Adding things to bunches of things vs multiplication

Question

"Suppose you bought four boxes of pencils having five pencils in each, how many pencils do you have altogether?" — "Nine." — "How come?" — "Because 4 plus 5 is 9." — "But you cannot add boxes to pencils!" — "Why?"

Indeed, why? Why you can multiply boxes and pencils, but cannot add? This is sort of self-evident for most adults, but how do you explain it to an elementary-school student?

I came up with an approach calling a box a "bunch of things" (Common Core likes to use the word "group"), so a box by itself has no meaning, what does have meaning is that it groups, combines, ties together several things that we are actually interested in, say pencils. If each box combines the same amount of things, then we can define and use multiplication as quick addition of the same number of things.

Similar approach can be used to explain why you cannot add two tens of flowers and five flowers as 2 + 5 = 7, because it is not clear seven of what we are getting. First, we "unbunch" two tens into one, two, three, ..., twenty flowers, then add five flowers to them, so we can count them, twenty five flowers. It just so happens that by having ten-based positional system we can simply write 5 to the right of 2 to get the correct number, it won't work if we had two dozen flowers instead of two tens.

Another phrase commonly used is that you can add "like things", things that are similar. All this kinda works, but does not feel perfect, does not seem rigorous, persuasive enough. Does anyone have a better idea, approach, script to explain to kids why adding apples to apples is ok, but adding apples to apple crates is not?

Answer 1

Roughly speaking, adding apples to apples is factoring.

One can reasonably interpret "four apples" as the result of multiplying the number "four" with the unit "apple" — in fact, I would even argue that you should think in this way.

So, when you add like things, what you are actually doing is invoking the distributive law to factor the unit out:

$$ 4 \cdot \text{apple} + 5 \cdot \text{apple} = (4 + 5) \cdot \text{apple} $$

But when you add unlike things, such as

$$ 4 \cdot \text{apple} + 5 \cdot \text{oranges} $$

there's nothing to factor out: no further simplification is possible. Note, however, that if we're willing to forget information, and just say apples and oranges are both fruits, we can then make a substitution

$$ 4 \cdot \text{fruit} + 5 \cdot \text{fruit} = (4 + 5) \cdot \text{fruit}$$

and then simplify.

The examples you consider are similar. If we add

$$ 4 \cdot \text{apple} + 5 \cdot \text{<crate of ten apples>} $$

we can't factor anything out. However, if we're willing to ignore the containers, we do have an identity:

$$ 10 \cdot \text{apple} = \text{<crate of ten apples>} $$

This means we can make a substitution into the original formula

$$\begin{align} 4 \cdot \text{apple} + 5 \cdot \text{<crate of ten apples>} &= 4 \cdot \text{apple} + 5 \cdot (10 \cdot \text{apple}) \\&= 4 \cdot \text{apple} + (5 \cdot 10) \cdot \text{apple} \\&= (4 + 5 \cdot 10) \cdot \text{apple} \\&= 54 \cdot \text{apple} \end{align}$$

so we also get the explanation of why you should multiply $5 \cdot 10$. (of course, you can actually compute the arithmetic anywhere in the calculation you like, rather than waiting until the end as I did)

Answer 2

Here is where it helps to get more concrete instead of more general. Have the student draw a picture of the problem and similar problems. First, you demonstrate drawing one box of pencils (a square) with five pencils inside (perhaps five tally marks) and make sure they understand the picture. Then ask them to draw four boxes of pencils (four squares) each with five pencils inside. When you ask them how many pencils there are at this point (by saying the original question again, and tying it to their drawing), they should get a correct answer (perhaps encourage them to skip-count if they are not good at multiplying yet), and at that point you should be able to ask them why they think adding 4 pencils + 5 boxes doesn't work to answer the question. Their self-explanation will probably be much better at making sense to them than any way we try to do it, because they will have processed it in context of what they already know about adding. You can always help clarify their wording at this point, and help them refine their statement so that they get at the crux of the issue: "pencils" and "boxes of pencils" are different "wholes."

To take it a step further if they still have a hard time explaining it, have them draw three pencils plus two boxes of pencils and repeat the process. How many pencils are there? Why didn't 3 + 2 work? Draw 3 pencils + 2 pencils. How is this drawing different than 3 pencils + 2 boxes of pencils?

Answer 3

The thing is, it is possible to add 4 boxes to 5 pencils.

