cheesemonkey wonders

cheesemonkey wonders

Thursday, June 2, 2011

Number Sense Boot Camp - Request for Feedback and Input

Maybe all of your Algebra 1 students showed up on Day 1 every year with a solid and fluent grasp of basic number sense, but mine sure didn't... and it scared the crap out of me. And then afterwards it haunted me, ALL   YEAR    LONG . . .

The stuff they didn't get was just mind-boggling to me:
  • subtracting
  • adding a negative number
  • the basic concepts of the real number line
  • fractions
  • measuring
  • counting
  • basic ops with fractions
  • absolute value (any related topic)
I mean, this is basic citizenship numeracy stuff, on the same order as basic literacy.

So since this does seem to be a general condition I am likely to encounter anywhere I am likely to teach, I decided to develop a "Number Sense Boot Camp" unit I could use to start the year off with, diagnose critical number sense deficits, use as an occasion for teaching basic classroom routines, give students a chance to dust off (or remediate) their basic arithmetic skills, and basically give us all a fighting chance of getting to some introductory algebra work.

Another thing that worked this year was stealing adopting game-like practice structures, such as those advocated by Kate Nowak in New York state and by the late Gillian Hatch in the U.K. As Gillian Hatch said, a game can provide "an intriguing context" as well as "an unreasonable amount of practice" in vocabulary, reasoning, procedural skills, generalizing, justifying, and representation than they might otherwise be inclined to do. As Hatch said, it also seems able to lead students "to work above their normal levels." As anyone who has tried any of Kate's practice structures can attest, there is something about introducing this playful element that really gets students to dive in.

One thing I did this past year that worked for many individual students was to do some specific work with the real number line. I made a printable number line and gave each person their own number line (downloadable from folder) and a plastic game piece to use with it as a calculating device.

Since the rudiments and rules of board games have such wide currency in our culture, most students found this a helpful physical metaphor that gave them both conceptual understanding and procedural access to basic counting, addition, and subtraction experience that had eluded them in their previous nine to eleven years of schooling.

These had the added benefit of conferring prestige upon those who had shown up for extra help and received their very own set (though I gladly handed them out to anybody who requested one).

Emboldened by my initial success, I realized could expand the idea of a number line "board game" to use as a basic structure for practice – both in using the number line and in many other basic number sense activities.

It even dawned on me that this could be made extensible by having different kinds of "task cards," depending on whether a player has landed on an even number, on an odd number, or on the origin (a decent justification for considering even- and odd-ness of negative numbers here ; go argue over there if you have a problem with this).

Players move by rolling one regular die and one six-sided pluses-and-minuses die (+ and –) (kids seem to need grounding in the positive and negative as moving forward and backward idea). Kids earn "points" in the form of game money, which could carry over and be used to purchase certain kinds of privileges (such as a "free parking" pass for a day when they don't have their homework to turn in).
Your thoughts?

Here are links to the different game boards, along with descriptions of each.
Basic Printable Number Line For Use With a Game Piece:

Printable Number Line Game Board With Spots For 3 Sets of Question Cards - 1-up version (for use with your basic at-home printer):

Printable Number Line Game Board With Spots For 3 Sets of Question Cards - 3-up version (prints a 24" x 24" poster at FedEx Kinko's--costs about 2 dollars):

I made Game Card files using Apple's Pages software (for Mac OS X) and MathType equation editor. You can use these as templates or make your own:

Here is a link to the folder itself if you'd like to share and upload your own documents or samples:

Please share any experience or insights you have with them. Enjoy!

Julia (@jreulbach on Twitter who blogs at has started a Number Sense Boot Camp page on the Math Teacher Wiki where you can share and find other Number Sense Boot Camp ideas and activities. Available at .

UPDATE - 14-Sep-11:
It's only been one day since I introduced the tournament of "Life on the Number Line" but I am already excited about how well this is working out. It is exposing ALL kinds of misconceptions and misunderstandings about adding a negative and about interpreting negative and positive as movement along the number line. Students are playing individually as a "team," and the team with the highest number of correctly worked problems will win 10 free points (2 problems using the 5-point rubric for each person) on next Friday's unit test.
     Since they are surfacing all kinds of misunderstandings about + and - movement on the number line, this is leading to vast amounts of mathematical conversation to get it figured out. So basically, they are teaching each other about adding negatives and subtracting negatives and interpreting that as movement along the number line. 
     I can see that each day it will make sense to give some daily "notes" at the start of class on clearing up common misconceptions I've seen the previous day in students' work so they can solidify their conceptual understanding as well as their procedural fluency a little more each day.
     Best moment yesterday: a girl looked up at me beaming and said, "This is way more fun than doing math!"
     I said, "Good!" but I was thinking, "You have no idea how much math you are actually doing!" :-)

Here are the game cards to use on the first day:

And here is a generic worksheet (front and back) you can print out and give to the kids to use as their template:

If you have only a ton of basic 1-6 6-sided dice, use Post-Its to make two (2) plus-and-minus dice for students to use with one (1) regular numbered die. This is a good task to give to a student helper. ;-)


Four final things:

Thing #1
This unit confirmed me for that kids really do need active, multi-day practice in "living life on the number line" to gain a sense of positives and negatives as directions WHILE AT THE SAME TIME they are developing a sense of positives and negatives as additive quantities. It's not enough for us to just wave the idea of life on the number line at students. It doesn't make sense to them. They really needed experience alternating between (a) positives and negatives as indications of directional movement and (b) positives and negatives as additive or subtractive quantities in the process of deepening their additive reasoning skills.

