Math Make-overs

Robert Kaplinsky wrote a post about open-middle vs open-end problems that got me thinking.
The punch-line is that there are simple make-over tricks that you can use to convert almost any problem into the type you need, whether closed- or open- (or half-open) middle, closed- or open-ended.

Note: some of this transformation thinking is clearly inspired by Dan Meyer's "remove the information" method.

I think this post is, mostly, intelligible without reading Robert's note, but everything will make more sense if you have particular problems (and problem-statements) in mind.

Closed vs open middle

Closed-middle problems tell the student how to answer the question, either directing them to a specific strategy or scaffolding a specific approach through explicit interim steps. Open middle problems leave it to the student to find (or select) their own strategy.

Perhaps there is a half-open-half-closed-middle where students are given a menu of strategies?

I conjecture that note that any problem can be made into either type. Specifically, the ones in Robert's post all can, including calculating 475/25. The main technique to change a closed-middle to an open middle is to remove any guidance about the strategy the students use to answer the question. Going in reverse is also possible (tell them what strategy to use, scaffold interim steps) but there are already too many closed-middle problem presentations, no?

Closed vs open-ended

Is there one right answer/end result regardless of which strategy is used? If so, then it is closed-end.

Again, almost any problem topic could be either closed or open-ended, depending on whether it has:
(a) how many interpretations are possible
(b) how much data is supplied and whether that data is consistent.

It is easy to be surprised by problems with multiple interpretations where we only expected one. Recently, Marilyn Burns wrote a post about fractions that seems to have one right answer, but actually depends on how we define our reference unit. Popular probability questions seem ripe for interpretation-based disagreements.

If the question has just the right amount of data or all the data is consistent, then it will have one common ending result and is closed-end. If there isn't enough data or "too much" (some data is inconsistent), then it suddenly becomes open-ended since students have to use other ways to fill in the gaps or make choices between inconsistencies.

In Robert's post, there is a question about hybrid cars that is open-ended because there are some assumptions the students need to make and there are elements in the data that aren't consistent, so choices need to be made. This could become closed-end by adding more data (removing the need for student assumptions) and doctoring the data to make it all consistent.

In contrast, his example of In-and-Out burgers could become open-ended by taking away data or making the data inconsistent. My favorite way to blow open the end would be to go way beyond the assumptions of simple extrapolation: let's order a burger with 1,000,000 patties! How would they price that?

By the way, you might think some questions simply can't become open-end. For example, calculate 475/25. Is it possible to transform this one?

Well...we didn't specify which base we are in. In base 10, of course, we get 19. In base 8, though, the answer is 17 with remainder 2! I'm sure cheeky students out there could find other innovative ways to interpret the question, if given a chance.

Which is better

Sure, we can change the problems from one form to another, but which is the better type of question? Open-middle, open-ended, of course!

Just kidding. All of them have their place and it depends on what you are doing and why. Though it conflicts with my personal preference, I even see a place for closed-closed questions where you want the kids to practice a particular strategy (closed-middle) and you need the consistency of a closed-end answer to quickly check that everyone has gotten to the same result.

Open-middle, closed-end can be good for generating discussions that compare and contrast strategies. That's logical, since the strategies are where you would (should) find differences from these problems. However, aren't some of the kids thinking: "we all got to the same place, I don't care that someone else took a different route." That was something I often thought as a student.

When you have enough time, I think open-open questions create really rich discussions that include strategy comparison. Our typical conversation is something like:

• Hmm, we got different answers!
• What did you do?
• I used method A and data Z
• Oh, I used method B and data Y
• If we used method A with data Y, what would happen? Is that even possible? Why or why not? Would we get the same as (A, Z)?
• If we used method B with data Z, would we get the same answer as (B, Y)? etc etc

You can see I have a bias for open-open questions. One last reason: kids get a lot of closed-end questions already, so I don't feel that I need to add more.

Factors and division

who: grades 3 and 4 at Baan Pathomtham
where: in school

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Sorry about the lack of pictures. This is a short and sweet note.

Dots & Boxes and Factor Game Mash-up

To start the year, we played a version of dots & boxes that integrates the factor game (here is one example). This is based on the game template from Mathified Squares Game that we used last year.
Instead of using dice to determine where each player can play, we introduce factors 1 to 6 at the bottom of the page and selectors.

As with the basic factor game, this version creates multiplication and division. This is the point we want to draw out for the game.

Homework

Play the game at home and write down 15 division equations that come up in the course of play.

*UPDATE* Having now played through this game fully, I really like this structure. Using the factor selectors drives some interesting thinking about common factors, especially during the middle and end-game phases.

