Ambiguity in math class

Math class is a special place. We've talked before about some of the special assumptions that are based into that context: teachers pose questions, students answer questions, all questions have answers, questions include all the necessary information, answers are usually "nice," problems can be answered with the tools students have (just been) taught, diagrams are indicative while the underlying true forms are perfect, etc.

Of course, not all math classes make these assumptions or leave them implicit, or are constant about which ones are in force, etc.

In this post, I want to pick up a thread related to the "one true answer" myth: problems that have multiple interpretations.

Example
You are driving from your house to a soccer tournament. The distance is 120 miles. For half of the trip, you drive 60 mph. For the other half, you drive 30 mph. What is your average speed over the whole drive?

Where's the ambiguity?
For the teacher who poses this problem, there is no confusion. Obviously, students are meant to calculate that it takes 1 hour to drive the first 60 miles and 2 hours for the second 60 miles. That means it took 3 hours for 120 miles, or 40 mph average speed.

The catch: what does "half of the trip" mean? As an alternative, it could mean half the time of the drive. If that feels contrived, consider the following natural statements about travel measured in time instead of distance:

• "The drive took 3 hours; we stopped for a snack half-way." In this case, time and distance are equally natural in normal conversation.
• "The flight took 6 hours;  I read half the time and slept the rest." In this case, time is the more common metric, but it wouldn't be considered unusual for someone to talk about the distance they flew.
• "We were gone for 2 weeks, half at the beach, half visiting our cousins." Here, time is the natural metric, while it would seem strange to focus on distance. However, a vacation spent hiking the Appalachian trail or cycling across country would shift the balance back to distance.

Sources of ambiguity

I came up with four potential sources of ambiguity in math questions:

Things that can be measured in multiple ways. 

This extends the idea from “half a trip” ambiguity about distance or time. J1 and I had a discussion a couple of weeks ago where we measured chocolate bars and cookies using three different metrics: mass, cost, utils. For example, which is more:

• 100 grams of chocolate that costs $2.00 and you value at 100 utils
• 80 grams of fresh baked sugar cookie that costs $2.50 and you value at 90 utils

In business, it is common to have to deal with the ambiguity of whether “stuff” is measured in physical amounts or monetary value.

Pronoun ambiguity

For example: Ellis had 10 strawberries. Ellis gave 4 to his father and he ate 2. How many does he have now?

Who is meant by each occurrence of "he"? In each case, it could mean either Ellis or the father which leads to 3 distinct answers: 6, 4, or 2.

I accept that this is an example of bad English, but we're in math class and never claimed to be masters of language (did we?)

Tense ambiguity

In the prior story it could be that giving the strawberries away and eating them happened before the state where Ellis had 10. Let's add some extra story context to make this alternative more clear:

Ellis still had 10 strawberries. He bought a pack of 16, but Ellis gave 4 to his father and he ate 2.

I think this alternative interpretation is more of a stretch, but I've seen cases where the uncertainty about when things were happening is more natural.   

Assumption of scalability

Joe can bake 2 cookies in 20 minutes. How long does it take him to bake 4 cookies? 400 cookies? 4 million cookies? 4 quadrillion cookies?

I saw one math class question that involved writing books, a task which is very unlikely to happen at a constant rate.

Your challenges

1. Find other sources of ambiguity that can infect (or add spice to) math class problems.
2. For N a positive integer, create a puzzle that has N distinct solutions based on (reasonable) alternative interpretations.

What is your function? More excuses to delay bedtime

J1 (5th grader, looking for an excuse to stay up): What are you working on?

J0: I'm writing a review of a book.

J1: The one we got from math circle (Martin Gardner's Perplexing Puzzlers and Tantalizing Teasers)? 

J0: No, the one about Funvillians.

From Natural Math!

J1: Tell them that it was fun!

J0: You really enjoyed reading it. I'll make sure to mention that. I was thinking that we should have used it as an inspiration to make our own adventures.

J1: You mean, like creating new characters with their own powers? We could have heroes who control fire and ice, some others that can go forward and backward in time.

