Category Archives: Middle School

Adventures in Common Denominators

At my high school, I host “math lab” on alternate days.  The lab is an open room where students can obtain math help during their lunch or free period.  I like this assignment because I get to see a lot of different kids during the week, gain some insight into the approaches of my colleagues, and get my hands dirty in many math courses.

Last week one of my “regulars”, who often leaves class early for sports, came to the lab for Algebra 2 help.  He missed out on a adding/subtracting rational expressions lesson, but had the notesheets handy.  We wrote the first problem on the board:

Problem 1

Nothing too fancy. But in math lab, I don’t know the students as well as my own, so I need to do a quick check for some background before diving into new material. A spot check for understanding of fraction operations was in order:

Add Fracts

The student approached the board confidently and “added” the fractions:


…sigh…. sometimes there isn’t enough coffee in the world. But all is not lost, and after a stare-down, the student recognized he had acted too quickly, and completed the problem correctly. This led to another problem. This time, I asked the student to just tell what me what the common denominator would be:

Problem 4

Without hesitation, the student knew the correct denominator to be 24. But why is it 24? The student could not defend his answer, but was absolutely sure 24 was the LCD. On the one hand, I am happy that the student has achived enough fluency with his number sense to confidently find the denominator. But, on the other hand, has lack of a process is going to hurt us now when we try to apply LCD’s to algebraic expressions.


When I start my lesson on adding rational expressions, I hand out index cards to every student.  I give students 2 minutes to repond to the following prompt:

How do you find a least common denominator? Provide directions for finding an LCD to somebody who does not know how to find one.

I collect all of the cards, shuffle them, and choose a few randomly to share under the document camera.  We will discuss the validity of the explanations, and use parts of the explanations to come up with a class-wide definition of an LCD.  Here is what you can expect to get back on the cards:

  • Some students will recognize that factors play a role, but won’t recognize that the powers of the factors are imporant.
  • Many students will attempt to use an example as their definition.  This allows for a discussion of a mathematical definition.  Is one example helpful in establishing a rule?  How about 2 examples?  How are examples helpful, if the reader does not know how to find an LCD?
  • Some students will provide a hybrid of the last two bullets.
  • If a student does provide a suitable definition, it’s time for you to play dumb.  Let the class assess the language and verify that the definition is, or is not, suitable.  In one class, I “planted” a working definition in with the student cards to see if they could identify a working definition.

The Least Common Denominator of two or more fractions is the product of the factors of all denominators, raised to the highest power with which they appear in any denominator.


I am curious how math teachers approach Least Common Denominators in earlier grades, and how these approaches translate to algebra success.  Here’s how a few online math sites approach LCD’s.

First up, mathisfun (search for “Least Common Denominator):


This is the approach I suspect many teachers take to help students find LCD’s.  It works for manageable denominators, but becomes cumbersome when we have 3 or more denominators to consider, and certainly is not helpful in our algebra world.  Also, if you get too confused, you can use the “Least Common Multiple Tool” this website provides.  I suppose it’s not an inappropriate method, but a more algebra-friendly process should eventually develop.

Next up, Everyday Mathematics at Home website:


The good news – we have a formal definition!

The bad news – you have to know what an LCM is to use it.

This is a more formal version of the Mathisfun example.  We could adapt it for use in algebra, but again, a definition of LCM is required here.

Khan Academy starts by using lists of multiples and provides and example with a trio of numbers for which we want the LCM:


The factor trees and the color verification that all 6, 15, and 10 are all factors of 30 is nice, but this example conveniently leaves out any scenario where a number is a factor multiple times, and this is the only example given.

Finally, let’s check out how PurpleMath tackles LCM’s, with a non-intuitive example:


Now we are getting someplace.  Not only does this method stress the importance of factors, it shows the importance of include all powers of those factors.  And I could transfer this method easily to algebra class!

Have any insihgts into teaching LCD’s, either for a fractions unit, or in algebra?  Would enjoy hearing ideas, feedback and reflections!

