Math games 7

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

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Skip counting warm-up

We've talked about warm-ups in several previous posts. Today, it proved to be a more substantial discussion than expected for a warm-up.

For skip counting, we go around the room with each child saying the next number in the sequence. This time, we started with some easy skip counting (2 and 3, based on requests of the kids in the class) and then something a bit more difficult (6 or 7).

Along the way, we saw some people getting stuck and knew that they would benefit from seeing some new strategies. When one child appeared to do a calculation very quickly, we asked them to explain their thinking. It turned out to involve splitting and regrouping:

We asked why they decided to split 6 into 4 + 2 and this was the explanation:

This discussion happened in each class and, in each class, we gave it the name of the student who explained it (Minnie and Jiping). For the rest of the day, when someone got stuck, their friends would offer encouragement and say "try X's method."

By the end of this warm-up, the second graders were excited enough that wanted to show off their technique for multiples of 9, so they spontaneously launched into that.

Parents at home: you can do the skip-counting warm-up driving together or at a meal time. Ask your child to show you how to get started.

If they are having difficulty with a calculation, first give them time to think through it.  Next, suggest tools they can use: write it down on paper, draw a diagram, use some objects. Finally, ask if they can break it into pieces that they know.

Ring your neck

Since some kids weren't in class last week, we began our games with a review of the new game from last week. It was a bit messy in second grade, but in first grade we got them to take turns explaining rules (one child explained one rule) and then they split into two teams to play a demonstration round.

Our intention had been to discuss strategy for this game, but the group dynamics didn't work well. These are the types of leading questions that encourage them to think about the structure and strategy of the game:

• Do you want to go first or second?
• If there is only one card left, do you want it to be your turn or your opponent? What about 2 cards? What about 3 cards? What about 4 cards? ....
• Is it "good" to take a card (do you expect to get points or lose points when you take a card)? What is the most points you could get? What is the least? What are all the possibilities?  What is the average?
• Do these things change as the game is played?

Parents at home: when playing games with the kids, ask them to explain the rules. As they are explaining, encourage them to show examples.  When you are playing this game, encourage them to add up their score each time they choose cards. Be patient when they need to take time doing the calculations and use the suggestions above when they get stuck.  When they aren't stuck, ask them to explain how they were thinking.

Finally, ask them the strategy questions.  It can be a fun discussion, especially when you don't know the answers!

Strike it Out

Players: 2
Material: paper and pencil
Set-up: Draw a number line and tick for each whole number. We gave the kids printed pages with number lines up to 20, but you should feel free to make longer or shorter number lines.
Start of play: the first player chooses a number and circles it
Each round: using the latest circled number, the player chooses another unused number, and forms a number sentence where the result is a second unused number. Circle this result and cross out the other two numbers.
Winning: the last player with a legal move wins.

This is an NRICH game and these pages show examples: student page and teachers' note.

Parents at home: This appears to be a more complex game as there are so many choices at each step.  When there are a small number of choices left, ask them to predict whether they can win. Also, ask them how many ways there are to form the number sentence and whether it makes a difference.

Homework

1. Play strike it out with parents, friends, and siblings
2. Play at least one of the card games we have taught this term.

Observations

First, we had a lot of fun playing these games with the kids this term. I think we found a collection of games that were fun for the kids and reinforced mathematical concepts appropriate for their current understanding. When playing the games, the kids were focused and engaged. Also, at least one of these (ring your neck) also has potential as a more subtle strategy game, though you have to reduce the value of the final bonus card.
When we weren't playing the games, we often found it difficult to keep everyone together for a discussion. This is something we will discuss with the teachers and think about strategies for next term.

Math Games Class 6

Who: Baan Pathomtham 1st and 2nd grade classes
Where: Bangkok, Thailand
When: after science and before lunch

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Pattern discussion

Homework from last week was to investigate the following patterns:

Pattern 1: Red - Green- Red - Red - Green - Red - Red - Red - Green .... (+1 increasing sequence of red blocks punctuated by single green blocks)

Pattern 2: Pink - Red - Pink - Pink - Red - Red - Pink - Pink - Pink - Red - Red - Red ...
(+1 increasing sequence of pinks followed by an equal number of reds)

As a specific  target in the investigation, we asked which color appears as the 100th term of the sequence and what is the closest position to 100 for the color that isn't the 100th term.

In class, we talked about how the kids approached this challenge.  Almost everyone took a brute force approach and wrote out the entire sequence up to 100. What else could they have done? Let's start with Pattern 1.

Most kids noticed in class that Pattern 1 has far more red blocks than green.  Given any large number, then, their best guess is that the block will be red.  They were also good about recognizing that their confidence in this prediction would be higher if the number was bigger. For older kids, this could be an interesting path for further exploration.

