Tag Archives: algebra

How Do We Assess Efficiency? Or Do We?

A problem on a recent assessment I gave to my 9th graders caused me to reflect upon the role of efficiency in mathematical problem solving. In particular, how much value is there in asking students to be efficient with their approaches, if all paths lead to a similar solution?  And should / could we assess efficiency?

The scene: this particular 9th grade class took algebra 1 in 7th grade, then geometry in 8th.  As such, I find I need to embed some algebra refreshing through the semester to dust off cobwebs and set expectations for honors high school work. For this assessment, we reviewed linear functions from soup to nuts. My observation is that these students often have had slope-intercept form burned into their memory, but that the link between this and standard form is weak or non-existant.  Eventually, the link between standard form and slope ( -A/B ) is developed in class, and we extend this to understanding to think about parallel and perpendicular lines.  It’s often refreshing to see the class see something new in the standard form structure which they hadn’t considered before.

The problem: on the unit quiz, I gave a problem which asked students to find the equation of a line parallel to a given line, passing through a point.  Both problem and solution are given in standard form.  Here is an example of student work (actually, it’s my re-creation of their work)….

linear problem

So, what’s wrong with the solution?  Nothing, nothing at all.

Everything here answers the problem as stated, and there are no errors in the work. But am I worried that a student took 5 minutes to complete a problem which takes 30 seconds if standard dorm structure is understood?…just a little bit.  Sharing this work with the class, many agreed that the only required “work” here is the answer…maybe just a “plug in the point” line.

My twitter friends provided some awesome feedback….

Yep, we would all prefer efficiency (maybe except Jason). Thinking that I am headed towards an important math practice here:

CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

It may be unreasonable for me to expect absolute efficiency after one assessment, but let’s see what happens if I ask a similar question down the road.

Confession, I really had no idea what #CThenC was before this tweet.  Some digging found the “Contemplate then Calculate” framework from Amy Lucenta and Grace Kelemanik, which at first glance seems perfect for encoruaging the appreciation for structure I was looking for here.  Thanks for the share Andrew!

Yes, yes!  Love this idea.  The beauty of sticking to standard form in the originial problem is that it avoids all of the fraction messiness of finding the y-intercept, which is really not germane to the problem anyway. Enjoy having students share out their methods and make them their own.

What do you do to encrouage efficiency in mathematical reasoning?  Share your ideas or war stories.

Put(t)ing Rational Numbers in Order

Many of my friends and followers have caught onto one of my guilty pleasures: my wierd fascination with The Price is Right (read about Price is Right and counting principles in this old post).  Here’s how a pricing game made for a fun review activity, and also made my life flash before my eyes (read to the end for that).

Here in Pennsylvania, we use the PA Core Standards.  For Algebra 1, here is a standard under “Anchor 1”:

A1. Compare and/or order any real numbers.  Note: Rational and irrational may be mixed.

Seems innocent enough.  Here is a sample “open-ended” task used to assess understanding on our state’s Keystone Algebra 1 exam:

image001 (1)

Exciting….now let me go over here and watch the paint dry….

But during the NCTM conference, a lightning bolt hit. I was checking out a putting game at one of the booths, and I suppose rational numbers were on my brain….Hey – Golf + ordering rationals = feels like Hole in One to me!

In the Price is Right Hole in One game, contestants place groceries in order from least to greatest by price.  The number of items they can order until they are incorrect determines where they putt from. After a quick trip to the sporting goods store to find a putting cup, and some time with a Google Doc, we’re all set!

To start, I created a Google Slides presentation with 6 games.  Each game has 6 numbers for students to put in order:

During the game, all students in the class had about 2 minutes to place the numbers in order.  They, we randomly drew our “contestant”, who came to the board to fill in the 6 boxes on the board.


Next, we went through the numbers from left to right, and determined how far the contestant had gone in successful ordering.

puttOn the floor, 6 lines were taped.  Line 1 was on the other side of the room, and the lines were closer and closer to the hole. If a student had 4 numbers correctly ordered, they were allowed to putt from line 4.  Two students were able to order all of the numbers and tried their putt from about 2 feet away.

Those who made their putts earned candy to share with their group.  In about 20 minutes, we got through 4 games – not bad for ending a Friday on a fun note.

But be careful! My last “contestant” – one of my less cooperative students and a sometimes hot-head – was able to putt from line 6 with the help of his group.  After missing the first putt, I reminded him that the game is really Hole in One – OR TWO, and had a second chance. Lining up the putt…he took it easy…and missed again.  This is when he raised the putter up and, for a brief second, it looked like the putter could end up flying in my direction.

“Sean, just pick up the ball and put it in the hole….here’s some candy…”

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.



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.

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.

Class Opener – Day 18 – We’re Going Bowling!

