I have always wanted to incorporate catapults into the quadratic unit, but frankly, the catapult designs I’ve always come across seemed complicated and daunting. So I was very excited to come across this design made out of craft sticks and rubber bands.

Here’s the link for video instructions:
Craft Stick Catapults

I don’t plan on buying the kits, as craft sticks and rubber bands are easy enough to get a hold of. As for the basket, I’m sure that could be made either with masking tape, or by simply gluing down a washer or a nut. As far as a good projectile, I’m thinking Skittles.

After building the catapults, students would then have to measure how far down they pull their basket (the height of the basket from the ground), the distance and time in the air of their projectile, and then model the quadratic equation.

The other thing that I really like about this design is that it makes it easy to adjust the settings. For example, since the catapults in the video are 9 craft sticks thick, students could easily change it to 8 or 10 craft sticks thick, and model those equations too. And then maybe come to some conclusion as to how many craft sticks they would need to have the projectile go x distance.
**11/10 update: Actually, I don’t think there would be a significant difference between 8, 9 or 10 craft sticks. Maybe just compare between 3 and 9 or something like that.**

Just mulling over ideas…

**11/18 update: I think this would have to be done outside to keep the skittles from bouncing around so much. Also, instead of changing the number of craft sticks, students could change the initial height of the catapult by first placing it on the ground, and then placing it on something higher, like a picnic table (there happens to be some picnic tables in the courtyard of our school).

## Math Notebooks

Around my third year of teaching, I started giving notebook quizzes and have kept this procedure ever since. (Not an “open notes” quiz, it’s basically just a notebook check). Many teachers I talk to don’t this, but I like notebook quizzes for two reasons. First, students are more diligent about taking notes and staying organized. Second, this is one of the ways my students are held accountable in my class for their Do-Nows or Warm-Ups, since I rarely collect or grade them.

A few teachers actually collect student notebooks, but I find the prospect of having to go through 35 notebooks very daunting!

A notebook quiz in my class consists of basic questions such as “What was the answer to do now #2 on the 17th?” or “In the notes section on the 14th, example 2 was a rectangle. One of the dimensions was (2x + 1). What was the other dimension?”

When I decide it’s time for a notebook quiz, I simply choose a student’s notebook, making sure it looks reasonable in regards to being dated, with organized do-nows, notes, etc.

The person whose notebook I’ve used gets a 100 by default. When the quizzes are done, we do a paper swap and grade them as a class. I don’t have to be organized, I don’t have to create a quiz in advance, and I don’t have to grade them.

This also has helped in regards to student absences since students are responsible for getting any notes they may have missed while they were out.

I only do this about once every 3 weeks and never announce them in advance.

NOTE: First, of course, I have to lay down the groundwork for the students during the first few weeks of class. Most students have poor note-taking skills and have to be taught to date their notes, label the sections such as “Do Now” or “Notes,” number their problems, etc.

## Letters to a Fellow Math Student

I used to never have much success with writing in math. The majority of what I got back read something like this:

“To solve a two step equation … first you subtract 2, then you divide by 6 and the answer is 3. That’s how you solve a two-step equation.”

So I ended up with stacks of papers that read like this, and really, what do I do with these? It simply showed me that the student knows the arithmetic of how to solve this particular two step equation. As far as it being a method for assessing student learning, I usually knew beforehand which students knew how to solve 2 step equations and which students struggled – simply from having walked around the room during independent practice.

Then I did one writing assignment that produced wonderful results. I told the students there was a Pre-Algebra class or another Algebra class in the building that was learning about two-step equations (or whatever topic I wanted them to write about), and they were having a hard time of it. So their assignment was to write a letter to a “fellow student” and share what they have learned and know about solving two-steps. I then gave them a list of requirements: they must use the words constant, coefficient, isolating the variable, explain how to check a solution, etc.

And those results were wonderful, I think because the students knew the letters were actually going to be read by somebody who might find them useful. Students who usually wrote big to make it look like they wrote more than they did were now trying to write smaller and smaller to cram more words at the bottom of the page. They added illustrations and elaborated and produced beautiful products.