4 boxes + 5 pencils = 9 things.

So start by showing the student what they can do with addition, and what its real-world application is. But then remind them what you really want to know - how many pencils are in the boxes (which they can easily count to see it's not 9).

And then you can show them how we can use multiplication to figure out how many pencils are actually in the boxes.

Answer 4

You can add, though it's more tedious.

"Suppose you bought four boxes of pencils having five pencils in each, how many pencils do you have altogether?"

First box has five pencils. Second box has five pencils. Third box has five pencils. Fourth box has five pencils. Altogether, then, we have $\underbrace{5 \text{ pencils}}_{\text{box} 1} + \underbrace{5 \text{ pencils}}_{ \text{box} 2}+\underbrace{5 \text{ pencils}}_{\text{box} 3} + \underbrace{5 \text{ pencils}}_{\text{box} 4} = 20$ pencils.

Or more conveniently, we can write $4 \text{ box } \times \dfrac{5 \text{ pencils }}{\text{box}} = 20$ pencils.

---

What you can't do is add one box of six apples, with 3 oranges, to get either 9 apples nor nine oranges, nor four apples, nor 4 oranges.

Answer 5

I wonder if it would help to have them make up the problems, instead of you? Clearly, these students are already discounting the meaning of math.

There is lots of research on the efficacy of well-led classroom discussions about math topics. (Deborah Ball has written eloquently about this.)

One book I loved, set at this level, is Little Kids: Powerful Problem Solvers, by Angela Andrews.

===

I see in one comment that you refer to 'him', so I now think you are working with one individual student. If he really doesn't see why you can only add like things, then he has already decided that math does not make sense.

If I were working with this student, I might focus on food. 1. You have 3 oranges on the counter and 2 oranges in the fridge. How many oranges do you have? 2. You have 3 oranges on the counter and 2 apples in the fridge. Can we put them together? How would we talk about them? (5 fruit.) After focusing on sensible problems, maybe you want to play with absurdity so you can laugh at silly answers together. You have 3 cars on the counter (they must be toys, not real, huh?) and 2 blueberries in the fridge. Can you put them together? (Maybe. As objects.)

Once he is clear on why adding only makes sense with like objects, then you can work on the multiplication issues.

Your student can make sense of this if you talk in language that works for him. You can find that through your connection with him better than we can find it through general principles.

Answer 6

1. When they are first learning multiplication, keep the numbers very small and allow them to do repeated addition. 2*4 and 4*2 are good ones to start with. Do 3*5 before 4*5. (Or a sequence of 1*5, 2*5, 3*5, 4*5, etc.)

2. Try to keep things simpler. Not boxes of pencils. But groups of pencils. Boxes of pencils is a bit of a word problem and a conversion problem.

3. Teach them the multiplication table via memorization and drill. Don't only approach multiplication in this manner...use concrete counting examples as well. But take a belt and suspenders approach. IOW, don't exclude learning of this kind. Having learned the table, kids are more ready to use it. Also, do not underestimate the joy in memorization and recall. Look how kids are with state capitals. Or how kids compete in games even simple drill.

4. Just persist and prevail. Don't assume they are as smart as you or as experienced. Repeat, repeat, repeat. That is the path to instruction more than "killer explanation". But of course, as in 3, use explanations AS WELL. But don't expect concepts themselves to magically unlock a stuck door. Some people even need to just learn by imitation, practice, and correction. (See coaching in sports and music.)

Answer 7

I am right now teaching Numbers to kids and I think it is the toughest part because Numbers are the true abstract thing in math. So I first taught them what 1(One) means and they got it pretty quickly. So now I am approaching bigger Numbers and I am trying to make them understand that bigger numbers are "COLLECTION" of smaller numbers. So the problem with "things" and representing them as "numbers" is that there is a transition between this real world, "things" and Mathematical Universe where those "things" are represented as numbers and this transition may not be as automatic and intuitive as that of a grown up. Addition can be used as an aid to explaining collection. There are additions where constituents are not visible after addition(7+2=9, or milk in milk) and add then there are additions where constituents are visible after addition(2 pencils + 4 pencils), these additions are collections of things. Also one of the things I try to make clear is there are various aspects of addition: 1. What is addition? 2. What is it used for? 3. What is the process of addition?

Difference between individual things, collections and parts of a thing are important and must be made clear to kids...