Globe-Trotting With

The Common Core places increased imporance on statistics in middle school, beyond the tasks of creating simple data displays often encountered in middle-school texts.  The new standards require that students be able to describe distributions, compare samples to populations, and design simulations:

  • CCSS.Math.Content.6.SP.B.5c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered
  • CCSS.Math.Content.7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

It’s all great table-setting for AP Statistics down the road, and working with authentic, interesting data.  In this activity, students use an online resource to perform a simulation in order to find the proportion of the earth’s surface covered with land (as opposed to water).  This is not a new activity, as a number of teachers suggest using an inflatable globe and some classroom tossing to reach estimates.  But I think the method below uses some web tools in a novel way, and encourages some authentic geography discussions.


Before diving into any simulation, I like to gather ideas from my students, to see if they have any initial estimates or backround about the problem.  Using sites like Padlet or TodaysMeet are great for encouraging and archiving discussions and participation, or you can go old-school and just record initial estimates on the board.  Asking an initial questions like: “Do you know the percentage of Earth which is covered by water” to start the discussion.  The start a discussion of sample size: “Would it be better to sample 20 points on earth, or 40 points, or 100 points?  What factors would be part of your decision?”


In this simulation, students will sample a random point on the earth’s surface, record whether the point is covered by land or water, and repeat for a given sample size.  For this, we will use the site, which generates random events, mostly things like numbers and dice, and their Random Coordinate Generator, which chooses a random location on earth and displays it using Google Earth.

Google Earth

The site works quickly, and the water/land data can often be determined without issue.  It is also easy to zoom in and out to take care of those “close calls”.  Quite a more accurate measure than the thumb-check data from the inflatable globes.  Very few issues arose in my trials with this method, but the biggest snag is Antarctica, which is land, but often appears light blue on Google Earth.  Also, a few rare occasions produced a data point above the North Pole on the map, which I discarded.  For your class, have each student generate a sample of size 10, and look at the proportion of land hits.  Below, I did 10 trials for 4 different sample sizes:


The next steps depend on the sophisitcation and grade-level of your class.  But in general, we want to know which sample size provides the best estimates.  How do you know?  Have students write explanations which defend a particular sample size.

Multiple sources (Circle graph from ChartsBin,NOAA Information) verify that about 29% of the Earth’s surface is land.  Do our trials verify this?  How often were our trials within 10% of this 29% mark?

Within 10%

As more data is collected, free site like StatKey can be used to generate appropriate graphs and statistics.



I see this as an improvement of an existing AP Stats exploration, and opening activity for Confidence Intervals for proportions, and extension into the behaviors of CI’s.  For those playing along at home, here are the calculations for 2 standard deviations (Margin of Error) for my given samples:


And the corresponding intervals, showing how often my sample proportions were captured within each interval:


4 Engaging Ideas From Twitter Math Camp

This past week, over 100 math teachers descended upon the Drexel University campus for Twitter Math Camp 2013.  It was a fantastic opportunity to meet people I had communicated with via Twitter for some time, make new friends, and share math ideas.  It’s a real rush to hang out with colleagues who share similar ideals on math instruction, and a commitment to improve our practices.  Check out the hashtag #tmc13 on Twitter to look back on some of the action and reactions, and find new math folks to follow.

While there’s so much to share from TMC13, I know there are many math friends who couldn’t attend who are looking forward to hearing about the goings-on, so in this post I share 4 ideas from this year’s Twitter Math Camp I am eager to try in my classroom right away.

EliELI’S BALLOONS – Followers of the blog know that I am a big fan of the Desmos online graphing calculator.  The highlight of the week for me, and I suspect for many, was having Desmos founder Eli Luberoff model a lesson using his creation.  Eli’s enthusiasm for sharing Desmos, and his sincere desire to work with teachers to improve the interface, are infectuous.  There were many “oooh” and “aah” moments from the assembled group, and a loud cheer for the “nthroot” command…yes, it’s a pretty geeky group!  (thanks to @jreulbach for tweeeting out the great picture of Eli showing off Desmos’ position when you Google “graphing calculator”)

Eli’s lesson idea has a simple and engaging premise:

  • Hand out balloons
  • Blow up the balloon.  For each breath, have a partner record the girth of the balloon
  • Consider the data set


That’s it.  No worksheet.  No convoluted instructions.  Eli walked us through an exploration of the data set using Desmos, using the table to record the data, and considered various function models: is a square root model?  Is it logarithmic?  The group eventually settled upon a cube root as the proper model – and how often in class do we encounter data best modeled by a cube root?!  Since the explanatory (air entering the balloon) is volume, and the response variable (girth) is linear, the cubic model makes perfect sense. Fun stuff.  But wait…there’s more!  Eli then analyzed the fit of curve by looking at the squares of the residuals.  Click the graph below to check out my best-shot recreation of Eli’s presentation, and play around with the fit of the curve by toying with the “a” slider.

