Mathish-ish

 Read Jo Boaler's Math-ish and thought it would be useful to pull out the valuable points and provide an overview of the rest.

Encourage Metacognition Through Eight Mathematical Strategies

This is from page 39-45. I'd note that this set of ideas is somewhat typical for the book: though described as methods to encourage "metacognition," these are closer to problem solving strategies.

This list is populated with ideas that will be familiar from other standard sources, like Polya's How to Solve It or Cuoco's Mathematical Habits of Mind. Nonetheless, these are still solid cues for a teacher/tutor to either help with the problem solving process (1, 2, 3, 5, 8) or to move beyond simply finding a right answer (3, 4, 6, 7.)

1. Take a step back
2. Draw the problem
3. Find a new approach
4. Reflect on "why?"
5. Simplify
6. Conjecture
7. Become a skeptic
8. Try a smaller case

Reflection cues

From page 48, a quasi-infographic with seven cues that are actually directed toward metacognition. 

1. What mathematical concepts did you learn today?
2. How is the idea you learned today related to others you have learned?
3. What opportunities did you get to struggle? How did that feel?
4. How could you use the mathematical concept in your life?
5. What different strategies or approaches to the problem were helpful to you?
6. Are there areas that you do not understand and would like more opportunities to learn?
7. Can you write your own problem for someone else to try to solve?

Boaler suggests asking students to answer one of these questions, of their choice, in place of standard homework problems. From my personal experience, it takes consistent effort to get students to take these reflection questions seriously.

Group work

Initially, I noted two blocks in this section (pages 52 and 53): a set of 5 roles during group work and 8 "mathematical ways of working." On returning to this discussion, I was more struck by the things that Boaler does not cover related to group work. In particular:

1. The critical importance of finding activities that truly require group engagement and cooperation.
2. The typical level of frustration that most students associate with their prior group work experiences.

For example, of the 8 "participation quiz" behaviors, only two of them somewhat depend on the group (underlined and bold):

• Recognizing and describing patterns
• Justifying thinking using multiple representations
• Making connections between different approaches and representations
• Using words, arrows, numbers, and color coding to communicate ideas clearly
• Explaining ideas clearly to team members and the teacher
• Asking questions to understand the thinking of other team members
• Asking questions that push the group to go deeper
• Organizing a presentation so that people outside [the group] can understand your [group's] thinking

Struggly

Boaler advertises a website, Struggly, that seemed worth investigating. It maybe hard to get a full sense from the demo activities, but I didn't see anything either bad or special here. If other people have used it, please let me know what is unique about Struggly (in comparison with ST Math, for example.)

Integrating math history

Boaler cites (page 86) several aspects in the history of Fermat's Last Theorem and Andrew Wiles' proof:

1. The vast amount of valuable ideas that emerged in the course of attempts to prove FLT
2. The fact that Wiles' initially announced proof had a gap
3. The way that professional mathematicians persistent through their work over extended periods of time

Personally, I do find it tempting to incorporate biographical and historical information in math classes, but I don't think I have found a way to do this that fully resonates with students or has the effect that I want.

What matters in pre-college mathematics

I'm on board with the idea that only a small number of key concepts from pre-college mathematics are really critical for students and a sharp distillation is helpful. Starting on page 95, Boaler identifies three candidates. For what it is worth, the source is David Coleman, the CEO of the College Board. In general, the College Board doesn't seem a force for good in education, but these three candidates seem plausible.

Number sense/arithmetic

An ability to calculate with basic arithmetic operations, an understanding of fractions, and a sense of estimation. Here is where Boaler really hits her "ish" concept, an acknowledgement that almost all real-world incarnations of mathematical concepts are approximate, rather than exact.  I would prefer to broaden this a bit and say that strong number sense should include an ability to switch between the approximate and precise and a recognition of the differences between them.

Data literacy/Data analysis and problem-solving

In this section, Boaler rehashes the (now) obvious point that data is abundant and students/citizens should have some facility to interpret and analyze data.  Some nice visualizations are included:

• Stephen Curry's 2015-2016 shot performance (page 124)
• NCAA Women's Soccer PSxG for penalty shots (page 125
• Examples from dear-data.com
• Student data representation from the student's life (page 127)

The youcubed data science course: https://hsdatascience.youcubed.org/curriculum/

I will try to find time to review that specifically.

Linear Equations

This section does not really justify why linear equations have a distinguished place in the top 3 concepts.  I may return to this to back-fill potential reasons, but most of what is cited (pages 129-131) is spurious, rather than real linear relationships.

Fractions

Visual representations of 1 ÷ (2/3):

In several parts of Math-ish, Boaler emphasizes the importance of understanding fractions, in contrast to a rote/algorithmic approach to calculating with them.  I think these visual representations are the best thing she offers as a step toward ways of understanding.

A context investigation

Note: I drafted this a while ago and never finalized the post. Reading it again, it seems fine and maybe interesting without additional work, so I'm publishing it.

Fake Math Models

Robert Kaplinsky wrote a note recently discussing fake math models and unnecessary context. This prompted an activity with the kids.

This issue seems to have come up a lot recently, so I've noticed a pattern: I really hate bad contexts.

Robert wrote: "it looks like the context is completely unnecessary to do all of the problems."
I would go farther: this context is harmful. The context creates a conflict between the specific new material (rational vs irrational numbers) and other important concepts (measurement and measurement error). Subtly, we are discouraging students from
(a) forming connections across topics. For my taste, surprising connections has to be one of the most beautiful and delightful aspects of math.
(b) using all of their ideas and creativity to understand a challenge.
(c) putting new mathematical ideas into a broader mathematical context (maybe I'm just repeating point a?)

