I teach a 7th and 8th grade math intervention class 3rd period with about 13 students. Only 3 of them are in my 5th period mainstream class, the rest have different teachers. This makes it a bit hard to coordinate homework at times, but I have a board with teachers names on it where they write in the assignment.
For the most part, I use the class as a place where they can support each other on their homework and I can do some reteaching. I also occasionally do an estimation, fraction talk, or number talk.
On Fridays, most teachers don't assign weekend homework, so that's my free day to do a cool lesson. We have made fraction strips out of construction paper, Desmos activities, puzzles, and sometimes a 3 act task.
I noticed students were having trouble with dividing fractions so I remembered a great task that Graham Fletcher has made called The Apple. It's a 4th grade standard, but definitely appropriate for any non-accelerated middle school class.
As you can see below, we first tackled the task with a too low, high, and just right estimate. The range was between 8 and 150. I was disappointed no students tried repeated subtraction or addition to get to the answer. I'll have to introduce that next time. Students realized they needed to divide, but couldn't figure out how to. Some students converted 3/8 to a decimal, then realized it didn't make the problem easier for them.
A side note, I love that this problem as a digital scale that measures in fractions, how perfect! Students had different ideas, and after some productive struggle they pieced together each of their ideas into an answer. Miguel said convert 5 1/4 to 21/4. Another student said you needed to flip the 3/8 and write it as 8/3. Then they multiplied across.
I showed them the common denominator method, and reminded them of the reason we multiply by the reciprocal when dividing by a fraction by modeling the Super Giant One from 7th grade.
I have the playing cards for the game 24. You are given 4 numbers, and have to make 24 using each number once and any operation. We did this one, 7, 7, 4, 1 as a group. We struggled together, and I gave them a hint for what times what is 24. So, they then came up with 7 plus 1 is 8, and 7-4 is 3. Multiply those together and you get 24. After they saw the challenge, they were super motivated when playing in partnerships which of course pleased me.
The following fraction talk is under the my favorite problems tab at the top of this blog. The quest is what fraction is yellow, and how do you see it?
As you can see, students came up with 1/3, 2/3. 1/6, 1 1/6, and 1/4.
The biggest misconception I can see with those answers is that the students aren't including the yellow part in their denominator. Also, most were not willing to share why they came up with those answers.
Miguel was able to explain that 2 red's make a yellow. So, I drew a diagram of his thinking as he explained it. He kept moving the red blocks up to make yellows. Therefore, he had a total of 4 yellow blocks. so, 1 yellow out of 4 yellow blocks is 1/4. Another student converted the yellow into 2 red, so he had 2 red out of 8 total red. I also showed them that since there's a line of symmetry, just like the one going down the middle of your face, you can see how much yellow there is of half of it, which is 1/4.
On another day, we returned to the same 3 act task web site for Rope Jumper. Kids always get a kick out of this one.
Once again we start off with low, high, and estimates before Act 2. I asked them what information they would need to figure out the number of jumps in 30 seconds.
For the 6th graders, they had trouble with this question, so I asked them what do you think is behind the blacked out part of the video?
Students reasoned it would be a timer, and a counter of how many jumps.
It's hard to read the writing below, but the main strategy was repeated addition here at first. Mavae reasoned that if it's 41 jumps in 7 seconds, he added 41 + 41 + 41 and 7 + 7 + 7 to get 123 jumps in 21 seconds. He realized he could add 1 more chunk of 41 jumps in 7 seconds to get 164 jumps in 28 seconds.
When it came to the extra 2 seconds, students struggled with that part, but said well just add 2 jumps for 2 seconds. This would be a great talking point that I could have used to connect the next method with this method, because the next method was unit rate.
Ayman divided 41 by 7 and got 5 point something. He rounded it to 6 and then multiplied it by 30 seconds. So, he basically found the unit rate, then multiplied.
Also, I missed an opportunity to show how you could have used proportions for the problem, but I usually steer the lesson in the direction that the students take it,
We also started Clapper, but didn't have time to finish it, so I'll write that one up later.