Math PD Assignment Katti
Wachs
The problem:
Grandma had made pies for a bake sale. She had carefully put
equal amounts of mixture in each pie tin and was now trying to find the weight
of the pies. She had a problem; she only
had one 200 gram weight and one 125 gram weight. She found that one pie
balanced on the scale with both weights and a quarter of the pie. How heavy was
each pie?
I selected this problem from the NRICH website. Although the problem is categorized as
appropriate for 4th graders, it seemed to me to be a sufficiently challenging
problem to work with.
Part1 . Planning:
This problem seemed to me to involve several
"cognitively challenging tasks." The problem is multi tiered in complexity. It
is a word problem, and as such, it requires the student to decide what is
relevant and necessary information by analyzing the problem, and to identify
the unknown variable (the weight of the pie), and to "reason abstractly
and quantitatively" in order to come up with a way to figure out the
answer. The problem also seems to require students to rely on some familiarity
with proportions, fractions and or algebra in order to solve it. Furthermore,
the problem seems to meet the Common Core Standard of being
"real-life" in the sense that it involves cooking, a practice shared
by many of us. That being said, I am not quite certain why it would be
necessary to figure out the weight of the pies after they had been prepared.
Also, if I were to use it with my adult students, I would probably change "Grandma"
to someone's name so as not to sound so childish. I might also change the units from grams to
ounces, for the terms to be more familiar and comfortable to the students
My Solution:
I began solving this problem by drawing it out because I am
more of a visual learner. I realized
pretty quickly that it did not matter that there were 2 pies, because the
problem was only asking us the weight of one, so I labeled and set up an
equation:
PIE
= 200 WEIGHT + 125 WEIGHT + 1/4 PIE
My next step was that I added together the two weights, so I
came up with this:
PIE=
325 WEIGHT + 1/4 PIE
At this point, I was able to seen clearer that the pie was
going to end up weighing more than the 325 grams on the scale. I then realized that I had set up some sort
of algebraic equation and that "PIE" and "1/4 PIE" might be
too confusing to work with, so I changed the equation to:
X=
325 + 1/4 X
I knew I wanted to find a way to combine the X variables, so
I subtracted 1/4X from both sides and came up with:
3/4
X = 325
I wanted to isolate the X, so I multiplied both sides by
4/3. The first time I did the multiplication, however, I forgot to carry the 2,
so I came up with 1280 as my answer.
Instead of checking my multiplication, I continued on with the problem
and tried to divide 1280 by 3. I came up
with 426.6 as my answer. I thought it
was a bit weird that the answer would not be a whole number but I tried to plug
it in:
426.6 = 325 + 1/4
(426.6)
After plugging in the numbers, I realized that that equation
did not work, so I started again with
X = 325 grams + 1/4 X
I followed the exact same steps but this time I ended up
with 1300 for 4 x 325, and I divided
that by 3, and got 433.3 grams.
I plugged that number in:
433= 325 grams + 1/4 (433)
I finally came up with 433=
325 grams + 108
Alternative Solutions
1. A more proportional
representation
A student might recognize that, if the problem is asking us
to think in terms of 1 quarter of a pie, it makes sense to represent the full
pie in quarters as well. One might set up the equation something like this.
à + à + à + à = 200 grams + 125 grams + à
(one quarter of a pie)
Then, subtract the quarter of the pie from the right side,
and from the left so that you are left with : à + à + à = 325
grams
Then, divide both sides by 3, in order to get the value of 1
piece of pie
à + à + à = 325
grams
________________________________ _____________
3 3
Finally, à = 108.3 grams
2. Working with percentages and reasoning
Given the information, we can assume
that 325= 75% of the pie, or 75/100
Thus, the
remaining quarter of the pie= 25%, is equal to 25/100
If we
divide the 75 % into 3, we will see that 25% = 108.3
Thus, the
pie (100%) is 325 (75%) + 108.3(25%)
3. Guess and check
A student could start to solve this problem by guesstimating
as well. He or she would know that the
pie weighed somewhat more than the 325 grams. She might start with a round
number, above 325 grams, that is easy to divide into quarters. A number like 400 grams would work well. She
might write out something like this
Pie = 400 grams
A quarter of the pie= 100 grams
Does a 400 gram pie =
325 + 100 grams? No it doesn't. No, the
pie seems to be too light.
Perhaps she might then try a higher number like 500 grams,
also divisible by 4.
Does a 500 gram pie = 325 + 125 grams? No, the pie seems to
be too heavy.
To me, this seems to be a much more challenging approach,
but one that eventually might lead to the answer.
4. Chart method
If the guess and check method was represented in a chart, I
suppose, one could pursue it a little more systematically:
|
Guessed pie weight
|
Quarter of a pie
|
Quarter of a pie + 325 grams
|
Conclusion
|
|
400
|
100
|
425
|
Total pie is too light
|
|
500
|
125
|
450
|
Total pie is too heavy
|
|
450
|
112.5
|
325+112.5=
437.5
|
Getting closer to even
|
Questions to help the
students along
I would start by asking
students:
What do we know about the total weight of the
pie?
> We know that the pie weighs more than 325 grams
> We know that the pies are an equal weight. We are using
one to help us find the weight of another.
> We know that one pie weighs 325 grams plus a quarter of
the weight of the other pie.
> We know that, if the above points are true, the 325
grams represent three quarters (or 75%) of the weight of each pie.
What do we know about the missing extra weight of the
pie?
>We know that it is equal to 25% of the total pie
If we are missing 25% of the total pie weight which, when
added to 325 gram equals total pie weight, what can we assume about the 325
grams?
What percentage of
the total pie weight does the 325 grams represent?
Challenges students might face solving this problem
a) Some students might have difficulty with their
calculations during the problem, the way that I did. Many of the ways to solve this problem
require a student to divide or multiply larger numbers and or
fractions/percentages. I would encourage students to check their math once they
believe they have found a solution.
b) I believe that students without any background in algebra
and the manipulation of variables might struggle with setting up this
problem. I guess I would try to make
sure that, whatever system they chose to use, they understood that the
proportional relationship between the missing weight, and the total weight of
the pie, as well as the idea that the pie must weigh more than 325 grams.
c) Students might struggle to know where to begin. I might encourage them to begin by labeling
and drawing any information they think might be important in figuring out the
solution.
I might also encourage them to start out with the guess and
check method. To begin modeling it, and also by giving them a more tangible
task, I might ask them something like: Could the pie weigh 600 grams? Why or why not?
What do I want students to get from working on this
problem/which math practices am I hoping to develop?
I think that this problem requires a certain amount of
reasoning and perseverance from the
student, without being totally daunting in its level of challenge. It provides the opportunity for students to
represent the problem algebraically, though it does not require it. It does
seem to encourage proportional thinking, as well as to place students in a
position where, even if using guess and check, they have to draw some
conclusions in order to progress.
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