I admit that the example only touches on these points lightly, but I suspect the accumulated weight over the course of a school math education is substantial.

If I were full-time in a classroom with a textbook, I'd be tempted to use it as follows:
1. create censored versions of all problems and examples (as you did)
2. work through the questions with the kids
3. Ask them what context they think the publishers originally included and why
4. show the published version
5. discuss (does the published version relate to the math, does it help them understand, does it add confusion, does it conflict with something they know, etc)

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

ideas for upcoming classes

warm-ups for all

WODB: (1) shapes book (2) wodb.ca

any: Traffic lights/inverse tic tac toe/faces game

good options here, mostly grades 1/2: some games

dots & boxes (maybe with an arithmetic component)

loop-de-loops

Grades 1 and 2

31 Game: could be used by any grade
Integer addition game 
jump to 50, chain of changes puzzles

close to 100 game: 

Equipment: A pack of cards with 10 and face cards (J,Q,K) removed.

Procedure: 

- Deal out 6 cards to each player

- Each player picks 4 cards from the 6 cards they were dealt to form a pair of 2-digit numbers.  The goal is to get the sum of the two numbers as close to 100 as possible but cannot exceed 100.

3 chips puzzle

Grades 3 and 4

Factor finding game (maybe warm-up?)
Factors and Multiples game
Contig for 3 and 4 (explanation).
Times tic-tac-toe: review for Grades 3 and 4
Fraction war for grade 4 (smallest card is numerator)
Multiplicaton models: worth making for grades 3-4 for solidifying concepts? Associated games
d 2

card subtraction

factor game

card on head game

Pattern revealed

Calendar tricks

Pico Fermi Bagel

Magic triangle puzzles

damult dice

4 dominoes puzzle

tables and chairs

(1) Dice game perudo

Equipment

- multi-player, 2-5

- Everyone gets the same number of 6 sided dice (full game they get 5, I would start with 3)

- Everyone has a cup to shake and conceal their dice

Basic Play

- Simultaneously, players shake their cups and turn them over on the ground or a table. They peak in to look at their own dice, but keep them concealed from the other players.

- starting randomly (or from the person who lost a dice in the last round), players make bids, for example: two 3s. 

This bid signifies that the player has 2 (or more dice) showing the value 3.

- the next player has two choices: 

• call/doubt the previous player's bid: if they do this, all players show their dice. If there are enough to meet the bid, the caller loses a die. If not, then the bidder loses a die.
• raise, either the number of dice or the value or both get increased 

Advanced rules

- Ones are wild, they count as any number toward the target bid

- If someone drops to their last dice, they start the next round. On that round, only the number of dice can be increased in the bid, not the value. Ones are not wild on this round

- After someone bids, the next player has a third option, to call "exact." If the bid is exactly matched by the dice, then the bidder loses a die and the caller gets an extra one. If the actual dice show either more or less than the bid, the caller loses a die.

remainder jump

we played this game before, but we could give them blank boards and let them create. See the last page here: http://ba-cdn.beastacademy.com/store/products/3C/printables/RemainderJump.pdf

https://www.beastacademy.com/store/products/5B/printables/GCF_LCM_Webs.pdf

another good game:

https://saravanderwerf.com/2015/12/13/5x5-most-amazing-just-for-fun-game/

We should still use the pyramid puzzle sometime:

http://mathforlove.com/lesson/pyramid-puzzles/

(1) double digit and double dollar:

We've done something like this, but I think there could be a good variation done trying to make 1000 baht, using 1, 2, 5, 10, 20, 50, and 100 baht units.

http://mathforlove.com/lesson/double-digit-and-dollar-digit/

(2) biggest rectangle. This could be used as a warm-up. For the older kids, they will probably have seen something like this, but I like the inclusion of perimeters that are even but not divisible by 4 and odd perimeters and the question about "smallest area" (here are 5 questions).

http://mathforlove.com/lesson/1009569184/

(3) some of these games are promising:

https://danburf.files.wordpress.com/2014/11/tapson-games.pdf

http://mathforlove.com/lesson/pyramid-puzzles/

Closest neighbor one-on-one

In my last post, I wrote about playing Denise Gaskins' closest neighbor fraction game with our 4th grade class. Yesterday, I spent time with J2 and used the game as a semi-cooperative puzzle.

This activity worked really well and the experience gave me some additional ideas about how to use the core ideas again with the 4th grade class.

Puzzle or game?

First, there were only two of us, one a kid and another an adult, so that background naturally makes the activity very different. As the key modification for play, we played all of our hands open and helped each other find the fraction in each of our hands that was closest to the target for that round. Then, we worked together to determine which of those two "champions" was closest overall.

Some of the consequences:

• the activity was not really competitive (see below)
• J2 had to do a lot more fraction work.

Let me explain the second point here. Because we were looking for the best play, J2 had to consider all of the combinations in his hand (20 choices). Some of those can be rejected quickly with simple analytical strategies depending on the target. Even this is good number sense thinking. Also, some combinations are close competitors and need to be analyzed more carefully.

If we were playing with closed hands, he could choose two cards, play a fraction based on them, and I wouldn't be able to say anything about whether those were his best options or not.

Second, while I write that "we worked together," as a sneaky dad, that means that I pretended to do work, while actually getting J2 to analyze my hand as well as his. Really, the only thing I offered was an alternative comparison strategy, once he had already worked through his own approach.

An example of some strategies

We found that some of the comparisons that arise naturally in this game are quite tricky, even for me. For example, quickly tell me which is closer to 1/3: 1/5 or 4/9?

We found that placing the fractions on a number line was a really helpful strategy for many of the comparisons. We also made very heavy use of the two strategies involving common numerators or common denominators.