More great new additions to Desmos are coming.  Thanks to Eli for letting us preview some of them!  Was a pleasure meeting you and hearing about your fascinating story.

GLENN’S PROBLEM POSING – Glenn Waddell is a colleague I feel I have a lot in common with, in that we have both experienced the frustrations of trying to “spread the word” to colleagues of the great new ideas, and strong need, for inquiry-based mathematics.  In this session, Glenn presented a framework for problem posing in mathemtics which can be employed equally-well with real-life problems (see the “meatball” example in Glenn’s Powerpoint, which was adapted from a Dan Meyer “math makeover” problem) or with a garden-variety drill problem.

The framework asks that teachers lead students in a discussion that goes beyond just the problem in front of us.  Think about the many attributes of a problem, list them, consider changes to them and their consequences, and generalize results.  Glenn suggests the book “The Art of Problem Posing” by Brown and Walter as a resource for getting started, which employs the problem posing framework.


Glenn led the group through an exploration of a quadratic equation, where we started by listing its many attributes.

Glenn Problem

Now we consider changes to attributes:

  • What would happen if there were a “less than” sign, rather than equals?
  • What would happen if the last sign were minus?
  • What is it were an x-cubed, rather than x-squared?

There’s no limit to the depth or number of adaptations, and that’s why I like this method of problem posing for all levels of courses.

Download Glen’s presentation on the TMC wiki, and explore the wiki to get the flavor of many of the sessions.



If you have never visited Mathalicious, go now….take a look at some of the free preview lessons, and you will become lost in the great ideas for hours.  THEN, make sure you sign up and get access to all of the engaging lessons.  Here is a company that is doing it right: lessons come with a video or visual hook, data which naturally lead to a discussion of tghe underlying mathematics, and just the right amount of structure to encourage students to contribute their thoughts and ideas.  At TMC, Mathalicious founder Karim Kai Ani led the group through two lessons.  A brief summary is given here, but I encourage you to check out the site and subscribe….you’ll be glad you did.

The “Romance Cone” – What is the appropriate age difference between two romantic partners?  Is there a general rule?  A fun lesson, “Datelines” on Mathalicious, where students explore a function and its inverse, without using those scary-looking terms.  I have been looking for an opening activity for our Algebra 2 course, which brings back ideas of function, inverses and relationships, and looking forward to trying this as a my first-day hook.  Also a great activity for Algebra 1.

PRISM = PRISN? – I have led my probability students through an exploration of false positives in medical testing for many years, and I like how this activity puts a new twist, and some great new conceptual ideas, on the theme.  “Ripped from the headlines”, this lesson challenges students to consider government snooping, and the flagging of perhaps innocent citizens.  If a citizen is flagged, what is the probability they are dangerous?  How often are we missing potentially dangerous folks in our snooping?  What I really liked here was the inclusion of Venn Diagrams, with sets representing “Flagged” and “Dangerous” people, where the group was asked to describe and compare the diagrams.  Fascinating discussions, and a good segue into Type I and Type II error for AP Stats if you want to take it that far.

This lesson does not appear to be available on the Mathalicious site yet, (update from Mathalicious – will be released in the Fall) but will be using it when it is completed!  Later that day, the TMC teachers broke into smaller groups to gain behind-the-scenes access to the Mathalicious writing formula.  Thanks to Kate and Chris for sharing, listening, and giving us all the opportunity to contribute ideas.

Math Review: Secret Phrase Scavenger Hunt

Another great night of learning at the Global Math Department last night, where Matt Vaudrey and Megan Hayes-Golding shared their ideas for Exam Reivew That Doesn’t Suck.  Enjoy the playback and admire Matt’s enthusiasm for teaching, and great ideas for keeping kids engaged.  And thanks to Megan for her continued willingness to facilitate and share.

