Saturday, December 31, 2016

Visual Patterns Warmup (@math8_teacher) (@fawnpnguyen)

So for this last week before winter break I switched up the warmups from estimations. Mr. Rodinsky was getting fed up with students coming into class with the correct estimate on their paper with no reasoning written. Some see it as a game and I tried to make the analogy that it is like telling someone the ending to a movie before they see it.

I found this visual pattern prompt on twitter, and it's posted under my Favorite Problems link at the top of this page. Here it is again:
After reading about Dylan Kane's ideas about Visual Patterns as an instructional routine, I set the timer for 3 minutes of independent think time where they could copy the patterns down if they wanted. Then 3 minutes to tell their elbow partner what they noticed about the pattern. Then we shared out.

Also, I purposely meant to introduce this pattern the day before introducing the border problem number talk, which I will discuss in the following blog post.

Many students shared that they saw the grey squares as the figure number, squared. So for figure 2, 2^2 is 4, and for figure 3, 3^2=9 the number of grey squares. Students also used what they knew about linear patterns to see that the number of red squares was growing by 4. Some reasoned that the figure before must be 4 so the rule was y=4x+4. Some students used that for question 3 and set 4x+4 equal to 64 to get 4x=60, and x equals 15 or figure 15 having 64 red tiles. 1st period didn't come up with this, but one eager student told me how he solved it first thing in the morning.

For figure 20 in question 1, students reasoned there were 400 grey squares because 20^2 is 400. They said it would have 84 red tiles, because 4(20)+4 is 84. Students reasoned that you could add those two together to get the total squares or you could take the figure number and add 2 to it, then square it.

The highlight was one student saying the rule was x^2+4x+4 also. I assumed and asked him if he had learned something outside of my class about dealing with (x+2)^2? The student replied no. The x^2 is the grey tiles, the 4x is the red growing by 4, and the plus 4 is the red corner squares. My jaw dropped. If you didn't figure out, I assumed the student was going to say the dreaded "FOIL" method (I'm a big fan of area model or generic rectangle first). I was incorrect in my assumption. The student also showed how to see which figure had 121 tiles by undoing the squaring on both sides of the expression to get 11=x+2 and getting x=9 for question 2.

So, for a 10 to 15 minute warm-up, it provided rich discussions and laid some foundation for the border problem the next day.

Wednesday, December 28, 2016

5 Practices for Orchestrating Math Discussions (Google Drive enhancement) @cpmmath

I have read excerpts from the book in the title and it has been mentioned at many professional developments I've been to. I have been trying to implement it on a daily basis when we start a new lesson from the textbook. The 5 practices are outlined in this PDF file. In summary it is:


  1.  Anticipating. Select a groupworthy task and anticipate how students will approach it (successes and common mistakes).
  2. Monitoring. Check in on groups to see what ideas are working and which aren't. I have made this much easier by downloading the Google Drive app to my iPhone, and taking photos directly into a folder for that day's lesson, ready to be displayed on my large display screen later.
  3. Selecting. Basically, once I take a picture of a piece of student work that is most likely part of the closure discussion. I also take a mental or post it note about certain sticking points that students argued and convinced each other on.
  4. Sequencing. In this case, the files are usually queued up in the correct order of the folder by the order I took the photos in, but in the case of the lesson below I sequenced the two tables of a system of equations, a graph with the points plotted, and the equations written with substitution (or Equal values method) used to solve the solution to the system algebraically.
  5. Connecting. This happens by questions I ask during the closure. I asked how does two tables help us solve this problem? Where does it show us the answer? (where the same two coordinates are). Then when looking at the graph: how does the graph show us the solution? (the point of intersection where the two lines meet). How does the solution show up in the algebra? (After solving for x, that's the x coordinate of the solution, and when substituted or checked the y value is the y coordinate of the point of intersection).
The groupworthy task is a problem called Chubby Bunny. A cat weighs 5 pounds but gains 3 pounds per week. A bunny weighs 19 pounds and gains 1 pound per week. When will they be the same weight at the same time? This lesson 5.2.3 from CPM's Core Connections Course 3 asks students to solve the problem multiple ways showing different representations. Some students went straight to the algebraic method. Some went straight the the graph or table. Students realized to make an accurate graph it would be wise to make a table to see how far up and to the right their graph would go and what intervals to choose.
1st period:

This student used labeled horizontal tables with the values labeled as years and weight.