The only catch is that this is a one time deal. In other words, you can only disguise a writing assignment as a letter once. Do it again and the quality quickly goes downhill.

Make sure you do actually give these letters to a fellow math teacher. It makes a wonderful assignment for the receiving teacher as students can check for errors, etc.

Finally, another thing this taught me is that if you want to improve the quality of a non-letter writing prompt, or any summarizing assignment in general, provide a list of requirements.

## Grading Tests, Quizzes and Classroom Management

I have rarely had the need to bring a stack of tests or quizzes home to grade. In fact, the majority are graded before class is over.

It helps that I can grade papers very quickly. And no, they’re never multiple choice or on a scan tron, I’m a firm believer of open ended questions when it comes to assessing students.

This is how I do it: I sit in front of the room on my stool as the students take their test or quiz. (Most of the time, I’m doing the test myself to create a key to use and to check, yet again, that there are no typos in the problems). They bring them up to me when they finish, and I grade them then and there. Here’s why I thought to comment on this procedure.

First is that the students love the immediate feedback. When my students first realize this is how I do things, they always comment on some teacher from a previous year who took weeks to return things to them.

But most importantly, it helps with classroom management. Students are restless after taking a long assessment, and they think they can whisper to their neighbor, whether their neighbor is finished or not. But my students know they’ll have to wait the next day to find out their grade if they make even one peep before the whole class is done. They hate this, especially when they know their graded paper is in my hand. I no longer have issues with talking during a test.

And I’ve never had a problem with a student rushing through just to get a grade. Students who take the whole period know that they can come back at the next class change and I’ll have theirs scored already, too.

Of course, I teach Algebra 1 and Pre-Algebra. I doubt I’d be able to do this with Calculus or Algebra II. And I do not in the least envy the essays English teachers have to pour through.

## Patterns, Linear Equations and Toothpicks

Toothpicks are a great tool for exploring patterns. Add mini marshmallows and now they take on a new dimension (just make sure to dry out the marshmallows for a day or so before using). An obvious use is to make those 3D shapes for geometry, but for the Algebra class, it’s a great way to introduce students to linear equations:

Toothpick patterns and linear equations
-Create a triangle using three toothpicks. Then add a second triangle next to it (which requires 2 more toothpicks). The second triangle should be upside down so that the two triangles together look like a parallelogram with a diagonal. Continue the pattern, keeping track of the number of triangles vs. number of toothpicks needed, plot the data and come up with the equation that models the data. Calculate how many toothpicks it would take to make 100 triangles, or 1000 triangles.

-Create a square using four toothpicks. Then create a second square next to it (which requires 3 more toothpicks). Then continue the pattern, keeping track of the number of squares and number of toothpicks. Plot the data, come up with an equation that models that data, use that equation to figure out how many toothpicks it would take to make 100 squares, etc.

-If you add marshmallows, you can now also create cubes or tetrahedrons, and follow the same sequence as above to explore more equations.

Options
-To add even more graphs to explore, or if your students are advanced, keep track of the number of toothpicks vs. the perimeter or area or volume (using toothpicks as the unit, of course).
-If you use marshmallows, keep track of marshmallows vs. perimeter or marshmallows vs. toothpicks, etc. There are too many variations, which is why I don’t have a student guide to go along with this.

Notes
I love any hands on activity that allows students to explore things on their own. This activity (or activities like this) gives students a foundation from which I can help build the meaning of “rate of change.” Unfortunately, the y-intercept portion does not work out too well, as 0 triangles should require 0 toothpicks, but the equation comes out to 1 toothpick. Regardless, since I use this as an exploratory activity, before any “lecture” on linear equations, then students never know to look for this. Plus the lowest value they graph is 1. And the point is to experience gathering data, graphing lines, figuring out how to come up with the equation and to see how the rate of change differs for each pattern.

## Scientific Notation Videos and/or Project

Scientific notation has always been one of the low points in the curriculum for me – I find it dull. After all, you’re basically just counting decimal places. But then one year I started showing things that were actually 10^8 kilometers or as tiny as 10^-5 mm. Here’s my final collection so far:

One year, I found some pictures of molecules that spelled out IBM. I think they even made a microscopic violin that actually played music (well, noise – we listened to the little soundtrack that accompanied it). But I can’t seem to find them anymore…

I don’t recommend going through all the links for 30 minutes straight in one class period. But these could be easily be integrated into your lesson. The powers of 10 video is always a good finale to the scientific notation unit.

