Building Enduring Mathematical Understanding, one lesson at a time.

Lurking, Learning, and Leaping

About three years ago, after a LONG career as a lurker, I dipped my toe into the waters of the #mtbos. (To be honest, I’m pretty sure that hashtag didn’t yet exist!) While my blogging and tweeting soon fell by the wayside, the lurking persisted, and here I am again, standing on the dock, ready to make the leap. 

For the past three years I have taught Algebra 8 and Math 8, but this fall I will be returning to where I started when I transitioned from high school to middle school, and teach two sections of Math 6 along with some Math 8! I am very excited about this change, for a variety of reasons: Our Math 6 classes are scheduled as a two-period block (yes – be jealous!) I am thrilled to work with our other sixth grade teachers, I feel that this age has the most potential to really change their perspective about what it means to learn mathematics, and I still get to work with my awesome PLC partner from Math 8 ( and support her as she takes on the Algebra.)

At the beginning of the summer I began to contemplate some of the changes I wanted to make for the this year: 

•Weekly homework assignments that include concepts from the previous week (not the current one!) and spiral back through earlier ideas, especially when related to upcoming concepts. 

•Modifying my SBG to include weekly assessments on four different standards, and like the homework, spiraling back through previous content.

•Embracing a problem-solving, conceptual, numeracy-based approach, making full use of groups of 3-4 students on my “monster whiteboards.”

BAM! On my Twitter feed I start reading about:

Make It Stick, and the ideas of “interleaving” and frequent small stakes assessments thanks to: @pamjwilson, @druinok, @mathymeg07, and others, as well as “the” @hpicciotto.

Resources to share (with teachers AND parents) regarding the difference between procedural and conceptual learning in mathematics:, and , thanks to a request by the ever-helpful @jreulbach, and replies by @MathMinds, @fawnpnguyen, @MathEdnet (and others. . .)

VNPS and VRG, thanks to: @wheeler_laura and others, referencing the work of Peter Liljedahl:

Jumping from link to link, I ran across a post from @AlexOverwijk after his session at last year’s TMC: He ended the presentation by putting something like this on the board:  


That’s the REAL leap I want to make this year!  

180 Days: Day 1-3

Day 1
I wrote my First Day plan about a month before school actually started, and surprisingly, it didn’t change much. (I was right on target when I said we would probably run out to time and not be able to even start the “My Math Stuff.”)

My Algebra students tried to count “Squares on a Chessboard” and became slightly overwhelmed by the enormity of the task after initially being sure it was just 64!


My Math 8 students investigated “Rice on a Chessboard” that will lead nicely into our first unit on Exponents and Scientific Notation.


Day 2
We spent a good chunk of this day making magnets, math portfolios to store their assessments and tracking forms, and Stars for Standards, but also did a “Get to Know You” activity (modified from an idea by MathMamaWrites) requiring students to think about positive and negative in the coordinate plane. Topics were randomly chosen from their “Know Me Notecards” and groups placed their magnets accordingly.


Day 3

In Algebra, we went back to the chessboard squares and made some more progress. Some groups studied the relationship between the size of the squares and the number of squares. Other groups looked at the size of the board and the number of squares. Patterns were emerging!


For the daily Math Greeting, students placed their magnets on the Venn Diagram regarding their thoughts on their relationship to math. This is an Algebra class. Some for the other periods were more interesting, but I neglected to take pictures :(

The glare (and an old pen) made it hard to read. Top circle says “enjoy,” left one says “understand,” and right one is “work hard.”

msSunFun: Musical Math Partners

One of the “math games” we play in my classes that I would like to share for

is “Musical Math Partners.”

It is quite flexible, gets kids out of their seats, and gives students an opportunity to use mental math skills.


Each student receives some sort of “card” depending on the topic of the day.

Some sort of “path” is designed for students to travel around the room without too many “log jams.”
Here is a diagram of how I make it work in my room:

Students follow a path around a “row” of four pairs of desks, but at the end of the row they may choose to turn either left or right and continue around that “row.” (Even though there are two arrows in the diagram, students are single file when walking between the desks.)

As the teacher plays their choice of music, students “travel” around the path – usually only 10-15 seconds. Once the music stops, students take a seat in the nearest desk and become “partners” with the person in the adjacent seat. (If you have more desks than students, you might want to identify some of the desks as “out of the game.” As it is, you will often have students that must continue to “travel” after the music stops in order to find a partner. If you have an odd number of students, the “leftover” person becomes partners with the teacher.)

Students “do the math” (more details below) – usually trying to finish more quickly than the other person – followed by a short debriefing regarding how they solved the problem, and then they trade cards so that they have different experiences with each new partner.

Possibilities. . .
The possibilities are only limited by your imagination / ingenuity. :)

Integer Operations
Each person holds a card with an integer. (A deck of playing cards work well for this: black = positive and red = negative.) When the music stops, the teacher randomly flips an operation card and students perform the operation from “left card” to “right card.” (I would only recommend division if you are interested in also focusing on fractions and mixed numbers.)

Fraction/Decimal Operations
Similar to the Integer Operations, only cards have fractions (or decimals.) I am fairly certain I would not include multiplication if decimals were involved, unless they were single digits. :) For subtraction, if students are not yet familiar with negative numbers, students can subtract the lesser value from the greater one.