We did find that it starts a bit slowly as players can make moves on distant parts of the board and decisions don't have clear connections to capturing squares. For the young kids, we recommend just pushing past that stage. For older kids, that can be an interesting (and difficult) strategic analysis.

Build the chair spatial reasoning (Gr 1 and 2)

who: Baan Pathomtham grades 1 and 2
where: in school

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Here in Thailand, summer is over and we are back to school! We are kicking off the math games and exploration class today with an activity from NRich (chairs and tables) that has a surprising depth. Also, keep your eyes opened for the hidden reasons why we are starting with this activity.

Build a chair

The starting directive is simple: use unifix cubes to make a chair. Here's an example, from NRich:

To start, we ask the kids to get 15 cubes each. What does 15 mean? How do they know they've got 15? Do they think they will need more or less than 15 to make a chair?

After they have built their chairs, how many did they need? If it was less than 15, how many are left over? If it was more than 15, how many more did they need to add?

Chairs for bears

Once we've all got one chair, can we make two more for the three bears from Goldilocks and the 3 Bears? We need a small one for baby bear, a medium sized one for mama bear, and a large one for papa bear.

As the final construction challenge, we ask them to make a table sized to accompany their original chair.

Homework

1. Based on the pictures above:  (a) find out how many cubes would be needed to build these shapes, (b) draw a 2d perspective of one of the shapes from one direction.
2. For those who have construction sets at home, try making chairs of different sizes. What things were similar to using the cubes at school, what was different?

Extension

Building off the three bears activity is a nice extension:

• What is the smallest chair we could make? How many cubes do you use?
• How would you make the next larger chair? The next chair larger than that? How many cubes are used for those?
• What about the tenth chair in this sequence? What would it look like? How many cubes would we use to make it?
• Same questions for the 100th chair?
• What is an equation for the number of cubes in the nth chair?

I'd note that these are challenging questions which go well beyond first and second grade. Also, there is no single correct answer, particularly as different students will have different ideas about what is required to be a chair or how the form should grow through the sequence.

Note: these questions follow the thinking of Fawn Nguyen's Visual Patterns.

Hive at the beach (quickie)

Have been at the beach for the last couple of days before school starts. We got to play with one of my recent impulse buys: Hive (carbon).

Chance for discovery

As an experiment, I started by laying out the pieces with the blank sides up. One at a time, I asked the J's to come in and tell me what they noticed. For each of them, at some point, there was a moment when they tipped or turned a piece over and discovered the insects. Their reactions were a real delight and they realized it was special feeling, so quickly helped rearrange the pieces and get another sibling so that they could have the same experience.

Making patterns

Having started with the blank sides of the pieces, it felt natural to the kids to sometimes play with these tiles just to make patterns. Here is one example:

Playing the game

Of course, there is also delight in playing the game itself:

From this detail, you can see that J3 (playing black pieces) is following an unorthodox strategy. At this stage, she is really learning the rules and sees the whole activity as a strange way to play together to create unusual patterns and combinations of the bugs:

Reversed inequality

Recently, Mike Lawler posted a challenge base on the first question from the recent European Girls Math Olympiad (side note: what is "European" about this contest now that a US team participates?)

I enjoyed thinking about this problem and wanted to come up with something related to do with our kids. One way to see this problem is that it appears to reverse the direction of the inequality between the arithmetic mean and geometric mean. My version for younger students involves exploring this inequality first.

Discovering the Arithmetic Mean-Geometric Mean Inequality

I had J1 and J2 select two dice from our pound-o-dice. Initially, J1 chose a d20 and d6 while J2 chose a d12 and d6. I asked them to make a table with 6 columns and 7 rows (we ended up adding more, so probably better to ask for 8 columns and 10 rows). I had them label the columns:

1. A
2. B
3. AxA +  BxB
4. 2xAxB

They rolled their two dice and put the values into columns A and B, the calculated the other two columns as labeled.

J1's first results were 16 and 5. After filling in the rest of that row, he paused and then switched to 2d4. This wasn't a problem for our investigation, but he did miss some calculating practice with more difficult seed numbers.

After filling in 6 rows of data, I asked them what they noticed.

Here were some of the observations:

• when A and B are the same, our calculated values are the same
• when A and B are different, our calculated values are different
• When A and B differ by 1, the calculated values differ by 1 (also vice versa)
• AxA +  BxB > 2xAxB
• We are only using positive integers

J2 really got into the spirit and asked for some more suggested columns. I told him to add:

AxA - BxB and (A-B) x (A-B). J1 also added the square of the difference.

