A middle school math teacher's quest to teach each lesson at least a little bit better than the lesson before.

Reinforcing Ratio Reasoning

I’m excited to be presenting a workshop at the North Carolina Council for Teachers of Mathematics (NCCTM) Convention in this week.  My session is Reinforcing Ratio Reasoning – Throughout the Middle School Math Curriculum.  If you’re attending the convention, please stop by!  My session is at 8:30 AM on Friday in the Turnberry room.

That’s awfully early for a Friday morning, but I hope to have a good turnout because I think developing ratio reasoning in middle school students one of the most critical things teachers can do to ensure success in higher level math.  Unfortunately, I think some teachers approach ratio and proportion by teaching students to set up proportions, cross multiply, and solve for the unknown value.  The End.  But in my experience, very few of their students — even the ones who can pretty consistently get the right answer this way — have much (if any!) understanding of how the quantities relate to each other.

Even after teaching ratio concepts (6th and 7th grade) in what I believe to be a very conceptual way, without ever even mentioning cross multiplication (much less teaching students how to use this “trick”), many of my students still really struggle to see the connections between ratio quantities.  This is especially true when those quantities aren’t “nice” numbers that are obvious factors and multiples of each other. When large numbers, fractions, or decimal values are thrown into the mix, their ratio reasoning breaks down — they start adding and subtracting rather than multiplying and dividing. They guess at the operation they should use, or at which number should be divided into (or divided by) the other.  That’s why I wasn’t at all surprised by the results of a 1986 study, in which researchers asked 12- and 13-year-old students to answer these questions (with blanks where the numbers would go):

The Price of Paint 

Paint is priced at per gallon. How much will ☐gallons cost?

It costs ☐ for ☐ gallons of paint. How much is the cost of the paint per gallon?

How will you solve each of these problems? (which operations will you use, on what part)

After students had determined how they would answer each question, the researchers gave students the same questions again, this time with numbers where the blanks had been.

The Price of Paint 

Paint is priced at $16 per gallon. How much will 0.85 gallons cost?

It costs $16 for 0.85 gallons of paint. How much is the cost of the paint per gallon?

How will you solve each of these problems? (which operations will you use, on what part)

The researchers reported that about 50% of the students changed their operations/strategies when the numbers were revealed.  This suggests that the students “may rely more on the quantities in a context than they do the context itself.”  I was curious to see how my 7th grade students would respond.  It turns out that over 50% of them changed their strategies once the numbers were revealed (some to the correct operation and others to the wrong operation), including about 1/3 of them who said they thought they needed to use a different strategy, but they weren’t sure what to change it to.

Keep in mind that this was about 7 weeks into the school year, and 7.RP is the only domain we had addressed in those 7 weeks.  And these are 7th graders who covered the 6.RP standards in 6th grade.  Clearly, these students need more work on ratio and proportional reasoning. But we can’t keep working on 7.RP at the expense of the other CCSS domains, so I need to find ways to incorporate ratio and proportion into other content standards as often as possible.

So rather than offering resources that primarily address the CCSS Ratio and Proportional Reasoning standards (6.RP, 7.RP, leading into 8.F), my workshop will focus on ideas for reinforcing ratio reasoning while addressing other content standards.  This is exactly what the CCSS Critical Areas are intended for:

  • Critical Areas should be used as a lens through which to view the Content Standards at a particular grade level.
  • Critical Areas are meant to help teachers plan meaningful learning opportunities for their students that connect throughout the school year and form a firm foundation on which to build concepts and procedures in later years.

I wanted to share my materials and handouts with my workshop participants, and I’ve uploaded them to Google Drive and decided to share them here as well, in case anyone else is looking for ideas.  So below you’ll find files and links to materials used in my session. First is the main handout, which outlines all of the content areas I linked to ratio and proportional reasoning.  The file includes hyperlinks to web resources I used. Below that is a link to the Google Drive file with copies of all the files I used.