Fraction/Decimal/Percent Comparisons
Each student has a card containing a number. (You could do all fractions, all decimals, or a mix.) When the music stops, students race to decide which value is greater.

Evaluating Algebraic Expressions
Each student has a card containing two pieces of information: an algebraic expression (as simple or complex as you wish) and a numerical value (also as simple or complex as you wish – including negatives, decimals, or fractions – but remember the goal is for students to evaluate fairly quickly.) When the music stops, students must evaluate the expression of the left card by the number on the right card and then vice versa.

Operations on Algebraic Expressions
Each student has a card with an algebraic expressions. When the music stops, teacher randomly selects add or subtract and students perform the operation from left to right. If the expressions are binomials, multiplication could also be included. (I am not sure I want students multiplying trinomials using mental math.)

Solving Linear Equations
Two options using cards throw the previous two “games” if the expressions are fairly simple.
Option 1: For the cards that have the expression and the number, students set the left expression equal to the number on the right card and solve, then switch to number on the left equal to expression on the right.
Option 2: For the cards that all have algebraic expressions, students set the two expressions equal and solve. (This would work best if the expressions were fairly simple, but again, it is up to the teacher depending on the level of students and the current unit of study.)

Measures of Center and Spread
Each student has a card containing a fairly small data set. When the music stops, the two data sets are combined, and students are to find median, mode, and range. Depending on the size of the sets and the values, you could also have students find mean or the quartiles and interquartile range.

Plotting Points
Each student has a card with an integer and each desk pair contains a coordinate grid. When the music stops, students point to the location on the coordinate grid (of course, the left card is x coordinate, and the right card is y coordinate.)

Pythagorean Theorem
Each student has a card with a value representing a side of a right triangle. When the music stops, the teacher either chooses “two legs” or “hypotenuse and leg.” Students find the remaining side. (Since relatively small values should be chosen in order for students to compute mentally, there would be some duplication. If students with equal values became partners, the second option could provide some interesting discussion!)

Distance on a Coordinate Plane
Each student has a card containing the coordinates of a point. When the music stops, student locate the two points and find the distance between them.

That’s about all for now. Feel free to add more ideas in the Comments. This is certainly an activity to use only after students have a fairly strong understanding of the concepts used in the tasks. Teachers need to be careful when choosing the values / expressions on the cards so that mental calculation is fairly accessible.

Made4Math Monday: Formative Assessment Forms

I definitely left some things “hanging” on my last SBG post, so I thought I would “kill two birds with one stone,” so to speak, and write about some of the ways I incorporate formative assessment into my daily routine.

A lot of people have written about “Exit Tickets,” and I guess, in a way, that’s what this is. I LOVE Sarah’s recent post at Everybody is a Genius (I’m not sure about everybody, but she IS a genius!) about the laminated dry erase ones :) I think I might try that for my more “reflective” tasks to end the period, but I like the “permanence” of the system that I have set up.

Here is my:


Question of the Day
After we have spent a bit of time on a concept (but before it is time for a Mini-Assessment (Quiz)) I use Question of the Day to assess where students are at on a concept and us it to guide my instruction for the next few days. I will put up a slide such as this:


at the end of class and given them about five minutes to work on it. (Depends on the task.) Rather than just using quarter sheets of paper (or anything dry erase) I wanted to have a tool that I could use to provide feedback and also to have students keep as a record of their growth. I came up with the following simple “form”:

I print out four copies and then use the “booklet” app on our copy machine so that it shrinks it down and prints all four onto an 8.5 x 11 sheet. The original copy has more “space” on the top section because when it shrinks down there is an extra “gap” at the bottom of the page. This provides opportunities for 8 QOD’s before the students need a new sheet. They will store them in the “pocket” of their Math Notebook once they receive them back the following day. I use colored markers to write their number at the top so it’s easier to sort and hand back to color or number groups. The specific standard (CCSS) is written on the form, and we update scores on their Tracking Sheets (see SBG post) fairly regularly. When the forms are “full” students store them in their Math Portfolio – a hanging file folder in a crate that contains their tracking sheet as well as other “larger” formative assessments. Since there is only one question (ok, there’s usually two to three parts though) they don’t take long to grade and record. I can also “sort” the forms to create “Just for Today” seats (also on Sticks and Seats post) to either differentiate instruction or provide support for students who have not yet mastered the concept.

Writing About Math
I have students write in math class quite often,

20120903-130039.jpg but I haven’t always spent the time reading and commenting on their responses as I should. This year we are implementing the Common Core State Standards, and there are more than a few that employ the use of the verb “explain” or “describe.” I am looking forward to challenging students to meet standard on those in addition to the more skill-based ones. I decided to modify my Question of the Day form to use it for Writing About Math prompts as well.

Sometimes the prompt will be tied to a particular standard and graded/recorded as such, but other times I am just looking into their thinking about a concept and wanting a way to provide feedback.

Recording System
Since I record multiple scores for each standard (and multiple standards on each assessment) I needed a way to organize that information so that I can see the progressions of scores in each area. I created a “grade book” form that allows for up to four scores (more if I sneak in a re-assessment score next to the original) for each standard with room for three standards on each sheet:

Once I have my class lists I will enter them on a blank copy, make four copies of that, and again use the booklet feature on the copy machine. This time I transfer them to the 8.5 x 14 size, otherwise it shrinks down more than I want. When folded they create a nice size for up to twelve standards. I usually don’t have more than that in any one unit, so I create a new “grade sheet” for each unit. When I am entering grades into the online Gradebook, I just have to look at the most recent column and update scores that have changed from previous assessments.