A unique sculpture greeted students as they entered class today:


There’s a lot of math goodness happening in this picture, but I don’t want to steer conversation in any particular direction right off. Time for some Noticing and Wondering! Students shared their thoughts on the back board:


Most of our class time today will be spent completing a jigsaw activity which guides students through many of the rich connections between Pascal’s Triangle, Combinations and the Binomial Theorem.  Knowing that I would eventually talk about Pascal’s Triangle (one of my favorite shares of the year!), I was hoping to see if we could generate ideas on triangular and tetrahedral numbers organically.  This visual opener did the trick. And while I ran out of time today for my Triangle chat, it’s in my pocket for tomorrow!

After sharing this experience on Twitter, Annie Fetter (the queen of noticing and wondering) chimed in with her ideas:

So many great ideas for packaging to be had here, but thinking I share it and leave it to my geometry colleagues to explore.

Class Opener – Day 16 – A Revealing Discussion of Factorial

I opened today with what I had hoped would be a rich discussion concerning a past contest problem, but turned into something more substantial.  Here is the problem:


I wasn’t too surprised when many, many students reached for their graphing calculator…this is what freshmen do. I was however surprised when I asked the class to volunteer their ideas on how they could simplify the expression – placing ideas on the board.  Some appear below, and we discussed why or why not the procedures were valid.

photo1 photo2

Many misconceptions regarding what it means to “cancel” in a fraction were revealed; in fact, the very nature of what it means to reduce was the star of our discussion.

2014-09-23_0003Later in class, I provided a hint which I hoped would provide some clarity with our factorial challenge. Some students immediately saw the link to the original problem and simplified. But how much closer are wo to finding the largest prime divisor? After simplifying, it was back to the calculators….which doesn’t really help much here.  Listing the remaining factors of 16! after dividing common factors of 8! leaves us with the clear answer : 13.

The moral of the story – those who never touch a calculator discover the mechanics of this problem much more quickly than those who take a sledgehammer to it with a Nspire CX.

Hitting the Home Stretch: Exponents, GCF’s and LCM’s

This is a busy stretch in my school year.  My 2 Prob/Stat classes are nearing the end of new material with PA Keystone Exams in Algebra 1 looming. For my College Prep class, about half have not yet taken the Keystone while the rest took it last year as 8th graders. In Academic, all students will take the Keystone in May.  Combine this with my AP Stats class taking their final exam this week, with the AP Exam next week, and my track and field meet responsibilities building as the season reaches its peak; it’s a hectic time of year.

In both of my Prob/Stat classes, we are beginning unit on polynomials.  The Prob/Stat class is a course we offer between Algebra 1 and 2. While the course contains much Prob/Stat material, we also clean up some concepts from algebra.  Unlike other algebraic units like systems of equations where there are many rich examples and opportunities to differentiate, the start of a polynomials unit often feels static. Here are two activities I have used this week for Laws of Exponents and GCFs/LCM’s.


This activity worked equally well in my college-prep group (for whom this was review material), and my academic group (where this was mostly new).  The file below contains 16 cards with numeric statements.  Break your class into teams of 2, 3 or 4.  The job of the group is to identify the true statements and the false statements.  For this activity I banned all calculators.

The letters on the cards are not anything the kids need to worry about, but allow me to easily check progress. The cards with vowels are all the false statements.

I was surprised at how much trouble my college-prep group had with building the correct false pile.  To reach some consensus on the false pile, I asked every group to hold up one card they knew to be absolutely, positively false.  Many groups mistakenly agreed that any number raised to the zero power was worth zero, which led to a class argument on who was right.

Tomorrow, we will look more closely at the falses.  In the file above, note that the cards are arranged in groups of 4. In the first group, we will review the addition rules for exponents; then the subtraction rule in the next group of 4; then the multiplication rule for the next group.  In the end, this felt much more satisfying, with increased engagement and peer discussion than simply listing rules on the board.


The speed dating concept is one many math teachers have stolen from the great Kate Nowack, and it worked perfectly in my Academic class to work through greatest common factors and least common multiples.  After doing just one example on the board, desks were arranged  into a pairs facing each other, down one long row.

Speed Dating

All students were given a card with a monomial.  They then worked with their partner facing them, and found the GCF of the two monomials.  The first time around, my co-teacher and I provided help to just about all groups.  After teams found their GCF’s, all students on the right-hand side stood and moved down one seat and worked with their new partner.  There were so many plusses to this activity:

  • all students were repsonsible for their own monomial
  • all students were engaged: no hiding behind a worksheet
  • students worked together, and with different partners each time

Some of the cards I handed out are shown here.  I tried to have a variety of cards which clearly shared factors, with different powers of x and y.


I was very impressed with how my class performed on this activity, and we moved onto a second round where LCM’s were found. This time I had students trade cards, and the left-hand side shifted down each time.

Let your kids work together, discuss and find patterns – the notes then write themselves.