If you’ve read this far and have ideas for other ways to incorporate ratio and proportional reasoning into other parts of the middle school curriculum, please share in the comments!  

Link to Google Drive folder containing all materials used in the workshop.  Note: for ease in sharing between operating systems, I’ve saved all files as PDFs.  But I’m happy to share the original files in Pages format (or exported into Word, which might cause the file to lose some of its formatting) if you find something you’d like to adapt for your own use.

I’ve seen a lot of MTBoS buzz about the book 5 Practices for Orchestrating Productive Mathematics Discussions  by Mary Kay Stein and Margaret Schwan Smith. It’s a great book that offers practical advice on getting the most out of discussions in the classroom.  But in order to have a great discussion, you need to start with a great question.  Where do you find good questions to ask?

One of my favorite resources for good, open-ended questions is the book Good Questions for Math Teaching: Why Ask Them and What to Ask, Grades 5-8.  (And there is a K-6 version as well.)  Here’s the description from the back of the book:

“Good questions”—or open-ended questions—promote students’ mathematical thinking, understanding, and proficiency. By asking careful, purposeful questions, teachers create dynamic learning environments, help students make sense of math, and unravel misconceptions.” 

My own copy currently has sticky notes marking a bunch of questions I want to use or that I’ve used in the past.  (Am I the only crazy person who will forget to use a resource if I don’t have a visual reminder of it?!)


The book is 195 pages long, and the questions start on page 17 — this is a book FULL of questions you can use right now in your classroom.  But if you’re looking for a resource for questions with an answer key, this isn’t it.   I love that, because teachers should be anticipating their students’ strategies and misconceptions (the first of the 5 Practices).  It’s so much easier to anticipate what students will do if you’ve gone through the process of answering the question yourself!  But the book does offer commentary after each question, outlining potential problem areas, giving justification for using the problem, or offering suggestions for implementing the question effectively.

The questions are organized by seven strands that should easily align with CCSS standards:  Number Relationships; Multiplication and Proportional Reasoning; Fractions, Decimals and Percents; Geometry; Algebraic Thinking; Data Analysis and Probability; and Measurement.  Within those strands, questions are subdivided into suggested grade levels (5-6 or 7-8), but don’t limit yourself to “your” grade level when identifying good questions to use!

This question led to a very prouctive discussion in my 6th grade classroom last year. It would have also made a great journal or INB prompt:

Which of the following problems has the largest product? Try to figure it out by solving as few problems as possible.  How did you choose which problems to do or not to do?

42 x 17

24 x 12

52 x 11

40 x 20

50 x 24

43 x 16

36 x 36

12 x 14

42 x 42

This week’s #MSSunFun prompt was to share a “go to” blog or website, but this book is such a good resource that I wanted people to know about it.  Where do you find good questions for discussions or journal prompts?


I just finished my first week of the school year.  This was the 12th first week of my career, and I actually slept well the night before school started for the first time ever!  I’m not sure why, especially since I’m teaching a new grade level (7th), at a new school, in a new school district this year — so you’d think I would have been more anxious than usual.  Maybe I was just extra exhausted from the week leading up to the first day.

My first day actually extended over two days.  At my new school we teach on an A-day, B-day schedule, which means that I only see my students every other day, for 85 minutes per class period.  Eighty-Five Minutes!  For the first time I can remember, I’ve actually had a couple of instances this week when I’ve finished class with a few minutes to spare.  I don’t think that will happen too often. The real challenge will be adjusting my pacing to cover all of the material I need to cover in only 1/2 of the instructional days!

