Year 11 have been looking at a variety of Paper 1 (non calculator) questions ahead of their exam on Tuesday. One question seemed to be challenging for the students in my class. They were given a number line from 0 to 2 marked at every quarter and they were to find the FRACTION that was halfway between two given fractions. Here is a sample of the answers year 11 gave. This is a foundation class but targets of 4s and 5s.
Most if not all of the students converted the fractions to decimals and then subtracted the two decimals but then struggled to halve 0.75.
I spent some time with a small group of students and we looked at splitting the scale further and in doing so the students spotted an easier way to find the half way point. What I found interesting was that students didn’t see that the halfway fraction between 1/2 and 1 1/4 was the same as the halfway fraction between 3/4 and 1.
It made me think about finding the numbers in the middle of these pairs of numbers
8 and 10
7 and 11
5 and 13
1.5 and 16.5
I made the following for the next lesson and I wonder if students can make the connections between the 3 questions
Order of Operations
I have used an ‘insert the brackets’ type question before but find students can sometimes just wait for the answer to just jump out at them rather than trying to fit in brackets to make the answer work (even if its on a trial an error basis). I like using the same expression so students are more inclined to try a few ideas. After using a similar question to the one below, one of my year 7 class said he had found some answers that weren’t in the cloud so i thought it might be nicer to give them an expression and ask them to find as many different answers as they could by inserting brackets.
I have been looking at Rotational Symmetry with year 7 and I was surprised at how much my high attaining students struggled with describing the symmetry of a given shape.
I created a task where students had to colour blue and red squares so that the rotational symmetry was either order 4 or 2. Students started with the grid of 3 by 3 and they completed that task but we talked about why some were possible and other weren’t and how you could know. Also, we discussed that if we had a solution for 2 squares then 7 was possible and the same for 3/6 and 4/5.
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.