J0: Is that how the Funvillian powers worked? I thought they needed to have inputs. For example Marge's power only works on two exactly identical objects.

J1: Sure, the current time is an input and the output is the time in 5 minutes. Or another one can do the reverse.

[pause, maybe he's starting to go to sleep?]
J1: Or... maybe we could make up some adventures where the Funvillians from the story have to solve their own challenges.  They could meet some villains... not Villians! (laughing)

J1: The one who can duplicate things ... what if that power could be used on people? After they were copied, would they all have to do the same things? For example, if I were copied and I raised my arm, would the other one have to raise his arm, too? Could they think different thoughts?

J0: Well, when they copied two toys, they could play with the toys separately. The toys didn't have to do identical things.

J1: Oh! But what if they were changed a tiny amount? Would they still be considered identical and could they get reduced down to one copy?

J0: I don't know. Where do you think the powers come from?

J1: maybe from living in their magical land. Probably when they have spent enough time there, a power develops.

J0: There, so that's what I'm going to write about. Thanks!

J1: Remember to tell them it was fun!

---

On Fridays for the last several months, my fifth grader and I have been spending 2 hours in the evening doing math together. By that time of the week, I'm not always feeling energetic enough to properly plan an activity or exploration. Looking to give myself a break, last week, I brought Sasha Fradkin's book Funville Adventures for J1 to read during the session.

He was engrossed and finished it with some amount of time to spare. Maybe 90 minutes of reading, leaving us 30 minutes to discuss.  He had read the addendum, so was already primed for talking about functions. In addition, he still remembered past conversations about "function machines" and programming functions. Using the characters as references, though, he found it much more intuitive to understand invertible and non-invertible functions. We talked about examples of arithmetic functions that were similar to different characters' powers and had fun giving examples of what would happen if different characters used their powers in succession.

The experience, so far, suggests that this is a helpful model for understanding functions, more human and vivid than what we'd previously done with function machines.

And remember, it was fun!
(now go to bed!)

Cryptarithmetic Puzzles for Grades 1 to 4

Inspired by a series of puzzles from Manan Shah, I decided to have the kids play with cryptarithmetic puzzles today. In addition to borrowing some of Manan's puzzles, I also used some from this puzzle page: Brain Fun. I've included some more comments below about the Brain Fun puzzles.

My main concern was whether the puzzles were at the right level. In particular, I was afraid that the puzzles would be too hard. In fact, I tried solving a bunch of them yesterday and actually found myself struggling. I'll ascribe some of that to being tired and sick. However, my intuition was to make some simpler puzzles of my own. In particular, I added:

• puzzles that have many solutions: I figured that many solutions would make it easy to find at least one.
• a puzzle that "obviously" has no solution. Now, obviously, the word "obviously" is a sneaky one in math, but I was pretty sure the kids could see the problem with this structure.

Grades 1 and 2

For the younger kids, I started with a shape substitution puzzle. This is one our family explored almost 2 years ago: Shape Substitution. I don't recall the original source.

Two reasons why I started with this. First, it has a lot of solutions, but there is an important insight that unlocks those solutions. Second, by using shapes, we can write possible number solutions inside them as we solve or guess-and-check the puzzle. This made it easier for the kids to see the connection that all squares have the same value, etc.

The second puzzle: BIG + PIG = YUM

Really just a warm-up practicing the rules and doing a little bit of checking that we haven't duplicated any numbers.

The third puzzle: CAT + HAT = BAD

Again, lots of solutions, but noticing leads to a good insight.

Fourth puzzle:  SAD + MAD + DAD = SORRY

This is a trick puzzle. The kids know that I like to tease them, so they are aware they need to look out for things like this. We discussed this in class and I suggested they give this puzzle to their parents.

Fifth puzzle: CURRY + RICE = LUNCH

When I translated this to Thai, all the kids laughed. I was sneaking a little bit of English practice into the lesson and then they realized that it was worth trying to read all the puzzles, not just solve them.