This student also has their graph completed with the point of intersection's coordinates labeled.

As you can see, this student used the Equal Values method. They may not hear this method later, so I've tried to tell students that it can also be called substitution, since whatever y is equal to is being substituted into the other equations y variable. This student forgot to omit the y= part in the first line, but thereafter did.

The next problem dealt with two schools, one that was shrinking and the other that was growing. Students had to make the jump to writing a negative sign in front of their growth and strictly using an algebraic method since the values were too large to put in a table and/or graph.

2nd period:
Neatly labeled vertical tables.

Properly scaled graph with two lines.

Steps clearly shown to solve the system.

The next problem solved: 20 years the two high schools will have the same population.

Vertical tables.

Color coded graphs.

Part of the lesson is asking students what x=7 represents. (The same weight in 7 years) I then ask how you could see what that weight is. When they are not sure, I ask them how they can be sure that x=7 is correct? (Check your solution, which leads to finding out what y equals in each equation)


The main points of the closure discussion were as follows: How did making tables help you find the answer? Students said it was when the weeks and weight were the same amount at the same time, 7 weeks and 26 pounds. Some students thought the answer might be 20 pounds because the bunny was 20 pounds after 1 week, and the cat was 20 pounds after 5 weeks. Their peers told them that it had to be the same weight after the same amount of weeks also. I prompted students to circle the solution in each table.

Then I asked the same question about the graph. Students said it was when their two graphes crossed, and the solution was at the point of intersection (7,26). Once again I asked what each number represented in the coordinates.

Finally, students described how they combined two equations into one equation. They then described subtracting x from both sides and so on. Like I said, students needed to be prompted and reminded to check their solution afterwards even though they knew their y value using the prior representations.

So, this is how I am running most of my lessons now. Circulating, finding common misconceptions, sequencing the order of taking photos of student sample work, prompting them to explain their work during the closure, or asking a volunteer to explain their work. Students absolutely love getting a photo of their work taken because they know the whole class will eventually see it during the closure (last 10 to 12 minutes)

Sunday, December 18, 2016

#clotheslinemath Slope Intercept Trial 1

I finally got a chance to try out the Slope Intercept clothesline math activity, introduced by Chris Shore and recapped via video introduction by Andrew Stadel. I took his advice and showed 0 in the middle with the other 4 benchmark numbers turned around. Students reasoned that to the left we should have -1 then -2, and on the right just 1 and 2. I showed students this image before this.

Some background information: I introduced this activity on a Friday in my Math 7/8 support class, but the 7th graders were in the back working on weekend homework and the 8th graders were done with their assignment due friday so I distributed the purple variable or you could technically call them parameters of each linear equation pictured on the following picture:

These students had previously worked on the mixed up algebraic expressions from Mr. Stadel's page where 2 colors of paper worked well. In this activity, in hindsight, it didn't work as well and I know for next time I would do the following: Make the benchmark numbers purple (-2,-1,0,1,2) and make m3 and b3 blue paper because it's a blue line. Make m2 and b2 on white paper because it would show up as black text and red paper (one student saw it as an orange line, which I said doesn't really matter because we know you're not talking about the black or blue line) for the m1 and b1 cards.

After seeing that above image, I started by asking what students noticed. Some said they saw parallel lines...(?) and another said all the lines are eventually going to intersect (I honestly did not notice that at first). This brought my attention to an idea. There are blank expression cards and I could easily write x on one and y on each and ask if we have an idea what these might be if Tommy thinks these lines will eventually intersect? (You can make the argument that x is 2 and y is 1 if you can convince others that the lines will intersect at (2,1)... but I suppose students could write and solve a system to see what it would be with the agreed upon values..?

