findingEMU

Building Enduring Mathematical Understanding, one lesson at a time.

Archive for the category “Algebra”

180 Days: Day 1-3

Day 1
I wrote my First Day plan about a month before school actually started, and surprisingly, it didn’t change much. (I was right on target when I said we would probably run out to time and not be able to even start the “My Math Stuff.”)

My Algebra students tried to count “Squares on a Chessboard” and became slightly overwhelmed by the enormity of the task after initially being sure it was just 64!

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My Math 8 students investigated “Rice on a Chessboard” that will lead nicely into our first unit on Exponents and Scientific Notation.

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Day 2
We spent a good chunk of this day making magnets, math portfolios to store their assessments and tracking forms, and Stars for Standards, but also did a “Get to Know You” activity (modified from an idea by MathMamaWrites) requiring students to think about positive and negative in the coordinate plane. Topics were randomly chosen from their “Know Me Notecards” and groups placed their magnets accordingly.

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Day 3

In Algebra, we went back to the chessboard squares and made some more progress. Some groups studied the relationship between the size of the squares and the number of squares. Other groups looked at the size of the board and the number of squares. Patterns were emerging!

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For the daily Math Greeting, students placed their magnets on the Venn Diagram regarding their thoughts on their relationship to math. This is an Algebra class. Some for the other periods were more interesting, but I neglected to take pictures 😦

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The glare (and an old pen) made it hard to read. Top circle says “enjoy,” left one says “understand,” and right one is “work hard.”

New Blogger Initiative: Mystery Number Puzzles

For the third prompt of the New Blogger Initiative, I have decided to choose Option 1, if not purely for the sake of wanting to learn more about how to use LaTeX, especially within a blog post.

My Algebra students will be diving into solving linear equations within the first week or so of the start of school. Since they have already had experience with 1-step and 2-step equations in sixth and seventh grade, I wish to quickly expand their repertoire into solving multi-step equations by first taking on those involving just a single “x” term. (Ok, seriously, I’m not going to use LaTeX for that! Would it just be x? Soooo frustrating not to know until it’s published!) (Aha! If I publish it as “Private” I can look at the preview of the actual online version – success!)

Initially, the tasks will not involve equations at all, but those goofy “Mystery Number Puzzles” such as:

I am thinking of a number. . . .
When I multiply it by 2 and then add three, multiply the result by 4 and divide by 6, then subtract 5, my answer is 1. WHAT is my number?!

Of course, if this task is just given verbally, it would be EXTREMELY challenging to solve – who can remember back that many steps?!

My lesson plan involves posting pages all over the classroom (in those lovely page protectors of course.) Students working in pairs would all start at a different spot in the room and complete a Scavenger Hunt. (I know I read something about this idea somewhere, but I don’t have the link. Edit: Found the link to Math-In-Spire thanks to @msrubinteach.) The bottom of the page will have a “Mystery Number Puzzle.” Once the pair has “solved the puzzle” they search around the classroom for that “solution” on the top half of another page. There will also be a “symbol” on the page for them to record. On the bottom will be a new puzzle to solve, and so on, and so on. There will be three different “sets” of six puzzles, so after six problems they SHOULD be back where they started. (Unless, of course, they solved a problem incorrectly and moved onto another “track” – uh oh!) The “symbols” they record will also be in the proper order if the problems were solved correctly. A class discussion involving strategies used to solve the puzzles would ensue.

After this opening activity, we will take a look at how to record this information mathematically. The first step would be to determine how to actually write down the original puzzle in mathematical form. After a brief partner brainstorming session the following “should” be agreed upon. (It will be interesting to see whether the use of parentheses will be remembered and/or emphasized.)

I am thinking of a number. . .
When I multiply it by 2: 2x
and then add three: 2x+3
multiply the result by 4: 4(2x+3)
and divide by 6: (here’s the tricky LaTeX part) \dfrac{4(2x+3)}{6}
then subtract 5: \dfrac{4(2x+3)}{6}-5
my answer is 1: \dfrac{4(2x+3)}{6}-5=1
WHAT is my number?!

It is important to note here that some students may alternatively use: (4(2x+3))\div6-5=1 instead of the fraction bar, and that is totally acceptable.

Next phase: pairs return to their “starting page,” flip up the page, and write out the mathematical representation on the top half of the BACK of the page using a dry erase marker (or dry erase crayon.) After returning their page to its original position, students do a Gallery Walk around the room and mentally conjure up the proper representation before peeking on the back side to see what other students wrote.

The last step involves representing the solution process mathematically. I am not “picky” when it comes to solving these types of linear equations. As long as there is only one term involving x (wow, it just starts to roll right out of your fingers) the entire process can be done using inverse operations. In fact, my goal would be for students to be able to write the solution for THIS equation in the following way: x=\dfrac{6(1+5)/4-3}{2}=3. I certainly do not expect that right off the bat, but I am very comfortable with something along the lines of:
\dfrac{4(2x+3)}{6}-5=1
\dfrac{4(2x+3)}{6}=1+5=6
4(2x+3)=6\cdot6=36
2x+3=36\div4=9
2x=9-3=6
x=6\div2=3
I would hope that students would begin to complete at least a few steps at a time while still showing the computation appropriately, such as:
\dfrac{4(2x+3)}{6}-5=1
4(2x+3)=6(1+5)=36
x=\dfrac{36\div4-3}{2}=3
As students begin to describe their steps/explain their thinking, I would introduce the term “inverse” to move along their mathematical vocabulary.

Finally, students return to their “page” and write their mathematical solution process using as few “lines” as they feel comfortable using, as long as ALL of the steps in the process are shown. Gallery Walk THIS time involves making sure each pair DID show each operation in the process!

(Ok, it’s not really over yet, but probably for that day. Given a multi-step equation, students will write – and solve – the Mystery Number Puzzle associated with it. Students will create their own puzzles and challenge others to solve them. THEN, what to do when the puzzle involves a step like: subtract your original number. . .? JUST when they’re feeling all proud and confident you set the bar a bit higher and off they go again!)

Final note: found a link on (someone’s blog – I think it was a new blogger – yeah, that really narrows it down- I was just so awestruck when I clicked the link that I never went back) post to an AWESOME WebApp called WebEquation, at least for iOs. (Maybe Android?) You can literally just WRITE a mathematical expression/equation/whatever and it will turn it into LaTeX and “Math Script”

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While I totally appreciate Sam’s link to more info on LaTeX syntax, I seriously would just go to WebEquation and handwrite the expression to find the proper way to “type” it. Unfortunately, you can’t just copy and paste the LaTeX script – it comes out in HTML or something. However, you CAN press on the LaTeX until it opens a new window showing the “Math Type” version and then click to save that as an image:

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(Ok, technically I cheated. It would not actually paste the image of the equation, but I pasted it onto a Pages document and then took a picture to insert here. Not sure it’s worth the effort here. It does make a nice work around for Pages and Keynote though!)

So. . . depending on the complexity of the expression, I might just skip the LaTeX altogether!

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