Yohaku: A Different Type of Number Puzzle

**Update: I have now created a website with these yohaku puzzles: http://www.yohaku.ca/

Like many folk, I'm a big fan of number puzzles. I always look forward to the Kakuro in the Toronto Star every Sunday: sometimes this will take me 5 minutes to solve, sometimes 5 days. I also love KenKens and have been seeing these used in classrooms as a great way of practicing number bonds and problem solving at the same time (always a win-win situation in my view). Sudokus are also OK but I won't go out of my way to work on one of these. I recently came across Multiplication Squares on the nrich maths site. This one particular problem struck me as having lots of potential of allowing students to practice and to problem solve both with additive and multiplicative thinking. I quickly created a few of these and, as I often do, I tried these with my own children.
I started with a 2 by 2 multiplicative one for my daughter:

The idea is to fill in the cells so the top two multiply to give 40, the bottom two multiply to give 14, the two left multiply to give 56 and the two right multiply to give 10. As she did this, I could see her use multiplication and division to fill in the cells. She also had to think about which pairs of numbers multiplied to give 40: in my opinion this is a far richer question than the more closed 4×10. I then gave her a 3 by 3 one: 

This definitely caused her to think more yet all the time she was using her number bonds: what three numbers multiply to give 42? To give 70? She had to stick at it for a while before she got it, trying a couple of possibilities in the process: much like I do when working on a KenKen or Kakuro.
At this point, my son wanted a piece of the action, so I gave him a 2 by 2 additive thinking square:

When I created this, I had a 2 in the top left hand cell: he started with 5. And that's when it suddenly hit me: there are an infinite number of solutions to these! I began to love these even more.
My son, seeing his sister work on the 3 by 3 multiplication, then asked for one for himself. I duly obliged:

As he persevered on this, it was clear that there was lots of lovely decomposition and recomposition going on. Then this happened:

I tried these puzzles in a grade 3/4 class for the first time yesterday. We began with a 2 by 2 additive puzzle which they absolutely gobbled up. When they finished, it was really powerful to say, "Compare your solution to your neighbour's" and then see them get excited when they had done it in different ways. 
I then wanted to see if they were ready for a 2 by 2 multiplicative puzzle so gave them this:

Now this is a class which has very recently begun learning about multiplication through the use of arrays and been recording their findings:

Without a doubt, they had to think more about this puzzle but the fact that they could think of 12 in different pairs of factors helped them.

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As a teacher, it takes hardly any time at all to create these and I can structure them such that if I want to focus on a particular fact, I can include that as a possible solution. I can also add an extra level by including some restrictions to the solution. For example, my son gave me this one to try:

 "I already know the answer," he told me. I looked at it and said, "Is it all fives?"
"Yes, but your not allowed to use fives!"
So this got me thinking. I can create a whole new set of puzzles with restrictions. These will require more perseverance to solve (you're welcome). For example:

Call me bias, but I really like these and as I haven't seen this type of puzzle before, I get to choose what to call them. 

So I have decided to call these puzzles: YOHAKU.

Catchy, eh?

I have created a website with these on: http://www.yohaku.ca/ Who knows, maybe people will like doing these and Yohaku will become really popular. Maybe one day they will be as big as Sudoku and KenKen! I'd love to know what you think.

Measurement to the Fore

I am sometimes concerned that our students' experience of Measurement consists primarily of memorising formulas and conversions at the expense of the the actual skills of estimating and measuring objects. It is all very well if a student can tell me how many millimetres are in three kilometres, but if they cannot measure the length of a floor accurately, then they have not learned about Measurement. 
My experience has been that as students use metre sticks, measuring jugs, and sets of scales, they develop such good measurement sense that they can see that since one metre is 100cm, then they will also understand that 5m is 500cm or that 725cm is 7.25m. In other words, they will use their measurement sense to make conversions. Those who rely solely on the staircase method for conversions will not develop their measurement sense so easily. 
As such, I like giving students activities which actually gets them measuring. It allows me to see if the are using the tools available to accurately measure objects. In two classes this week, I began the following task:

Design a container to hold twelve golf balls.