I passed out m1 and b1 to one table, m2 and b2 to another, and m3 and b3 to a third. I wonder if I should only make an x and y clothesline card if they ask for it, or if I have 8 table groups just give each table one clothesline card to bring up in turns after taking notes on a personal whiteboard of where they see it placed in comparison with the other ones. That's what I will do for the next time.

Students knew that b was the starting point or y intercept of the graph and they knew m would be the growth. Aidan reasoned that since the black line doesn't increase or decrease it's growth must be zero, so he put m2 under 0. By the way, all of these are subscript numbers and the students immediately noticed that also.

Emma reasoned that the blue line was increasing so it must have a positive slope so she put m3 under 2. We didn't discuss why 2 may be a better choice than 1 (future conversation). She also reasoned that the y-intercept was on the negative side of the y axis so it must be -2, so she put b3 under -2.

Students struggled with differentiating or figuring out the b values for the 1st and 2nd lines. They especially struggled with the growth of the first line.

The good news is I will revisit this activity with these same students and the rest of the class in the mainstream class and they should have some valuable contributions to make.

Also, for the next time I pasted the graphic of the graph on a google doc 4 pages in a row, and then printed it and changed it to 4 copies per page so I can photocopy and chop the graphs for students to paste into their notebooks and take notes of the activity.

Students also had the option of attaching a second parameter to a first using a clothespin.

Update:
To extend this lesson I could introduce the screen shot of a desmos graph and ask if they are still correct. I restricted the domain so students could draw on a hard copy or prove they intersection using substitution. Open it up to notice and wonder.


Tuesday, December 13, 2016

Fractions Busters w Video Hook

Thanks to SVMI PD a few years ago who introduced me to this awesome clip from the movie Little Big League..
I like it for many reasons. First of all, it has humor. Secondly, it shows how the players weren't afraid to TRY the problem and be wrong, something we want our students to be.

This is the infamous paint problem, and I remember Jo Boaler talking about it in her summer math course she offered a few years back. My purpose wasn't for them to figure it out with their own method, or for them to sue the work formula. One person paints a house in 3 hours, then other one in 5 hours. How long would it take for them to paint it together?

So first we defined the variable x. What are we trying to find out? The time it takes to paint the house together. I asked them how much of the house would the first person paint in one hour? (1/3) So, that's 1/3x. The second person would paint 1/5 of the house in 1 hour, so 1/5x. Then I asked how much of the house they'd want to paint together? The whole thing, so those added together should equal 1. Then I asked them to solve it (1/3x+1/5x=1) I also asked what are you dividing by when you multiply by 1/3? (3) So this equation can also look like x/3 + x/5 = 1.

Some students realized the LCD of 3 and 5 was 15, so they converted both fractions to fifteenths. Some didn't realize they could then add the terms together. Others realized you could, and did. Then they were stuck at 8/15x=1. Some realized you could multiply both sides by 15 to get you 8x=15. Some put the 1 over 1 to make it look like a proportion and then to use cross products. Few students realized they could divide by 8/15 on both sides and multiply both sides by 15/8.

A few students in my first 2 classes remembered a method called Fraction Busters to eliminate the fractions in the equation that they were introduced to in 7th grade. Many had forgotten this method. I highlighted this method by using the Google Drive app on my iPhone to take a picture to my Google Drive and view it with the whole class and have that student explain their method.

Then students practiced on some of the problems in the section of CC3 section 5.1.2:
This student did not eliminate the fractions.
This student saw it as a proportion to solve for x at the end.
This student remembered the fraction busters method without me telling them.
In this practice problem, students realized all the terms had decimals. So, all terms on both sides were multiplied by 10 to make them whole numbers. The Estimation 180 we did at the beginning was really helpful because they estimated the capacity of a soda can. Most students estimated 8 or 12 ounces. The answer was 12.5, so the percent error ratio was 0.5/12.5. I asked students how to write the fraction without decimals. Some students said multiply by 10 to get 5/125. Others multiplied by 2 to make it 1/25. My colleague expected students to multiply by 8 to make a denominator of 100 but I didn't expect any of my students to do that and none did.
Here it wasn't completely necessary, but here all terms could be divided by 10 to make it simpler.
Here was a 2 step equation that they figured out.