An option is to scrap the videos (until maybe after the projects are done) and turn this into a project. Have students find 10 things that are sized in the negative exponent range, and 10 things in the positive exponent range, then they must arrange their pictures on a poster board from least to greatest, labeled in both scientific and standard notation of course. (You’ll have to be more specific or they’ll just find planets and cells around the same size).  Or they could power point their presentation and convert it into a video…

Another thing I do to introduce students to small and big numbers (well, big numbers really) is have the students figure out how long it takes for a million seconds to go by, a billion seconds and finally a trillion seconds. (12 days, 32 years, 32,000 years). This lets them at least appreciate the difference a couple of zeroes make. (By the way, you’d think I’d get used it it, but it always amazes me how many students actually attempt this problem by repeatedly adding 60!)

## Get the Math

I was pretty excited when I saw that they were going to make a new show about math in the real world. The series is called Get the Math, and so I immediately went to their site to see what they had to offer. There are, so far, three videos. One is about fashion, one about music and the one I watched was about math in video games.

Since I only watched one video, you can assume that I was very disappointed. The video took five or ten minutes (I can’t remember how long it was) to very generally say “we use math to make video games.” They spend most of the time talking about how they get ideas for new games, and the creative process in general, but not much about math. They do comment on the coordinate plane and linear equations, but only briefly, and only because they made a very 80-ish looking video game just for this video so that they could mention the coordinate plane and linear equations. Since I have some programming experience, and am quite a nerd as well, I know there’s a lot more math going on in the background.

Take, for instance, the Pokemon games. Here is a tiny piece of the algorithm used just to determine whether a pokeball will catch a pokemon (I have four kids…).

I think they water these math shows down way too much.

I’ll end up taking a peek at the other two videos, but I really do wish these shows would consider hiring a math teacher as a consultant to tell them what math teachers really need.

**Okay, so I decided to give it another go and watched the fashion one. Not much math there, but I thought the industry standard of a 220% markup was interesting, that fact may come in handy for a future activity…

## Integers and a deck of cards

One of the easiest ways to practice integers is with a deck of cards. Take out all the face cards except the aces, which count as ones. Working in pairs is ideal, but groups of 3 or 4 will work. Below are some ideas for card games, but please be advised that card games are a waste of time if your students are struggling with integers. If a student can’t add 2 + (-7), they’re not suddenly going to be able to add them just because the numbers are now on a card. There are more efficient methods for guiding below level students to understanding integers.

These games are only good for students who have a good grasp of integers, but just need to work on speed. Plus they make a great end of the year or “just before Christmas break and we don’t want to do anything” activity. Or if you are tutoring a struggling student one on one and are actually going to play with the student yourself.

WAR
The winner is the one who ends up with the deck of cards or the one who has the most cards when time is called.

One Operation

Deal the cards evenly between the players. Each player lays down two cards. They must add (or subtract or multiply, depending on your instructions) and whoever has the highest value wins. (You could modify to lowest value). For advanced students, change the number of cards laid down to 3 (or 4) instead of just 2.

All Operations
This time, students may use any operation to get the highest value. For example, a 9 of hearts and a 3 of spades added together gives a -6, subtracted gives a +12 or -12, multiplied gives a -27 and divided gives a -3 (division never wins). The best option in this case is to subtract for the +12. The other student has an 8 of clubs and a 2 of spades. Multiplying them gives a value of +16, so the second student wins. (Students must inform each other which operation they have chosen. If the second student had chosen subtraction, for instance, then the first student would win the cards).

TRIPLES
The winner is the one who no longer has any cards in their hand, or whoever has the most triples when time is called.

