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How Much Pie?
Task
After a class potluck, Emily has three equally sized apple pies left and she wants to divide them into eight equal portions to give to eight students who want to take some pie home.
 Draw a picture showing how Emily might divide the pies into eight equal portions. Explain how your picture shows eight equal portions.
 What fraction of a pie will each of the eight students get?
 Explain how the answer to (b) is related the division problem $3 \div 8$.
IM Commentary
The purpose of this task is to help students see the connection between $a \div b$ and $\frac{a}{b}$ in a particular concrete example. The relationship between the division problem $3 \div 8$ and the fraction $\frac{3}{8}$ is actually very subtle.
$3 \div 8$ is the number you multiply 8 by to get 3.
$\frac{3}{8}$ is the number you get by taking 3 copies of the unit fraction $\frac18$.
So $3 \div 8$ is defined in terms of multiplication, and $\frac{3}{8}$ is defined in terms of unit fractions. It is not obvious that these two numbers are the same, so students need opportunities to see that they will necessarily always be the same. Note that if $b$ people share $a$ pies equally, then each person will get $\frac{a}{b}$ of a pie by the same kind of reasoning shown in the solution below.
This task is probably best suited for instruction or formative assessment.
Attached Resources
Solution

Below is a picture of how Emily might divide the three apple pies into eight equal portions. Here each color (yellow, orange, red, green, blue, dark purple, light purple, and light blue) represents one portion. So each portion consists of three pieces of pie and each piece of pie represents $\frac{1}{8}$ of a full pie:
Because these pies are all the same size and they are all apple pies, Emily does not need to give each student one piece of each of the three pies: two or three pieces of the same pie could go to one student. This picture, however, shows clearly that the pies have been divided into eight equal portions. If multiple pieces of a particular pie were to go to the same student, it would be necessary to analyze the picture more closely and count how many slices of pie each student received to check that it has been divided evenly.

As the picture shows, each portion consists of three slices of apple pie. Since these slices represent $\frac{1}{8}$ of a pie, this means that each student gets $\frac{3}{8}$ of an apple pie.

If 3 pies are divided into 8 equal portions, then 8 of these portions makes 3 pies, a fact that is clearly illustrated in part (a). We can write this in symbols if we use a question mark to represent the amount of pie in one portion: $$8 \times ? = 3$$ When we know a factor and the product, we can find the other factor by dividing: $$3 \div 8 = ?$$ So one person's portion is whatever we get when we divide 3 by 8. In part (b), we saw that one portion is $\frac38$. So that means that $$3 \div 8 = \frac38.$$
How Much Pie?
After a class potluck, Emily has three equally sized apple pies left and she wants to divide them into eight equal portions to give to eight students who want to take some pie home.
 Draw a picture showing how Emily might divide the pies into eight equal portions. Explain how your picture shows eight equal portions.
 What fraction of a pie will each of the eight students get?
 Explain how the answer to (b) is related the division problem $3 \div 8$.
Comments
Log in to commentlhwalker says:
over 5 yearsFantastic! As a high school math teacher, I am so impressed with the conceptual depth I have taken for granted all my life. Two suggestions:
a) A possible intermediate step 3. might read, "3. If the students do not arrive on this conclusion on their own, ask how they would share one pizza with eight students."
Rationale: we do not want to train the students to wait for us to tell them the answer. It is best to ask strategic followup questions to "pull" responses from the students.
b) On 5. the word "simplify" is used and I am thinking we will be using "reduce"
Kimberly says:
over 5 years"Reduce" is now considered an inaccurate term for showing fractions in simplest form. Please consider transitioning to using "simplify" instead. It is important to use precision in our terminology.
lhwalker says:
over 5 yearsThanks for clarifying. I was going by some of the statements like page 4 of Expressions, "The standards avoid talking about simpliﬁcation, because it is often not clear what the simplest form of an expression is, and even which form is “simpler”" Is there an official explanation somewhere about the word "reduce?" I am surprised by that as eliminating a common factor seems best described as a reduction.
Kristin says:
about 5 yearsThis is an excellent topic of conversation. The problem that some people have had with using the word "reduce" is that some students think this means the value of the fraction is less (because it is "reduced," as in a sale) when in fact it is an equivalent fraction and has the same value.
On the other hand, there is a potential problem with "simplify" as wellwhat is simpler in one case may or may not be simpler in another. Sometimes the fraction 6/8 is simpler than 3/4, for example, when you want to add it to 5/8.
I have heard some people advocate for using the phrase "lowest terms," but I don't know that there is a single descriptive phrase that will always be clear to students until they are very proficient with equivalent fractions in all their forms. In the end, I think what matters most is that we realize that the language we use to describe equivalent fractions can be confusing, largely because the concepts are deep and difficult, and that we need to be prepared to explain what we are asking for, and why we are asking for it, from different perspectives.