We had a great conversation in all classes about remembering to multiply all terms by 5. Some students forgot to multiply the 1 by 5 on the left side which was a great discussion.



Order of Ops WODB, 5 Practices with 3 Act

I had a substitute give all my students the following Which One Doesn't Belong prompt on a quarter sheet of paper to all my students as a warm-up. I saw some of my 8th graders got some wrong, so I assumed my 6th graders struggled as well. So, I passed out some brand new copies of it. They spent a lot of time on it.

As you can see below, most students' first instinct is to simplify it all from left to right. In the bottom right they really want to subtract 6 to 20 rather than subtract 6 from 38.
We discussed some of the mistakes. But before doing so, for one of them there were almost 5 different answers. I made the analogy that if people in New York get one answer and people here in California get a different answer, we have to agree on a method so we are not getting different answers for the same problem. This is why mathematicians came up with the order of operations. We also confronted the myth that you add before subtracting because "A" comes before "S" in PEMDAS.


We then dove into Graham Fletcher's 3-act task "Gassed." To be honest I did not prepare for alternative incorrect solution methods I did reflect and know that if a student got an answer by adding or multiplying incorrectly I could ask for an estimate or how much 2 gallons of gas would cost, or 3? When I asked them if it seemed reasonable, they thought so. Therefore, I have to have a backup question to that initial one.

After watching Act 1 of the video, students wondered how much money was given to the gas station. I told them that I never go inside, I just swipe my card because I want to fill it up all the way.

Part of a 3 act is then asking students what they want to know to answer the initial Act 1 question. They came up with asking how many gallons could the car hold and how much does it cost per gallon. I showed them these clues (9.52 gallons and $2.09 respectively).

Many students just added the numbers together. Others unsuccessfully got past multiplying one factor by the hundredths place (not understanding the placement of the factor multiplying by the tenths place).

This student for some reason thought that the $0.23 shown in Act 1 had something to do with the answer.

This is the student that was able to estimate the answer was near $20. As you can see above, they did not place their 3rd row of multiplication correctly lined up below the ones place.

Once again, struggling with 3 digit by 3 digit multiplication.
This student ended up lining it up correctly, as you can see with the zeros with the line through them representing the placeholder zeroes. They made one small calculation error for 9*5+1, coming up with 43 there. This affected the rounding.
This person did not make any calculation error here.


The best part of the conversation was when they lined up the decimal in their product with the decimals in the factors. Then they said, 1,989.58?? That is WAYY too much money. So, instead of telling them a rule, I asked them where it made SENSE to place the decimal. Then they quickly said after the 19. Unfortunately, we didn't discuss rounding to the nearest penny because we were looking at a students work that miscalculated it slightly. They did come up that their answer was off by a penny.

Students also loved watching the money display increase quickly. This relatively simple problem provided a rich discussion in our 6th grade math support class. Thanks Graham!

Sunday, December 11, 2016

Productive Peer Feedback in Gallery Walk (Student samples)

As a follow-up to my NCTM blog post (#2) about eliciting productive peer feedback, I used the same Google slides to remind students what type of feedback isn't useful and what type of feedback could move the learning forward. 

Students had 2 days to complete their posters, and on a minimum day schedule we had a gallery walk where they looked at all the posters then settled on 1 poster to stay at to look at it in more detail and give feedback.

In one class one particular group was off task and unproductive so there poster was lacking a lot compared to the rest of the posters. Only 1 person went to their poster to give feedback, and I hung all the posters in my room. I told them that it should be a reminder to them of the experience and motivate them to put forth their best effort the next time they are given a poster project.

