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Different Variable Usages

14 Jan

I remember there being a time when I had identified at least 3 different ways to use a variable, but for now, I can only think of two.

  1. Using variables to represent an unknown.
  2. Using variables as a pattern summary.

Well, w/ Common Core right around the corner, here are a couple of activities I loved, but rarely found the time to do well in the past. However, with my double Algebra this year, I have twice as much time w/ them to go over the same material so I got to work on these much more thoroughly… and I LOVE ’em even more!! I truly believe the lack of understanding variables in the different ways is what makes Algebra hard and not something that should make solving problems easier.

1. Variables to represent an unknown.

Usually used in solving simple equations. x + 5 = 9. “Some number plus 5 will give me 9”. I did a LOT more mental math to teach this part and I really do believe it went significantly better than ever before. Even up to equations like -2x + 6 = 0.

Another big way of using variables as an unknown that we neglect often, is in word problems. So this year, I have used a couple of worksheets that my former professor created which contains just word problems. After going over them pretty thoroughly for the first time this year, I fell in love with it even more because my students came up w/ different ways of writing the equations! Here we come, Common Core!

One example:
Lunch Money. Daisy’s mom gives her lunch money every weekday. On Monday and Tuesday she gave her an extra dollar. The rest of the week, Daisy received the regular amount. For one week, she received a total of $42 for lunch. How much money does Daisy get for lunch each day normally?
Here are the 3 different answers my students came up with:
Once I got them to start drawing pictures to represent the problems prior to this, they did this one on their own. =)
Execution Tips: Only pass out the first two pgs of Amazing Stories first (no answers). We did the problems, one by one, up to “Pocket Change.” Starting “Lunch Money” (the posted example), they were supposed to do it by themselves.
Once they started working by themselves, after a few minutes, I projected all the possible answers so that they had to find their equation (or the equivalent) up on the board. They got a copy of the answers in order to finish the worksheet at home for homework.
The next day, I gave them the follow-up worksheet of “Life of a High School Student” for them to get a few more practice problems on their own, without having a bank of answers.
I saved a few problems from “Life of a HS Student” to use as an exit ticket and on future quizzes.
                 2. Variables as a pattern summary
This is the type of variable usage where all those darn rules come from. For example, if 2^3 * 2^5 = 2^8
4^9 * 4^7 = 4^16
what does x^m * x^n = ?
Anyways, here is the activity on practicing this type. Again, I love it all the more due to the multiple solutions that are possible. I copied down my entry below.
Materials: Cubes. Preferably not too small. And the worksheets.
Students are then asked to build certain objects that build on top of each other and asked how many cubes it would take to build a structure of 10 layers… then 100 layers.
For example:
First question after this would include: “How many cubes will it take for 10 cows? Explain.
Next they would be asked, “How many cubes will it take for 100 cows? Explain.
You can then summarize the patterns by writing an equation, with the input (x) representing how many cows, in this case, and the output (y), representing the number of cubes.
TimingWe started Pattern 1 and 2 in the last half hour of one class, continued the next day for about an hour, and then they were doing Pattern 5 by themselves by the next day’s 50 minute class. Amazing how many students who do not understand Algebra actually are really good at figuring these puzzles out.
Anyways, to get a better feel, you have to take a look at the multiple solutions that these problems have to offer. I have written out what my students had come up with this year. Yep, these were my intensive students coming up with these different solutions. We also took time to go over how to write a good explanation also. I went around giving them feedback and having them re-write a lot of what they had written…
Again, here are the links to the two types of variable practice.
Aside

Commutative Property

18 Oct

I made some flashcards for practice of the commutative property of addition/subtraction. I created them in 2 separate parts, the first part WITHOUT variables and the other WITH variables. (possibly creating a third to help them see products w/ parentheses as well.)

I’ve also made a Google site to start collaborating w/ other teachers in my district who teach the same lower level Algebra. We’ll see how it goes. For now, it’s a great place for me to post my useable lessons to share. =)

Totally Underrated!

11 Oct

It’s INTENSE alright. Not the advanced kind, ladies and gentlemen, the other kind. With all the love and frustration I have for this class though, it’s right up my alley. =)

I basically have the privilege of teaching the 9th graders weakest in Algebra for a block period, 5 days a week!! I’m seriously very happy to be teaching this class, though I was thrown in with almost no materials or prep.