Final Note: QOD and WAM are not always relegated to “end of period tasks.” I also use them at the beginning of the period. After I collect them we discuss possible responses. I am excited to use “My Favorite No” either when we do them at the beginning of the period, or the following day as a re-cap / remediation activity!

New Blogger Initiative: Mystery Number Puzzles

For the third prompt of the New Blogger Initiative, I have decided to choose Option 1, if not purely for the sake of wanting to learn more about how to use LaTeX, especially within a blog post.

My Algebra students will be diving into solving linear equations within the first week or so of the start of school. Since they have already had experience with 1-step and 2-step equations in sixth and seventh grade, I wish to quickly expand their repertoire into solving multi-step equations by first taking on those involving just a single “x” term. (Ok, seriously, I’m not going to use LaTeX for that! Would it just be x? Soooo frustrating not to know until it’s published!) (Aha! If I publish it as “Private” I can look at the preview of the actual online version – success!)

Initially, the tasks will not involve equations at all, but those goofy “Mystery Number Puzzles” such as:

I am thinking of a number. . . .
When I multiply it by 2 and then add three, multiply the result by 4 and divide by 6, then subtract 5, my answer is 1. WHAT is my number?!

Of course, if this task is just given verbally, it would be EXTREMELY challenging to solve – who can remember back that many steps?!

My lesson plan involves posting pages all over the classroom (in those lovely page protectors of course.) Students working in pairs would all start at a different spot in the room and complete a Scavenger Hunt. (I know I read something about this idea somewhere, but I don’t have the link. Edit: Found the link to Math-In-Spire thanks to @msrubinteach.) The bottom of the page will have a “Mystery Number Puzzle.” Once the pair has “solved the puzzle” they search around the classroom for that “solution” on the top half of another page. There will also be a “symbol” on the page for them to record. On the bottom will be a new puzzle to solve, and so on, and so on. There will be three different “sets” of six puzzles, so after six problems they SHOULD be back where they started. (Unless, of course, they solved a problem incorrectly and moved onto another “track” – uh oh!) The “symbols” they record will also be in the proper order if the problems were solved correctly. A class discussion involving strategies used to solve the puzzles would ensue.

After this opening activity, we will take a look at how to record this information mathematically. The first step would be to determine how to actually write down the original puzzle in mathematical form. After a brief partner brainstorming session the following “should” be agreed upon. (It will be interesting to see whether the use of parentheses will be remembered and/or emphasized.)

I am thinking of a number. . .
When I multiply it by 2: 2x
and then add three: 2x+3
multiply the result by 4: 4(2x+3)
and divide by 6: (here’s the tricky LaTeX part) \dfrac{4(2x+3)}{6}
then subtract 5: \dfrac{4(2x+3)}{6}-5
my answer is 1: \dfrac{4(2x+3)}{6}-5=1
WHAT is my number?!

It is important to note here that some students may alternatively use: (4(2x+3))\div6-5=1 instead of the fraction bar, and that is totally acceptable.

Next phase: pairs return to their “starting page,” flip up the page, and write out the mathematical representation on the top half of the BACK of the page using a dry erase marker (or dry erase crayon.) After returning their page to its original position, students do a Gallery Walk around the room and mentally conjure up the proper representation before peeking on the back side to see what other students wrote.

The last step involves representing the solution process mathematically. I am not “picky” when it comes to solving these types of linear equations. As long as there is only one term involving x (wow, it just starts to roll right out of your fingers) the entire process can be done using inverse operations. In fact, my goal would be for students to be able to write the solution for THIS equation in the following way: x=\dfrac{6(1+5)/4-3}{2}=3. I certainly do not expect that right off the bat, but I am very comfortable with something along the lines of:
I would hope that students would begin to complete at least a few steps at a time while still showing the computation appropriately, such as:
As students begin to describe their steps/explain their thinking, I would introduce the term “inverse” to move along their mathematical vocabulary.

Finally, students return to their “page” and write their mathematical solution process using as few “lines” as they feel comfortable using, as long as ALL of the steps in the process are shown. Gallery Walk THIS time involves making sure each pair DID show each operation in the process!

(Ok, it’s not really over yet, but probably for that day. Given a multi-step equation, students will write – and solve – the Mystery Number Puzzle associated with it. Students will create their own puzzles and challenge others to solve them. THEN, what to do when the puzzle involves a step like: subtract your original number. . .? JUST when they’re feeling all proud and confident you set the bar a bit higher and off they go again!)

Final note: found a link on (someone’s blog – I think it was a new blogger – yeah, that really narrows it down- I was just so awestruck when I clicked the link that I never went back) post to an AWESOME WebApp called WebEquation, at least for iOs. (Maybe Android?) You can literally just WRITE a mathematical expression/equation/whatever and it will turn it into LaTeX and “Math Script”

While I totally appreciate Sam’s link to more info on LaTeX syntax, I seriously would just go to WebEquation and handwrite the expression to find the proper way to “type” it. Unfortunately, you can’t just copy and paste the LaTeX script – it comes out in HTML or something. However, you CAN press on the LaTeX until it opens a new window showing the “Math Type” version and then click to save that as an image:

(Ok, technically I cheated. It would not actually paste the image of the equation, but I pasted it onto a Pages document and then took a picture to insert here. Not sure it’s worth the effort here. It does make a nice work around for Pages and Keynote though!)