With that, they noticed two more things:

• AxA + BxB was larger than (A-B)x (A-B)
• AxA + Bx B = 2x Ax B + (A-B)x(A-B)

You can see that they also added some negative numbers on J1's sheet and challenged one of these observations.

A picture proof

To finalize our exploration of the inequality, we used tiles to build intuition for a picture proof of this identity:

AxA + Bx B = 2x Ax B + (A-B)x(A-B)

This is one of the pictures we liked the most. In this case, A = 5 and B = 2. Along the left side and the bottom are rectangles AxB (the green + yellow and the blue + yellow regions). These two rectangles overlap in a BxB square (the yellow region) which they also highlighted with the 4-color square. The remaining red square is (A-B)x(A-B).

Side note: In our discussion, I was delighted that J2 would correct me whenever I slipped and said "greater than" instead of "greater than or equal to" when discussing the inequality.

Two quick algebra extensions

For kids who have already done a little algebra, proving the identity using the distributive law should be fairly straightforward. The other little extension is to show that our expression A²+B²≥ 2AB is equivalent to the arithmetic mean-geometric mean inequality.

4 Coins challenge

Recently, we were at lunch and the kids asked for a challenge to entertain themselves. I had just read this Numberplay column and had Lora Saarnio's Four Coins Problem in mind.

The basic rules are to choose values for coins in a new system so that any value from 1 cent to 10 cents can be made with a single coin or with two coins.

For our older two, this was a great on-the-go puzzle without pen and paper. Later, we got into some extension questions that required more careful organization of their observations and attempts.

Mostly, I want to report some of the extension questions that we found interesting. The key point: this problem is very accessible (J3 could also work on a simplified version) but it is ripe for a Notice & Wonder discussion that will reveal increasingly complicated and challenging extensions.

Note: we used names of neighboring countries, but none of the claims in these scenarios are true (as far as we know).

Number fussiness
In Burma, the president really likes the sequence 1, 2, 3. Is there a coin system we can suggest that includes coins with all three of those values?

No duplicates
We noticed that, for all the coin systems we created, there were always some values that were duplicated. What we mean is that there are two ways of using the coins to make that value. For example, in the coin system 1345, the following values can be made two ways:

4 = 4 = 1+3
5 = 5 = 1+4
6 = 3+3 = 1+5
8 = 4+4 = 3+5

Is there a coin system where all the values can be made only one way? If not, why not?

Two rival countries
Cambodia and Laos really love to compete with each other. As a result, they want coin systems where there aren't any shared coin values. For example, if Cambodia has a coin with value 5, then Laos can't have a coin value 5.

Is it possible? If not, what is the smallest overlap possible and how many possible coin systems achieve this smallest overlap?

Note: when we started this investigation, we had already designed a new system for Cambodia, so our starting question was whether there was a new system for Laos that wouldn't overlap this particular system for Cambodia, how much overlap was unavoidable, and then how to change both systems to minimize the overlap.

More values!
If we want to extend to amounts up to 20, how many coins are needed to have a compliant system (any value possible with only 1 or 2 coins)?

We haven't pursued this to a final answer, but the J's quickly recognized that, for any compliant coin system covering up to 10, they could create an 8 coin system that would cover up to 20. However, they also realized that four coins would be too few. This means that the minimum required to cover up to 20 is 5, 6, 7, or 8.

Coin triples
In the More Values! extension, we allow more than four coins in the system. The other way to relax the original constraints is to allow more than 2 coins to build a value. What can we achieve if we allow three coins at a time? Could we use fewer than four values in the system and still cover all amounts from 1 to 10? What about 1 to 20?

Chocolate cake reference

This was Nigella Lawson's recipe, but that page currently doesn't include the ingredient amounts, so I'm reproducing it here. This was a test ahead of J3's birthday, so she helped. Since it seems successful, we will be making it again and I will include more pictures of the process.

Heat oven to 170 deg C

MIX:

• 3/4 cup + 1 tbsp flour
• 1/4 tsp salt
• 1/2 tsp baking soda

1/2 cup water (boiling)

6 tbsp cocoa powder

COMBINE into a thick paste and allow to cool (doesn't need to come fully to room temp)

2 tsp vanilla extract

ADD to cocoa paste and stir

2/3 cup olive oil

1 cup sugar

3 eggs

COMBINE in mixing bowl and mix on high (directions said 5 minutes, we mixed for 2-3)

COMBINE wet ingredients and mix thoroughly

COMBINE dry mix and wet just to incorporate together.

Use small amount of oil to grease a 9 inch baking pan.

Pour in batter and bake for 45 minutes (I rotated the cake every 10 minutes).

Another dessert reference to store, unfortunately dairy: Caramel Sauce