Now, I’ve taught Algebra a ridiculous amount of times considering I came from a 4 by 4 school back in the days. However, I stopped teaching it for the last several years due to intense overload and hatred of life from teaching Algebra 1A up to 4 times a year. Now that I’m back, things seem more clear and I feel like the fogginess of how to order teaching/introducing Algebra is starting to get clearer.

We’ll see though…

Anyways, first insight of the year (and feel free to tell me what you think), but I think the COMMUTATIVE PROPERTY and the ORDER OF OPERATIONS are waaayyy underrated!!

On the commutative property

  • I think when students don’t recognize you’re distributing -3 in 4-3(x+9), it’s a lack of knowing the commutative property
  • I think when students can’t seem to decipher which term is negative or positive in simplifying expressions like 4 – 6x + 9 – 5 + 3x, I think it’s a lack of knowing the commutative property
  • I think when students can’t seem to take 4/7 of a number on their calculator, it’s a lack of knowing the commutative property
  • I think when students don’t recognize that they can simplify a product of fractions before multiplying across, it’s a lack of knowing the commutative property

On the order of operations:

  • Very related to most of the previous issues, I think the order of operations in these damn textbooks don’t emphasize enough how multiplication and division are essentially on the same level and addition and subtraction are on the same level. Therefore the commutative property should NOT be only the commutative property of addition and multiplication by this point, but apply to all 4 basic operations!
  • The focus in Algebra should not be on the FIRST order of operations but now seeing the LAST order of operations. (The light bulb was turned on for me by this author and book.) Students should be able to chunk long expressions by the last order of operations.

Lastly, more things that I highly underrated: writing expressions from patterns and connecting mental math to the algebra.

Lessons I’ve got so far on focusing on these to be continued…

Movies in Math. Math in Movies.

18 Nov

Here is a website with a compilation of short movie clips on math and physics that you can freely download.

http://www.math.harvard.edu/~knill/mathmovies/index.html

I use the Shrek clip as the introduction to inverses and contrapositives in Geometry.(This one was also on YouTube.) The PowerPoint I used is in the Box widget.

Perpendicular Lines “… flip it and reverse it…”

15 Nov

This is Missy Elliot. She will help us w/ our lesson today on perpendicular slopes. (If you don’t know this song, no worries, all your kids do. And if you really don’t, it will be SOOOO much cooler if you still sing/ rap the lyrics at the end!!)

  1. Give each group/pair 1-3 lines to graph.
  2. Teach them what perpendicular means.
  3. Have them pick one nice point on the line and graph a line perpendicular to their original line.
  4. Ask them to figure out what the relationship between the slopes are of their two lines.
  5. When students look like they’re on step 4, start playing the pertinent part of Missy’s song on repeat (only up to 0:40 in this Youtube video). Point out that this is their big hint. Start singing the important part around the 3rd time (“put your thang down flip it and reverse it…”)
  6. Wait for the big “OHHHH… I get it!” to start spreading throughout the room.
WARNING: The song is quite dirty. I really mean it when I say only play the pertinent part, especially since we’re asking them to listen to the lyrics for the hint… Also, I wouldn’t recommend actually showing the video. Too much of a distraction.

M&M Catapult

2 Jun

I’ve been eye-ing this for awhile now so I was so excited to run this!! SO excited! I first read about this from Sweeney Math and seriously fell in love with it because the math in it is not forced. I really dislike those forced math projects, though I must admit, I use them sometimes anyways for engagement purposes.

Student’s Goal: To build a consistently-shooting M&M Catapult and find the equation of the M&M’s trajectory when shooting from the floor. Using that equation, calculate where the M&M will land when shooting from on top of a table.

Teacher’s Goal: Again, this was for my post-Calculus students so I really wanted them to understand some of the physics behind what we were doing. I created a few pre-questions to prime them a bit. I really wanted them to be able to see the integration between all the different maths over the past several years. (Pictures of problems below)

Copy of my modified lesson plan is to the right in the Box.net widget. As a note to that copy, I gave them one front-back page at a time. In other words, they had to get pgs 1-2 completed before they getting pgs 3-4 and they had to finish pgs 3-4 before getting pgs 5-6.