Finally, you can see in this example that J2 is comfortable mixing decimals and fractions, for example converting to 1/2 to 3.5/7 to aid some comparison:

Our grid

Through our play, we filled out this grid, taking turns putting in our best results and congratulating each other when our hand was the ultimate champion for that round:

Competition and Strategic thinking

I was particularly pleased by one comment J2 made about this overall game: "this is mostly luck, how well we can play depends on the cards we get." This comment came after one round where he had several duplicate cards in his hand, reducing the number of distinct values he could play. We've discussed elsewhere my goals of helping the kids think about game structure, so I always love it when they bring those ideas up themselves.

Some thoughts about competition. While we played this game non-competitively, I'm not opposed to competition nor do I think that this game always needs to be played non-competitively. Ultimately, my litmus test is how to play in a way that is the most fun. If I were a more serious educator, I suppose I would also consider which way is the most educational, too.

It won't always be obvious what is the best way to play each game. In this case, I got to benefit from the prior experience with the class and my close knowledge of J2. Many times, I'll tell the kids that there are several ways to play and we'll try them out together, then review the experience.

Among other things, this is why I love handicap games like Go. By adjusting the starting advantages, we can create scenarios where it is very competitive and very fun, even though the players have very different levels of experience and current strength in the game. And also, there are things we can do together when we want a non-competitive activity.

Ideas for going back to class

From this time with J2, here are my ideas about taking the game back to the 4th grade class are:

1. Spend a lot of time on fraction comparison strategies before we play
2. Reduce the number of cards dealt to each player
3. play as teams
4. convert to open hands with a lot of talk about why we chose particular plays

An actual puzzle

As a reward for reading down this far, here's an actual puzzle related to the closest neighbors fraction game:

During the round where the target is 1/2, Jay plays 6/6 = 1. Was that her best play? How do we know?

Remember, we are playing our game with a single deck of playing cards and each player is dealt five cards

The target for the next round will be 3/4

Burning 2 identical values might allow Jay to increase the diversity of her hand, especially if she happened to have 3 or 4 sixes.
Hey, life doesn't promise that all puzzles will have solutions that can be wrapped up in a nice neat package, does it?

My Closest Neighbor Fraction game

Denise Gaskins recently flagged a post about a good fraction game: My Closest Neighbor. I tried this out in class today.

A pre-test
First, I wasn't sure whether the level of the game would be right for the kids. I was considering it for the 3rd and 4th graders, but had some alternative activities planned in case. To start, I posed the following questions:

• Which is closest to one-half: 1/3 or 2/5? The third graders really struggled with this, so I left it alone and went to my plan B games. The fourth graders were all confident on this one.
• Which is closest to 3/4: 5/11 or 11/12?  This was a challenge for the fourth graders, but I thought it would be ok to play the game.

In our discussion of the second question, we explored two strategies:

1. making a common denominator
2. comparing with reference numbers

The common denominator is a bit of a pain, since 11 is prime, though at least we have the fact that 4 is a factor of 12. One student soldiered through this approach, but it was difficult for the other kids to follow.

For the second strategy, we made use of some observations that were more elementary for the kids:

(a) 5/11 < 5/10 = 1/2

(b) 3/4 is halfway between 1/2 and 1

(c) 11/12 < 1

Combining these, it was easy for us to draw a rough number line, place 5/11, 1/2, 3/4, 11/12, 1 and see that 11/12 must be closer.

The game
We played three rounds: target 0, target 1/3 and target 1/2. I think this game was very challenging for the kids.  Everyone had to work to figure out the best play from their hand and didn't always make the right (local) choice. For example, whether to choose 5/8 or 5/9 for a 1/2 target.

Once everyone had played, the challenge was still just starting. They had to figure out who was closest. I structured the discussion by helping them figure out which plays were lower than the target and which were higher. For the ones that were lower, they could put them in order and only needed to consider the highest. Then, we worked on the ones higher than the target and got the lowest of those.

In the course of this discussion, we added a third strategy to the ones listed above:

• making a common numerator

Summary thoughts

Fraction comparison like this was still too difficult for the kids to make an engaging game. If I were to do it again, I would change to make it more of a puzzling exercise, removing competition and any sense of time pressure.

Once the kids gain a bit more experience, though, I think this game has some nice features. It is particularly good for practicing fraction sense, and the multiple rounds allow some scope for strategic play.

Our math curriculum

Sasha Fradkin (who writes one of our favorite blogs) asked a question about the curriculum we use. My reply was getting long, so I decided to make it into a separate post.

Do we use an existing curriculum or are we making our own?

We are doing a mix. My wife prefers to have a linear curriculum as a guide and fall-back, in case there wasn't time to plan anything more customized. She currently uses:

• RightStart/Abacus Math: ok, but not exceptional curriculum, highlight is the extensive use of physical manipulatives.  
• Beast Academy workbooks: wrote more extensively about this in a review before. I think these create good jumping-off points for really fun conversations. 
• DreamBox: for consolidation of standard skills, our enthusiasm for this is waning, rather than waxing right now (noted in same review as BA). 

We also use the CCSS math standards as a reference. I periodically check against the standards to see whether we are missing anything. If so, I will go to the Georgia Standards of Excellence, read through their activities for the related unit, and pick a couple that seem fun.

If I were forced to use only a single source, GSE would be my recommendation.

About Georgia Standards of Excellence
As far as K-5 math, this is a really awesome resource with a ton of great activities. For some reason, we find the webpage organization a bit confusing, so here is our quick recipe.  

To get to the great activities, I click the expansion menu in the right-hand box for the grade level of interest, then look at the curriculum map for that grade. I find the topic of interest, then click the link for the unit that covers that topic. The unit doc includes a lot of teacher background, which I mostly skip and focus on the activity descriptions.