In class, we talked about some other ways to explore these patterns. After completing this post, I will write up an outline of what we discussed.  Part of the homework this week is for the kids to share with their parents what they learned from this exploration.

Ring Your Neck (New Card Game)

Materials: 2 players, standard pack of playing cards (all 52) and a piece of paper to keep score.
Set-up: Deal out 13 cards face down in a circle in front of the players
Play: Players alternate picking up either 1 or 2 cards (their choice) from whatever remains on the
table 13 cards
Scoring: After all 13 cards have been collected, players add up the cards they have collected according to the following values:

• card 7 has value -7
• card J has value -11
• card Q has value +12
• card K has value +13
• card A has value 1
• all other numbers have their face value

Most importantly, the person who collected the last card (or two cards) gets + 50 bonus!

To play the next round: Collect the first 13 cards in a discard pile, then deal face down the next 13 from the deck.

Homework

1. Talk with your parents about what you learned in the pattern exploration
2. Play Ring Your Neck 4 (or more rounds) with your parents and show us the scoresheet

Math Games Class 5

Who: Baan Pathomtham 1st and 2nd grade classes
Where: Bangkok, Thailand
When: after science and before lunch

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Special note for class parents:
Thanks to all the parents who have been playing the math games with the kids. They were really excited to talk about it and it shows how much they appreciated working on the challenge with you.

The homework this week is to investigate some patterns. We haven't found a great translation of this word into Thai as there seem to be different words for each of the slightly different meanings. As you help the kids with their work, you can encourage them with the following questions:

• What do we notice? 
• Why is this happening? 
• What do we think is happening next? 
• What can we do to test our prediction? 

It will be great if you can spend some time talking about these issues, even if you don't feel that they got to a particular "answer."

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Today, we had a slightly different lesson plan:

• Skip counting
• Talking about the snugglenumbers game from last time
• Exploring some patterns
• Playing indian addition poker

Skip counting

For each class, we gave them a number to skip count and P said an initial number.  For example, 

skip count by 2, starting with 0. We would go around the class at least twice. Today, we skip counted by 2, 3, 4, and 1/2.

This was intended as a fast warm-up, so we didn't choose anything that was particularly challenging. Even so, it proved to be good practice for nearly every student.  Also, there was a moment of deeper interest in the 1st grade class when we were skip counting by 4 starting with 1 and someone asked: "will we get exactly 100?"  That led to a good discussion of the pattern and then the class was really interested to keep going to see if they were correct.

Snugglenumbers game

The kids were all enthusiastic about this game which, frankly, has surprised us a bit since the game play is very limited.  They wanted to talk about whether they won or their parents.  For grade 2, we posed the following questions:

• Are there any numbers that are very helpful? if so, can you say which are the most helpful and why?
• What is the maximum possible score? Does it matter whether you are playing with one deck or two?

We didn't have an exhaustive discussion for either set of questions. For parents at home, I would encourage you to discuss these questions with your children.  Full disclosure: J1 and I had already discussed the question of the maximum possible score last night, so I had sworn him to secrecy for the class today.  As it turned out, we didn't really discuss this in grade 1.

Patterns

This activity was shown to us by David Ott, the K-12 math coordinator at ASL, one of whose recent exploits can be seen here: Family Math Night.

Set up: using construction blocks, I set up 5 patterns, the hide them in rolled-up paper, then hide those in a small bag.

The big reveal:

One pattern at a time, one cube at a time, I reveal the colors. At most steps, I will ask the kids for a show of hands to vote which color they think comes next.  Rotating around the room, we then ask for some explanation of the choice. I will usually also tease them a bit to encourage them to think more broadly. When they have all settled on a particular pattern and agree (which I don't force or encourage) then I whose the whole stick and ask them for a continuation, for example, what color is the 20th or 40th or 100th cube in the sequence.

The reaction

The kids absolutely loved this activity.  They were really excited to predict what would come next, eager to argue their view in the voting and then really emotional when the next cube came out and either matched their expectation or disproved their prediction.

Here were the patterns we used today:

Simple tools to cause mathematical delight

Your challenges:

What is the color of the next cube for the pattern on the far left?  If the glare makes it hard to see, it is pink-red-pink-pink-red showing. What is your reasoning?  How many other possibilities can you think of for that color and what is your justification?

If the pattern in the middle stick is repeated, what is the color of the 100th cube? What if I tell you that the pattern I want to extend is increasing whole number red series with a single green separating the runs of red?