Having 85 minute class periods allowed me to pack in a smorgasbord of first day activities.  I started and ended the class with a name tag idea I got from @rachelrosales’s blog, Purple Pronto Pups.  I’m so glad that her school year started earlier in August and that she blogged about her awesome First Day name tags!  For several years I’ve used name tags, but these name tags have an interesting addition.  The front of the folded 8.5 x 11 card stock has the student’s name, but the back is cut into three sections.  Each day for the first three days of class, students write an “I notice” and an “I wonder” statement about me or my class.  Then I respond to every student’s noticing or wondering.  All 138 of them.  (This is one time I’m relieved to be on an A-day/B-day schedule.  I’m not sure I could handle responding to 138 students all in one day! Especially not for three days in a row.)  But I really feel like I’m already getting some great information about my students and what they value.

After the students created their name tags, I introduced myself with my Me, by the Numbers keynote.  Then I explained their homework assignment, which was their own Me, by the Numbers list.  I really enjoy reading what students have to say about themselves.  It’s a great way for me to make connections with them and to start getting to know each of them — not the easiest thing to do with so many students whom I only see once every two school days!

Then we moved on to Fawn Nguyen’s Noah’s Ark Problem.  I introduced the idea of noticing and wondering about this problem, and asked students to put the challenge into their own words.  (What *I* noticed in going through that process is that a lot of kids missed the fact that they were supposed to figure out how many seals should replace the question mark.  If I hadn’t gone through that process, I think I would have been busy explaining to nearly every group that 1 polar bear doesn’t answer the question!) The kids worked with their seating groups of 3-4 students.  I gave each group a large whiteboard, dry erase markers, and a copy of the problem put into a plastic sleeve so they could use a dry erase marker on the outside.  They were SO engaged.  They were SO challenged.  They were SO disappointed to not have an answer by the time class was over.  It is SO awesome when students don’t want to stop working on a problem!

I heard a great quote from one of my students as her group was working.  She was having trouble following another student’s logic, so she said, “I’m still a little bit confused.  Can you explain how you got that again?”  This actually gave me the idea to start a Communication Quote of the Week.  I’ll have to blog about it once I get all of the details worked out, but the gist is that I’ll highlight things I overhear students say (or something a student writes) that will serve as exemplars for others. If my little plan works, I’ll have students trying really hard (by using effective communication techniques) to be one of the students quoted!

We finished up the first class with students’ noticing and wondering on the back of the name tags.

Some of the things they noticed:

  • I notice you are very organized.
  • I notice it’s important to you that we explain our thinking.
  • I notice you expect us to treat each other respectfully.
  • I notice you really want to get to know your students.
  • I notice our class is noisy.
  • I notice our class is quiet.   Those last two were from students in the same class!  I agree with the previous student. 

Some of the things they wondered:

  • I wonder if you give much homework.
  • I wonder why you have a Texas A&M cup on your shelf, since you went to the University of Florida.
  • I wonder if we will do a lot of group work like we did today.
  • I wonder if we will work on many problems like today’s problem.
  • I wonder if math class will always be this fun.   

I actually got that last ‘wonder’ from quite a few students.  First days don’t get much better than that!

I can’t wait to read about other people’s First Days (or First Weeks) through this week’s #msSunFun theme!


Today’s Sunday Funday topic is Icebreakers. I guess Icebreakers are typically activities that allow participants to get to know each other — so I’m not sure if this really counts as an Icebreaker, but it’s what I did the first day of school last year to introduce myself to my students and to learn more about them.  I think I heard about Me, by the Numbers in a Global Math meeting last summer, but I’m not sure who presented it.  Maybe it was from someone’s blog.  I need to keep better records of who I’m stealing from!

I started with a Keynote of numbers that would let students know a little about me.  I included my favorite number (42 – the answer to everything!) and then a countdown from 10 to 1.  I hesitated to use my favorite number as one of “my” numbers because that when students use their favorite numbers as one of their numbers it typically doesn’t tell much about them unless they have a reason it’s their favorite number and/or elaborate on their choice.  But I had to include it because 42 is a number they’ll hear a lot this year, because when students ask me the answer to a question, my answer is almost always 42 or a variation of 42 (e.g. 4.2 miles, $42,000, 420 unicorns).  It’s my way of saying I’m not going to give them the answer!