Games and explorations
My personal preference is much less structured. I really like games and explorations and spend a lot of time exploring math activities on the MTBoS. Most of what I do with the kids is inspired by something I saw while doing my own play.

That said, there are some sources that are so good, we are essentially going through all of their activities:

1. Mathpickle. Cannot say enough positive about this.
2. Peter Liljedah's numeracy activities. Now that I notice them, I bet his good problems, card tricks, and resources pages will all have gems as well. 
3. NRICH and Wild Maths.

Teaching goals for English Language Arts

These are our current objectives for English language study with our children. These are very high level goals, but we feel it is important to write these down so that they can guide our detailed choices. Ultimately, our hope is to guide the kids to be independent learners.

Strong Readers (in-bound communication)

• Able to use reading as a tool for further learning. This means they must:
• Enjoy reading
• Have a large vocabulary and tools to build their vocabulary
• Good comprehension and tools to analyze what they are reading
• Familiarity with sources of information
• Read a wide variety of material: topics, authors, styles, forms

Writing (out-bound communication)

• Able to communicate ideas clearly and effectively (writing and speaking)
• Comfortable with the mechanics of writing: vocabulary, spelling, grammar, punctuation, physical writing and typing
• Learn a writing process: research, brainstorming, forming ideas, organizing ideas, drafting, revising

Conduit for other content

We will make use of language activities that also teach them:

• History: having data of history to learn from the past, ideas of historiography and perspective
• Science: technical jargon, tools to understand and develop scientific frameworks
• Philosophy and comparative religion: what are the great questions, different perspectives, forming their own values and understanding those of other people
• Current events: understanding the current context of their lives

Develop skills that support other language learning

• Grammar frameworks
• Methods for learning vocabulary
• Motivation

A talk for parents about math at our school

Today, P led a session for the other parents at the school. We wanted to share the material and some links for those who weren't able to attend.

Agenda:

1. What does it mean to be good at math? What are we trying to achieve?
2. Key concepts we are using in teaching: Concrete-Pictorial-Abstract, Concept Progressions
3. Examples
4. What should parents do at home?

Good at Math

To start, P asked the parents, what does it mean to be good at math? Some of the answers:

• can add, subtract, multiply and divide
• calculate quickly
• able to estimate 

The range of opinions was good to see. Many ideas fell within the traditional answer: being good at math means being able to calculate precisely and quickly. We were especially pleased to hear skill at estimating as one of the ideas.

Our additions:

• Thinking logically. For example, if a certain thing is true, what else is true? 
• Looking for patterns and relationships; forming connections with other things they know.
• Asking questions about what they see, especially investigating structure
• Persevering

Admittedly, these are necessary for many other subjects. Math is a particularly good place to develop these skills because there is much greater objectivity and right/wrong are often clearly distinct. In this domain, the power of reasoning and independent thought is stronger than the power of authority.

Process of Learning Math

For this discussion, we focus on two key concepts in the way we teach and study math: (a) the Concrete-Pictorial-Abstract modes and (b) multiple models in progression and contrast.

C-P-A

"Concrete" means using physical objects. For example, a pile of 15 beads can be a concrete representation of the number 15. Taking 2 beads in the left hand and 3 in the right hand, then combining them is a concrete experience of adding 2 and 3.

In this mode, children are able to see, touch, move, examine, smell, and hear mathematics.  

"Pictorial" shifts to pictures on the page or board. For example, a picture of a room showing a vase with 2 flowers and another vase with 3 flowers can lead us to identify 5 flowers all together.

In this mode, children are able to see, construct (by drawing themselves), obliterate (by crossing out), and add color (by coloring, naturally) the mathematical objects.

"Abstract" is where we shift to symbols. For example, 2 + 3 = 5. This is a sequence of 5 symbols that don't offer any clues to their own meaning.  

In this mode, children are able to imagine and to move beyond physical constraints or necessities.

When new concepts are introduced, they generally go through each stage, starting with concrete, then pictorial, then abstract. This doesn't mean that abstract is superior, however. The ability to move back and forth, to give specific examples, to draw diagrams, to demonstrate a concrete model is also very important.

Multiple Models

Complementing the three modes, we also try to have multiple models, different ways of seeing, new concepts. There are two great resources that nicely illustrate this.  First, the models of multiplication posters from Natural Math:

4 of 12 models at Natural Maths

For the discussion, P gave examples of the equal groups model (sets per each), repeated addition, array, number line, and area.

I strongly encourage you to take a look at all 12 models in their poster, so here's the link again:
http://naturalmath.com/2013/09/12-models-of-multiplication/

The second resource is Graham Fletcher's series of progressions videos: Addition and Subtraction,
Multiplication, and Division.

For this talk, we presented abridged versions of the content in the multiplication and division videos. For your ease and viewing pleasure, here they are.

and the division video:



What to do at home
1) Ask questions
Parents can relax about being the source of knowledge. Don't worry about "teaching" or having the right answer. Instead, develop habits to cue thinking and their use of problem solving strategies:

• How do you know?
• What pictures could help us?
• What do you notice? What do you wonder? This works especially well if the parent serves as scribe writing down the kids' ideas.

"How do you know?" does three things. First, it is one way to escape from the child's questions "is this right?" Remember, the power of reasoning is stronger than the power of authority. We want to reinforce this by side-stepping calls to authority.

Second, it is a cue to get them thinking about their own thought process, which aids learning.

Third, this opens a potential discussion about different ways to attack the problem. Comparing and contrasting multiple strategies is a powerful tool for deeper learning.

