Category: Algebra

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Algebra

Inverse Function Partner Share

We’re working through functions in my college-prep pre-calculus class; meaning a more rigorous treatment of domain, range, and composition  ideas than what students experienced in earlier courses. As I was about to start inverses last week, I sought an activity which would provide some discovery, some personalization, and less of me rambling on.

These are the times when searching the MTBoS (math-twitter-blog o’sphere) leads to some exciting leads, and the search for inverse functions ideas didn’t disappoint – leading me to Sam Shah’s blog, and an awesome discussion of inverse functions which I turned into a sharing activity. A great list of blogs and MTBoS folks appears on this Weebly site.

To start, I wrote a function on the board, and asked students to think about the sequence of steps needed to evaluate the function:

The class was easily able to generate, and agree upon a list of steps:

  1. Square the input
  2. Multiply by three
  3. Add 1

From here, I asked the class to divide into teams of 2. Each partnership was then given two functions on printed slips (shown below) to examine: list the steps of the function, and provide 3 ordered pairs which satisfy the function.

THE FUNCTIONS:

inverses

Notice that the functions are arranged so that A and B in each set are inverses.  Partners were given two different functions, but never an inverse pair. So a team could get 2A and 4B, but not 3A and 3B.

My plan was to complete this entire activity in one class period, BUT weather took hold. They day we started we had a two-hour delay, and the next two days were lost due to snow, then a weekend. SO, the best-laid intentions of activity, sharing and resolution became activity…..then 5 days later.

As we started the next class day, I asked students to review their given functions (and re-familiarize themselves), then seek out the teams who had the other half of the function pair and share information. So a team which had 2B sought out 2A, and so on.

After the sharing, a classwide discussion of the pairs was then seamless. Students clearly saw the relationships beteen the inverse pairs and the idea of “undoing” steps, and we could now apply formal definitions and procedures with an enhanced understanding. Also, by sharing ordered pairs, students saw the domain-range relationship between functions and their inverses, and this made graphing tasks much easier. I’m definitely doing this again!

Finally, notice that pair 2A / 2B features a quadratic / square root. While we didn’t dive right in at the time, this set the trap for a discussion of one-to-one fucntions and the horizontal line test the next day.

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Algebra Class Openers Technology

Class Opener – Day 74 – PolyGraphs

My 9th graders have only about 2 weeks left with me before their final exam. Most of them will move on to Algebra 2 next semester, so my strategy with them has been 2-pronged: ensure we are produtive with new material and put them in a “happy place” to make a seamless tranisiton to Algebra 2. With a unit review today, and a pre-holiday-break quiz Monday, this was a perfect time to test-drive the new FREE PolyGraph activity from my friends at Desmos, along with the awesome work of Dan Meyer and Christopher Danielson. The Parabola activity sounded perfect for my class, though there are also activities featuring linear functions, rational functions and hexagons.

My freshmen have limited understanding of quadratic functions. While we have encountered some useful vocabulary regarding parabolas in my class (intercepts, vertex, domain and range), these students have not had a formal unit in graphing them yet. I was curious if students could transfer what they already knew to a new scenario. I was tempted to do a quick review of vocabulary before sending kids to the lab, but thought better of it. I want gut reactions.

In the activity, one student acts as the “picker” and chooses 1 parabola from a set of 16. The “guesser” then asks yes or no questions to help narrow down which parabola was chosen. “I don’t know” is also available as an option, if the question is not clear.

polygraphBetween games, students are given challenges to help guide their understanding of vocabualry and “good” leading questions.  I found these “intermission” questions to be extremely helpful, and noticed that the quality of the questions students asked improved after participating in them.

Some obeservations about my students in this activity, which we did for about 30 minutes.

