## Monday, November 28, 2016

### 3 Act Tasks in Math Intervention

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.

## Wednesday, November 23, 2016

### NCTM MTMS Blog Post #1: Positive classroom culture

Cross posting this. The first post in a 4 part series I wrote for NCTM's Math Teaching in Middle School Blogarithm blog.

### #bottleflipping 2 day Desmos Lesson

After reading about water bottle flipping and a Desmos activity being created on it:

I took a look at a few teachers experiences with it including  the blogs of Elizabeth Raskin, Jon Orr, and Trish Poulin.

I decided to merge some of the ideas, with the first day using Mr. Orr's hook by trying to throw to flip a full water bottle and experience failure in front of the whole class. They screamed, "too much water, you need it to the third line!"

I showed them activity builder Mr Orr made with the 4 choices, and most selected I believe it was yellow. I asked them how we could figure out the best amount of water to have? Some suggested measuring with a ruler, and others said measure the water in it. So, having graduated cylinders that filled up to 100 milliliters, we used those. They made a table of 100% down to 0% with 10% increments. I asked them how we could see how much water for 90% full.

The water bottle is labeled with 16.9 ounces and 500 milliliters. I asked students what they would rather use. They said milliliters because it's a round number.

They said take 10% of 200 milliliters, and subtract it from 500. That left them with 450 mL. Then they'd pour another 50 mL out for their next trial. They would do 10 trials of each and record their successful flips in the data table of the activity builder.

As they complete the table, the points appear on their graph. So, seeing their small sample, the activity asks what it would look like for the whole classes data? Here's their predictions:

2ND PERIOD
GASPARD MONGE
It's gonna look like a decreasing straight line.
PIERRE DE FERMAT
It stays at zero until 30 % and then gradually declines.
CHRISTIAAN HUYGENS
the class is gonna look like a little hill
ARCHIMEDES
it will increase then decrease
ARTUR AVILA
It will start at around 5 and decrease slowly then increase a little bit. Then it will decrease and then very slowly down to 0.

3RD PERIOD
JAMES MAXWELL
It will be mostly a straight line but between the 20 to 40 percent filled the chart will peak up.
ATLE SELBERG
It will look similar because it most likely people will have more flips when the water bottle is less full and less flips when the water bottle is full for everyone.
SOPHIE GERMAIN
I think it would look like this where there are zero flips in all percentages of water instead of smaller amounts such as 20,30, and 40%.
DOROTHY VAUGHN
Stacey: I predict that at first, the graph will increase, then start decreasing. Sebastian: I predict that the most flips will occur between about 20-40% full Zurin: I predict that the graph will increase upwards a little when it has 50-20% of water and decrease to 0 again. Kaitlyn: I predict that the graph will increase similarly during 30-40% full.

5TH PERIOD:
ALAN TURING
We think there will be more successful flips in the middle region
TERENCE TAO
First it will start to increase than decrease .
SHIING SHEN CHERN
I think our class´s data will look like a zigzag since that´s how our group data looked like.
MARY ROSS
I think it will gradually increase after 80% full and it will be highest at 30% because it is not too full or too empty.
JEAN-PIERRE SERRE
we think that the data would be low because, the chances of a bottle flipping and landing on the bottom is slim.

DORIS SCHATTSCHNEIDER
Around the 50%-40% mark, that's where you're most likely to land more flips than any other percent.

& interpreting their data after seeing the whole classes graphs:

2ND PERIOD
GASPARD MONGE
View Graph
Around 30% the bottle is able to be flipped the most.
PIERRE DE FERMAT
View Graph
Mostly everyone got the most flips when the water bottle was 30% full.
CHRISTIAAN HUYGENS
View Graph
This is how I expected it to look, because it looked like a little hill. It was more flat though.
ARCHIMEDES
View Graph
when it got to 50% full, people started to land their bottles WE LOVE MATH ❤

ARTUR AVILA
View Graph
The highest number of flips is 31 at 30%. The lowest number of flips is 0 and the graph increases, decreases, then stays at 0 then increases and lastly it decreases and stays at 0.

3RD PERIOD:
JAMES MAXWELL
View Graph
The class did well in the first half compared to the last half in the experiment.
SOPHIE GERMAIN
View Graph
Conclusions we can draw are that they get a lot of flips at smaller percents. Though it is surprising that people landed the bottle at higher amounts such as 60% and up.
DOROTHY VAUGHN
View Graph
More flips occurred around 20-40% full but there were a few times when there was a success over 50% full.

