Category Archives: Algebra

Class Opener – Day 63 – Function Addition

Not all of the class openers go off as intended.  I suppose if they did, and I had a magic formula for engagement, then I’d be living on a dessert island by now with the money I’d made off bottling the secret.

After a short quiz today, a lecture/exploration on operations on functions would begin. It’s not the most exciting lesson of the year, but there are some “ooh” and “aah” moments as students experience new functions.  Domain and range also frame the discussion – and we finally move beyond “all real numbers”.

Even though we were starting with a quiz, I wanted a visual to get students thinking about function behavior, and start to make some conjectures about addition and subtraction of functions. The Desmos graph here was animated and rolling as students entered.

Click the graph to play on your own, and the animated gif below gives you a flavor the the motion.


After the quiz, I hoped to generate discussion regarding the graph.  What did the students notice? Any interesting patterns?  How are the graphs related? Can we gain some insight by looking at a table of values?


I was hoping students would eventually notice that the ornage function was the sum of the green and red, or at least note the “betweenness” of it all. But with the rush to get notes and discussion started, this opener ended up on the back burner.  They’re not all winners….. All is not lost, though, as I’ll come back to this one tomorrow to build some connections between our notes and the homework.


Class Opener – Day 49 – A Magical Prime?

Today is student choice day, where I look through the Edmodo submissions from my students and choose a short and snappy opener.  Chris B. submits this number “magic” as a video worth sharing with the class:

What’s going on here? Is it just plain coincidence? Dumb luck? The devil’s work? 37 is certainly not a number we encounter too often in our daily math life.

A student in my afternoon class quickly picked up on the mystery:

37 times 3 is 111, and they all have 111 in them

Good enough. Converting a division example from the video into multiplication helps verify the claim”

777 = 37 * 21
111 * 7 = 37 * 3 * 7

It’s a quiz day, but I leave the class with the following challenge:

Develop a similar pattern for a longer string of identical numbers

They are pretty easy to find, such as the one below for five-digit strings. And Chris provided a low-stress math challenge to get us into math thinking mode before our quiz.


Class Opener – Day 47 – Visualizing Absolute Value Inequalities

We’re moving through our algebra review, and today is absolute value day! I’m not sure the students are as thrilled about this as I am –

absolute value

My students have seen absolute value equations and inequalities before, mostly as a stand-alone unit with a series of special rules to memorize. But I find that students have rarely been asked to think about solutions to inequalities as the comparison of values between two functions. So instead of re-hashing some old rules, let’s fire up the netbooks and look at some graphs!

The Desmos desmonstration here (it’s clickable) allows students to experiment with the parameters of an absolute value function, and compare to a constant function.  Before diving into some specific problems, I allowed a few minutes for partnerships to play and try to summarize any relationships they saw.  Very few saw an immediate link to what we have already been working on – inequalities – and the best was yet to come.

To start thinking about specific inequalitiy problems, I asked students to set the sliders so that they represent the following problem:

The graph then lets us analyze the relationship between the absolute value function (dashed green) and the constant function (dashed blue).

absolute graph

Time to find out if my students see the link between the graphs and the inequality. Groups were given 2 minutes of “table talk” to discuss:

How does this graph allow us to find solutions to the given inequality?

This was not a quick discussion. Many students were eager to participate and provide ideas, but many went back to pencil and paper, rather than analyze the graph.  Soon, with some students approaching the board, links beween the green and blue functions were found.  But, if scaffolding is needed, think about these prompts:

  • When is the green “above” the blue? What does this indicate?
  • When is the green “below” the blue? What does this indicate?
  • Where do the green and blue intersect?

Finally, students began to understand the meaning of the black and purple lines on the graph – representations of the “greater than” and “less than” solutions sets.

In the end, I find that using technology to analyze the visual relationships between functions allows for a deeper understanding than algebraic maniupulation alone. Yet, I am often surprised when students don’t know that this is a valid (not “cheating” or somehow dirty) method of solving an equation. To assess what parts of this lesson “stuck”, I plan to give the following opener tomorrow.  Solve for x:

Wondering how many will immediately whip out Desmos on their cell phone….hoping they do!

Class Opener – Day 45 – Adventures in Standard Form

After a first half of the semester filled with Probability and Statistics goodness, my freshman course now shifts to algebra topics, and a bridge between topics many have not seen for 2 years as many of these students took algebra 1 in 7th grade.  The next few days will be a blitz of past ideas: slope, linear functions, inequalities and absolute value.  Today, one of my favorite pictures from Dan Meyer‘s fun site, 101qs, is quite a conversation starter:


Why did they built it like that? How do they eat in that house?  How do they get in the front door?  What’s the bathtub like?  All are questions generated by the class.  You can enjoy more interesting questions generated by guests to the 101qs site. And we are off are running with slope!

I find that my students coming to the HS from our middle school have been trained well in navigating slope-intercept form for linear equations.  There are some stumbling blocks with fractions, and I need to do some hand slapping to keep kids away from their calculators, but I am mostly satisfied with where students are with slope-intercept form.