"What pictures could help us?" is a cue to move between the Concrete-Pictorial-Abstract modes. If available, go to Concrete by asking about objects or physical models that are related.

"Notice & wonder" is a deep topic. One key idea is that, by serving as the scribe, we parents demonstrate that we care about the ideas that the kids have, we know they can contribute to solving the problem. This also gets us listening and understanding their perspective.

Notice & Wonder also involves skills that the kids will strengthen with practice, starting with superficial or (mathematically) irrelevant ideas and eventually moving on to thoughts about patterns and structure.

 For more about notice and wonder, please take a look at Annie Fetter's talk, 

2) Talking numbers to develop number sense
Two examples of number sense. Say we bought 8 bags of snacks and each bag cost 17 baht:

• I know that the total cost is not 1000 baht based on understanding order of magnitude.
• I know that the total cost is not 137 baht based on the pattern that all multiples of 8 are even

Like learning a language, number sense takes practice, it requires frequent exposure, and is built up by drawing children's attention to numerical and mathematical ideas.

Specific activities to do at home include estimating and measuring (length, time, weight, volume, etc).

Make math a part of everyday life by asking questions about what you see around and asking them to find examples of the concepts they are currently learning.

3) Play games
We play a lot of games at school and ask the kids to play them at home with their parents as homework. The games don't go stale when we move on, parents can play old games again. Some games are very calculation heavy. These are great opportunities to flex the Concrete-Pictorial-Abstract muscles.

Other games (or explorations) are much more about strategy. These are a great place to practice the other questions, especially "what do you notice?" and "what do you wonder?"

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.

Polishing a ... (trying to fix a textbook lesson)

Graham Fletcher called out a bad textbook in his recent post, Placing a hit on Pseudo-Context.

He was inspired to create a related lesson. Here is another, super low effort attempt to salvage something from the original textbook. What do you think?

1. What is our scenario?

Okay, obviously, this sets us up to make boxes to hold DVDs. What do the kids think? Why is Mike doing this? What should he consider?

Seems natural that the teacher should have on-hand a bunch of DVDs and construction material.

2. So, Mike made some progress constructing his box:

Some questions for the kids to debate:

• What shape did Mike choose?
• What dimensions?

3. Here are the choices Mike made:

• What do you notice? 
• How do his choices compare with the ones you made?
• What do you wonder ?

*UPDATE* I talked through this post with J1, my 8 year old. Here are some of the things he explored and talked about:

• How would the DVDs inside the box be arranged? Stacked on top of each other, multiple stacks, with some type of dividers between them or not?
• How big are DVDs? He measured the diameter of one, then stacked 10 (in protective covers) to measure thickness.
• How many DVDs does Mike have? How does the desired structure of the box change if there are 5, 20, 100, 1000 DVDs? This also linked to practical considerations around finding and extracting a particular DVD from the box and what would be suitable materials for the box (paper, cardboard, wood).
• Since he measured our discs in cm and the book had inches, we had a conversation about conversions.
• As I'd guessed, getting him to think about the dimensions of his own box before seeing the textbook dimensions made him wonder about symmetry.
• Note: this prompt really didn't get him to think much about the volume of the box. I think this is because there isn't really a natural trade-off between the dimensions for DVDs, so it is natural to think of the base area compared with a single DVD and the height based on the number of DVDs. Graham's sugar cubes was much more direct for engaging the concept V = L x W x H.

Random 100 board and some subtraction review (math games)

In grades 1 and 2, we (re)introduced the card subtraction game from one of our earliest classes.

Since this is our first class of the new (calendar) year, we also did one of the calendar tricks again in all of the classes (the first one on this page).

Repeating activities is something that we rarely do. Usually, we have been introducing a new game or activity each class, possibly following up on that activity in the following week. However, we realized that it takes more time for the kids to extract all the ideas or practice. Also, since these things are often still challenging, their interest level is high.

Four in a row

PK came up with this game, but it is based on an idea she saw elsewhere. If we recall the original source, I'll link to give it credit.
Players: 2
Material: randomized 100 board (we used 0-99 randomly arranged on a 10x10 grid); 4 dice (we used 4d6)
Turn: Player throws the dice, then combines the values using arithmetic operations to create a final result. They mark the corresponding cell in the grid with their color.
Winning: The first player to get four adjacent cells in a straight line wins (NB: diagonals also allowed)

We also asked the kids to write down their number sentences for each move, below the playing grid.

Here are the 4 grids we used (random order created using pencilcode similar to the 6th graders final assignments last term!) There's nothing magical about these, but we wanted to mix up the "easy" and "hard" numbers as well as give some additional variation for each play.

Homework

• Grades 1 and 2 should play the subtraction card game at home
• Grade 3 should play the 4-in-a-row game and explore the calendar trick. Can they figure out a way to find the square their friend chose? 

another race to 100 game

Today's game at the math classes was not particularly well liked, but we are including this note for completeness and future reference.

Race to 100

how many players: 2-5
material: 1d6, 100 board, position markers (the kids made their own out of play-dough)
start: all players start on 1 on the 100 board
turns: each player's takes a separate turn. They roll the dice, then move their piece up the 100 board some multiple of the dice value (up to 10x).
winning: first player to get exactly to 100 wins

This game practices multiplication, skip counting, and factoring. Here are some example questions to stimulate thinking about game strategy:

• Would you rather have your piece on 99, 98, or 96?
• What about 71 and 70?
• If you are on 88, what are your chances of winning on the next roll?

Game reception
The kids found this game fairly easy. In retrospect, perhaps we should have played this game before the Times Square factors game.