5TH PERIOD:

ALAN TURING
View Graph
we thought that the middle of the graph would rise
SHIING SHEN CHERN
View Graph
The amount of water is the dependent variable in this experiment, and that the less the water, the higher chance of getting a perfect flip.
MARY ROSS
View Graph
we thought that the 30 would be the only one that would move up
JEAN-PIERRE SERRE
View Graph
most people got the bottle to flip correctly on the last ones
DORIS SCHATTSCHNEIDER
View Graph
The number of flips increased around the 40% mark and starts to decrease after 50%.

Here are their whole class graphs in order of 2nd, 3rd, and 5th period:

In each class there appeared to be some outliers at around 80%. Though you can't deny that 30 or 40% is always the highest, usually 30%.

On day 2 we had a competition where students switched off flipping bottles for a minute. They had 3 trials. I asked them what we could do with their trials to represent their overall performance. They said to find the average. So, they explained to add up all the successful flips and divide by the number of trials, 3.

Then I borrowed ideas from other math bloggers and asked how many could they land in 1.5 minutes? Make a table and graph their average if they continued flipping at a constant rate for various amounts of minutes. Write an equation. How many lands in 10 minutes? How long to make 100 lands? And if you were given a 10 land head start, how long to land it 200 times?

It was a great opportunity to revisit proportional relationships.

The highest average in each class was in a final flip-off at lunch, with all participants in the finals getting a donut, with 1st through 3rd place getting more than 1 donut.

## Monday, November 14, 2016

### 2 Day Paper Airplane Statistics

All credit for this lesson goes to Julie Reulbach who posted her favorite middle school lessons on twitter and this statistics lesson was one of her examples.

I showed students the video and then asked them what I should do first. I made sure they noticed that after each step the step number appeared next to the crease. Then we went into our cafeteria where earlier my colleague Mr Rodinsky and I had marked off 1 meter increments in 3 columns to a little beyond half court of a basketball court. It went to about 15 meters.

Students paired up and flew their plane. Then they helped each other measure it. Some measured in meters, other measured in centimeters.

Upon returning to class, I asked students how we could represent our 3 trials accurately. Some students suggested getting the mean average, by adding the flight distances and dividing by the number of trials, 3. Other suggested the median. Others suggested the mode. Most students did not have a mode. We also discussed converting back and forth between meters and centimeters.

Some students threw it backwards on accident. As you can see in the picture below, students reasoned that the minimum flight, -300, would be subtracted from the maximum flight 1035, which results in subtracting a negative, which is equivalent to adding a positive.

In math support we did the border problem (you can find it on the functions unit on youcubed.org or just google it). A 10 by 10 square with just the border shaded in. How many squares without counting 1 by 1? After 2 explanations of 40, one student realized the whole thing is 10 by 10, or 100 squares. He wanted to subtract out the middle square, which he said was 8 by 8. That resulted in 36. Students revised their thinking with 40 and said subtract 4. They couldn't say why, but they realized it was overlapping. Other answers were 32 and 28.

I then did a second example with a 6 by 6 square. Answers there were 20, 22, 24, and 26. They reasoned it would be 6+6 and 4+4.

I made sure my colleague knew the nuance of histograms. a bar here represents all values 100 or greater, but less than 200.

On day 2 I had students finish getting their mean and median. I wanted to compile a class list, so first each student shared their 3 flight distances. Then they shared either their mean or median, because as we saw, you wanted to share the bigger number, especially if your mean got much lower with a negative number. Medians would disregard a low outlier.

After composing a class list with a number being the mean or median from each student, I asked what we should do next. They said put it in order, it'd be easier to look at and to find the median.

I then asked students if they remembered what a histogram was. Many said a bar graph, but there were ranges of numbers. I added that the bars are next to each other, and have no space between them. They said to number the y axis by 1's to see how tall it would be.

After constructing this as a class, I asked what else we could do. They said a box plot. I reminded them that they needed 5 important numbers to construct it. They told me: minimum, lower quartile, median, upper quartile, and maximum.

I said that they should discuss in their group those 5 numbers. After a few minutes, they reported the results out. Then we constructed the box plot as a combination with the histogram because the x axis was already scaled. This also allows you to have 2 representations at once. Clearly below, 1st period was skewed left.