So. . . depending on the complexity of the expression, I might just skip the LaTeX altogether!

msSunFun: Homework Hassles

Somewhere along the clogged up Google Reader, I missed the topic for this week’s:


It didn’t take more than opening up Flipboard to find out that the dreaded “Homework” is what’s on the menu today.

I have taught Middle School Math for the past four years. Until this year I have always taught sixth along with “something else.” Beginning Wednesday I will have three classes of Algebra (8th graders) and three classes of Math 8. I must admit that I am not exactly looking forward to the “Homework Hassles.”

I use Standards Based Grading in all of my classes, and one of the tenets that I feel strongly about is that a student’s grade is based purely on their understanding of math concepts and not on “participation,” “effort,” “behavior,” or “homework.” Therefore, even though I assign homework fairly regularly in my classes, it does not factor into their overall grade. Let me rephrase that: There are no “points” from homework involved in their grade, but I do feel there is a fairly strong correlation between a student’s homework completion rate and their overall grade. Unfortunately, (surprisingly enough,) not all middle school students necessarily have that same philosophy.

From my (albeit brief) observation of student behavior, sixth graders are much more “regular” in regards to homework completion. I think that whatever habit they developed in Elementary School tends to stay with them as the begin their life as a middle schooler. At some point, for some of the students, “grades” start to get in the way. If an assignment for another class counts as a part of their “grade” then it surely takes precedence over their math homework that “doesn’t really count” for their grade. By the time they are eighth graders, the percentage of the student population who find “more important things to do with their time” has definitely increased. Instead of completing a set of practice problems that your teacher has assigned in order to help you learn and retain the math concepts, school has become a series of earning “points” (or not earning them) to reach a desired “grade.”

Enough ranting, here is my “multi-pronged” approach for this year.

Math@Home Logs
For at least the first month of school, all students will be required to fill out a log, tracking the date, the assignment, time spent, whether it was completed, student initial, parent initial, and teacher initial.

This will be kept in the front “pocket” of their math notebook. I will stamp (or not stamp) daily and collect the log on Fridays. A “bulk” email will be sent home to any students who were negligent in completing the assignments and/or having their parent/guardian sign off on their log.

If a student has been successful at completing their homework for the first month of school they will be excused from completing the log unless their habits begin to change.

No Homework Notice
As I circulate and check off (stamp) homework, student who did not complete the assignment (didn’t start, didn’t finish, or didn’t show process) will be given the following reminder:

and will then be required to fill out a GoogleDocs form detailing the reason why they didn’t complete the homework along with a plan for completion.

I can print off the spreadsheet whenever necessary. As students do complete the assignment I can delete that row from the sheet.

Morning Math / Learn @ Lunch / Afternoon Academy
These are just my fancy ways of saying before school / during lunch / after school. Each Monday, students with assignments still missing will need to choose 15-60 minutes (depending on how much is missing) of time in which they will come to class and work on their missing work. (I have a cool grid on my board all ready to go for this. Students will each have a magnet with their name that can be placed in the location of their “first appointment.” No picture today – maybe later this week when I am at school.)

Making Homework Accessible and Appropriate.
Why have homework? What is my goal for my students when I assign it? These are important questions to consider when making decisions about a lesson each day. One goal I have for this year is to make sure I don’t assign problems “too early” in the concept development cycle. Just because a topic was introduced in class that day, doesn’t mean that students are really ready to “practice” on their own. Instead an assignment might focus on previous skills that will be helpful in solidifying the current concepts. This will take careful consideration, since I am not always sure how much progress will be made in class each day.

For some units I design quite a bit of the homework assignments myself, trying to incorporate some “puzzle/problem” solving activities that require students to come up with strategies that will, in the long run, help develop their understanding of math concepts. (See the example in this “Tricky Tables” post.)

In addition, I hope to make textbook homework less “rote” and more “reflective” at least SOME of the time. A few posts by David Coffey that I read recently share how to “flip homework” so that students are really analyzing a set of problems instead of just “cranking out answers.”

I certainly don’t have the “magic bullet.” I forced myself to refrain from reading today’s posts before composing this one, but I am looking forward to finding some ideas that will be useful. :)

Tech Talk: Keynote + Absolute Board = Awesome!

Continuing my series on iPad Apps. . .

When I first received my iPad a year ago last spring, I was DESPERATE to use it in the classroom. My school laptop was getting old and cranky, freezing up at the most inopportune moments. I have used Power Point in the classroom for quite awhile – not to “tell” students information, but to “ask” them questions. I searched through the App store quite extensively, looking for something cheap to do the job, but nothing looked too promising, so I bit the bullet, forked out $9.99 + tax and bought. . .