Sources: I think I made up all of these puzzles (original authors, please correct me if I'm wrong).

Grades 3 and 4

The older kids already had experience with these puzzles. We did refresh their memory a bit with BIG + PIG = YUM

I asked them to give me the rules and explain why those rules made sense. As with most games, I want to communicate that we're doing things for a reason, but those reasons can be challenged. If they think it makes sense to do it a particular way, we're open to their ideas.

Second puzzle:  SAD + MAD + DAD = SORRY

Same discussion as for the younger kids. When prompted, this was pretty easy for them to spot, but they weren't naturally attuned to think about whether a puzzle had solutions or how many. This led me to take a vote on all the puzzles at the end to see who thought the puzzles would have 0, 1 or many solutions.

Third puzzle: ALAS + LASS + NO + MORE = CASH

A puzzle from Brain Fun. I think this is one of the easier ones on that page. Again, a bit of English practice.

Fourth puzzle: LOL + LOL + LOL + .... + LOL = ROFL (71 LOLs)

This was from Manan. I think it is one of the easier ones in his collection, but it looks daunting. Turns out none of the kids in the class were familiar with (English) texting short-hand, so my attempt to be cool fell flat.

Fifth puzzle: CURRY + RICE = LUNCH

Again, everyone was delighted when I translated this one. We're in Thailand, after all, so at least one puzzle had to be about food.

The key exercise

The final assignment everyone (all four grades) was given was to make up a puzzle for me to solve. I was thinking it would be nice to have one in Thai, but we decided to keep it in English as further language practice.

Manan wrote a nice post about having kids design their own puzzles. If it goes well, this is actually the activity that ties a lot of the learning messages together: they think about structure, they think about what allows multiple or single solutions, they apply their own aesthetic judgment, they use their knowledge of the operations, they are empowered with an open-ended task that cannot be "wrong."

We'll see how it goes. At the very least, I expect a lot of work for myself when their puzzles come in!

An extra sweetener

Two kids asked if we could use other operations than addition. That prompted me to put this on the table (also from Brain Fun):

DOS x DOS = CUATRO

Brain Fun Problems

The first time I'd seen the Brain Fun problems, I added them to a list and called them "basic" (see this page.) When I actually went to solve them, though, they didn't seem so easy.

Big confession time: I actually looked at some of the solutions.  However, I was disturbed to see that the solutions involved extra information that wasn't included as part of the problem statement! For example, in THREE + THREE + FIVE = ELEVEN, the solution assumes that ELEVEN is divisible by 11. This seems to be the case for several of the puzzles involving written out arithmetic:

TWO + TWENTY = TWELVE + TEN (assume 20 divides TWENTY and 12 divides TWELVE, I wasn't clear about whether any divisibility was assumed for TWO and TEN)

I'm not sure if similar assumptions are allowed/required for any of the others.

Maybe I shouldn't complain, since this assumption creates an additional constraint without which there could be further solutions. Perhaps part of the reason it doesn't sit well is aesthetic. In the 3 + 3 + 5 = 11 puzzle, 3 doesn't divide THREE and 5 doesn't divide FIVE.

Lastly, there is a typo in the final puzzle of the Brain Fun page. That puzzle should be
TEN x TEN = FIFTY + FIFTY

Perfect Play for My closest neighbor

Joe Schwartz at Exit10a wrote a fraction comparison post that prompted me to write up more of my experience and thoughts on this game.

Let's find perfect play
This week, I intended to use the game one last time with the 4th graders as an extended warm-up to our class. The challenge I presented:

If we got super lucky and were given perfect cards for each round of the game, what are the best possible plays?

My intention was to spend about 20 minutes on this. Depending on how quickly it went and the kids' reactions, I considered giving them a follow-up for a short homework: what are the best plays if we include all cards A (1) through K (13)?