We gave each pair of students a single golf ball and told them that they could choose to use any further tools they might need from around the classroom. Right from the get go, all students were engaged in the task. They quickly began thinking about how they should arrange the golf balls. Some initially decided to design a bucket (but quickly abandoned this idea!). Others looked at a 12 by 1 arrangement; or a 2 by 6 arrangement; or a 3 by 4 arrangement. A couple of groups even thought about a 2 by 2 by 3 box. As they began to measuring the golf ball we noticed a few misconceptions:

1) Measuring from the edge of the ruler and not the zero mark.
I reckon most teachers have seen some of their students make this mistake or a variant of it (e.g. measuring from the 'one' mark). One way to correct this is explicit teaching: demonstrate to the student that measuring this way leads to inaccuracies and have them move the object (or the ruler) so that they measure from the zero mark.

2) Measuring the circumference and not the diameter

Maybe some students 'see' this attribute (circumference) more readily than the diameter. Having measured this, one group multiplied by 12 to get 158 cm. We asked them how they would use this value to design their container. When they seemed unsure, we showed them a metre stick to help them realise that this measurement was greater than a metre. At this point, they began to realise that they were in a bunker so we prompted them with 'Is there a better measurement to use?'

3) Spheres are difficult to measure!

We could have given the students a similar task with objects that are much easier to measure (e.g. design a box that will hold 12 juice boxes) but that being said, I think it was great that they were challenged; it brought out some creative approaches. For students who really struggled, I put a ruler either side of the golf ball and they measured the distance between these two rulers.



4) Only thinking two-dimensionally
This was fairly common and we dealt with it using this prompt: 'OK, you are phoning me, the manufacturer of these containers. Tell me all the information I need to know to make your container accurately.' This got the students to think about the height of the container.



The next step for both of these classes is to take their plans and actually make the boxes out of card. However, before this happens, the students need to know the old adage, 'Measure twice, cut once'. Most of them measured the diameter as 4cm when really it is about 43mm. They need to factor this exact measurement into their design (and consider if they need to add a little extra) before they actually begin construction.
Once all these boxes are constructed (again, this will involve a lot of hands-on measurement), they can argue which of the boxes is best for the job. I already know one student who is convinced that the 1 by 12 box will be best as it will fit in nicely alongside the clubs in the golf bag: it will be interesting to hear the other students' thoughts on this!

       *       *      *

I once gave students the task of designing a pasta box that would hold 1000 cubic centimetres. The choice boiled down to a 10 by 10 by 10cm cube or a 10 by 5 by 20cm rectangular prism. Many students made the argument that the cube was the better box as it had less surface area and therefore would be cheaper to manufacture. One student though argued against the cube as it was difficult to hold and therefore pouring the past into a pot of boiling water would be an issue. Another student argued that the cube had a smaller 'front' than the rectangular prism (100 compared to 200 square centimetres) so it would not be as visible on the shelf of a grocery store which would impact sales. 
It goes to show that sometimes the mathematically best solution is not the best solution practically. 

The Subtle Art of Taking a Break

There is, I believe, a common misconception that Maths is a subject which consists entirely of questions that need to be answered immediately. Whilst I acknowledge that that efficient fact retrieval is a huge advantage in solving problems, our students will be at a huge disadvantage if they never experience questions which force them to stop and rethink. Our students need to know what to do when they reach that cul-de-sac. One strategy I think we might do more to encourage is to get our students to actually take a break from the problem.
A couple of months ago, I was in a meeting and doodling away when I accidentally sketched this shape:

I wondered what its area would be, thinking it would be a trivial problem. Five minutes playing around with it didn't yield anything so I put it to one side. I revisited it several times, each time not making any progress. I knew that I could take a purely algebraic approach (using co-ordinate geometry) but where's the fun in that? I was looking for a beautiful proof. Then one day, I took the problem out again and suddenly saw the answer:
 And there have been many times when the solution to a problem suddenly materialises after I have taken a break from the problem. Once I was stuck on this problem from a Number Theory course I was taking as part of my Masters of Mathematics for Teachers at the University of Waterloo:

Show that for any positive integer, n, there exists n consecutive values, none of which are prime.

