Lesson 3
How Many Groups?
3.1: How Many of These in That?
 We can think of the division expression \(10 \div 2\frac12\) as the question: “How many groups of \(2\frac 12\) are in 10?” Complete the tape diagram to represent this question. Then find the answer.
 Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then find the answer.
3.2: Representing Groups of Fractions with Tape Diagrams
To make sense of the question “How many \(\frac 23\)s are in 1?,” Andre wrote equations and drew a tape diagram.
\(\displaystyle {?} \boldcdot \frac 23 = 1\)
\(\displaystyle 1 \div \frac 23 = {?}\)

Andre wasn’t sure how to deal with the remainder.
 Diego says, “The answer is \(1\frac13\), because the remainder is \(\frac13\) of the rectangle.”
 Jada says, “I think the answer is \(1\frac12\). Since we want to find out ‘how many \(\frac23\)s there are,’ we should compare the leftover part to a group of \(\frac23\). The remainder is \(\frac12\) of a group.”
Do you agree with either of them? Explain or show your reasoning.

Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer.
 How many \(\frac 34\)s are in 1?
 How many \(\frac23\)s are in 3?
 How many \(\frac32\)s are in 5?
3.3: Finding Number of Groups

Write a multiplication equation or a division equation for each question. Then, find the answer and explain or show your reasoning.

How many \(\frac38\)inch thick books make a stack that is 6 inches tall?

How many groups of \(\frac12\) pound are in \(2\frac 34\) pounds?

 Write a question that can be represented by the division equation \(5 \div 1\frac12 = {?}\). Then, find the answer and explain or show your reasoning.
Summary
A baker used 2 kilograms of flour to make several batches of a pastry recipe. The recipe called for \(\frac25\) kilogram of flour per batch. How many batches did she make?
We can think of the question as: “How many groups of \(\frac25\) kilogram make 2 kilograms?” and represent that question with the equations:
\(\displaystyle {?} \boldcdot \frac25=2\)
\(\displaystyle 2 \div \frac25 = {?}\)
To help us make sense of the question, we can draw a tape diagram. This diagram shows 2 whole kilograms, with each kilogram partitioned into fifths.
We can see there are 5 groups of \(\frac 25\) in 2. Multiplying 5 and \(\frac25\) allows us to check this answer: \(5 \boldcdot \frac 25 = \frac{10}{5}\) and \(\frac {10}{5} = 2\), so the answer is correct.
Notice the number of groups that result from \(2 \div \frac25\) is a whole number. Sometimes the number of groups we find from dividing may not be a whole number. Here is an example:
Suppose one serving of rice is \(\frac34\) cup. How many servings are there in \(3\frac12\) cups?
\(\displaystyle {?}\boldcdot \frac34 = 3\frac12\)
\(\displaystyle 3\frac12 \div \frac34 = {?}\)
Looking at the diagram, we can see there are 4 full groups of \(\frac 34\), plus 2 fourths. If 3 fourths make a whole group, then 2 fourths make \(\frac 23\) of a group. So the number of servings (the “?” in each equation) is \(4\frac23\). We can check this by multiplying \(4\frac23\) and \(\frac34\).
\(4\frac23 \boldcdot \frac34 = \frac{14}{3} \boldcdot \frac34\), and \(\frac{14}{3} \boldcdot \frac34 = \frac{14}{4}\), which is indeed equivalent to \(3\frac12\).