How did it go?
In the end, the basic activity took the whole class. These comparisons were difficult for the kids, so we spent time talking about each different strategy for comparison:

1. common denominators
2. common numerators
3. distance to 1
4. relationship to another benchmark number. Like 1/2 in Joe's 4/6 and 8/18 example, a benchmark is a "familiar friend" that should be relatively easy to see it is larger than one and smaller than another. In practice, 1/2 seems to be the most popular benchmark. 

For visualization, drawing on a number line seemed to work best.

I did not assign the full deck challenge as homework. Instead, we gave them some more work with fractions of pies and bars.

What have I learned?
This game is really effective at distinguishing levels of understanding:
(0) some kids are totally at sea. They don't really understand what this a/b thing means, how a and b are related, etc. These kids struggle with the first round of the game when the target is 0, when the idea is to just want to make their fraction as small as possible.

(1) Some kids have got a basic understanding of the meaning of the fraction and can play confidently when the target is 0 or 1. They might still be weak about equivalent fractions. Trying to play some spot-on equivalents when 1/3 and 1/2 are targets is a give-away.

(2) familiar with some frequent friends: kids who can tell readily whether their plays are larger or smaller than the target for 1/3, 1/2, 3/4.

(3) proficient: have at least one consistent strategy they can work through to make a comparison

(4) fraction black-belts: using multiple strategies, already familiar with many of the most common comparisons.

What would I do differently?
Generally, I think it is valuable to spend more time and more models directed at the basic understanding of what fractions mean. The kids who were at or close to stage 4 have, over the years, been seeing diagrams of pies, cakes, chocolate bars, number lines and physical experience with baking measures and fractional inches on measuring tapes and rulers. Oh, and also actual pies (mostly pizza), cakes, cookies, and chocolate bars discussed using fractional language.

More locally, for this game in a class of mixed levels, I would

• lean toward doing this more as a cooperative puzzle
• re-order the targets for the rounds as 0, 1, 1/2, 3/4, 1/3, 2 (note: I don't have strong feelings about where 2 fits in this sequence)
• I also would consider allowing equivalent fractions to the target as winning plays

Secret Numbers (Addition Boomerang variants)

We have been enjoying Mathpickle activities in our math games classes recently. This week, the 3rd and 4th graders will be playing with some of the more advanced Addition Boomerang variations.

During our planning, we came up with one extra pointer to tie the activity more closely with multiplication. Also, we had ideas for variations and wanted to record our notes so we can use them again in the future.

Tie with multiplication

In Gord's video explanation, he sometimes records in the center of each loop how many times that loop has been used. We emphasize this and write some related equations to help draw out the connection between the repeated additions in this activity and multiplication.

The first way we do this is by making tally marks inside the loop every time that branch is chosen. At any time, you can pause and write down an equation for the current total in a form

AxN + BxM = Total

where A and B are the values of the loops, N and M are the number of times each loop has been used.

Alternatively, we can show a "completed" round of throws by simply writing the number of passes for each loop in the middle and, again, write out an equation showing the total as the sum of two products.

Secret Number Variations

We start with a basic addition boomerang lay-out, either with 2 or 4 branches, both players (or teams) share a common set of addends and take turns adding on to a common running total. In our variants, the players choose and write down a secret number that helps inform their target for the game:

• Version A: players pick a number between 70 and 100. This is their target for the game and they win if the common total hits that value, whether the target is reached on their turn or their opponents turn.
• Version B: players pick a number larger than 15. They win if the total hits a multiple of their secret number. For example, if they choose 17 and the running total hits 51 (aka 17 x 3) then they would win. If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.
• Version C: players pick any number. They win if the total hits a multiple of their secret number that is larger than 60 (not equal to 60). If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.

Version B is, I think, the most directly playable.

Possible issues
I'm not sure how to deal with the case where both players choose the same secret number.

For Version A, it will be interesting to see what modification kids can find that will deal with the fact that it is very easy to miss any particular target. In the basic game, once the total is larger than your target, there is no hope of recovery. There are several ways to address this. I would be eager to hear any rule sets that kids create and hear about the experiences.

In Version C, I wonder if choosing 2 as the secret number is too strong a move?