  • Students didn’t have vocabulary to describe parabolas which “open up” (a>0) versus those which “open down”. The question “is it a smiley face or a frowny face?” was used by more than one student and led to some side discussions of what this meant.
  • Students also recognized that parabolas could have different widths, and describing the differences between these was more challenging. Questions like “Is it wide?” or “Is it narrow?” are helpful for identifying some extreme curves, but without a baseline for what a “regular” curve looks like, this leads to some confusion over which parabolas should be eliminated.
  • In the first round, few student mentioned the x or y-axis in their questions. Later, I noticed these became valuable tools for elimination.
  • Questions which attempted to use the vertex showed mixed success. “Is the vertex positive?” is unclear, but these attempts improved with more game plays. Similarly, attempts to describe domain or range often needed more work.
  • Students can be sneaky, and mine are no exception. Some students attempted to bypass mathematical conversation by asking “Is it in the top row?”.  Nice try – until they realize the parabolas are mixed up. Also, sometimes students were assigned to play against students sitting right next to them. Not ideal, but workable.

Here’s where I would go with this, if my next unit was on a formal discussion of quadratics: copy the student-developed phrases like “smiley face” nto a document. As we encounter those ideas formally in graphs, develop more math-specific language, match them up with the student descriptors, and improve the document. I want students to take ownership of their descriptions, and allow for their self-generated language. Hopefully, this builds richer connections to the vocabulary.

At the end of each class, I had students complete a Google Form evaluation. I appreciate the feedback from students who took this task seriously!

What did you enjoy most about this activity?

  • This activity was really really FUN!! I liked it because I was able to interact with my classmates. I had fun as well as learn.
  • It was interesting to see the language people used to describe the graphs.
  • i really liked that it was an interactive activity that we could do with our class mates. It really allowed us to think about math in a fun way!

How did this activity increase your understanding of parabolas?

  • I learned that a “smiley face” is positive and a “frowny face” is negative. these math terms will probably be useful.
  • I had to really think about the vocabulary and what I was saying while asking and answering questions
  • It forced me to think mathematically and use many math terms to figure out the answer.

Down the road, I think it could be fun to have one class code and invite students from a number of schools to join in. Knowing that their partner was somewhere in the room caused some goofy behavior, and I wonder how much more focus they would have if their partner was from a different school. In the end, I appreciate this activity because it is fun, forces students to think mathematically, and has clear entry points for class discussion after leaving the computers. Finally, can we have students use their already-existing Desmos accounts for logging in? I like that the data from all students is shared with me, and would be even better if their activity data is all in one place.  Awesome job team!

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Algebra Class Openers

Class Opener – Day 73 – You Can’t…Because You Can’t

In the past few days, my 9th graders have worked through a chapter on polynomials: multiplying, factoring, solving, simplifying. There’s a lot of process here, and often my fear is that students attempt to memorize short-cuts (such as the old stand-by…FOIL) without fully understanding the reasons WHY procedures are valid. It’s an easy “out” to tell students they will need procedures for their next class – I drink from this well sometimes – but I need something more my students. I want them to be able to clearly articulate and verify, using precise vocabulary, the rationale for all steps they take in math class.

In today’s opener students were presented with two problems on the board I had “solved”, and were asked to comment on my procedures:

algebra

The problem on the left is one we had completed yesterday in class, and a number of students noticed that one of the solutions, zero, was missing. I asked students to identify reasons why my solution gave a different solution set:

Because you don’t get the right answer.

Because zero is supposed to be an answer.

We’re not quite getting to the heart of the matter….I asked students to look over my solution to the problem on the right and comment.

You have to subtract the 5 over first.

You need to set it equal to zero.

In each case, students were fixated more on what I should have done, rather than what was presented in front of them as a solution.  Time to re-direct the conversation – I asked students to think about each step I had done in the problems, and tell me specifically which step was in error. This is a much more uncomfortable experience.  For each problem, the steps “feel” right. In both of my classes, the breakthrough eventually came, with some coaxing:

Students: You set both factors equal to 5. You need to set them equal to zero.

Me: Why can’t I set them equal to 5? The equation equals five.

Students (eventually): Because if two things multiply to make zero, one of them must equal zero.

Now we are getting someplace. The zero-product property is often taken for granted in this unit, but it is a powerful little engine. Name two numbers which multiply to make a product of 5….is it guaranteed that one of the two numbers MUST be 5? Nope. Zero is the hero. Hoepfully, some new conenctions were made regarding the nature of zeroes here.

The problem on the left was a much tougher nut to crack. The conversation eventually focused on the “other” solution – zero – and the perils of dividing by zero. Definitely look for more “devil’s advocate” moments as we explore rational expressions further.