Mrs. Royster – By the Numbers

After I going through my Keynote, I asked students to come up with 5-10 numbers (any numbers — they didn’t have to be in a countdown format) that would help me get to know them.  In the school where I taught last year, students had MacBook laptops that they could use for this assignment, so it was easy for them to include a picture of themselves on their Me, by the Numbers page.  They printed their pages and I kept them to read and to quiz myself on names and faces.

I had to keep reminding students to choose numbers that would really tell me something about themselves.  This was a lot harder for students to understand than I anticipated.  I got a lot of numbers like 1 – the number of pets I have and 3 – how many meals I eat every day.  But I also got some really great responses, such as 1 – is the number of parents I live with.  My parents are divorced and I live with my mom and my sister. And 26 – the number of times I’ve been to Disney World. (I should have anticipated the fact that she’d miss a week of school to make that 27 times!) Another one I loved, in hindsight, is 10,000,000,000,000 – is a lot of doughnuts.  At first that one seemed like a really random, unhelpful number to share.  But as I got to know that student I realized it was a great example of his off-the-wall personality.

Sample Student Work

This year I’ve changed school districts and I’ll be teaching students who don’t have laptops. After teaching with the laptops for 5 years, it’s going to be a challenge to adapt my lessons to the lack of technology access. Of course students can list their numbers on paper, but having that picture was really helpful for me to learn students’ names.  Most of my students will have access to computers at home, but I guess I’m about to find out how well they do with assignments that require technology!

Do you have suggestions for how to increase student-to-student interaction for this activity?

Pi Day Stations

I love celebrating Pi Day in my 6th grade classroom.  Students rotate through stations throughout the class period, which makes this a timely post for @jreulbach’s Middle School Math Sunday Funday theme!


Pi Day is packed with activities, so we get started as soon as students enter the room.  I pre-assign students to groups so that they’re ready to rotate between the stations.  They have a packet of information and directions to take with them as they rotate between these 5 stations:

1- Pi Making Contest
2- Pi Digit Detectives (requires laptop)
3- Calculating Pi
4- Pi in the Past and π tattoos
5- Eating Pie

My packet is customized for myself and my students, but below are links to the files (Pages and Word versions), in case you’re interested in adapting for your own use.  (The packet refers to Angel, which is our school district’s learning management system, in case you’re wondering. I highly recommend giving students access to bookmarks for the websites they’ll use, for the sake of time)

Pi Day station activities (Pages file)

Pi Day station activities (Word file)

The Pi Making Contest is usually a favorite, but it isn’t what it sounds like!  Students string pony beads (small cheap plastic beads) onto plastic craft lacing, competing to see which student can (correctly) represent the most digits of Pi within a 2 minute time limit.  The fist year I did this, I had students string their beads onto pipe cleaners, but I can’t find extra-long pipe cleaners and some of my students would run out of room on their pipe cleaners before time was up. So I switched to 18-inch lengths of plastic craft lacing, tied in a double knot at the end so that the beads can’t slide off.  I give students a printed list of the digits of Pi (a full page of them — I don’t know how many digits, but way more than they could ever do in 2 minutes).  They also have a “key” for what color of bead represents each digit, 0-9.  The key shows all of the colors representing 0-9, strung onto a length of pipe cleaner.  I make it extra-difficult for them by not labeling each number.  I just label the 0 end and the 9 end, with the numbers 1-8 represented by the colors in between.  I also have a pre-made “answer key” to check students’ beads.  Line up the answer key next to a student’s beads and check to see if they match, removing all of the beads from that point on if a student makes a mistake.  As students rotate through the station, the group winner’s string of beads is labeled with that student’s name and set aside so that we can determine the class-wide winner.  I always have a parent volunteer at this station, because the competition can cause students to get pretty rowdy! (For you non-crafty folks: plastic craft lacing is the stuff kids weave to make lanyards.  Here are links in case you’re not sure what I’m talking about: pony beads and plastic craft lace)


Pi Digit Detectives requires internet access.  My students have laptops, but if you have a few desktop computers in your classroom, that would work too.  Students search for strings of digits that are meaningful to them within the digits of Pi.  Here are a couple of websites where students can do this:  One Million Digits of Pi and Pi Search. I require students to write the string of numbers they search for and record where in pi it is found, if it is found at all.