Materials Needed:

  • box of fat popsicle sticks
  • one clothespin per group
  • multiple glue guns
  • a wooden board to attach the catapult to (I asked the woodshop teacher who had a bunch of cut up desks available)
  • 2-3 bags of M&Ms (they crack with every launch so students kept wanting new ones)
  • tape measure
  • masking tape to hold M&M. I didn’t have any so we cut small Dixie cups I had laying around for geometry.
  • timer(s). I used my Flip because each launch lasted around one second and I can be accurate to within 1/30th of a second with it.

Overview of Project

  1. Build a catapult that shoots consistently to the exact same spot.
  2. Time how long the M&M is in the air for and measure the horizontal distance the M&M shoots
  3. Use a bit of physics and the vertex form to figure out the equation of the M&M’s trajectory
  4. Give students the height of the desk they will launch from. They calculate where they need to place the bull’s eye if they were to shoot if off that desk!

Tips and Hints

  • I gave my students freedom to build the catapults as they saw fit. I just gave them the hint that it should launch to the exact same place every time AND that they should know at which angle it launches if their M&Ms were flying crooked.
  • Doubling up the “diving board” makes each launch a bit more consistent b/c it gets rid of the bend.
  • Try to run this outside or in the gym. It was raining and windy in sunny So-Cal the days we were doing the trials so we had to limit the launches to within the classroom. This makes the launches limited to the height of the room, making the times shorter than I had wanted.
  • Try to avoid using the stopwatch the shorter the launches are.
  • Make the target fit into about 1 square feet. I’d say the bull’s eye was about 2 in radius. I had several groups get a bull’s eye on their first shot!

… the rest is in the file “MM catapult” to the right.

Egg Bungee

21 May

The screaming egg of a good bungee

None of these projects are my creation and as much as I would love to give due credit for this one, I really can’t remember where I got this from! It has also been adapted and tweaked a lot before getting to what it is now. I have attached the handout to Box.net on the right. One copy is what I give to my Algebra 1A students but since the goals for my former Calc students were a bit different, I modified it and attached that one as well.

Student’s goal: After gathering data on the stretch of the bungee chord for every rubber band added for up to 100 cm, they must use their linear model to predict how many rubber bands will be needed to bungee the egg off the top of the bleacher of 200 inches. The goal is to bungee the egg as close to the ground without having it hit the ground.

Teacher’s goal: For my Algebra 1 class, it is to be able to use a linear equation/model to solve/predict a solution. Not only does the slope of the line show up in how much stretch per rubber band, but the y-intercept shows up in the length of the egg in the harness w/ zero rubber bands.

For my former Calculus students, however, more than just coming up with the linear equation, I really wanted them to be able to formulate what information they needed to gather and be able to come up with a strategy of finding out how many rubber bands they needed for the bleachers while working within their constraint of 100 cm.

Materials needed: Raw eggs (bungee participant), plastic eggs that can be filled (test dummies), candy (test dummy innards), scale to weigh ounces, non-ziplock plastic bags (harness), rubber bands (bungee chord), short tape measure (I used Ikea paper rulers)

Overview of project:

  1. Weigh out test dummy to equal the weight of the real egg
  2. Gather how much egg stretches with each rubber band added
  3. Plot points of rubber bands vs stretch distance
  4. Come up with a linear equation (by hand for Alg 1, using regression line for Calc)
  5. Use equation to predict number of rubber bands for bleachers

Notes:

About project:
They usually come out short because they get scared and take off too many. Encourage them to stick to their numbers. The stretch in all the drops also makes their data show the drop to be longer than with the fresh set of rubber bands. *Buy of bag of all the SAME sized rubber bands.* I tried to vary it to see what this class would do with it and it turned out varying too much.

Scientific Error:
I’ve done this before with my Algebra classes, but this time, I had a whole new goal, which completely changed what I was expecting. For one, I required a write-up of the lab with tons of explanations because I was more interested in the process than the actual equation and result. With this new goal in mind for my Calculus students, I found a very interesting discovery-

My students don’t understand what scientific “error” is. I asked them to explain where there might have been some error while gathering data in this project, as in, explain where they might have lost some accuracy in gathering data and coming up with their math model (like from all their approximated measurements). They, however, kept saying things like “we used 38 rubber bands when we should have used about 20 more.” When I tried to explain what error really meant, they still didn’t get it in the re-write of their write-up. This is so interesting to me because I believe they’re thinking about “error” as they have seen it in their previous years of math, the wrong answer. They kept trying to explain away how many rubber bands would have given them the correct jump instead.