Indian addition poker

P introduced this game two weeks ago and blogged about it here: http://3jlearneng.blogspot.com/2014/11/math-games-class-3.html. As she wrote at the time, this is a really good game.  We played two rounds at the end of each class and I was really struck by how much respect the kids showed to themselves and each other.  They took their time calculating, didn't get frustrated, and those who discovered their number faster were encouraging for the other students. To recap: everyone is dealt a card that they hold on their forehead and the teacher reveals the sum of all the cards.  Students then try to figure out which card they have.

In second grade, we played with a full deck, making J= 11, Q= 12 and K= 13 (A=1 in both classes). In first grade, we took out the face cards so we had A through 10.

One other modification is once there are 3 or fewer students left and they seem to be stuck, we tell them the sum of the cards from the remaining players.

I would encourage families with several kids to try this at home.  An alternative version is to have everyone take a hidden card, then reveal sums of 3 cards.  Feel free to experiment and see if you can find another version you enjoy.

Homework

We asked students to work on the two more complicated patterns, the increasing reds with green punctuation and the increasing pink and red stick.  The questions they were asked were:

1. What is the color of the 100th cube? How do they know?
2. If the red/green pattern has a red cube in the 100th place (a strong consensus among the classes) then what is the place of the closest green cube to 100?

Note: these questions can be answered by brute force, building the sequence or writing it down. What we would like to encourage is, of course, looking for patterns. For example, for the red-green pattern, do they notice anything about where the green blocks appear?  Can they make predictions about when another green will occur?  In the pink-red pattern, can they say when a block of reds has ended, do they notice anything about those places?

Math Games (Class 3)

Who: Baan Pathomtham First and Second grade classes

Where: at school
When: mid-morning

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Here is the third installment of our math games series.  Despite shortage of staff (Josh is traveling), it's a successful session -- the kids loved it so much they begged for more after they finished their first round.

Indian Poker Addition Game
Source: www.mathsticks.com

This game is a twist to the indian poker.  Although it's more fun and challenging with more players, for younger kids, I find that 3-4 players work out best.

Equipment: A pack of cards with face cards (J,Q,K) removed.
Here's what we do:

- Deal one card face down to each player.  When given the signal, they hold the card outward on their forehead so they cannot see their own cards but only see their friends' cards.

- The card dealer announces the sum of all the cards. 

- Each player guess his or her number.

We have 7 children in each class so they have to add up 6 cards and then subtract from the announced sum.  For children who struggled, I gave additional hints -- I told them the sum of his or her card plus two other friends' cards so they only have to add up 2 cards.  

Close to 100

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.

Counting with 6 hands

Please read for the *questions* below as I'd love to have your thoughts in the comments.

Who: Baan Pathomtham 1st grade class (J1's class)
Where: at school
When: 2 hours Tuesday morning

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We (P and J0) got a chance to spend the morning talking about subtraction with J1 and his classmates. It was an opportunity to see some differences between talking math one-on-one (or one-on-two) and a larger group.  Here are a couple of tidbits from the discussion.

Practice with poker chips
Of course all children need to be familiar with the standard gambling implements: dice, cards, and poker chips. The first two are already well known, so we did an activity with poker chips this time. How would you count all the chips in the picture?  Well, what if "you" were actually a group of 3 first graders?

Here's the strategy one group implemented, spontaneously, as far as I could tell:
(1) divide the chips into equal piles for each child
(2) count the remainder in the center of the pile (in this case, one chip, so the count was trivial and done without an explicit effort)
(3) take turns putting one new chip into the pile
(4) all count together as the new chips are added

Actually, this was their second strategy. At first, it was a free-for-all with all three trying to count all the chips and messing up each others division between counted and uncounted chips. 

Note: counting the chips was just accidental to the activity we were doing, so this shared counting strategy was just a cool thing we noticed along the way. If it had been more central, I would have talked with them about the equal piles at the start (which links with multiplication) and why there was a remainder (which links to the division algorithm).

Enthusiasm
Most importantly, the kids were all really excited and enjoyed working on math. They liked asking mathematical questions about a picture we presented, had fun doing calculations, trying new modeling tasks, and playing the mathematical game.

This confirms, once again, that enthusiasm and curiousity are things we (usually) kill during the educational process.  Not at our school!

Explaining
Given their enthusiasm and apparent facility with the calculations, I was surprised that they struggled to explain their calculating strategies. I can't tell if this is a language issue, if the calculations they were asked to describe are so ingrained that they don't consciously think about them, or if they don't really understand what they are doing.

My key take-away: I will focus a lot more of my discussion time on getting the J's to talk about how they calculated something, see if they can draw a picture, and see if they can explain using a concrete object.

Extensions
As preparation, P and I talked about 3 models of subtraction: taking away, differences, and counting back. P made two comments:
(1) Word problems are harder than straight calculations (said while we were discussing what types of problems to use to have the kids investigate the three models)
(2) "Counting back is such a waste, I always knew the answer through another method and had to artificially demonstrate counting back."