For Calculating Pi, students measure the circumference of round objects (like round lids from food containers) with ribbon, then measure the length of the ribbon and the diameter of the object using a ruler.  Ribbon works better than yarn because it doesn’t stretch when it’s measured.  Students divide their measurements to approximate pi. For the sake of time, I allow calculators at this station. I’m always surprised at how much help they need to measure with rulers, so this is a station where a parent volunteer is helpful.


The Pi in the Past and π tattoos is where students research the history of pi.  I find that students tend to think that the mathematics they learn today has always been around, so it’s interesting for them to see how the idea of Pi has evolved over time.  This station is the other station where my students use laptops, but if your technology is tied up with the Pi Digit Detectives station, you could print out information for students to use when answering questions.  While students are working on answering the questions, they get their choice of a Pi tattoo.  I order Pi tattoos from academictattoos.com every year.

Of course Eating Pi is a very popular station.  I ask parents to send in pies and suggest that they cut the pies before they send them.  Parents also send in napkins, forks, plates, and water bottles.  I use a google doc for parents to sign up — that way I don’t end up with 1 pie and 1000 napkins, and it cuts down on the “what do you need me to send in” emails, too.

Usually students don’t quite finish the work for some of the stations, so I assign the rest for homework.  I always let them know this ahead of time so that they work diligently while they’re in class!  :-)

I’ve always been fortunate to have a great group of parent volunteers.  I couldn’t pull this off without 2-3 volunteers for each class!  Even with help, it’s an exhausting day, but SO worth it.  I’m always looking for great ideas for Pi day, so I’d love to hear what you’ve done in your classroom!

This week’s Middle School Math Sunday Funday prompt is helping students that are behind in math.


It’s always a challenge to find ways to get students caught up when they’ve fallen behind, and for me this year has been more challenging than ever.  I’m teaching at two schools — two classes at one school in the morning and two classes at the other school in the afternoon.  This means that I’m only available to give extra help before school in one location and after school in the other, which often doesn’t match up with my students’ availability outside of school.

I teach in a 1:1 laptop school district, so since my students all have their laptops outside of school hours, I’ve looked for ways to use technology to help them get caught up. My favorite method right now is to make video tutorials using the Doceri app on my iPad, and then upload the video to my class resources in our district’s online learning management system. Although my in-class lessons typically involve hands-on activities, rich tasks, and in-depth discussions, I’ve had to let go of the idea that I can always provide those kind of experiences for students who need to get caught up. This causes a great deal of anxiety for me, but I’ve come to realize that sometimes we just need to fill the gaps however we can, and filling a gap in a less-than-ideal way is better than not filling it at all.

There are so many video tutorials (for just about any math concept) available online, but when I preview them, they almost never cover exactly what I’d like for them to cover, or use exactly the vocabulary I’d use, or show exactly the examples I would have chosen.  (Okay, confession here: I’m a bit of a perfectionist.) And maybe the “perfect” video tutorials exist out there somewhere, but in less than the time it would take you to find it, you can make your own! 

Doceri is easy to use.  It records your voice and your writing on the screen, but it doesn’t record a video of you. (Bonus points for being able to make tutorial videos in your pajamas!) You can change the background patterns (there are some great ones for math, like various sizes of graph squares and isometric graphs). You can import files as a background. You can set up “pages” ahead of time so that you don’t have to write it all out as you’re recording.  I’m sure Doceri can do a lot more than what I’ve discovered so far, but it certainly does what I need it to do! I made a quick video to explain some of the things I like about Doceri.