For example, suppose n=4, I have to show that there are 4 consecutive numbers which aren't prime. I can do that: 24, 25, 26 and 27 are four consecutive numbers which aren't prime. But I had to prove that it works not just for 4 consecutive numbers but for any amount of consecutive numbers. I worked on this for quite some time but without making significant inroads so I took a break, and as it was late went to bed. My son woke me up at half past three in the morning asking for a glass of water. I got this for him, tucked him in and then headed back to my bed. In the six steps it took me to get back to my room, in the middle of the night, I suddenly saw the answer. In fact I saw it so clearly that I knew that I didn't need to write it down anywhere.

Now I am sure there will be some readers who might think, "You got stuck on those problems? But they are easy!" And, now I know the answer, I do wonder why I got stuck. But the fact remains that for whatever reasons, I did get stuck, and consciously taking a break somehow reset my way of thinking.
And I know that I am not alone in this. In the wonderful book about the great mathematician Paul Erdos, The Man who Loved Only Numbers  another great mathematician, Ron Graham, explains how he had a "...flash of insight into a stubborn problem in the middle of a back somersault with a triple twist." 
I would love to know the neurological reasons why this happens. Is it a case of the brain thinking too hard about the problem (as result missing some vital information) and then,  after a break, a rejuvenated, more relaxed brain sees what should have been seen all along?
Whatever the reasons for this, as a teacher, I need to model what I do when I get stuck. I need to get students to understand that sometimes the best way to crack a problem is to leave it alone. I can tell my students that if they get stuck with a question (on a test for example) to leave it, do some other questions and then come back to it: they might then take a fresh, more productive approach to the problem.
But also, in the same way that English teachers will habitually share with their students what book they are currently reading, maybe we as Maths teachers can share with our students what Maths problem we are currently working on. This, for example is what I am working on right now (courtesy of the University of Waterloo CEMC's Problem of the Week.

A Concrete-Diagrammatic-Symbolic Development of Division

It is no secret that students find division the most difficult of the four operations to understand. I don't just mean the procedure of the long division algorithm, I mean the concept of what division is as an operation; how it relates to the other operations and how it is used in different contexts.
Here are some thoughts on how we can get our students to truly understand division, based on the Concrete-Diagrammatic-Symbolic continuum. Whilst one of the end-products of this will be that students can use an algorithm to divide two numbers, I truly believe that just teaching an algorithm will not get our students to understand what division is. It must be pointed out too, that a huge part of understanding division necessarily involves seeing how it links to multiplication. Graham Fletcher has created a great video on this. And we should also remind ourselves that knowing your multiplicative facts is more than knowing the answer to 7×8. It is also knowing the answers to such questions as 24÷a=8 and a÷8=7 or even 48÷a=b.

Concrete Division
This can begin in primary grades. Kids might well have real-world experiences of sharing situations (e.g. sharing candy, dealing out cards, cutting a pizza into equal slices). They might not have real-world experiences of grouping situations though so it is a good idea to give questions like 'If an egg box hold 6 eggs, and you have 24 eggs to pack, how many egg boxes are needed?' or 'There are 24 kids in our class and they need to put in groups of three. How many groups will there be?' It is vital that students experience this tactile sense of what division (either sharing or grouping) is. As they become more familiar with this, we can give them problems that will involve remainders so that they can consider what effect this has on their answer (e.g. if an egg box holds 6 eggs, how many are needed to hold 32 eggs?)

Diagrammatic Division
With enough experience of this, they can then represent the division action using diagrams. By this, I don't mean that they need to draw pictures of the actual objects that are in the question but rather use this method which I call Spoke Division:
Suppose you have to do 517÷4.
Firstly, write 517 with four spokes radiating outwards:

Now think of a friendly number that you could put into each spoke. In this case, 100 seems to be a good choice. After taking four hundreds out we are now left with 117:

Now think of a friendly number that you could put into each spoke. Some students might say 10, some might say 20.  Both of these work but will take a little longer. I myself will use 25:

Now we have 17 left, so I can put four more into each spoke leaving a remainder of 1.

Since each spoke has 129, we can say 517÷4=129 R1 or, if you prefer, 129.25.

Initially, I'd be careful about what numbers to use; friendly at first, then building complexity. What I like about this method is that it connects with the students' concrete representation of division and, as such, still feels like division. There will come a point when the spokes method can be developed into something more powerful.

Symbolic Division
Although I learned a version of the standard algorithm growing up, it is not the one I would initially show my students. Instead, I would use the following method, often called partial quotients. 