My kids use an Apple laptop at home, so I was familiar with the program. (We actually bought a “five-user pack” of Keynote/Pages/Numbers way back when. . . I cringe when I think of how much we spent. Who knew what would happen to the App industry?!) The iPad version doesn’t quite have as many features as the Mac version, but an upgrade sometime during the past year brought the two closer together. I have been able to email my old PowerPoints to myself and open them in Keynote. Occasionally there are a few glitches – borders on a table don’t show up, or a cool font isn’t available – but for the most part it has worked out well. I have different slide backgrounds that I use for different activities in the class, and have added to my collection with the ones in Keynote. Working on the iPad is more enjoyable than on the laptop, and moving or modifying slides is a snap. You can “nest” slides under one another, so this summer I combined all of my slides for each day of a unit together, with a general lesson plan as the “top” slide.


Clicking on the triangle by the slide will “collapse” all of the slides underneath.

Drawback #1

One glaring absence in Keynote is the ability to use superscript (and subscript, but I don’t need that NEARLY as often.) It is really unfathomable to me that this is not available in the font modifications. HOWEVER, I have found a work around. Remember when I said I emailed PowerPoints and opened them within Keynote. The superscript STAYS so I just have a “fall back” slide that I go to when I need to use an exponent. I can change the font/color/size and the superscript will change along with it! (Strange, but true!) I use a similar “shortcut” with square roots, as the keyboard shortcut for the radical symbol is non-existant, but I’ve copied it over from PowerPoint. Since the highest level of math I teach is Algebra, it’s not as if I need a full-fledged Math Editor (although it would be nice!) I am playing around with Mathbot and TeXit and learning a bit of LaTex so that I can paste in some more complex equations later on this year.

Drawback #2

I always though it would be nice to be able to “draw” on the slides! It is, after all, an iPad App! There is really no “freehand tool.” You can created shapes and lines and curves, but you can’t just “scribble something out” on a slide. At first I was really bummed about this, but I have come to realize that maybe I wouldn’t really want to, since I would need to use the slide again the following period. It would be kind of a pain to make multiple copies of each of the slides I wish to write on, so maybe not such a deal-breaker. (Although, it still would be an awesome feature just for CREATING the slides, but I doubt that Apple is listening.) So. . .from the multitude of options that I have downloaded, played with, and even used for awhile, the winner in the end is the FREE App. . .

Absolute Board!

It really IS free!


The way I use this in conjunction with Keynote on my iPad is that I will take a screen shot of the PowerPoint slides I plan on using so that they are stored in the Camera Roll. When I pull up Absolute Board I can quickly pick the slide and it will fill the screen. (For awhile I did it ahead of time, but it doesn’t take any longer to grab the slide than it does to select it from the pages stored in Absolute Board.) I can zoom in and out, change pen size and color, and write down a solution process as a student shares it aloud. (I have found that it is a BIG time saver to have me record rather than a student “write” on the iPad. There are other opportunities in class for them to write.)


The “marked up” slides are then saved in Absolute Board, and I can pull them up later. I am not sure what the limit is on the number of “pages” you can store. Every once in awhile I will “purge” the old drawings, but I can also save them to my photo album if I wish!

These two together make a great combination for me!

Made4Math Monday: Sticks and Seats

One of the benefits of working part time for a NUMBER of years was that I could find the time to be a “parent volunteer” while my own kids were in Elementary School. I must say that it was a valuable experience as a teacher as well. Most of my teaching experience up until that point was at the high school level, so many of the classroom routines were new to me:) One of those ideas was “picking with Popsicle sticks.”


The question is. . .how do you pick sticks when you have six different classes? Do you have six different cups of sticks? After a few years of “refinement” I have a method that works well for me. I picked up a package of big, brightly colored “craft sticks” at the dollar store. There are actually six different colors, but I generally only use four. I wrote the numbers 1-8 on the bottom of the sticks and I am good to go for a class of up to 32 students. Each student is assigned a color and a number (which they remember quite readily after the first few days.) I, on the other hand, have a “cheat sheet” that I post on the board and “borrow” during class so I can actually call on the student’s name instead of just the “color-number combination.”

Here is my “Wizarding World of Harry Potter” butterbeer cup that I will being using this year. (I am finally retiring my University of Minnesota cup with the lettering all worn off, but I am still using it to hold my little six inch rulers.)


Here is one of the (currently empty) “cheat sheets” that will be filled with names once our class lists are finalized.


(Ha! I just noticed the Yellow column twice. I must have used my class that only had three colors last year and copied and pasted a new column. It’s supposed to say Green!)

Soooo. . . what if you have more than 32 students in a class? I’m glad that you asked. There are a number of different ways that you can deal with this issue. Since there are more than four colors available, you can certainly use more colors and fewer sticks of each color if you wish. Last year I had a class of 34 students and I included two blue sticks (in addition to the red, orange, yellow and green ones) numbered 1-2. If you have far fewer than 32 students (less than 24, for instance) you can eliminate one color all together, and never “pick” that color during that class. I have an additional “rule” that I follow as well. If I pick a stick that does not “belong” to anyone in the class (not just because they are absent) then I am the one who must answer the question! (Occasionally I have a class with lots of “unclaimed sticks” and I end up “calling on myself” more often than I wish, so I will put a limit on how many times I can answer in a period.)

