Rotation Help

Fellow math nerds: help me out here. I was describing how to rotate a figure 270 degrees clockwise to someone. I told them to reflect the figure over the y-axis, then switch the coordinates to fix the orientation. They told me this was the formula for rotating a figure 90 degrees counterclockwise. Well, yeah. . . . what's the difference? Is there a different formula you're supposed to use for a 270 degree clockwise rotation vs. a 90 degree counterclockwise rotation? I'm at a loss. . .

Project Based Learning

I recently found out that I'll be moving from middle school to high school next year. I'll be piloting a new program with another teacher with Algebra I and Algebra II students in the same classroom.

Our high school is a "New Tech" school, which means it's focused on project and problem-based learning. The tentative plan is to do projects, lead small group workshops, and generally be "less helpful," (to borrow the words of Dan Meyer). We'll have a 1:1 student to computer ratio. No sitting in desks in a row, no lectures, no traditional homework assignments . . . . have we lost our minds? I'm excited, and also a little terrified about how this is all going to work.

We're going to be using a program called ALEKS as the main backbone. I've never used it before, but have heard great things about it. ALEKS is an "artificially intelligent assessment and learning system." To me, it takes standard based grading to a whole new level. Students have to show mastery of a concept before it's added to their "pie." I can also set up assessments on a regular basis - if they get something wrong, it gets taken back out of their pie, so they're constantly reassessed on what they know. And the best part is that the computer grades it all for you! Talk about freeing - I can't wait to have more time to spend developing projects instead of grading papers.

Comic Strip Math

I stole an idea from The Exponential Curve this week. I had my students create comic strips to show how to graph a line. Some of them turned out really good - it didn't scan all that well, but here's a sample:

Illuminations

I often find myself at the NCTM Illuminations website on Sunday nights, trying to come up with an idea for Monday's lesson. They have a lot of good ideas for lessons and activities. I don't usually use the worksheets that come with them, but they're a good starting point and are pretty easy to adapt to fit my classroom.

Last night, I was looking for an activity to review slope-intercept form with my students. I found this. Students place ordered pairs on a graph as battleships. They draw numbers out of a hat to use as the slope and write equations of lines that will "sink" the battleships. NCTM's suggestion was to have students do the activity with a partner and then check each group's paper for accuracy at the end of class. I knew I wouldn't have that kind of time, so I changed it a whole class activity instead. I had each row of students choose a point to place their battleship. I put the numbers for the slope in a bowl and had random students choose one at a time. We worked as a class to figure out which battleship we could sink. They had a little bit of a hard time figuring out where to start on the y-axis, but got the hang of it after a few examples.

Standards-Based Grading

Think Thank Thunk just wrote an interesting post about Standards-Based Grading. Here's a few pieces:

Prob­lem: Kids want to play games to get points in order to get an ‘A’. This is a prob­lem because it puts empha­sis on accu­mu­lat­ing points and not on what the points are sup­posed to rep­re­sent: learn­ing. You must migrate your sys­tem of grad­ing away from grad­ing every sin­gle assign­ment sum­ma­tively (that is assign­ing a sta­tic grade for every­thing a kid does), and towards grades that are indexed by content.

Stu­dents could not care less about their score on “Quiz 5″ from last month; they don’t even know what was on that quiz. Don’t put that in your grade­book. Put the indi­vid­ual ideas that that quiz assessed in your grade­book, so that the stu­dents know what it is you care about.

If you switch to Standards-Based Grading, the "What can I do to raise my grade?" question will become much easier to answer. It's not about accumulating points, or doing extra credit to raise your grade. Instead of telling them to study harder for the next test, or giving them an extra credit worksheet (don't get me started about the kids that ask for extra credit!), you can be specific and look at the areas they need to improve.

I've been pretty blessed at my school since I started Standards-Based Grading. The vast majority of my students love it and would never go back to the traditional quizzes and tests that I used to give. I actually have students asking me: "When do we get to take another concept check?"

I'm just sorry I didn't start this sooner.

Team Challenges

I tried a new group activity in class today (inspired by Riley Lark's post at Point of Inflection ). I took one of my quizzes from last year (I no longer give traditional quizzes and tests now that I've switched to standards based grading - if you haven't heard of it, you really need to be reading dy/dan) The quiz had 8 questions on it, so I put the students in groups of 4. Each student had to solve all 8 problems, but I only graded 2 problems on each paper. Example: problems 1 and 2 from student A, 3 and 4 from student B, etc. I called it a "Team Challenge." Somehow that's incredibly less threatening to middle school students than calling it a Quiz.

Pros: I was really impressed to hear several students explaining their answers to their partners. I even heard two lower achieving students debate which formula to use to find the area of a shape. They were looking in the book for help, getting their notes out, asking each other questions: it was a miracle! For the most part, the students were on task and it forced them to work together to make sure they all agreed on the answers they submitted. (Added bonus: grading 7 group submissions vs. 28 individual ones)

Cons: A few students simply copied the answers their group came up with. How do I avoid this? I know they probably don't know how to solve the questions and don't want to ask for help - how do I change this? I debated making each person explain one problem to me, but that would take forever . . . Any ideas? At least they copied correct, complete answers. Better than doing nothing I suppose.