See how it connects nicely with diagrammatic division. Also notice how it allows the student to use friendly numbers to get the answer. It is not so easy to use friendly numbers in the standard algorithm.

With regards the more abstract standard algorithm, it should be pointed out that different countries have different versions of what this looks like (see this entry in Wikipedia). I myself learned something which I believe is called short division and spent a long time focusing on single digit divisors. When I first saw long division, it seemed (to me at least) to involve an unnecessary amount of writing.

Here in Ontario, not too many teachers seem to have seen 'short' division. It does require the user to mentally compute the remainder at each step (e.g. there are 6 sixes in 40 with a remainder of 4). Will this be tricky for students? I don't believe so, especially if we gradually build up the complexity of such questions.



Double digit division is problematic (unless the double digits are 'friendly': the advantage of working with single digit operators that standard algorithms usually have vanish when trying to do something like 7054÷82. Mentally, I'm trying to do 705÷82. It's doable for sure but potentially time consuming and open to error. Yet using partial products, a student can use friendly numbers:

Some students are fine at using the standard algorithm for double digits. For those who aren't, get them to try partial products; my experience is that this is a game changer for these students.

This concrete-diagrammatic-symbolic development of division takes a long time, years even. It is not to be rushed unnecessarily; I am not convinced that the best way of teaching students about the operation of division is to jump straight to the algorithm of division. 

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Recently, I gave a Math Night for parents at one of the schools in our Board during which I shared our Board's vision of what good Math education looks like. Afterwards, a parent approached me seeking some guidance as how he could help his daughter with long division. He said, "They have to do something like 'Dragons Must Suck Blood' and, to be honest, I don't understand what any of this means. And neither does she." As we chatted more, I gathered that the Dragons Must Suck Blood was an mnemonic to help students remember certain steps of the algorithm (Divide, Multiply, Subtract, Bring Down): clearly it wasn't working. At this point, rather than explain the algorithm in a different way, I showed him how I would develop the concept of division and how this needs to be in place before we try to make sense of the algorithm. It was neat seeing his eyes light up when I showed him the spokes method and the partial products methods and hear him say, "I actually understand those ways!"

How Many Eggs Does it Take to Make a 3-Egg Omelette?: Understanding the Problem

One of the things that we are actively trying to do in our Board is to get our students to fall in love with problem solving. But how can we achieve this? Simply giving students problems to work on isn't enough; if the students don't know how to solve a problem, then what, as their teacher, do I need to do to help?

Early in my career, I would 'help' students by basically doing the problem for them. So when a student asked me, "What do I have to do?", then I'd reply , "Oh, easy. Just multiply the 45 by 213 then subtract the original 1234." And on to the next student. Then the next. And so on, rescuing as many as I could. Now I realise that all I was doing was telling them what calculations they needed to do: I was not developing their problem solving skills.
I am at the point now where, whenever I solve any particular problem, I think very carefully about what I am doing to solve the problem, and how I can model these actions to my students. George Polya's four-step problem-solving model has been valuable in helping me think through this. Polya lists the four steps as:

It is important to realise that Polya's model is not a linear model. Instead, it should be expected that we go back and forth between the stages in order to make sense of the problem, and to try different strategies and to optimise our solutions. 

In a previous post, I mentioned at how students think they look back at the solution when in fact they do not. What has been of greater interest to me recently is this:

Do students really understand the problem?

And if they don't understand the problem, what do we do? This came about initially from giving the following problem to some Grade 4 students:

A farm packs eggs into boxes of 6 and boxes of 8. Every box they pack has to be completely filled. There are a 100 eggs to pack. Show how this can be done.

As a class, we read the question. We re-read the question. We spoke about how we can pack into boxes of 6, or boxes of 8, or both. We agreed that we understood the question. And so the students set out to solve the problem.
Or not, as it turned out.
Some engaged very earnestly in long division (100÷6=16R4, 100÷8=12R4) but didn't know why they were doing this. Others still added 100+6+8 to get 114. One or two did 100−6−8 to get 86.
One student had written 11×8=88, 2×6=12, 88+12=100. 'Great!', we thought and asked him how many boxes were needed. "Ermm, one hundred?" came the reply.