One of my favorite test reivew activities is a Secret Phrase Scavenger Hunt.  The problems here are from a review of inequalities, but can be easily adapted for many grades and courses.  Full disclosure: while looking through files for this example, I was shocked to discover that this Word document is one of the oldest files in my network drive, from September 2000!  Maybe I need to edit stuff more, or maybe this activity is just plain perfect.

Here’s the idea: take a sheet of probems, and assign each problem a “secret letter”, so that problems completed in order will spell out the “secret phrase”.

Inequal sheets

Make an index card for each answer, and tape them around the room or the hallway.  I usually place the letters on the back of the cards, but for this activity they appear on the front.

Cards on board

After the sheets are handed out, teams complete problems and may get up at any time to hunt for answers.  I usually assign students to teams for this activity, and it is interesting to observe different approaches.  Some teams will complete all problems together, then hunt for all solutions.  Other teams will complete some problems, hunt for solutions, then go back to work.  Another approach is to split up the problems – “divide and conquer”.  This apporach often leads to Civil War as one or two students in a group will invariably make enough errors to bog down the process.

The winner is the first team to come to me with the “Mystery Phrase”.  To keep the ball rolling, I will often give award to the first 2 or 3 teams to find the phrase. Here are a few tips for setting up your hunt:

  • Don’t be afraid to make your phrase something goofy.  After students fill in a few letters, they may try a “Wheel of Fortune” approach.  Unpredicatable phrases avoid this some.  For my inequality sheet, I chose the phrase “TWO HAWAIIAN UKELELES” – not easy to guess.
  • Make harder problems the “key” letters in your phrase.
  • Adding a few “distractor” cards – cards that are not solutions to any problems – also works nicely.
  • If students come to me with an incorrect phrase, I do not tell them where they went wrong.  It is up to the group to re-visit their problems and troubleshoot mistakes.

Hope you and your class enjoy the phrase hunt!

Let’s Build Some Bridges!

You’re either a genius, or the biggest idiot here.

– a colleague

Can’t I be both?


It’s the first year of Keystone testing here in Pennsylvania, and everyone is adjusting to the fun changes.  And by fun, I mean time-consuming,  nightmarish organizational hoops to jump through provided by our wonderful state government.  This year, many of our 8th graders get zapped with the testing equivalent of Haley’s Comet: state grade-level testing, along with grade-level tests in writing and science….followed by the cherry on the sundae, this week’s Keystone Exam in Algebra 1.  It’s a shock they ever have time to actually, you know….learn stuff.  Hoping our kids don’t suffer too much bubbling withdrawal at the end of this week.

We have about 400 8th graders here, and many of them will take the Algebra 1 test this week.  Some, around 80, took algebra 1 as 7th graders and have already passed the Keystone.  Meanwhile, 40 or so are in a pre-algebra course and will take the Keystone next year as 9th graders.  So, what to do with 120 students, while their grade-level friends endure a state test.  For two days, and 4 hours, I have been given carte-blanhce, an emtpy slate, to keep 120 8th graders entertained.  And money!  Well, some money anyway.  What would you do?

My BridgeTomorrow morning, 120 8th graders will meet with me in the auditorium to learn their fate.  I have split the kids into 26 groups of 4 or 5 and gathered supplies for a popsicle-stick bridge-building contest.  The concept and many guidelines came from the site TryEngineering, which provides many neat and simple tasks for kids to encourage creativity and discovery.  I have worked with a colleague from our high school, who teaches an intro to engineering course, whose students found some great resources to share with the kids to get them excited about the project.  Two short and snappy videos they found from MIT feature simple bridge designs, with Lego-men being experimented upon:  Part 1 and Part 2.

SuppliesThe supplies are simple:  each group will receive 200 popsicle sticks, a glue gun, and glue sticks.  Teams will only receive the glue gun after they have drawn some sketches and discussed a plan for their design.  Most of today was spent organizing 26 boxes of sticks, and getting groups ready.  Groups will be graded on their design, how much load their bridge will hold, and how well they work together as a team.  And about those groups….all groups have a similar mix of “advanced” kids and “pre-algebra” kids, which I have assigned beforehand.  This mix led to the “genius or idiot” comment above from a colleague.  Yep, this could go badly.  But, it could go great!  Its too tempting to not try!