I rarely use the sticks for “total cold calls” in class. Quite often students work on a problem, discuss it with a partner, and then I pick a stick to share their response. Other times I will have students work on five or six problems and as I begin to pick sticks, the first person chosen is allowed to decide WHICH of the five or six problems he or she wishes to share. (If no choice is made, I will choose!) By the time the last person is chosen, at least we have already discussed the other problems – even if the most challenging one was left for last. However, this is certainly not always the case. Some students definitely WANT to share their response to the toughest problem! I do not allow students to “pass,” even if they have not completed the problem. I will instead ask questions, questions, and more questions, to help them reach a solution. I will also call on raised hands after a problem has been shared if students wish to add more information, an alternative process, or an alternative solution. I generally leave the stick out of the cup for the rest of the period. I am not sure this is a “good thing.” Some students then “relax” knowing they won’t be called again. Others are disappointed that they won’t get “picked” another time.

I have to chuckle sometimes at the responses I get when I start to pick a stick. (Some times I will have already grabbed it, just waiting for the time to call.) Since they know their color, some are happily anticipating that it might be them, while others are nervously hoping it WON’T be them! Often the people up front will see the number as the stick is drawn (apparently I’m not very adept at hiding it) and actually KNOW the particular student before I can even look it up! Occasionally a student professes “disbelief” because they (let’s be honest, it’s usually a “he!”) “called it” that he would be picked. Oh, sixth graders – I will miss them this year :(


Why all this hassle just to assign a stick to a student? I have ulterior motives. :) The color-number combination is also often used to assign seats! On a daily basis, students enter the room and need to figure out where they are sitting and who they are sitting with for that particular day. (I alluded to this in my First Day post, but here is the full text version.) Some days the desks will be in groups of four and they might be sitting with their number groups or “half” of their color groups (evens and odds or highs and lows.)

Other days they will be sitting in two’s where they are generally paired up with someone else in their color group.

Since there are usually six or seven other people in their color group, this partner will also change from day to day. Part of my reason for posting the lists is that they act as a “cheat sheet” for the students as well.


You may have noticed signs above the class listings that notify students the groups for the day. I have laminated (double-sided so I can just flip for a new option) all of the different seating choices available. For the “color pairs” there are seven different signs that all have each number paired with a different “partner.” (Now there’s a math problem for you!) I also have laminated signs for each number group that I set out on the groups of four, and for each color group that I place out to determine the “row” or “section” for that color. These are all stored in a basket right under the section of the board I use for group assignments.


Just to make things even MORE confusing, I also have a few additional ways of picking the groups for the day. One is “Find Your Match” (pairs or trios) that I described in my post on Math Cards. The other is “Just for Today” groups (pairs, trios, or quads.) I often use this option after a formative assessment or during review activities where I purposely group students so that at least one person in the group has a strong understanding of the concepts. I sort their formative assessments and put them in groups, then jot down the names on a blank seating chart in a page protector using an overhead pen (I guess they still have their uses!)


I assign new “color groups” every quarter. (I used to do it more frequently, but it can be time consuming, and with eight people in group, plus the other options, they get quite a variety throughout the quarter.) Initially the groups are usually alphabetical. By the second quarter I put some work into assigning groups. I often have the highest performing students with the same number (or two to three numbers) so that I can choose to use “Number Groups” when I want to differentiate a bit. (Those groups would “receive” a more challenging problem than some of the other groups.) I am also aware of when I end up with “high-low” pairings so that I either take advantage of built in “tutors” or at least I do not plan an activity in which partners might be “competing” against each other. A definite part of my lesson planning involves deciding how students will be grouped for the day. Finally, for the last quarter I usually allow some student input regarding who they would like to have in their group. I take requests of 2-3 people for each student and I can usually place them all in a group with at least ONE of the people they requested, often with two (or another in their number group.) Again, THIS is a challenge as well!

Back to Sticks

There were blue sticks I used in my class of 34 last year. When in “Number Groups,” they just created a group of five. When in color groups pairs they were my “Wild Cards.” They took the place of students who were absent for the day or sat together as a pair. (I also randomly drew sticks that they would switch with so each day there were different student’s sitting in the “Wild Card” seats.) For odds/evens or high/lows, if everyone was present, I would have them join a “convenient group” with the most extra space to form groups of five. I would also probably do this for a class of 25 or 26 instead of having four color groups with “lots of empty seats.”

Last year I came up with a new way to use the sticks during group work. If the students were in Number Groups, I would walk around with one stick of each color in my hand. When stopping to check on a group or to answer a question, I would place the sticks behind my back, mix them up and pick one to determine who to call on to either ask or answer a question. If students were in Color Groups, a collection of sticks with different numbers could be used. (Although I also used an octahedron, but this sometimes resulted in MANY rolls if I was at an “even” group and the numbers kept coming up “odd.”)

Whatever For!?

Sticks: I am TERRIBLE at calling on “purely random” students, so the sticks help me to do that on a more regular basis. (We also have class discussions that don’t involve sticks – it depends on the particular activity.)

Seats: I want every student in the class to be comfortable and familiar working with every other student in the class. We all have our strengths and weaknesses that we bring to a group and I want student to be aware of that fact, and looking for opportunities to share their strengths while acknowledging and working on their weaknesses. Not every student “enjoys” working with every other student in the class, but they know it will change the next day. Sometimes we will stay in the same groups for two consecutive days, but really no more than that. Some students swear to me that they have been with the same partner waaaay too often because they “happen” to end up partners in find your match (over and over. . .), they are IN the same color group so they are in pairs and quads with that person, AND I even put them together in a “Just for Today” group!! Oh, the injustices of being a middle school student!

