Tuesday, January 10, 2017

NCTM Blog Post #4: Replacing a Textbook Lesson

Sunday, January 1, 2017

Border problem (@youcubed @joboaler)

I also saw a video of Cathy Humphreys demonstrating this task.

It all starts with the image below (I found it by googling "Math border problems," make sure you don't forget the plural form and math or you'll get a bunch of articles on immigration policy). This is a picture of a square with a side length of 10. The border has been shaded orange. How many orange squares do you see without counting one by one? 
And then the magic happens...

In the particular class pictured below, students got answers of 37, 36, or no answer at all. I put 40 up their because I think some thought that but were too afraid to share it.


As you can see, there are 5 distinct methods that students in this cass used. The most common I believe is the one in the bottom left where the multiply 10 by 4 and subtract 4 because the corners were overlapping. I made sure students understood that it was subtracted because each corner was counted twice. This leads to 4x-4.

One student worked around that by saying there's a spiral of 9 square lengths around the border. It's hard to draw, but you can see it in the upper left. Therefore they got 9*4, which leads to 4(x-1).

Another said there were 10 squares on the top and 10 on the bottom. Therefore the left and right sides had 8 each, so 10*2+8*2. We didn't write that one as an algebraic expression.

Karin said she saw the square in each corner for a total of 4 corner squares, and 2 rows of 8 and 2 columns of 8.

Finally, Sebastian saw it similar to the visual pattern from yesterday, as a square with a total of 10^2 or 100 squares subtracted by the inside white square which is 8^2 or 64. 100-64 gets you 36. This leads to x^2-(x-2)^2.

I think I'm going to revisit this warm-up again, but slightly different as a Contemplate then Calculate instructional routine that I saw David Wees present at a Global Math department webinar. I can't find it right now but I'll post a link once I ask him.

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!