So, tomorrow we start building bridges.  My coach friend Gayle and I built a bridge on our own, which you see above, and we were quite proud of ourselves.  If time permits on Wednesday, we will test the strength of the bridges.  Our bridge snapped at 7000 grams.  But I am confident the kids will do a better job.

Looking forward to a fun, but chaotic, time the next 2 days!

Load Test 1

Load Test 2

Probability Openers – Separating the Possible from the Plausible

A recent problem I reviewed from the Mathematics Assessment Project (MAP) caused me to refelct upon the coverage of probability in our classrooms.  The website provides sample tasks and assessment tools for schools and districts as they adjust curriculums to match the Common Core.  In the standards, probability begins to take center stage in grade 7:

Investigate chance processes and develop, use, and evaluate probability models.

Probability is treated like the ugly step-sister in many math courses: ignored, shoved to the side.  Look at standardized testing results, including AP Statistics, and you will find that probability standards often produce weak results.  The isolated fashion with which we treat probability is certainly not helping.  Let’s develop strategies to not only re-think probability, but to encourage communication of ideas and develop understanding.

I have taught probability at many levels: as an 8th grade teacher, as an AP Statistics teacher (and reader) and as the author of a Prob/Stat course delivered to 9th grade students.  This year, I taught my first college course in Statistics, where the problems with probability persist.  The picture below is from my college Stat 1 class, which you can also see on the great site Math Mistakes, by Mike Pershan.  Visit and provide your input:


Here are two activities I hope you can use in your classrooms to help fight the probability battle.


The Core Standards provide a framework for our students base knowledge of probability:

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Before we can start diving into formulas and fractions with probability, we need students to understand, and be able to express, whether an event is likely or not likely.  For simple events, like flipping a coin or drawing a card, this can be an easy discussion, but what happens when we start talking about complex events, like flipping multiple coins, tossing 2 dice, or choosing a random car from a parking lot?  Download my Probability Opener, which asks students to assess events and compare their probabilities.  Answers are provided.  The events start off innocently enough, but soon meander into more complex tasks, where students are asked to identify the more likely outcome:

You roll the dice and move your piece in the game of Monopoly…

A:  You end up on Boardwalk

B:  You end up in Jail

This is also a great activity if you have a classroom clicker system, or for Poll Everywhere if cell phones are allowed.


This activity is an adaptation of a Spinner Bingo problem from the Mathematics Assessment Project site I mentioned at the top of the post.  Visit the MAP site for not only the task, but a rubric and samples of student work.

The problem presents a scenario where students are asked to assess a spinner game, and bingo cards created for the game.  Read the files given on your own, but here is a quick summary:

  • 3×3 bingo cards are filled out, using non-repeating numbers from 1-16.
  • Two spinners with 8 equal spaces numbered 1-8, are spun, and the sum computed.
  • Players mark off their bingo cards if the sum appears on their card.

This is a nice scenario, so let’s adapt it and use it as our unit opener.  Having students play a game, and develop and justify a strategy, is a great way to get started.  Here is a lesson guide I have written to help you get started.  Also, the site Unpractical Math provides a virtual spinner applet you can use.  It’s often easier to just dive in and play, so here is a brief video demo of the game and how you can play it in your classroom:

Please let me know if you use either of these activities, and would appreciate your feedback. Thanks.

Follow-Up on Math Term Expungement

In a comment from my recent post “3 Phrases From Math Class we Need to Expunge“, Tina from the blog Productive Struggle shared a Google Doc she has been assembling of terms and “tricks” we all could evaluate in our math courses.

Tina is requesting 3 categories of entries:

What tricks do you hate when students shout out?
What words do your students use without understanding?
What notation do you wish students started using earlier?

Here is the link to the document: Tina’s Google Doc

My favorite so far is Tina’s idea to introduce subscript notation for sequences and series earlier in math courses.  It always surprised me how much trouble that was for my 9th graders….silly almost.