There were some correct answers. One student used a hundreds chart really well to count by eights to 96 (12 boxes) then realised this was not going to give a solution so took one box of eight off to get 88 (11 boxes) then counted up by sixes from there to get to 100 giving a total of 13 boxes. Another realised that 2 boxes of six and one box of 8 would pack 20 eggs. Repeating this five times would mean 10 boxes of six and five boxes of 8 would pack 100 eggs (or 15 boxes in total). These students had made sense of the question.

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Another colleague drew my attention to a question that stumped some of her students:

How many eggs does it take to make a three-egg omelette?

Were these students not understanding because they were reading the question too fast and had missed the all important 'three'? Or were they wondering what an omelette was (let alone a three-egg omelette)? When I tried this in a different class, I was actually pleased that one student asked me what a three-egg omelette was: if you know what this is, then you can answer the question!

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Another way to see if students understand the question is to give a nonsensical one. The following one was first used by Professor Kurt Reusser in his fascinating 1986 research (and later used by Professor Katherine Merseth in her 1993 research) and also referred to by Robert Kaplinsky in this post. I wondered how these students would react to it:

There are 125 sheep and 5 dogs in a flock. How old is the shepherd?

Now these students are very much urban so I wanted to make sure they all knew what sheep were, what dogs were, and what shepherds were. They assured me that they did! So on with the problem.

We saw all manner of answers: 

• 125+5=130
• 125-5=120
• 125÷5=25 (the most common)
• One or two said a random number like 63 or 41 and I wondered if these students were guessing because they knew that it was impossible to tell and they just wanted to give an answer.
• A few said that it was impossible to tell.

So what do we do with all this? Well, as a class, we asked them what information do we need (the shepherd's age) and then what information do we have that could help us get the age? At this point, most students began to realise that we had no such information therefore we couldn't answer the question. "It was a trick question!" one student asserted.

Yes. Yes, it was.

So after this class visit, myself and the three teachers I was working with began thinking about different types of questions that we could give our students. We came up with the following:

1) Not-Enough-Information Questions
A classic is 'How much does it cost to run a car for a year?' or 'How much does it cost to travel from Toronto to Halifax, Nova Scotia?' or Dan Meyer's Super Bear question. For this type of question, information is deliberately withheld from the students: they have to think about what the question is asking and therefore what information they need to get. Our thinking is that these tend to be open-ended questions.

2) Too-Much-Information Questions
These are questions such as 'How much does it cost a 40-year old to carpet a 10m by 8m by 3m room with square carpet tiles that are 30cm long and cost $5 each?'. Some students might realise that the 40 is a red herring but still might do something with the 3 in the 3 metres.

3) Just-the-Right-Amount-of-Information Questions
I sometimes call these 'Goldilocks questions', not too much, not too little, just right. One such example is 'If 2.4kg of potatoes cost $7, how much would 6kg of potatoes cost?'. These can be single-step or multi-step problems. Our feeling is that this type of question forms the vast majority of 'problems' that students are asked to do.

4) Self-Answering Questions
These are ones such as the 'How many eggs does it take to make a three-egg omelette?' from above, but also include: 'Which is heavier: a kilogram of feathers or a kilogram of stone?'; 'How long does a thirty-minute bus ride take?'; 'Who wrote Beethoven's Fifth Symphony?'; and one of my Dad's favourites which was 'If one-and-a-half herrings cost three ha'pennies, how much would six herrings cost?'

5) Nonsensical Questions
The 'How old is the shepherd?' is an example of these as is this one: 'A town was founded in 1847 and is at an altitude of 254m above sea level. What is its population?'

Part of the challenge in getting students to 'Understand the Question' is to get them to think critically about all the information that is given to them in the question. 

But what if students encounter nothing but 'Just-the-Right-Amount-of-Information' questions, especially if they are single step problems? 

Is there a real danger that our students will be conditioned to just take the numbers that are in the question and select an operation without doing any deep thinking? And if they don't do any deep thinking, how are we developing their problem solving skills? As part of our inquiry, we are going to make sure that we give our students  a variety of these types of questions. Then, if we ask the students right at the start 'Do you understand the question?', then we might get them thinking more carefully about the information that is contained within the question.

Our list is not meant to be exhaustive by any means, and it is probably the case that there are better ways of classifying problems already out there. We'd love to hear what you think about this and what strategies you have used to get students to understand the question.