Thing #2
Right before we started, I had the bright idea to give every group TWO +/- dice and ONE six-sided number die. If you don't mind my saying so, this ended up being a master stroke because it forced students to think about rolling (–)(–)(3) and rolling (–)(+)(3) and every possible combination thereof. This one thing alone might have done the most to deepen their sense of additive reasoning and of +/– as directions of movement.

Thing #3
Here's a link to a zip file that contains ALL of the game cards I created for this unit (on the math teacher's wiki): Game Cards- ALL

For all those who have asked and those who are thinking of asking, I'll say that my school uses the California edition of the McDougal Littell Algebra 1 textbook (by Larson, Boswell, Kanold, and Stiff). For this reason, the game cards are targeted at each of the lessons in Chapter 2. However they are not tied to that textbook and could easily be used with any curriculum or textbook (just sayin').

Thing #4
I'll have to take a photo of the final game boards our instructional aide mounted and laminated for us. They are a true work of art!


  1. I really like these ideas. I teach an algebra 1 support class, and more often than not I've found that the kids can do algebra but their 'basic math' skills are so incredibly poor it's like they are trying to tread water with a 10 lb brick.

    Not sure how easily these games could be incorporated at the end of the year, but if used from the start and as part of class culture, I can see them being very successful. Do you have any other plans to create/list/pilfer more activities of this type and put them in a list anywhere? I will happily send on any I can find. Mmm, summer project.

    Thanks for sharing. This makes me really excited about support next year as a way to liven up the class and get them more confident with their basic skills.

  2. I like your metaphor of the kids trying to tread water while wearing cement shoes. That's how mine were too. So it seems like a dramatic intervention is required -- one that presents a very different set of classroom structures and expectations. And the beginning of the year is definitely the only way to get this kind of shift under way.

    So yes, I am definitely planning to create/list/pilfer more activities like this in addition to fleshing this one out. So it would be great to have a partner in crime for this, as it were. ;-)

    I pushed myself to get this project started because I want to work on three areas of personal professional development this coming year: (1) building a stronger classroom culture, (2) creating better algebra support activities like these, and (3) integrating primary source texts into discovery and writing activities. And the key to #2, I have realized, is integrating them into #1.

    Combining classroom culture and refreshing (or remediating) basic skills seemed like a natural pairing. Plus putting that kind of classroom structure into place at the beginning of the year strikes me as a wise investment of time, energy, and resources.

    So watch this space for further developments!

  3. I use a lot of these activities and ideas with my lower ability students in years 7-10 (11-14/15 year olds).

    Activity 1 is very similar to my tug of war activity which was an (re)introduction to adding and subtracting integers and was then extended to negative integers. This was very popular with my year seven and eight students who didn't see this sort of activity as doing maths.

    I like activity 2 a lot and may have to pilfer the idea myself come September.

    Once students are used to the idea of using games in maths they can come up with and produce some brilliant ones themselves. I've had a version of monopoly with teachers instead of streets on the squares with brilliant questions written by the students as well as the pretty board amongst others.


  4. Thanks for your comments, @HappyMathsGeek. I'll post the PDF of the game boards so you can print them out and use them.

    - Elizabeth

  5. I LOVE the number line with the game piece. What a great tactile representation. I work with LD students and inevitably I'm fighting their lack of number sense. It takes so long to get mastery of operations with signed numbers, I'm definitely stealing this idea and would absolutely love the PDF of the game boards.

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  7. I love the boot camp idea. If you can get students up to speed at the beginning you can save time (and trouble) all year long. Students have tragic number line skills from fractions to negative numbers. I'm going to give each student a laminated number line this year so they can write on it with a dry erase marker too. They need to know where numbers ARE bc I think it helps them understand what numbers are. I have a great card game I found called Zero which uses negative number and absolute value. The kids loved it.

    1. I love this (4-year-old) idea! I'm going to laminate them and tape them to their desks!!!

  8. Following a Twitter conversation with @jreulbach and @wmcneary on this general subject, I dug out another source of good ideas I wanted to share on developing and/or deepening a sense of place value. In Chapter 2 of Steven Leinwand's Accessible Mathematics, he outlines a way to do a daily spiral review... but what really caught my eye was the way he integrated a daily place value question or two. His examples are questions like, "What number is 1,000 less than 18,294?" and "What number is 100 more than 18,294?"

    This approach can also be extended to include decimals and fractions, as in, "What number is 0.01 less than 9.102?" and "What is 1/10 of 450?"

    What I like about these questions is that they encourage students to develop a mature sense of place value that is embedded in more mature language than most of them use as they enter Algebra for the first time.

    Now I'm thinking about building a card deck for the boot camp game boards (the number line) that poses these kinds of questions.

    Another thought I've had about the game-as-review is that students should have to compare answers and convince each other of the rightness (or wrongness) of a given answer. Having an answer key is less important than being able to check and/or challenge one's own work -- and to justify your thinking using appropriate mathematical language. :)

  9. Hi! I know it has been awhile, but I was trying to access these links and they are no longer working. Is it possible to update them to the working links? Thanks!