After discussing this with a friend, I think maybe this type of error analysis, as they see in science, is exactly the type of “error” my students should be thinking about in math too. For example, if they get a wrong answer, instead of calling that finality the “error,” the focus of the actual error should be where that error was made back in the process of things.

**New goal for my classes** Have a few regular routines where I give them full credit for explaining where the “error” was made in the process of the math. Maybe I can even coin it to say something like, “the theoretical outcome is 3. One student’s actual outcome came out to be 4. Where could there have been some error?” It’s very scientific this way, no?

Write-up
I had forgotten how much my students needed structure on explaining mathematical process. They need at least two re-writes before they start to understand what is expected of them. I really wish I could have them journal so much more frequently than I actually do in our two month terms.

Flip
I love this thing for my classes. So far, not only did it give me a better Calculus Camp video this year, it makes projects so much more fun. The snapshot above was taken off a Flip recording. We determined how low the eggs fell by video recording it and then analyzing the video frame by frame. In the next project, the M&M Catapults, we use the Flip as a timer. It’s much more accurate than stopwatches this way.

Our very first bungee

One sweet landing

Quadratic Shapes and Songs

17 Apr

I was demonstrating what the solutions of an equation like ax^2+bx+c = 0 could look like graphically.

I asked the class, “What is the shape of THIS?” as I circled the ax^2+bx+c in the equation up on the board.

Amidst a few yells of “parabola!,” I heard a very serious response of “an oval“.

I tried rewording it to “what does this graph?” instead, but it was too late. The sentence came out choked in laughter as I saw the round eyed small student who said it. He was thinking about the actual circle that I made around the formula!! As I realized this, I couldn’t stop laughing so hard that I was crying. Thankfully, my class is not the type to take offense but laugh at me and my crying instead. Cute.

Then yesterday, I taught them the Quadratic Formula song to Pop Goes The Weasel along w/ the sign language for it. I’ve never had a class sing so loud! I’ve had a class of 40 sing this before and it was never this loud. They were like little kids singing their hearts out louder and louder! I might not have been a happy choir teacher w/ the result, but I was one happy math teacher!

Other random things:

After singing them the Q. Formula song and telling them that they will have to sing it too, one kid responds, “But I don’t know the lyrics!”

My response, “My point exactly! The whole goal is to learn the lyrics!”

Then when asking them to write the Q. Formula three times by covering each up while they write the next one, one kid goes, “Aww man, can’t I just write it once to the third power? It’s a math class isn’t it?”

Calling for conceptual help!

7 Oct

I know I’m a nerd when a question like the following is my hot topic of the day:

Why is it OK to replace a row in a matrix with the answer from adding two rows?”

For example, in solving for:

x + y + z = 0
x – y + z = 2
x – y – z = 10

a matrix like this can be used:

1   1   1   0
1  -1   1   2
1  -1  -1  10

The next step would be to add two of the rows and replace one of the equations

1   1   1   0
2  0   2   2
1  -1  -1  10

But WHY is this allowed? It’s not like I’m replacing the row with an equivalent equation…

After some discussion and thought, a coworker and I came upon one conceptual theory. Conceptually, what we are doing is replacing an equation of a line with a new equation that got rid of one (or two) of the variables. Each of the variables, in this case, represent an added dimension.

Therefore, what I think we are doing when replacing a row is that we’re getting rid of dimensions and using a model in a lower dimension to represent the original line.

For example, this is how I see it:
Picture a line sticking diagonally out from your computer screen. This line is definitely in 3D. Now imagine taking a picture of that line coming out of the screen. This picture on your camera now made a 3D line into a new 2D line!

It’s kind of like taking a 3D object and taking a picture of that object. That picture is a 2-dimensional representation of the 3D object.

