Using a nonsense maths question

On Monday, the children and I had an excellent maths / critical thinking lesson, based around a couple of YouTube videos that had popped up on my YouTube homepage.  (You know how it is -- you watch a few videos on a topic, and then YouTube suggests more videos on the same theme.)

The first video showed up because I had been watching Sal of Khan Academy introduce Common Core (the US maths standards) in a series of videos.  YouTube then suggested lots more videos about Common Core, including this one.

In the 3-minute video, the authors introduce a maths question, which they claim is a US 4th Grade Common Core question, asking a few "random people on the street" to have a go at answering it.

Of course, it is all fun and funny, because the random people fail completely.  These stereotypically-standard middle-class USers flounder around with half-starts and guesses to a question supposedly for 9-year-olds.  Most finally give up and conclude that the question doesn't have enough information to be meaningful -- that it is a nonsense question.

While nothing is said directly by the video authors, I assume they are trying to imply that Common Core is Bad and Wrong, as it includes crazy nonsense questions like this one that no normal adult could answer.

I am not going to get into a discussion here about whether this question is a genuine one, or a genuine Common Core one, or whether Common Core is Bad and Wrong.

What I want to do here is show that the question asked in the video is actually an excellent question to ask children, and it can form the basis of an excellent lesson.  In my opinion, it is definitely not a worthless question.

If you haven't clicked on the YouTube link above, here is the question:

Juanita wants to give bags of stickers to her friends.  She wants to give the same number of stickers to each friend. She is not sure if she needs 4 bags or 6 bags of stickers.  How many stickers could she buy so there are no stickers left over?

For those of us who have recently been doing maths at around the 4th Grade level, the question seems structurally pretty familiar.  There are stickers and there are friends, and we want to share the stickers evenly between the friends and have no stickers left over.

A lot of 4th Grade maths work is about getting familiar with using the basic multiplication and division facts, and learning to divide by one-digit numbers with and without remainder.  Word problems verbally similar to this one are introduced to check that the students understand the meanings behind the equations (one of the big aims of Common Core is to ensure understanding, not just rote memorisation).

But there are a few weird things about this particular question:

1. We don't know how many friends she has.
2. We don't know the relationship of bags to stickers -- is Juanita buying stickers and then putting them into her own bags, or is she buying bags of stickers directly?
3. How does the 4 or 6 bags fit in?

It is obvious that if this was a standard 4th Grade maths question we would say, as the adults did in the video, that there is important information missing.

So, knowing that this was a weirdly-worded question, albeit structurally similar to questions that Mulan is familiar with, I introduced the question to her and Miya, then sat back and waited to see what they would do.

Just like the adults in the video, Mulan wanted to know how many friends Juanita has.  She puzzled for a while over the ambiguity of bags/stickers, as well as the 4 or 6 bag thing.

But then Mulan came up with her own solution, and one that I hadn't thought of.

Mulan said that Juanita could rip the stickers to divide them evenly among her friends.  Then they could draw in the other parts themselves.  So, it really didn't matter how many stickers she bought or how many friends she has.

This solution is typical Mulan, seeing sharing, compromises and communal DIY pen-and-paper activities as the way to go.  She would not see it as important to buy more of something to make it even between everyone, but just jointly use whatever they have got to keep things fair.

I immediately agreed.  Solution number 1.

But then I challenged Mulan further by adding a new requirement that they want to keep the stickers whole and so won't rip them.

At this point the three of us discussed it together for a few minutes.  I can't remember exactly what any of us said, but one thing I did want to emphasise to both Mulan and Miya was the importance of not being fooled by distracting information.

We observed together that the 4 or 6 bag point was not phrased as a definite requirement, but simply that Juanita was unsure of what to do.  I hoped the girls would see that people can easily get into the habit of scanning a maths question for numbers and then thinking that any number mentioned must be part of the calculations.  But it need not be; extra, unnecessary numbers may be sneakily put in to test our understanding of the question.  And we all agreed that the 4 or 6 bags thing was surely there as a distraction.

Mulan then returned to the problem of the missing information about how many friends there were.  She said that, since she couldn't rip the stickers, she would buy as many stickers as there were friends.

I then suggested that we could let x stand for the number of friends that Juanita has.  Mulan quickly caught on, and said that then she could buy multiples of x stickers.

Solution number 2.

I then said that there was one more possible solution that I could see.  When they stalled, the girls asked for a hint.  So, I directed them to the idea that there is one number of stickers in which there will always be no remainder, no matter how many friends there are.

Still no bites.  So I wondered out loud if it is always necessary to buy things.

At that, Mulan's grin grew wider with understanding, and she said that Juanita could buy 0 stickers.  0 stickers divided by any number of friends will always have 0 remainder.

Solution number 3.

Mulan liked the question so much that she wrote it out by hand to show the cousins.

Yesterday, with great delight, Mulan and Miya presented the question to two of their cousins (ages 11 and 9).  Without any adult initiation or involvement at all, the four of them discussed it together in a very systematic way.  I didn't catch all of the conversation, but I overheard Mulan clearly and accurately articulating the points that we had made the day before.

At the same time, I asked the question to Mama.  Mama immediately said that Juanita could buy all the stickers in the shop.  After all, there would then be no stickers left over in the shop.

Brilliant.  Solution number 4.

When our two discussion groups came back together, I pointed out Mama's new solution.  11-year-old cuzzie immediately said that she had said the same thing.

So, there you have it.  Excellent discussions and four possible solutions from a maths question that at first glance looked like silly nonsense.

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The second YouTube video we watched on Monday was this one from Derren Brown.

For those who don't know, Derren Brown is a UK TV personality who has been doing TV shows for several years centred around hypnotism, mind reading, etc, but from a psychological / scientific perspective.  After the impressive trickery, he points out some of the main psychological techniques.

This particular video that I showed to the girls showed up the unconscious aspects of advertising.  (I thought it fitted in with the theme of appearances and question misdirections.)  It is pretty impressive.  I highly recommend it.

After watching the video, Mulan made the connection between this and the political advertising that we are seeing around our home these days (local body elections).  This then turned into a discussion about how the politicians use advertising to try to influence us unconsciously to vote for them.

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UPDATE 29/10/2016: I see over at Math Mammoth they mention this question (Example 2 in the section titled Bad examples of "common core" or "new math").

They focus on the 4 or 6 bags of stickers sentence, linking to a conversation on The Math Forum and agreeing with Bart Goddard, who writes:

Presumably she's giving one bag to each friend (although this is a bit ambiguous, too) so she is expecting either 4 or 6 friends to show at the meeting. (I suppose that there's a set of twins who have a habit of not RSVP-ing, but crashing the party anyway.)

This means that they treat the question as a common multiple one.  Since common multiples of 4 and 6 are 12, 24, 36, ..., they think that any multiple of 12 is the correct answer.

While I can see that this is one possible interpretation of the situation -- that the reason she is not sure how many bags she needs is because she is friends with unreliable twins -- I think it is important to remember that this was not what was actually written there.

What they have done is give a possible interpretation, which includes additional made-up information that has consequently constrained their answer.  But this possible interpretation is not a necessary one (Juanita's uncertainty may have been for other reasons than what they have presumed), and so the constraints they put on the answer are also not necessary.

To put it another way, I think it is important to always read exactly what is written in a question, and not invent extra requirements through our own presumptions and interpretations.  The Math Mammoth answer is wrong because it adds requirements and constraints which were not part of the question, as written.