All Operations
Each player gets seven cards, the rest are face down in a stack to draw from. They can set down triple sets when added, subtracted, multiplied or divided.
For example, a 6 of hearts and a 2 of spades could equal -4, 8, -8, or -3 (answers must be 10 or below since the highest value card is 10). So the student could set down the 6 of hearts, 2 of spades and an 8 of spades as a valid set.
To start the game, the first student lays down all the triples in his hand. He then draws a card to see if he can make a triple with the new card. If he can’t, it’s the other student’s turn. They continue taking turns until all cards are drawn.
As the students are playing, I’ll walk around the room and ask them to explain their triples.

25
The ideal size for this game is small groups of 3 or 4. Deal the whole deck of cards to the players. Players should keep their own cards face down in a stack. The first player picks up a card from her stack and lays it face up in the middle. Let’s say she shows an 8 of hearts (-8). She must say “pass,” because the total thus far is not 25. The next player then reveals a card from his stack, and lays it down next to the first player’s card. In this example, let’s say he has a 9 of spades (+9). The total so far would be +1, so the second player must now say “pass.” This keeps going until a player reveals a card that makes the total = 25, at which point the player who laid the card down would shout “25.” If a player says “pass” when the sum is 25, the first of the other players to shout “25″ wins.
It’s a good idea for students to keep a running sum written on a sheet of paper, as this can get pretty long. Or to make the game easier, simply make the target number lower.

TARGET
For this game, it’s best to leave the face cards in. Black face cards are 10, red face cards are -10. Each player gets dealt 8 cards, the rest of the cards are face down in a stack to draw from. Flip the top card over, that becomes the target number. The first player draws a card and then sets down any pairs of cards that when added or subtracted equals the target number. Then it’s the next player’s turn. This goes around until a student runs out of cards or all the cards have been drawn. The player that runs out of cards first is the winner.

## Order of Operations Relay Race

This is an activity that my students have always enjoyed so I thought I’d post the idea here. It’s perfect for the last fifteen or twenty minutes of class. (It takes up an unnecessary amount of printouts to make this into an actual funsheet):

1) Split the class into 5 or 6 teams. This actually works best if your desks are in long rows or in circles. For the rest of the directions, let’s assume the students are in rows.

2) Make a long-ish order of operations problem, preferably one with more than six or seven steps. You’ll need to write this same problem onto 5 sheets of notebook paper (or however many teams you have).

3) Each student is allowed to complete only ONE step. To keep track, it’s best if students use colored pencils.

4) When the student finishes his/her step, he passes the paper back (the last person in the row has to run up and give the paper back to the first student) until the problem is completed.

5) The first team to finish wins … almost. The teams that aren’t done must turn the paper over while you check the answer. If it’s incorrect, write the work on the board. Once the error is found by the students, the remaining teams can resume their problem solving until a team comes up with the right answer.

For my below average classes, each student was allowed to help the student behind them. This takes some of the burden off any student who isn’t confident with the topic.

The first run-through takes a few minutes and has a bit of confusion, but the second time around is a lot more fun for the students. Make sure you have 3 or 4 problems ready to go. Oh, and to keep it fair, if the first person started the problem in the first race, make sure to let the second person start the problem in the second race, and so on. Also, a jolly rancher prize for the winning team tends to make the students work a bit more carefully.

## Go! Animate

I played around with some animation gadgets today. The first one I found was Go! Animate and here is my goofy attempt (The beginning is annoying because Jack Sparrow says boo hoo way too many times – in the Preview, he kept moving his mouth for about 15 seconds before it would go on to the next line, so I was trying to fill that time lag. Apparently, when you press save, it edits all that out, but I didn’t know that):

GoAnimate.com: Solving Proportions. by algebrafunsheets

It was relatively simple but the results were a little less than impressive. It could make a good summary project, students could make an animation about how to solve an equation or graph a line, the possibilities are endless. And the free basic package would be all that students would need, you just type in the dialogue and it does everything else for you.

Best of all, this would only take up a portion of one class period because of how simple it is. The hardest part is summarizing everything in ten dialogue scripts. If you have a computer lab or access to laptop carts, this might be something to look into.

Here’s another video made by someone else. I don’t know why but it cracks me up. It’s about direct variation.

GoAnimate.com: Kitchen Cooking with Hopeless Dad by pcramer