Standard form, meanwhile, is quite a different story.  Asking students to convert from slope-intercept form leads to painful moments: moving terms, and multiplying to rid ourselves of fractions.  But it also allows for entry to a new idea – leveraging relatioships with standard form and developing a new formula for slope, m = – A/B.  Developing this via some examples, and letting a few crackerjack students summarize this finding for the class, opens the door for a new method for finding the equation of a line.  Now, when presented with a slope and a point, we have two options.

OPTION 1: find the equation in slope-intercept form and convert to standard form. Messy, and some nasty fractions can appear!


OPTION2: use what we have now discovered about slope and standard form to build our equation directly in standard form, and solving for C.


“Why didn’t they just show us this in middle school?!!!”  Well, maybe you weren’t quite ready then, or maybe standard form isn’t the star of the show it needs to be. In any case, today was a great day to combine old skills with some new explorations and keep things feeling “fresh”.  Tomorrow, the payoff will continue when we look at parallel and perpendicular lines, as homework tonight expands on today’s theme.

Class Opener – Day 38 – Are Any of My Students Compatible?

Today’s opener was inspired by a movie correlations activity I have used in AP Statistics, and Cathy Yenca’s awesome activity which brings this idea down to the Algebra level.

For my freshman class, I wanted to students to “discover” the role of the correlation coefficient r – how it acts as a measure of the strength of the relationship between two quantitative variables.  To begin, 10 potential vacation / off-day activities were listed on the board:

  • Ski
  • Go to Beach
  • Amusement Park
  • Baseball Game
  • Broadway Show
  • Camping
  • Washington DC Tour
  • Shopping Day
  • Big Concert
  • Cruise

Students were each asked to rank these activities from 1 to 10 (10 being most desirable) and using each number only once. The class then moved into partnerships with my suggestion that they work with someone they maybe did not know so well in class, and compared results.  With an odd number of students, I worked with a student to share interests.  Results for each activity were plotted as ordered pairs, with each partner contributing their number score.  Students plotted their points on graph paper, while my student partner and I used Desmos – and quickly discovered that we have little in common.


galleryFrom there, students learned how to use graphing calculators to analyze the data – making the scatterplot and finding the best-fit line.  The partnerships also wrote this mysterious new statistic – r – on the bottom of the graph and shared their graph in the board.  Through a gallery walk, the class examined the graphs and tried to conjecture the meaning of r.

This worked better than planned, as the class quickly made some key observations:

  • Pairs with stronger relationships have “higher” r values.
  • There are no r-values greater than 1.
  • r can be negative if people answer opposite each other.

Definitely will add this activity to my arsenal every year!

If you are interested in the activity for AP Stats, you can check out the Google Form we use, then some instructions for processing the data in this video:

Class Opener – Day 29 – Geometric Series

Aren’t infinite geometric series cool?  If you just shouted “yes”, then you are potentially as geeky as I am. A “proof without words” from MathFail kicked off today’s discussion:

I wasn’t quite sure what sort of observations I would receive from my class. But just enough ideas were generated to get us going:

There are an infinite number of triangles down the right side.

All those triangles on the right add up to the half-triangle on the left.

Both are great starts for what I hope my students will learn today. A video I made in my driveway continued the ideas of geometric series and their infinite terms.

A few students wanted to argue that the sequence in the video was arithmetic, but some meaningful debate yielded agreement that geometric made more sense.  Groups then worked through a similar problem involving a Superball being dropped, leading to terms and total distance traveled.

seriesMany groups employed a “brute force” method to find their answers. Using the Desmos calculator (many students chose to use the iPhone app), we found value in developing the equation and using tables and summation symbols to find solutions. This was my first time usign Desmos with this particular lesson, and it was an awesome addition, which added value to the need for writing a clear function to define your situation.

Class Opener – Day 20 – Infinite Chocolate

How is that possible? Tell me the answer?

Some of my students haven’t picked up on my sneaky side yet. There are no free answers in my class, including this visual which greeted them today:


Some students had seen this before, but few could figure out the mystery of the infinite chocolate. In my afternoon class, one student took charge, showing the subtle differences in the sizes of the pieces as they are reconnect…a future math teacher in the making. Today’s opener wasn’t intended to connect to anything course-related; it’s just a fascinating geometric mind trick, and great for generating math conversation right away. You can Google this problem and find a number of versions, many which explain the illusion, but we ended this opener with a video which shows some potential geometric shenannigans.

Today I desired a short and snappy opening hook, as my goal was to get students to the boards right away to work on binomial theorem problems. This was the second day students viewed videos and took notes for homework, and the response has been outstanding. Classes the last two days have been energetic, as the group doesn’t need to hear me drone on….they heard that at home. The focus today was terms in a binomial sequence – enjoy the video notes here.  Also, pay attention for the rough edit at the end due to my mistake….was more fun to leave that in than to edit it out.