Word problems seemed, to me, the natural way to motivate using a particular model for subtraction. For three quick examples:

• You started with 5 cookies and ate 3, how many are left? This is taking away, obviously.
• Don has 27 poker chips and Tanya has 13. Who has more and how many more do they have? Differences, naturally.
• Walking along a straight line, you go forward 6 meters and then back 2 meters, how far are you from your starting point? Counting back suits this one.

*Question* is this the wrong way to use alternative models? Is it necessary to force them to use "unnatural" models to demonstrate proficiency (for example, using take-away to resolve the differences question)? Does this create difficulties for problems involving alternative missing values in the same types of questions (i.e., you started with 10 cakes and now have 3, how many did you give away?)

Counting back is the same as the movement model, which I called "forward movement" in my post on addition models. This model leads nicely and really easily to emphasizing the role of 0, negative numbers, and subtraction of negatives. Looking a bit farther ahead, it links with vector addition by just extending our operation to more dimensions. Taken along another path (ha, the puns!) it can be used for modular arithmetic (replace directed movement on a straight line with directed movement on a circle).

Adding and Subtracting Games

Who: J1's first grade class
When: in class time or extra time around school
Where: at school
Why: to accompany lessons on adding and subtracting
What will we use: cards (playing cards or number cards), dice, possibly some packaged games

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These are ideas for enrichment and deepening activities, to be discussed with the teacher. Do you have any thoughts on which are the best games? Please comment below. 

Future posts will repeat this exercise looking for activities that support time+calendar work and another around money.

Games
(1) Blackjack:
Game A: Use number cards instead of playing cards, no double down
Game B: With playing cards, can use simplified traditional rules (no double-down).
Game C: To focus on adding by a particular number, redefine the face cards to that value, e.g., 9, to make the frequent additions more challenging.

Note:  B or C can be modified to fix the value of ace as 1 instead of allowing the 1 or 11 option.

(2) 21 (as with blackjack, these can be played with traditional playing cards or number cards)
Game A: deal out n cards to all players (n should be between 4 and 7, depending on number of players). Players take turns putting down a card and adding to the sum until the next player can't stay under 21. The last player to play collects the cards which have been played.  Count number of cards played at the end as points.

Game B: Same as A, but each player can add or subtract the value of your card from the accumulating value of the pile. The value of the pile has no bounds and gets collected by the player to land exactly on 21.

Game C: modify the value of the face cards (if using traditional playing cards).

(3) Strike it out: described on this NRICH site and blogged about here

(4) Number bonds of multiples of 5
We blogged about this game here: http://3jlearneng.blogspot.com/2014/06/a-game-to-practice-number-bonds-of.html

(5) Sum swamp/Chutes and Ladders
For those who don't know Sum Swamp, players roll three dice, two with number values and one with addition/subtraction operators. They perform the indicated calculation and move that number of spaces along a path toward the finish line. In other words, it is chutes and ladders, but with the operation die added.

Game A: play with 2d6
Game B: play with other pairs of dice. I think Sum Swamp will work up to 2d10, but the path is probably too short for larger dice. Chutes and Ladders can probably work up to 2d20, but I would impose a rule that the winning player has to land exactly on the final square (or maybe within 5?)

(6) Dotty Six

Game A: describe on this NRICH page

Game B: change the target to complete each box to 10 and use tally marks instead of dots

Game C: change the target to another higher number (I suggest either 15 or 20) and use tally marks.

(7) Pass the peas, again, NRICH
Though the kids will probably end up throwing dried food at each other?
Game A: as described
Game B: use 2 dice and then multiply the die value with the value of the number square it lands on, then subtract those from your residual figure.

(8) Subtraction squares

Challenges
(1) Eggs in a basket: NRICH
(2) 5 steps to 50: NRICH. Their version only uses 10s and 1s, but the step sizes can be changed. Also, they don't specify what dice to use, so I would prefer 2d10 to get 0 to 99 as the range of starting values. I made a pencilcode program to explore this (here) but now see that I missed the point about using dice to start, so maybe an interesting constraint is formed by using 2d6?

Investigations
(1) Play with a number balance (as pictured on this page)
(2) using a digital scale: weigh pairs of objects separately.  Figure out how much you think they will weigh in together and then check.
(3) magic boxes: another NRICH activity

note on materials:
Dice: [n]d[m] = use m-sided dice and we need n of them. For example, 1d6 is a single six sided die (usually with standard 1-6 markings), 3d8 means three octahedral (8-sided) dice.
Traditional playing cards: 52 card deck, two through 10, jack, queen, king, ace
TPC numbers: 40 card deck using ace (1) through 10
number cards: special card deck using numbers, we got ours through Abacus Math