Potential extension
The game is nicely suited to analysis by working back from the higher positions and/or analyzing a simpler version of the game. This may be a nice exercise for our programming classes, especially as we have recently been working with arrays.

math games class catch up

It has been a while since I posted a summary of our math games, so this is just a quick catch up to summarize what we've been doing:

Grade 1

Addition war

2 players

pack of playing cards (A to 10)

Deal out all cards to both players and keep the cards face down in a stack. Each round, both players turn over the top two cards and add their values. The player with the higher sum wins and collects the cards in their points pile.

If there is a tie, those 4 cards are kept to the side as a bonus for the winner of the next battle. Repeat this with ties until there is a winner for one round.

After playing through the original stack, look to see who has collected more cards in their points pile. That person is the winner.

For a more challenging version, use face cards and assign values J = 11, Q = 12, K = 15.

Solo addition bridge

2 - 4 players (we played with 3)

pack of playing cards (A to 10 for beginner game, add face cards for extra challenge)

Deal out 5 cards to all players. They pick up these cards to form their hand. Proceeding clockwise, each player lays down one card from their hand, going twice around the group. Each player adds together the value of the two cards they played. The highest sum wins and collects all the cards played. That is one "trick."

After each trick, deal out 2 cards to each player to refill the hands to 5 cards.

The player who won the last trick is the first to play a card on the new round.

When there aren't sufficient cards to deal equally to all players, deal the hands equally (all players start each round with the same number of cards) and keep the remaining cards as a bonus for the player who takes the last trick.

We played with 3 players and, for the advanced game added 2 jokers to make a deck of 54 cards. Based on popular consensus, the jokers were assigned a value of 1,000,000. Interestingly, the extremely large value meant that the players were reluctant to play their jokers and, twice, both players kept them to the final trick so that the winner was actually decided by the higher of the second card!

Group addition bridge

4 players working as pairs (partners) with the partners sitting opposite each other

pack of playing cards (A to 10 for beginner game, add face cards for extra challenge)

Generically, play is the same as solo addition bridge, but each round the partners each play one card and the team that has the higher sum wins the trick. While there are still reserve cards in the deck, hands get refreshed up to 5 by dealing a single card to each player. The leader for each trick rotates clockwise so that everyone gets a chance to be first (second, third, and last) to play.

Second and Third Grade

Multiplication Blind Man's Bluff

3 players

pack of playing cards using A to 10 (A counts as 11)

One player deals a single card to each of the other players. They hold that card up to their forehead. The dealer announces the product of the two cards. Then, the two players try to figure out the value of the card on their own forehead.

Role of the dealer rotates after each round.

We played this as a cooperative exercise. To make it competitive, you can award points to the first player to get their card value.

There are two ways to make this more difficult. Adding face cards with made up values is one way. Instead, we had the dealer give one player two cards, add those, then multiply that sum by the value of the other card. 

Yet another step is to deal each player two cards, then multiply the two sums.

When playing this version at home, J1 came up with the idea of giving clues to figure out the value of the two individual cards. This was a really interesting activity because it got him to think about what characteristics help specify the two cards and which clues actually don't provide new information.

For example, if I know the sum of my two cards is 11, does it help me to know that I have one odd and one even number?

To make a standardized version, the second round of clues is to tell each player the product of their values. 

Largest Difference/Smallest difference

many players (at most 9 per deck of cards, fewer with advanced versions)

pack of playing cards A to 9

Deal out 4 cards to each player. They then form two 2-digit numbers and subtract the smaller from the larger. The player with the greatest difference wins that round.

For slightly greater challenge, deal out 6 cards (for two 3-digit numbers) or more (forming 4 or 5 digit numbers). Again, the aim is to form two numbers with the same number of digits that have the greatest difference.

For a much more interesting game, we shift the goal: now, we try to find two numbers with the smallest difference (larger minus smaller). After playing a bit, we had some good conversations about what the students noticed, what strategies they used, and whether there was always a unique answer.

Multiplication Pig (variation of addition Pig)

2 dice (we used 2d6)

2-3 players (or more, grouped into teams)

Players start with 200 points and try to work to 0 (or below).

Each turn, the player rolls both dice. If neither is a 1, they multiply the two values and add this to their score for the round. They can either choose to roll again or take their score for the round and subtract that from their cumulative score.

If two 1's are rolled, then  their overall score is set back to 200. If one 1 is rolled, then their score for that round goes to 0 and they lose their turn.

Variations come from varying to characteristics of the game:

- Start with 0 overall points and, each round, add the points for your round to try to break a target (practices addition instead of subtraction in forming the overall target)

- Add the two dice instead of multiplying (shifts the practice to addition instead of multiplication)

- Use dice other than 2d6, possibly more dice or differently shaped dice (note: the overall target and/or penalty conditions might require some adjustment)

Some PIG observations

Dice games are loud games, compared with card games. I think this is because the value of the dice is revealed to everyone at the same time.

Based on expected values, the optimal decision whether to keep rolling to bank the points for that round depends on how many cumulative points you have and your score for that round. However, we observed that the students chose to bank their points very early, relative to an expected value maximizing strategy. I think this is because their experience with the game causes them to over estimate the likelihood of rolling a 1 (or two 1's) and/or to underestimate how many points they can earn on a single roll because of the multiplication.

Projects: Finale (programming class 18)

Who: Baan Pathomtham Grade 5
Where: at school
when: Monday morning for 2 hours

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This was our last class of the term, so we spent the time working on finishing the projects.
For each project, there are many potential extensions, but everyone has made a complete program (or very, very nearly complete).

Programs can all be found here: Class Projects.