Overall, I think it was a success. I think I'll try it again next week.

Dot Paper

Dot paper is a great way to create area problems for students to solve. The best part about it is that it forces students to figure out which measurements are important, instead of just giving them a diagram already labeled.

A word of caution: some students will just count the dots instead of counting the spaces between the dots. It's a good idea to do at least one sample problem first.

Writing in Math

I tried a new writing activity in class. (I have to admit, it was mainly because I knew the lesson was a little short and I would need to fill an extra 5 or 10 minutes) Anyway, I was actually pretty impressed with how it worked out. I'm starting a chapter on area and volume. The book immediately jumps to the area of parallelograms and trapezoids. Maybe your kids are different, but I knew mine would barely remember how to find the area of a rectangle or triangle so I thought we better spend a day reviewing those. Before we took notes, I had the kids fold a paper in half: hot-dog wise in case you're wondering :-) On the left side they had to answer 4 questions about area and perimeter - just so I could see what they remembered from last year. I gave them a couple minutes to write down what they knew and then had them set it aside. At the end of class, after taking notes and working on problems with a partner, I had them take the sheet back out and write down whether their original answers were correct. If not, they had to write down the correct answer. I just walked around the room collecting them - it was a super short and easy way to see which students understood the lesson and which ones still needed help. I think I might try it again . . .

Connect Four

Today we played Connect Four as a review game on the SmartBoard.



It didn't go as well as I had planned. I thought it would be fun, but they spent more time arguing over where to go and accusing me of favoring one team over the other. Maybe next time I just shouldn't play it on a Friday.

Do you FOIL?

How do you teach your students to simplify (x+3)(x+4)? I used to tell them to use "FOIL," mainly because that's the way I learned it and that's the way the other teachers all taught it. The problem I have with FOIL is when they get to problems involving trinomials. If they've memorized the word FOIL, they have no idea what to do if there are extra terms.

Instead of using FOIL, I've found it works much better to start with simpler problems like (x)(x+4) - this way when you get to (x+3)(x+4), most students correctly suggest that you just need to distribute twice. First distribute the x, then distribute the 3. Why have them memorize FOIL when you could just show them that they're just distributing twice? I've found that my students have a much better understanding of the concept and will be able to tackle harder problems that have more terms.

I know there are other teachers still using FOIL, so I do tell them about it at some point, but most agree with me on the need for it.

Merry-Go Round

Here's a review idea that my kids love to do. I type up review questions, one per page and label them A, B, C, etc. and tape them up around the room. I assign each student a letter to start at - they answer the question with their group (each kid brings a piece of notebook paper and something to write on with them). When it looks like most groups are ready, I have them rotate to the next letter. They keep rotating until they get all the way around the room. When they're done, we go over the answers together as a class.

The student really enjoy it - it gives them a chance to work together and they like being able to get up and walk around the room as they work. It's also nice for me because I know which letters are harder than others so I can anticipate which groups are going to have questions.

This also works great for solving equations that have several steps. I have the groups do one step at a time. When they rotate, they have to check the previous steps and make sure it's correct so far, then complete the next step in the problem. They keep rotating until all the problems are solved.

Reviewing for Standardized Tests

Our state's standardized test is coming up soon. To help my students review, I'm stealing some questions from the Massachusetts Comprehensive Assessment System. Their department of ed's website allows you to search previous test questions by grade level, subject, even all the way down to content strand if you want to be super specific. You can also search for multiple-choice vs. open response, calculator vs. no calcualtor etc.

Here's some sample problems from MCAS:

Hannah’s mean score on four mathematics tests is 92.75. What is the sum of the scores of Hannah’s four tests?

A. 368
B. 370
C. 371
D. 372

Ms. Gleason is opening a new restaurant.
•She has enough booths to seat up to 40 people.
•She is ordering tables to fill the rest of the seating space.
•Each table can seat up to 6 people.
a.If t represents the number of tables Ms. Gleason orders, write an expression to show the total number of people that can be seated at booths and tables.
b.Write an inequality that could be used to determine t, the number of tables Ms. Gleason needs to order so that she has enough seating at booths and tables for at least 125 people.
c.Solve the inequality from part (b) to determine the number of tables Ms. Gleason needs to order. Show or explain how you got your answer.



Check it out - it's a great resource. If only all states made it this easy to search for standardized test questions.

1st Post

I've been teaching math for five years and finally decided to start my own blog to post ideas and (hopefully) get suggestions from the other pros out there who know way more about teaching than I ever will.

I'm about to start a unit on graphing linear equations with my pre-algebra class (8th grade). This is a unit I've always struggled with - quite possibly the one I get the most "when are we ever going to use this?" comments on. So this year, I'm trying to start more with real world applications and then transition to equations with x and y. Hopefully this will help. Here's some images from the lesson I made to use with my SmartBoard.