Today during a year 10 lesson, students were faced with a calculation: 81 – 45 during a worded problem. They reached the answer 36. One student was waiting for the other students to reach their answer and he said “All of the numbers are in the 9 times table”
We stopped to look at why that was and we came to the conclusion that if you had 9 nines and you subtract 5 nines then you are left with 4 nines. Students explored which calculations this worked for and they summarised that two numbers that are in the same times table had a difference that was also in that times table.
I then went home thinking about generalisations and how this isn’t always apparent
I want students to notice that when you subtract multiples of the same number then the difference is also a multiple of that number.
I then hope to move onto 3 and 3 squared and how you cannot simplify this.
I think the slide below is a little complicated but i want students to compare the two ideas.
My daughter and I like to play a game called Penguin Pairs (memory I think people call it) but with matching penguins. It was a game that I played as a child and really enjoyed. If you don’t know it: each player takes it in turns to select two cards and turn them over to see if they match; if they do, they keep the pair but if they don’t they place them back faced down. The winner is the player with the most pairs.
After we have finished we have to figure out who has won. When my daughter was younger I’d announce, “you’ve won!” or “oh well, you lost but see if you can win next time”
Then a few months ago I didn’t say anything. My daughter said; “Have you won?” and I replied with “why don’t you tell me who has won?” She thought for a few minutes and announced she had, but without looking at the number of pairs. I said “are you sure?” So she reached across and started lining her pairs up with mine and looked at me and said “no, you won” I said “how do you know?” and she pointed to the extra 2 penguins and said “because I don’t have any to put here” (pointing at the gaps)
The next time we played it was in a smaller space and on a little table rather than the floor. We were faced with the same problem at the end: who had won?
Space was limited so during the game my daughter and I had made piles of our pairs rather than place them side by side. So when the game had ended we both piled up our cards and we looked at the two piles. She announced “we both won because my pile is the same as yours. They are both the same tall”
Last night we played the game again after a month of not playing it and we played it on a gymnastics mat. Without discussing it we had both placed our pairs in a line along the edge of the mat. I’m not sure who made the decision to do this but we ended up two lines side by side. She said that her line was the longest so she was the winner. I did question whether a longer line meant she has won so she counted the lines to double check. Her line was 11 penguins and mine was 5 so she had won.
I was team teaching with a trainee today and she asked a question similar to the bottom middle question. The students found the unknown angle to be 70° and she asked ‘how did you know you could halve 140° to get the unknown angle?’ Students gave the answer ‘you take 70° from 180° and then you always halve whats left’ The trainee questioned that ‘do you always halve? What was it you saw in this question that let you know you could do this?’
So It made me think about questions that could have answer of 70° sometimes, always or never.
Some questions that use Mean or i suppose they actually require students to realise that in order to find two numbers that have a mean of 5 the total of the two numbers must be 10. Students then select the two values that satisfy the totals.
The first game shown here is Square number Run. The premise is to make a path across the grid so the total is a square number. The first to reach the other side wins.
This can be adapted. Maybe it works better to keep the counters on each square you pass the other player cannot use that square after. I’d love to know how the game goes when you use it.
Here are all the games in ppt form
connect the factors
square number run
The next game is called Prime Pairs and each played places counters on adjacent squares so the numbers add up to a prime number. They keep taking it turns to do this until someone can’t go, the person who places the last pair is declared the winner.
The next game is loosely based on connect 4. Players announce a number (less than 100 – this could be changed but i figured if they multiply every value in a row then all the numbers in the row can be covered) and place counters on numbers that are adjacent and are factors of that number. The winner has the most squares covered.
UPDATE! I decided to trial two of the games ‘prime pairs’ and ‘connect the factors’ with year 7 today. They seemed to go down well and students seemed to be engaged. It did invite some interesting discussion about strategy (if there was a row of 4 numbers in Prime Pairs then students told me they placed the counters in the middle in order to stop the opposing player having a go)
After they played each game I gave the students 5 minutes to reflect on the games and how they might improve them. A selection of the most popular improvements are below. This will form a discussion for the next lesson
‘can you have diagonals in the prime pairs game?’ i was considering allowing diagonals and see if they notice anything. I might let them create their own grid to play with.
‘Should we only allow 3 or more shaded for the factors game?’
‘How can we have got a row of 3 instead of two in this situation?’
‘What number(s) have 12, 9, 2 as a factor?’
‘Can you find a number who’s factors add up to the number?’
I wanted to make a question to practice Venn Diagrams with year 11.
Using colour will help the students describe the regions in class. I thought it might help getting instant feedback (using whiteboards) on which region they thought fit.
The object of this puzzle is to fit rectangles on the grid that are unique, have different areas and all of the areas are multiples of 3.
I wanted a task where students consider multiples and factors. They need to know which length have a product that is a multiple of 3 and also consider that all the areas have to fit on a 9 by 12 grid so the areas all need to add to 108