Minor Pitfalls

Last year my sixth graders were my first class of the day. The buses arrive by 7:30 and class starts at 8:00, so more than a few of them would “hang out” in class for awhile. Then, I started noticing some patterns. Within a “column” of eight seats, the pairs can choose their spots on a first come – serve basis. Some students would quickly “claim” spots so that they could be just across the aisle from their “BFF” – who they were not actually in a group with (on purpose, from my point of view) but wished to sit near them anyways. I began to be quite careful about where each color groups was assigned, or where each number group was placed to try to avoid the “cliques.” I was not always successful. This year I have 8th grade Algebra students first thing in the morning. I am not sure they will “hang out” in class before school, but if so, I think I will wait until closer to 7:55 to make the group placements for the day!

Whew! “Leadership Team” meeting this morning, working in my room all afternoon, and I still finished this post at a decent hour – West Coast Time! Still nine more days ’till students arrive – unless you count out Open House on Wednesday, but I think I’m ready for that :)

Reflections: Lurking to Learning

Today is my one month “blogversary!” I don’t have much to add to the conversation on Advisory for msSunFun, but I do have an item on my To Do page that I would like to tackle: My journey to becoming a blogger.

I first “lurked” onto the “mathedublogger” scene just over SEVEN years ago! At the time I was teaching at an Alternative High School, and we had just been “chosen” to participate in a 1-1 program beginning during the 2005-6 school year. We (the teachers) had some training at the end of the school year and then took our laptops home over the summer. Wow! The Internet in your lap! During the hours/days/weeks spent searching for online tools and resources I ran across some “bloggers” talking about what they do in the classroom – especially in regards to technology. It was still a fairly new idea from my perspective. Every once in awhile I would wander back and take a look at what was happening. Often one site would link me to another, and so on, and so on. . .

Meanwhile, I moved to one of the middle schools in the district and was no longer involved in the 1-1 program. However, our Computer Lab teacher shared GoogleReader with us, and my lurking seriously went up a notch or two. My tiny iPod touch became my window into the “mathedubloggosphere.” I don’t have the data to back it up, but I really do believe that the MathEd blog scene has grown exponentially – meaning that the growth was actually quite gradual to begin with, but then started to take off! I remember running across dy/dan (what a cool blog name, I thought) before he WAS dy/dan! I “lurked” as some of my favorites, Continuous Everywhere but Differentiable Nowhere, f(t), and MathMamaWrites built their followings. ThinkThankThunk next slammed onto the scene along with his SBG cohort Point of Inflection. Even though most of the bloggers were teaching at the high school level (or higher) and I was now working with sixth graders, I could relate to a lot of what they were saying – especially in regards to SBG. I thought, hmmmmm, maybe I should do this. However, I am the LAST one in a large group of people that I don’t really know to actually speak up. . .And I wonder why my own kids are so shy. . .

The next “jump” in my lurking occurred when I got my iPad and found Flipboard, that I shared in this post. It makes scrolling through blogposts pure pleasure! I would look at the blog rolls of the bloggers I was reading, check out new writers, subscribe to their posts, and so on, and so on. . .

After returning from our vacation this summer, I read some posts about Twitter Math Camp. Seriously? These people just planned their own “retreat/workshop/conference!?” Wow! More new bloggers were added to my feed. . .and then one afternoon I was weeding out in the yard, listening to my iPod when I heard the lyrics, “This is your life, are you who you want to be. . .” and something just clicked. Yeah, I can do this! You know, it makes so much sense to do it NOW, when I am just going back to teaching full time after being part time for 16 years. Sure, I have PLENTY of time n my hands! Oh well, I downloaded the WordPress app (recommended by Sam) and signed up for Twitter to boot! After all, I had already picked out a name and everything. The rest, as they say, is history. . .but not really what I expected.

It has been a roller coaster ride over the past month. I started out with ideas about what I wanted to “say,” but I find myself “saying” a lot of other things too. msSunFun came onto the scene shortly after I first started. “Ok, I’ll try that.” Then I noticed the Made4Math Monday. “Suuuure, that too.” Sam posted his New Blogger Initiative. “Hey, that’s me! I’ll sign up.” All of a sudden I have more ideas to share than there are hours in the day. How to keep up? Oh yeah, and school is starting up soon, too! My Start, Stop, Continue post includes blogging goals that I hope I can keep. I even made a To Do’s page to keep myself honest.

Posting comments on other blogs was one of my first “baby steps” after I started my blog. I really appreciated it when the blogger would then reply back. Especially now that I added all of the NBI bloggers to my feed, I find myself overwhelmed with how much there is to read. I want to make “meaningful comments,” as opposed to “I really love that idea.” Maybe that’s more of a “Twitter” response to a post.

Twitter has been hard to “jump into.” I once tweeted that I felt like the “new kid at school,” just listening in on other conversations – except they may have happened hours ago! It is very strange sometimes. I will “reply” and then realize that the person may have already moved waaaay beyond that part of the conversation. I don’t know how people can even BEGIN to follow as many people as they do!