For teachers of CPM, this is Core Connections Course 3 4.1.1 where they are exploring the 4 representations of a quadratic tile pattern.
What I liked about this is the student made a connection between how they wrote their rule, and how another group wrote their. She also elaborated how they got their rule (by making the pattern a rectangle).

Here you can see they complimented how the group showed the growth of the tiles by shading.

This student likes drawing a lot, so I decided to show how she got creating with her characters showing positive and constructive feedback.

This student shared their opinion and also why a graph can be more helpful than a table.

Here they provided some suggestions on how to make their graph more complete and easier to read.

This student noticed how a table group didn't connect the points on their graph and explained why, since it was a discrete graph and there's no such thing as a decimal figure number (that was on the poster).

One group made their y axis scaled by 1 (labor intensive) and this student commented that it inspired him. Haha.


NCTM Blog Post #3: Cooperative Learning Strategies

SF Math Teacher Circle

The first SF Math Teacher Circle was held yesterday, December 10th, at Proof School. It was nice to see some familiar faces that had attended the Oakland PCMI professional development a few weeks ago. We joked how our spouses were asking if there is a math event every weekend. Nope, but just recently.

Paul Zeitz, who organizes the Proof School, organized the event. It was a rainy day and I took BART to avoid any parking issues, and they were very generous with coffee, pastries, and bagels provided as well as some Chinese food for lunch.

They started with a pledge to not give answers out to spoil it for the rest of the group. We also were seated randomly with cards.

We started off with a group activity from the awesome Get it Together book. It was an activity I hadn't done before, which was about a sequence. As you can see below, the clues were: a person had 10 acres of land after the first year of the big drought, in the summer of 1914 she had 410 acres of land, after every fall harvest she bought all the fields that shared a fence, and finally each field was a 10 acre square, that shared a fence with 4 neighbors.

We worked separately then shared ideas. I used a diagram as you can see below, and kept track of the acres with a table. We had a nice discussion about how she had 410 in the summer, so it was BEFORE she had bought more in the fall. So, at the END of 1914, she had 610 acres. From there I just labeled the years going backwards, seeing that the starting point in 1908 was 0,0 when the drought happened.
Then Avery Pickford presented on some voting topics. I like how he started with a table of values and asked us to notice and wonder. We realized each column had the numbers 1 through 5 in different orders. He then added the labels to the table to see how the columns were different sized groups of people and the rows were the food preferences ranked on a scale of 1 to 5, 1 being their favorite.

The big question was, how could you satisfy the most people? It was interesting, because different methods produced different winners. John in our group did a golf score method where he multiplied the rank by the number of people in the group, then summed those results for each type of food, and burritos had the lowest score. Another method is taking the average of each score, which ends up not being too fair. There were also some other methods we discussed.

We then looked at how voters would vote based on what type of candidate there was in a 2 party system and a 3 party system. 

Finally, Paul presented on some math games. It reminded a bunch of us of the 21 flags survivor game from PCMI. Game 2 was Basic Takeaway. Start with 16 pennies and remove 1, 2, 3, or 4 pennies. Our strategy was to go first and take 1, leaving your opponent with a multiple of 5 to choose from. The person who grabs the last penny/pennies is the winner.

Game 3 was Don't be Greedy. Basically you start off with pennies, but you can take any amount that is not all the pennies. Your opponent can then take that amount, or less.

We also investigated a Cat and Mouse Maze that seemed to be never ending, unless the Mouse got to to the top left corner and was trapped.

The last one we talked about was breaking the bar. If you have a 8 by 6 chocolate bar like a Hersheys, whoever makes the last "legal move" or break is the winner. Basically, every time you break a piece, you are left with 1 more piece. So, the game has 48 moves, so I BELIEVE you'd want your opponent to go first on this one.

Once it was lunch time, I took off because I had to pick up my 6 month old from my parents who were watching her, but it was a fun day. It reminded me that I've got to prepare some Get it Together card sets for my support classes as well as my 8th grade classes.