How can I visualize dividing by a fraction as partial and full groups?

I can't visualize all these "full groups" and "partial groups." Can someone please picture them? Please see my questions red in-line. Please don't use the identity that $\dfrac{a}{b} \div \dfrac{c}{d} \equiv \dfrac{a}{b} \times \dfrac{d}{c} $.

Because $10 ÷ 4/3$ isn't an integer, I changed the numerator from 10 to 6.

Let's take a step back and consider what does dividing mean in general. There are a lot of ways of interpreting division, but one way is to say that (for example) 6 ÷ 4 means "if we have 6 items, and we split them into 4 equal groups, how many items will we have in each group?"

Let's see what happens if we try to apply this to fractions. 6 ÷ 4/3 means we divide 6 into 4/3 equal groups, and we ask how many items is in each (full) group. But what does it mean to have 4/3 of a group? 4/3 is 1 ⅓, so it'd make sense that 4/3 means you have one full group, and then a second "partial" group that's 1/3 the size of the full group.

Think about how you would physically take items to do this. You could say that "oh, I need a total of four 'partial' groups, since three of the partial groups make a full group. At the end, I'll put three partial groups together to make a full group, and then count how many are in that final full group."

If each partial group is $1/3$, then "three of the partial groups make a full group" because $3 \times \dfrac{1}{3} = 3$. And you need 4 partial groups because $4 \times \dfrac{1}{3} = \dfrac{4}{3}$. $\color{red}{\text{1. Then what? What does "count how many are in that final full group" mean? }}$

In turn, what does multiplication mean? Again, multiple interpretations, but one of them is that 5 × 3 means "there are 5 items in a group, and there are 3 groups; how many items do you have total?" Again, let's apply this to fractions. 6 × 3/4 means that you have 6 items in a group, but you only have 3/4 of a group. How many do you have total?

In this case, how would you physically act out this multiplication? You might say, "oh, there should be 6 to a full group, but I have only 3/4 of one. I guess I could make that by making a full group, splitting it into four parts, and throwing away one of the parts."

But now notice the actions you're taking for both division by one fraction and multiplication by its reciprocal. In both cases, you're breaking the 6 up into 4 parts, and then counting only three of the four parts. The reasoning is a bit different (in division, you were thinking about the number of "partial groups" and how many partial groups are in a full group, while in multiplication, you're thinking about how to break down a number into a part of a group which involves first breaking it down into a lot of equal small parts and then putting them back together into a bigger part-of-a-group) but the same 3 and 4 numbers are there in both cases.

$\color{red}{\text{2. I'm nonplussed. In division, how was I counting "how many partial groups are in a full group"?}}$

3. Isn't u/Viola_Buddy wrong to use "group" here? What's a more correct term? "Group" is for Group Theory, but undoubtedly I'm not doing any Group Theory here, when I'm having difficulty with division!

posted about 3 years ago

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