When I was lurking it was so much more of a passive experience. Now that I am “in it,” I have been learning sooooo much more from others. I am reading posts and tweets from Middle School Math teachers who are out there in force, (@jruelbach, @4mulafun, @fawnpnguyen, @mr_stadel, @Borschtwithanna, @mathbratt, the list goes on, and on, and on) and I didn’t really know about them before. I LOVE the posts and twitter conversations with/between those involved in Math Education (@delta_dc, @mathhombre, @ChrisHunter36, @trianglemancsd) that make me think more deeply about learning mathematics! I feel a kinship with other “newbies” like me (@danbowdoin – although he is on the fast track to blogger stardom, @G8rAli – who should really start a blog, @aekland – who has such thoughtful posts, @ray_emily – who has an abundance of enthusiasm, and Pai Intersect – who I haven’t seen on Twitter, but has great insights on his blog.)

I vacillate between thinking that “nothing I have to say has any value when compared to all of the ideas that others have shared” and “oh, I really want to chime in,” or “I think I should share that. . .”

I am surprised at how much I have learned about myself and my teaching from writing posts for the blog. @ray_emily tweeted earlier today: “I’m finding I have a new clarity / fresh eyes on a topic after blogging about it.” I entirely agree! I am especially looking forward to learning even more as I blog about my experiences in the classroom :)

New Blogger Initiative: Integer Context Cards

For Week 2 of the “New Math Blogger Initiative” (or is it an “initiation?” hmmmmm) I decided to learn how to embed a document using Scribd and show something that I am proud to share!

If you read my Made4Math post from last week (Math Cards) you know I like “multi-purpose” tools. I haven’t been teaching Middle School too long, in the grand scheme of things, and when I first had the opportunity to introduce Integers to a class of very low sixth graders (all Level 1 on the state test) I knew that putting them into context would make all the difference. So. . . what are some contexts for integers? Well, there’s temperature, and altitude, and money. . .? I brainstormed long and hard and came up with quite a list.

Without further ado, my very first Scribd document!

Integer Context Cards

(Hmmmm. Rats! The font changed when I uploaded the document. The original was in Herculanum, a very cool looking ALL CAPS font, so now it appears that I don’t know my capitalization rules – oh well.)

There are a total of eighteen different contexts with six cards for each one. Some are “stretching it” a bit, but still reasonable. Note: for altitude, one of the cards is for Arlington, Washington where my school is located, so you might want to “personalize” that one. :)

I made one copy of each page on a different color of copy paper to make them easier to sort and then had them laminated, cut, and paperclipped in sets of six. (I only made one set for the entire class, but you can certainly create duplicates, especially if you have a very large class.)

Order, Order, Order

Phase 1: After a brief exposure to a few of the cards, we started our first activity with the cards. My students were already assigned to groups of six. We actually went out into a space in the hall for this.) Each group received a different set of cards, passed them out amongst the group members, and silently “raced” to put themselves in order from least to greatest. Once a group was done, they ALL had to raise their hands. After I checked for accuracy, they turned in their set and grabbed a new one. (The first year I did this I checked the groups off on a master sheet, but the following years I just trusted their memory – “We already did that set.”) The goal was to accurately get through as many sets as they could in the time allotted. Often enough, groups were in too much of a hurry to read carefully enough to identify the “key words” that signified whether their value was positive or negative. Getting a “no” when their hands were raised definitely encouraged them to take their time a bit more.

Phase 2: The next activity took it just a bit further. Groups (of three this time) had a sheet of number lines. Each time they received a new set of cards they had to “fairly accurately” plot and label all six values (along with zero if it wasn’t in the set) on a number line. Choosing a scale was challenging for some of the sets (especially with the first group of students.)


(See the cool font? Oh, well.)

Integer Operations in Context

A few weeks later in the year, my seventh grade class was working on operations with integers. I printed the Integer Card sheets four to a page and created little packets for students to share. (See the image above.) Using a “Think Pair Share” type of model, I posed questions using the people, places, or things on the cards and students had to write a math sentence, model the problem on a number line, and find the value (first on their own paper, then with a partner on a mini-whiteboard.) A big key was writing the math sentence as opposed to just finding the value. I wanted them to make the connection so that when they saw a “naked numbers” problem they could try to connect it to the contexts we worked with in class.

We initially focused on addition and subtraction situations:
(**The white text is shown first. After the Think-Pair-Share on whiteboards I reveal the number sentence and diagram and move onto the next problem.**)

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Next we moved onto subtraction in “finding differences” contexts, as well as multiplication and division. I dropped the “number line” requirement, although we did end up sharing it on some of the more challenging multiplication/division situations. (Again, the white text for each problem is given first, TPS, then the yellow answer is revealed, discussed, and we move on to the next problem.)

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In subsequent years I used the Integer Operation activities with my sixth grade classes as we introduced this seventh grade concept in the Spring after the state test. During “Review Activities” time in class I had small groups create original “stories” along with their associated number sentences, from the cards, but I think I would like to make that a more integral part of the initial learning as well.

Ahhhhh, context! Even if it is really “pseudo context,” in this situation students have something to grab onto that helps them make sense of the integer operations as opposed to a mercurial “set of rules to follow.” On the other hand, it is THROUGH these experiences that students begin to create their OWN rules about the patterns involved in integer operations, but only when the concepts “make sense” to THEM! I think we are finding a bit of EMU right there. :)

Note: There are multiple slides for each operation, and if I got my act together I might copy and paste them all into one slide show and try to attach them, but that would take a higher level of understanding on Scribd than I currently possess, as the slides are all now on my iPad, and I am just sharing screenshots.

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