Where: at school

When: mid-morning

Today was our opportunity to play some mathematical games with the younger kids at J1's school. It was fun and we played two games that the kids can take home to play with their families for further investigation.

Apologies, no pictures: we were too occupied to take any snaps.

**Warming-up**

We start each session with some quick question warm-ups. Going around the class, everyone gets a question in turn. These are meant to be fairly easy, but get them actively engaged.

Examples:

- questions about days of the week e.g., what day comes before Monday?

- months of the year, e.g,. what month comes after December? How many days in June?

- continuing number sequences, e.g., What comes after 19, 20, 21? What about 35, 34, 33? What about 75, 73, 71?

- Skip counting, e.g., counting to 20 by 2 starting at 0. Counting up by 2 starting at 1.

- What are some ways to make 10 with addition?

**Making mathematical observations**

After warm-ups, we gave the children a picture and asked them to make mathematical statements or ask mathematical questions about the picture. The types of statement depend on the picture, but examples include:

- counting specific objects, e.g., There are 5 apples in the picture. My favourite is something like "There are zero footballs in the picture" or another object that is totally unrelated to the picture.

- comparing numbers of objects, e.g,. There are more people than lions.

- number sentences: there are five children and three adults, all together there are 8 people

- comments about shapes

This is an idea we got from Mathematics Mastery. For younger grades, they have developed a series based on fairly tales that are really good. Naively, I had thought it would be possible to take random photos off the web, but most just aren't that detailed or there is limited variation.

**Game 1: Euclid**

Our first game was from Let's Play Math: Euclid's Game. The rules are simple:

- start with a 100 grid (we used 60 for the 1st graders)

- First player chooses a number and draws an X over it

- Second player chooses another number and draws an X over it

- players take turns crossing out numbers that are the difference between two numbers already crossed out.

- last player with a legal move is the winner

We played several rounds on the whiteboard, J0 against the kids and also half the class against the other half. We recommend using contrasting colors to make it easier to see any patterns that emerge.

At the basic level, this is just practice subtracting two digit numbers. We found that both grades were struggling a bit with this, so the extra practice was useful. For the next level, ask about patterns, during and at the end of the game:

- can you see what pattern of squares we are crossing out?

- how do you know if there are any moves left?

- who do you think is going to win?

- what is the largest number we are going to cross out?

- what is the smallest number we are going to cross out?

On their own, the first grade class started to speculate about whether they could know in advance who would win.

We gave out 100 sheets so the kids could play at home with family and friends.

Look: your very own 100 grid! |

**Game 2: Don't Make a triangle**

This game comes from Math4Love and is called Don't make a triangle. All you need to play at home is a pencil and paper.

We actually didn't play the Math4Love version, but instead showed them a simpler variation where players take turns connecting the starting dots and they try to avoid forming a triangle with vertices on the starting dots. Here is a progression of variations and explorations we considered:

- variation 1: students start with 6 dots, take turns connecting pairs. the one to make a completed triangle first loses (a segment drawn by either player counts)

- variation 2 (as suggested in Math4Love): start with 6 dots, students use different colors and only a triangle with all sides their color count

- exploration 1: will it make an interesting game if you are trying to be the first to complete a triangle? Test this for variations 1 and 2

- exploration 2: is there a winning strategy? do you want to be the first or second player?

- exploration 3: try these games and questions with a different number of dots

**For future sessions:**

There is a lot more we can do to explore the two games we introduced today. I plan to spend at least part of the next session on a bit more of an investigation into Euclid's Game. Here are some other ideas we may explore

(1) three chips puzzle

(2) Always truthful/always lying logic puzzles. there are many of these, here are two examples:

a) Tom always tells the truth, Dick sometimes tells the truth and sometimes lies, Harry always lies. You don't know who is who, but start to ask their names. Their answers:

- First person: I'm Dick

- Second person: I'm Harry

When you ask the third person, what answer does he give?

b) A family has 2 sons, one who always tells the truth and the other always lies. Their house is next to a Y- intersection. Walking by there house, you get confused about which direct you need to go (left or right). You go to their house and one son comes to the door. What can you ask that son so that you find out the correct way to go?

(3) Nim variations

- 1-2 Nim

- 1-2-3 Nim

- Nim on 10 frame

- Poison

- points for taking counters and bonus for ending condition

(6) Einstein/elimination puzzles

I did a lot of these around this age, so am curious to see how the kids find them. I just found this collection, so will see if any seem suitable.

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