Building Enduring Mathematical Understanding, one lesson at a time.

Archive for the category “Linear”

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:
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:
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”

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:

(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!


Lesson: Tricky Tables Part 2

If you haven’t yet read the first post in this series, please do before continuing here.

We left off with the students/interview panel having completed an activity involving “filling in” the missing values in tables that increase or decrease at a constant rate. I neglected to tell you that,on the day prior to this activity in class, students were assigned the following homework:


I had also handed it out to the interview panel to work on briefly before the rest of the “lesson.” So, coming in, most of the students had “tried out” these kinds of problems in advance.

Student Strategies
What were some common strategies (both on the homework and the slide problems?) Guess and check was popular with the eighth graders – especially since most of the rates were whole numbers. The further apart the “known values” drifted, the more challenging it became for them to apply this method. However, some continued to stick with that strategy, even after others shared more efficient methods. Just “hearing about” another way to solve a problem doesn’t automatically change your mindset if you are not ready.

Many focused on how far apart the two y values were, and then students (and interview panel) started counting the “spaces” between the values. If there were three spaces between the y values, they divided the difference in the y values by three. Then they realized that didn’t work and adjusted up to dividing by four (often without really thinking about why.) One member of the interview panel came up with a formula for find the amount of increase or decrease: d/(g+1). We then had to define the variables in the discussion portion: d = difference in the y values and g = “gap” (the number of spaces in the gap.)

Next Steps
In class (and during the interview) we pulled back and added some terminology.


I chose to use “rate of change” at this point – reinforcing the idea that we are interested in how “fast” the values are changing. The “non-example” was an eye-opener for some, as they were entirely focused on the y-values and not really looking at x at all.

The next section deals with activities we did in the classroom. The interview “mini-lesson” wrap-up will resume further down.

Expanding the Concept
Before moving on to finding more challenging rates of change, we moved to how this relates to graphs and story situations. In each of these activities I use a “Think-Pair-Share” routine.


Most (but not all) students were able to “see” the rate of change by looking at y-coordinates for consecutive x-coordinates. (My words, not theirs!) They often (but not always) started at the y-intercept (not a “known vocabulary term yet”.) Those who are not as comfortable with reading coordinates off of a graph struggled more. Marking in the points on the graph and identifying coordinates definitely helped. (I use Absolute Board to draw on screen shots of the slides, or shine the projector on the whiteboard instead of the screen and “write” on the image.) We segued from there to defining another term:


I am sure this is not the “classic” definition of slope people are looking for, but it follows from how we had been approaching the problems.
Next up was connecting to story situations:


Students whipped through these like lightning. For future planning I want to stretch them a bit by addressing situations like the ones given below, but maybe not until we reach the next level.
(1) Stu had $100 in savings. Four weeks later he had $200. On average, how much had he saved each week?
(2) On March 3, Sherry put an iTunes gift card on her account that was worth $100. By March 8 her balance was down to $70. On average, how much did she spend each day?
Disclaimer: These are all very much “pseudo context” but at this point we are developing concepts and not “problem-solving.”

At this point I want them to pause and reflect on what they understand and know how to do. Most are feeling quite confident regarding this concept. The biggest challenges were working though finding a process to “calculate” the rate of change, and some are still not there (yet!) Keeping the values fairly simple has allowed them to determine this process on their own rather than being “told what to do.”
Student writing prompt:


Interview panel writing prompt: (no discussion of slope, graphs, or stories)


Writing about math is always a challenge. Even if the students feel comfortable “knowing what to do,” it is still hard to put it into words.

Students have taken the first steps towards developing Enduring Mathematical Understanding regarding the concept of “rate of change” or “slope.” Even if it is just the idea of “how much it goes up to down by,” the concept will be expanded and refined in the days to come.

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