Plan of work today:

• Titus: (1) instructions for the game (2) webcam version?
• Win: (1) correct flow, (2) add fractions?
• Boongie: (1) how do players win? (2) is the penalty condition working the way you want?
• Gun: (1) penalty for hitting the walls, (2) winning condition, (3) finish maze 

Examing a graph and 35 game (math class 12)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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Weekend outing

Our school went to a play over the weekend, somewhat about chickens. Unfortunately, there were no mathematical chickens, but maybe someone will draw one this week?

Dice turn into a graph

The homework last week was to play damult dice with subtraction, targeting a score of exactly 100. We had discussed some observations about the game and had left open the observation that a lot of scores seemed to be multiples of 3. Over the week, I made a little pencilcode program to show the scoring distributions and thought this would be another good focus for the kids to make observations, conjectures, and pose questions. The distribution we discussed today was the max scores distribution, so you can play along here: Scoring histograms.

Once again, this turned into a good conversation:

• Why are some bars red and some blue?
• What is the meaning of the two rows of numbers on the bottom?
• How does this relate to our dice game?
• In the top row of numbers, why is 2 the smallest? Why is 72 the largest? Why is 3 missing?

For homework, we gave everyone a copy of the possible scores histogram and assigned them to write an observation or question about the graph.

35 Game

This is another card game. I'm not sure of the source, but will link back here when I find it.
Material: Pack of standard playing cards (J=11, Q=12, K=13), manipulatives for adding (we used 100 charts) are also helpful.
Set-up:Deal out all cards to the players. In the first version, it doesn't matter if people have different numbers of cards. Players don't look at their cards.
Play:Going clock-wise around the table, the player turns over one card and adds its value to the running total, if that result is less than 35. If the sum is 35, the player scores a point and the running total resets to 0. If the sum would be more than 35, then the player subtracts the value of their card from the total instead of adding.
Example: if we have gotten to a total of 26 and the next player puts down a 9, they would get a point. If they turn over a 10 or face card, then they subtract that value from 26.
Tips:

• When the card value is too large, encourage the kids to state the addition equation anyway. For example, "30+10 = 40"
• Before playing a card, ask what value would take you to 35.
• Occasionally ask for values larger than 13 and ask if there is a card that would let you get to 35 on the next play.
• For a more involved version, deal equal numbers of cards and let player's choose which card to play on their turn.

Homework

1. Write down at least one observation about the damult dice possible scores histogram
2. Play the 35 game with a parent or sibling until someone gets to 5 points and wins

We noticed . . . (math class 11)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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Warm-up patterns Today, we started skip counting by 4 (first grade) and 6 (second grade). We are extending the damult dice game today and the kids benefit from as much practice as they can get.

What did we notice (dominoes)?

We split the class into pairs (the extra student went with P) with the following instructions: talk with your partner about what you noticed or discovered about the dominoes puzzle, write down one sentence and be prepared to share with the rest of the class. I assigned the partners and intentionally broke up traditional pairs. I made sure to have some boy-girl mixing and also paired students with different levels of historical engagement. Many kids had questions about what types of ideas they were meant to be discussing, so we gave them the following prompts:

• What were your strategies for solving the puzzle?
• What did you find for the smallest side sum?
• What did you find for the largest side sum?
• How many solutions did you find?
• How did this compare with the triangle puzzle? What was the same, what was different?

After 5 minutes (longer on request by the first graders who were still writing), we regrouped to discuss.
With this structure, we had a really good conversation. There were a lot of ideas shared and some good debate about each of the points. Some highlights:

1. Some students claimed to solve the puzzle by randomly ordering the paper dominoes we made last week. They were surprised when I praised this approach as they were expecting to be criticized. We discussed that the whole strategy was actually: make manipulatives, create a possible solution, check whether the solution is valid, reorganize into a different configuration. P emphasized that this is widely valid, if they are working on something and get stuck, one thing to try is to use a tool to help them, not to suffer in confusion.
2. For each answer we discussed, there were usually multiple ideas. This gave kids an opportunity to talk about what they found and how they found it.
3. Comparing with the triangle puzzles was a very rich vein for conversation. The contrast helped them identify structure in the puzzle that, otherwise, would have been either too subtle or too obvious to mention. For examples, the dominoes force the numbers to sit in pairs, the square arrangement has 4 sides vs 3 for the triangle, the domino puzzle had 8 numbers vs 9 (or 6) for the triangle puzzles, and there were some repeated numbers in the domino puzzle while the triangle puzzle had distinct digits.

What did we notice (damult dice)?

We repeated the discussion process again, this time based on playing damult dice. Again, this was a really good discussion, covering a range of ideas:

• the smallest result possible (2= (1+1) x 2)
• the largest result (72 = (6+6) x 6)
• strategies to make the largest result with any given roll
• strategies to make the smallest result with a given roll
• why do we get so many multiples of 3? (more on this below)

"New" Game: exact damult dice

We turn again to Math4Love's page on Damult Dice. In the comments, there were many suggestions for extensions to the basic game. The one we introduced today was simple: you have to hit the target number of points exactly and you are allowed to subtract the result of your dice roll instead of adding it. The basic calculation remains the same: roll 3 dice, add two, then multiple the sum by the third.

This is a very simple extension, but it makes the game much richer. Gone is the simple strategy of accumulating as many points as you can on each roll, replaced by something a lot more subtle. I'm looking forward to hearing what the kids find.

Food for thought: Multiples of 3

Playing damult dice last week with my own son, we noticed that maximizing the result often gives us a multiple of three. The challenge: why is this the case and how often do multiples of three actually occur?