This object was ORIGINALLY in 3-dimensions

This object was ORIGINALLY in 3-dimensions

Taking a picture of it, however, makes this new representation 2D

Taking a picture of it, however, makes this new representation of the same cube, 2D

I think the same thing is happening with these lines. When solving a system of equations like

x + y + z = 0
x – y + z = 2
x – y – z = 10

what we are doing when looking for a “solution” is finding the point of intersection of the 3 lines. So in order to do this, I think we’re taking “pictures” of the lines to simplify our 3D model to 2D and then to a 1D representation that reveals the point of intersection.

So again, this is only what I’ve theorized so far. I can picture it and I’m looking for a way to visually prove it. I’ve never actually taught this before in Algebra 2, always due to lack of time. I finally decided to teach it this year and would love any kind of feedback!

QUESTIONS:

  1. Does this mean I can replace a row only with a new equation that gets rid of a variable/ dimension? What does it mean if I replace a row with a combined equation that does NOT get rid of a variable? Is this like taking a picture from a different angle not parallel to one of the axis?
  2. If all this is true, why does COMBINING the two equations give you an accurate “picture” of the 3D model?

An exciting MISTAKE

5 Oct

I usually start off the systems of linear equations section with a CBR activity where in their group they have to use two TI-graphing calculators and two CBRs to match 4 different graphs. It’s a great activity because it gives them exactly what the graph looks like in real time. This means that they can actually see the creation of the abstract graph while watching what is being represented.

This year though, I came upon a very interesting mistake that I too quickly put off at first. I kept thinking about it while they were doing their activity and in the end, I was convinced it was a brilliant mistake and one that must be addressed.

But first, here is the actual activity for those who might want it. I was actually debating whether I should even include this activity this year due to time constraints, but as soon as I heard the discussions that arose from just the first question, I knew it would be well worth it. Then of course, there was this “mistake” that made it THAT much better. =)

Picture 1
Picture 2
Picture 3

The Mistake
The actual mistake was made while one group was working on the second graph with the parallel lines. We had been using tape measures taped to the floor in order to create accurate graphs.
440412085_1532177044_417002507_1254329337873.jpeg

When I turned around, I saw that two people were walking away from their tape measures with the CBR pointed towards them. I was baffled. So I went over to ask what it was that they were doing and why they were doing it.

The response I got was “It looks like that, so we’re making it look like that.”

Ah~!! At first, I thought it was just a funny mistake. When I asked them to explain what the graphs actually meant and compared it with how they did the previous one, they all started to laugh too upon realizing what they did.

BUT, like I said before, the more I thought about it, the more I thought it was a brilliant mistake to address. So I took pictures of the tops of their heads at the end of the period to present the mistake to the rest of the class the next day.

So the next day, one of the things I did was to have the class actually think of what it was that the group was thinking. (In the Powerpoint, the multiple heads actually move one by one across.)
Picture 1

Picture 2

Then, in order to address the issue and have the class really understand why this would not give the right graph, I had them interpret 3 points on the graph and what they meant. Not only that, I drew on this slide, where you saw the x and y coordinate as actual DISTANCES on the graph.
Picture 3

I then had them re-think how this could have been done w/ the girl walking the same way, but simply moving the CBR, which would have looked something like this: (again, the powerpoint actually had them move instead of this mess of a screen capture.)
Picture 4

What I loved about this mistake:

  1. It made sense. When I had asked my students what they thought this group was doing, I almost immediately saw the wave of understanding pass through the class as they started to discuss it.
  2. The concept of seeing each point on an x-y plane having TWO values, TWO meanings per point is a tough concept for students to grasp. When they saw the tape measure as the y-axis, they ignored what the x-axis meant in also existing.
  3. Seeing the dissection of each of the coordinate point as a horizontal distance and a vertical distance is a concept that is seen all throughout math!! I suppose in any kind of graph reading, but I can think of unit circles, iterations, geometry, and even just slope! (which is probably another added reason to why they have such a hard time seeing it still.)

This is one of those things, like most things in this class as it turns out, that might not show up on a CST exam. I don’t even know how to test for understanding myself. I just found it fun and interesting to address. Of course, I believe it was crucial too. But *shrug*, what isnt?

(In case you want the activity, including the instructions on how to transfer the files, see Box.net widget to the right. The file is titled “CBR Systems” and the calculator instructions is called “TI Combining Data”.)