Math Proof

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad
has a total of 280 pieces of fruit. There are twice as many raspberries as blueberries,
three times as many grapes as cherries, and four times as many cherries as
raspberries. How many cherries are there in the fruit salad?

-Our first step we need to do is split our fruit into groups, what we know is that there are
a total of 4 groups:

1.Blueberries 2.Raspberries 3. Grapes 4. Cherries

-Second, we need to examine our word problem and pull the key details that we know
from it so we can apply it to these four groups.
What we know is that:
*There is a TOTAL of 280 pieces of fruit, so we know that
Blueberries+Raspberries+Grapes+Cherries=280
And remember for this problem we want to find out the total number of cherries,
but to do so we need to find how many fruits there are in each group.

But how do we know what numbers we should apply in our groups?

-Third we look back at our word problem to see if we have any information on these
groups. Let's look at our word problem again and now underline key details we can use:

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad
has a total of 280 pieces of fruit. There are twice as many raspberries as blueberries,
three times as many grapes as cherries, and four times as many cherries as
raspberries. How many cherries are there in the fruit salad?
-Fourth, what do we learn from these sentences?

twice as many raspberries as blueberries

From this first sentence we know that the total number of raspberries is twice the total
number of blueberries. Which in a equation can look as so:
Raspberries(total)= Blueberries(total) X2

Now we will examine the next sentence:

three times as many grapes as cherries

What this sentence tells us is that the total number of grapes is equal to three times the
total number of cherries. We can now also write this in an equation:
Grapes(total)=Cherries(total) X3

Now we will look at the third sentence:

four times as many cherries as raspberries

What this tells us is that the total amount of cherries is four times the total amount of
raspberries. We can also write this in an equation:
Cherries(total)=Raspberries(total) X4

-Fifth, we will write these equations on our chart we created in step one:

1.Blueberries 2.Raspberries 3. Grapes 4. Cherries

As highlighted, the division and multiplication of the same number cancels eachother
out. This leaves us with our final equation of: R/2= B

-Sixth, What do we do with these equations?

Well, we are going to plug them into each other so we can solve for the number values
of each group, so let's look at all those equations together and our total sums equation

Blueberries+Raspberries+Grapes+Cherries=280

↑ ↑ ↑ ↑
B=R/2 R= B X2 G=C X3 C=R X4

By looking at this you can see we can’t plug them all in here or the numbers will still be
a big mess! The next step we have to take is comparing the four equations together and
plugging them into each other first. Let’s look for some equations that share the same
variable first so we can find them and create a new equation!

-Step seven, we are going to plug two equations that share the same variables into
each other. What we are going to do first is plug our raspberries equation into our
cherries equation, These two equations being:

R= B X2 C=R X4

By combining equations we are going to create new, What two variables do you see
both these equations share……….
The both share an R
So this means we are going to plug our first equations into the second one where we
see R, as the first equation is the value of R

C=R X4
↑
R= B X2
This now creates our new equation:
C=(B X2) X4
⇡ ⇡
We can use the distributive property to simplify our
equation and combine and the 2 and the four, by multiplying them together

This makes our final equation now……

I have highlighted this equation so that we can remember that we will need to find it and
use it later again.

-Step eight would now be creating one last new equation.

Again we will examine our chart and decide on the two equations to combined:

The only equation we have not used yet is our grape equation: G=C X3

To create a new equation I am going to combined this with our *red and blue equation*
I have chosen to do this as our red and blue equation created a new value of C, and
rather than being left with 3 variables if we plugged this into our cherry equation we will
only be left with two variables if we plug the grape equation into our red and blue
equation.

We want to focus on simplifying our equations and not making them longer.

C=8B ← Our red and blue equation

G=C X3 ← Our grape equation

G=C X3
↑
C=8B

Plugging this in will give us:

G=(8B)X3
↑ ↑

We will then use the distributive property once more to combine the 8 and the 3 by
multiplying them together. Now forming this final equation:

G=24B We can call this our purple and blue equation

We are now going to use these equations to solve our overall total equation.

-Step 9, plug it all in to find the different fruit group totals!

Blueberries+Raspberries+Grapes+Cherries=280

What we need is to plug in equations which will keep our variable value low
We will plug in our raspberries equation →R= B X2
Our red and blue equation → C=8B
And our purple and blue equation →G=24B

By plugging these in we will only have B variables which means we will solve for our
total blueberries first: Blueberries+Raspberries+Grapes+Cherries=280
↑ ↑ ↑

R= B X2 G=24B C=8B

This new equation will be:

B+2B=24B+8B=280

We will combined all B’s 35B=280

/35 /35 And divide by 35 to simplify

-Step 10, plug blueberry value into other equations to solve.

The equations left to plug into:

R= B X 2 → R= 8 X 2 → R= 16 Raspberries

G=24B → G= 24 X 8 → G= 192 Grapes

C=8B → C=8 X 8 → C= 64 Cherries

*This gives us our final answer that there are a total of 64 cherries*

=64

To check our answer we can plug our totals into our total sums equation

Blueberries+Raspberries+Grapes+Cherries=280
↑ ↑ ↑ ↑

8 16 192 64

This gives us 280=280 which means our answers were correct and our math proof
was successful!

8 16 192 64 =280
 Reflection

How does going through this exercise help you to better teach the concept of Math
Proofs?

By completing this math proof it has helped me see how important attention to detail is
when teaching math. Of course looking at this problem we can see in our brains how to
plug it in, but to explain it to someone who has never done it before takes a lot of
thought and consideration. Even when teaching how to solve this short math problem
which may take someone 5 minutes to solve, it would take us 30 minutes to maybe
even an hour to complete, as you could imagine there would be so many questions from
younger aged students when we actually are teaching this.

What struggles did you encounter?

Some problems I encountered were really thinking about how I can explain my decision
to make that step and why. For example, when I began plugging in equations into each
other to create new equations I really had to think of my reasoning for this as I realized
many students would ask well why didn’t I plug my blueberry equation into my raspberry
one as they share the same variables?

I had to really think about this for a minute, as I knew that wouldn’t be the right step. So
I had to backtrack in my math proof and see how I could explain this to them. In prior
steps I created my blueberry equation by using my raspberry equation and transforming
it. I would then show them this step in my math proof so they can see I did not plug
them in as they are the same equation.

I think I would say my biggest struggle was having to think about all the questions I
would get asked while demonstrating this math proof, and trying to account for all of
them but being concerned I may not have noticed a few of them.

How can you help your students overcome their struggles?

To help students overcome their struggles I think it is very important to be very detailed
in your math proof. If a student becomes lost or confused during the lesson, the math
proof can be used to show them further reasoning and allow them to take a few steps
back and really digest the information they are learning.
Also after seeing how much time would be invested in conducting this math proof, in
order to try to lower stress and confusion in the classroom I believe it would be best to
work at a slow place and limit the number of math proofs used each day to 1 or 2.