Send me a message and projects 3 (programming class 17)

Who: Baan Pathomtham Grade 5
Where: at school
when: Monday morning for 2 hours

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Reminder, this is our standard lesson plan for the rest of the term:

1. New or review concepts
2. Exercises related to the projects
3. Project work

New/Review

Today, we had three new concepts: while loops, objects, and message passing. This was clearly too much for everyone to understand everything, so I focused on the while loops. My reason for introducing the other ideas and short programs was to give Titus some tools for his project.
These exercises and discussions were based on these two programs from the pencilcode guide:

• Shared Memory
• Message Passing

While
Talking about while gave a natural opportunity to talk about for again. We had two segments to the conversation, first comparing while vs for, then deciding which is best for different scenarios.

• Do you know how many iterations you want, before you start to loop? If so, use a for loop, if not, while.
• Eating: while hungry, eat. If you use a for loop, you may either still be hungry or explode!
• brushing teeth: while mouth feels dirty, brush your teeth.
• Putting on socks: for [1..2] put on a sock
• Cleaning vegetables for stir fry: while veggie bowl has food, clean and chop veggies
• Adding eggs to a cake: for [1..numEggs], crack an egg and add it to your wet ingredients
• etc 

This was a really good conversation as each kid had a chance to think about the funny outcome if you used a for when while was more sensible.

Shared Memory
We didn't talk extensively about the concept of an object. Like while/for, objects would pair naturally with arrays, but we also haven't talked a lot about them. For now, the kids should understand that the object is something like a chest of drawers with the drawers given their own names.

As our exercise, we extended the pencilcode guide program to use a button and add an extra value to create this program where the turtle's angle of rotation is entered through the input boxes and the distance moved each step is increased by the button:

Message Passing
Like shared memory,we didn't spend a lot of time on this new concept. However, inspired by Kan wondering why the program stopped responding to his button clicks, we slightly modified the guide program to use a while loop instead of the for loop. Our version is here.

Project exercises

Next week, I will focus the practice/exercises/review on concepts we've learned that are being used in the projects.

Projects and homework

As a reminder, I have stored copies of their work up to the start of today's class in this folder: Project Directory. The homework this week:

• Kan: finish drawing his maze. Next module is to decide what penalty he wants when the turtle runs into the wall. This is the same assignment as last week. Unfortunately, he overwrote his maze drawing program sometime during the week.
• Boongie. implement the penalty when the turtle runs into the wall, the turtle bounces back to a previous shell in the maze. There are several ways to do this and I'm curious to see his approach.
• Win: create functions to ask subtraction, multiplication, and division questions. Another area to think about are gradations in how hard the questions are.
• Titus: Use the shared memory and message passing code ideas we learned about today to check to see if the player chooses a match.

Dominoes and Damult Dice (math class 10)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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Warm-up patterns

Today, we started skip counting by 3, 4, and 6. This was because I wanted to play a game involving multiplication and I wasn't sure whether the kids were ready. They are familiar with skip counting and understand the linkage with multiplication, but clearly need more practice to become fully comfortable.

New Game: Damult Dice

The new game comes from Math4Love. Like NRICH, they are a consistent source of good material for our activities. I wanted this to be a more structured game than usual, so we established the following set-up:

• Class is split into two groups (3 and 4 students)
• One player in each group rolls three dice
• that student chooses two dice to add together. The third is for multiplying
• By skip counting, the team works together to multiply the two numbers.
• that result is their team score for that round and gets added to their total.

This game worked very well. It was especially encouraging to see them thinking about which combinations would yield the largest result and remembering prior results.

Old homework: The Triangle Puzzle

Homework from last time was to work on the triangle puzzles. I was curious to hear what they found. Nearly everyone found a solution to at least one of the sizes (6 gaps or 9 gaps), so they all got to share something about what they found.

• For the 3 space triangle, were there any solutions? No one had really thought about this, but during the discussion a couple of second grade students realized that all the numbers had to be the same for the sum of sides to be the same. I let them share their ideas about why.
• For the 6 space triangle, students found answers with sides summing to 9, 10, 11, and 12. 
• We talked about why 9 was the smallest (6 has to be linked with numbers at least as big as 1 and 2) and why 12 is the largest (1 has to be linked with numbers at most as large as 5 and 6).
• For the 9 space triangle, students claimed to find answers with sums 17, 20, 21 and 23. For an extra investigation, they can confirm these and try to find solutions with sums 18, 19, and 22.

For the 9 space triangle, I noticed a relationship between two of the students' answers, so drew these on the board:

Admittedly, I made some adjustments to highlight the relationship. You see that the numbers in corresponding slots add up to 10, right? In a sense, these are complements. If
you look closely, there is one other relationship, but I'll leave that for you to discover.

A dominoes version

I was lucky to find a direct complement to our triangle activity on NRICH: 4 Dom. This time, the challenge is to arrange 4 dominoes into a square with three numbers on each side where the sum on each side is the same. We got the kids to make their own dominoes by cutting out and coloring little strips of paper. That gave them a hands-on tool to explore three questions, all assigned for homework:

1. Allowing sides to have different sums, what is the smallest sum you can make with these 4 dominoes? How do you know it is the smallest?
2. Allowing sides to have different sums, what is the largest sum you can make with these 4 dominoes? How do you know it is the largest?
3. What arrangement makes all the sides have the same sum? What is the sum? How many solutions are there?

Homework

We gave them four pieces of homework. This seems like a lot, but the game is very short and most of the kids answered several of the dominoes questions already in class:

1. Damult dice: play the game with someone in your family, first to 200 wins. For each roll, write down the equation you are calculating, for example (6+1) x 2 = 14
2. Dominoes puzzle: answer the three questions listed above. In short, what is the smallest sum that a side of the square could have, what is the largest, what sum and arrangement works so that all sides are the same?

Congratulations for getting this far. Here is something pretty for your efforts. Please post in the comments any mathematical ideas this picture gives you!