Check out Doceri!

And on a related side note: I’ve also created Doceri videos to use on days when I’ve had to be out of the classroom.  I’m not a lecture-then-practice sort of teacher, but it’s hard to find substitutes who are able to teach 6th grade math and I really hate wasting a day giving my students busywork to do while I’m out.  So having video tutorials for my students to watch before practicing a new skill (or reviewing an old one) at least keeps those days from being a total waste!

Differentiation is an ongoing challenge for me.  I teach math to gifted 6th graders, but with our district’s broad definition of “gifted,”  the range of my students’ mathematical knowledge and general aptitude is pretty wide.  I have to be honest and admit that I don’t do the best job at differentiating assignments and lessons for my students.  It is so is hard!  I’m happy to  have a success story to share from this week, though.  Just in time for Julie Reulbach’s #msSunFun assignment to blog about differentiating homework!


Using rich tasks is a method of differentiation that really works for me.  If you give students a meaty problem with multiple access points, lower-performing students can still get into the problem while high-achieving students can really dig into the problem and take it to the next level!  One of my favorite resources for rich problems is nrich.maths.org, which is where I found the problems I used this week.  (I professed my love for nrich at the #GlobalMath meeting last Tuesday, February 5th, so if you missed it, check out my own Favorite — plus favorites from other teachers — by watching the recorded session at Global Math Department at BigMarker.com).

Back to the problems and how I differentiated for homework.  We’ve been working on mean, median, mode, and range, but I didn’t want to just hand my students sets of numbers and ask them to find the mean, median, mode, and range of each set.  Enter nrich’s problem “M, M, and M“:

There are several sets of five positive whole numbers with the following properties:

  • Mean = 4
  • Median = 3
  • Mode = 3
Can you find all the different sets of five positive whole numbers that satisfy these conditions?
Can you convince us you have found them all?

If I also tell you that the range is 10, can you identify my numbers?

I assigned this problem for homework, knowing that I wanted to spend class time the next day discussing the idea of how you’d know that you have found all possible sets.  I wanted every student to come to the next class being able to participate in the discussion and having something to contribute. But for some of my students, just finding a few sets of numbers that satisfied the mean, median, and mode requirements was enough to challenge them significantly.  On the other end of the spectrum, I expected that a handful of students would be able to tackle the full problem and identify EVERY possible set.  So for homework I gave a minimum number of sets that students had to find (5), but I challenged them to find more than that, and I told them I’d really love it if they found all possible sets.  Of course they all wanted to know how many sets were possible, and of course I wouldn’t tell them! (We’re over halfway into the school year and some students still think I’d answer a question like that?!)

The next day they came into class BEGGING to know how many sets were possible. I had some students come in with the required 5 sets, many come in with more than that, and a few come in with every possible set.  I was able to quickly pull out the students who’d found all (or most, or in a couple of cases, too many) sets so that they could work together and not spoil the solution for the other students.  Students then worked in groups of 3 or 4 students to come up with a list of all of the sets possible, and nearly every group came up with all correct sets in the end!  For the groups that started with most, or all, of the correct solutions, I asked them to spend their time working to prove that their lists were complete.

For the next night’s homework, students tackled the Final Challenge that goes with the M, M, and M problem.  It really was a challenge, and even after our “how do you know if you’ve gotten all of the possibilities” discussion, many students had trouble wrapping their minds around what the challenge question was even asking.  I asked students to find at least 3 sets of numbers that would work.  Just like before, many of them found more than that, and a few found all of them

How many sets of five positive whole numbers are there with the following property?

Mean = Median = Mode = Range = a single digit number

When I give one of these “Do at least this much, but I’d love it if you did more” homework assignments, it’s always interesting to see who will try to go beyond the minimum requirement.  Of course I have the usual bunch of high-achievers, but every once in a while one of the other students will really surprise me with the effort that they’ve put into a non-required assignment.  Those are the ones who really make my day!