the groundwork for the episode presented in this lesson.
T e ac he r pr e se nt s P o ur Y o ur se l f i nt o I t - A dd L i ke
De no mi nat o r s | 12 min
Present Matific ’s episode P o ur Y o ur se l f i nt o I t —A dd L i ke
De no mi nat o r s to the class, using the projector. The goal of this
episode is to use addition and subtraction with given quantities in
order to reach a specified amount.
The major change from previous episodes of Pour Yourself into It
is that this episode uses quantities that are mixed numbers.
Each screen begins with two or three jugs, each of which can be
filled with water by dragging the jug to the spigot.
Note that jugs can be emptied by dragging them to the plant.
In the example below, the two given quantities (jug volumes) are 1½
and 1 (where volume is in liters).
Note: The examples mentioned in this portion of the lesson plan
can be found using presentation mode.
The specified amount that must be reached is ½ (liter).
Thus, the class will need to figure out how to arrive at ½ using
some or all of the values 1½ and 1.
Note that this amount will also need to end up in the correct jug
(here, the 1-liter jug).
E x a m p le :
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In the example above, there are likely several ways to arrive at the
correct result.
At first, it is fine to encourage a more exploratory approach, but try
to push the class toward a more analytical approach to solving
these problems as the lesson progresses.
While we are trying to get ½ liter of water in the 1-liter jug, the
equivalent abstract question is: How can we make ½ from 1½ and
1?
Some students may notice that 1½ - 1 = ½. Ask them how that
equation can be used to reach the goal.
As shown below, this equation can be equivalent to filling the
1½-liter jug, then using this water to fill the 1-liter jug.
E x a m p le :
From here, we have the desired quantity of ½ liter. However, it is
not in the correct jug.
Empty the 1-liter jug and refill it with the ½ liter that remains in the
1½-liter jug. Thus, we will have ½ liter of water in the 1-liter jug.
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As mentioned previously, this is just one solution. Your students
may find others, especially if they are taking an exploratory
approach at the beginning.
Encourage your students to tie in arithmetic with the solutions
they find.
Consider the following alternative approach: fill the 1-liter jug,
pour its contents into the 1½-liter jug, refill the 1-liter jug, again
pour its contents into the 1½-liter jug (only ½ liter will pour in,
completely filling the 1½-liter jug), then have ½ liter of water
remaining in the 1-liter jug, as desired.
This solution should be connected to the equation 1 + 1 – 1½ =
½. This connection may not be obvious to your students at
first, but building this bridge will help them for future volume
riddles!
The example below involves three jugs, which means that there are
vastly more possibilities for the process. Again, you can approach
this loosely, but the primary aim should be establishing repeatable
strategies based in sound arithmetic principles.
Consider starting by asking your students: Can we make from ,
, and ?
There is not an obvious solution to this question. While many
starting points exist, one possible route is to consider
changing the mixed number into the fraction , and changing
to .
Doing so allows you to view the problem as: Can we make 3
from 6, 5, and 4?
This is because we are now adding and subtracting fractions
with like (common) denominators.
E x a m p le :
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Taking the above-mentioned approach might yield the idea that 5 +
4 – 6 = 3. In terms of the given jugs, this equates to
.
Thus, fill the 1-liter jug and the -liter jug with water, then attempt
to pour both into the -liter jug. The first jug you pour will empty
completely, but the second will only pour enough to fill the -liter
jug, leaving of a liter, as desired.
Again, remind your class that other solutions will exist in most
cases.
Even using the above solution, you may find that you need not
change mixed numbers into fractions, provided your class is
comfortable working simultaneously with whole numbers,
fractions, and mixed numbers.
Because there are many solutions, encourage the unique thinking
and processes that your students come up with.
Ask them to explain why their process works, again trying to
move away from mere guessing.
Additionally, challenge them to find more concise routes to the
solution, since many solutions will have extraneous steps or
repeated processes.
Most importantly, challenge your students to find the connection
between the process of filling (and pouring) jugs and arithmetic.
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S t ude nt s pl ay P o ur Y o ur se l f i nt o I t —A dd L i ke
De no mi nat o r s | 12 min
Have the students play P o ur Y o ur se l f i nt o I t —A dd L i ke
De no mi nat o r s on their personal devices.
Circulate, answering questions. Continue to develop creative,
analytical strategies that are based on arithmetic involving
fractions (and mixed numbers).
Encourage your students to write down the arithmetic that
corresponds to the filling and emptying of their jugs. This can
help keep track of the amounts in partially-filled jugs.
Consider having your students work in pairs, so that they can share
and compare strategies, as well as come up with more efficient
solutions.
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C l ass di sc ussi o n | 5 min
Discuss any challenges the students faced while working
individually.
Ask the class for responses as to how they dealt with any
common issues their classmates brought up.
Remind the class that the exercises in the episode are certainly
not simple—especially because of the added visual element—but
that having comfort with addition and subtraction of fractions
can help a great deal.
If a sizable portion of the class still appears to be struggling with
adding and subtracting mixed numbers, whole numbers, and
fractions, consider working through some examples without a visual
or scenario.
While connecting concepts to the real world is a nice way to get
your students to buy in to the importance of an idea, learning the
basic structure of the arithmetic is often simpler without the
added complexity of a real-world connection.
Some examples to consider (in roughly increasing order of
difficulty) are:
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