Category Archives: Number Sense

Some of my favorite Geoboard activities

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I love using math manipulatives.  For some kids, using visual logic, or a hands-on approach can help to remove barriers to understanding, and can take some of the intimidation and fear out of learning about difficult concepts.  Using Algebra tiles to teach polynomial multiplication and factoring quadratics help to reinforce an area model of multiplication, and completing the square – well it does make more sense if you actually complete a square.  (And if you get the lab gear designed by Picciotto, you can complete the cube too!)  I’ll post about my experience with these sometime soon.  For today, here are some ways that we have used Geoboards in my Algebra classes.

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My Algebra students had a blast using geoboards to find triangles and squares.  The Math Forum has put together a set of very rich ideas for using geoboards, and we worked over a few days to discover line segments, triangle area, and practice with the pythagorean theorem.  I asked students to use the geoboards to find solutions, and then to translate their ideas into pictures on dot paper by drawing all of their solutions.

I had some challenges handy for some students who were ready. Here are some extensions to the challenges posed by the Math Forum, which my students grappled with.

  • How many different squares can you find in a 5 pin x 5 pin board?
  • How many different triangles can you find in a 5 pin x 5 pin board?
  • How many total squares can you find in in a 1 x 1? …2 x 2? …3 x 3? How about nn?
  • How many total rectangles?  IMG_1322

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There was some beautiful and deep thinking going on, and this is a highly differentiable activity.  Students who are at the concrete level could physically count squares, while students who were ready to generalize could find the cubic (I guess that’s not too much of hint), which describes the number of squares in an area.  Teachers, feel free to drop me a note or tweet at me for my solutions.  In an effort to keep the problems from being Google-able, I haven’t included them here.

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Geoboards are excellent to physically see how  slope works as well.  For next year, I’ve ordered the 11 x 11 pin versions.  Stay tuned!  How have you used Geoboards in Algebra class?

 

One of my favorite number activities is the 4 4s

I heard a colleague say recently, that math was good for three things:  “…making predictions about the world, making models of the world, and because math is beautiful.”  One of my favorite number activities, the 4 4s is low entry, and high ceiling, and the mathematical context by itself gives us a place to explore the beauty of numbers and relationships.

Mind your 4s

I created this floor to ceiling white board last year in my classroom with 4 x 8 paneling. Kids LOVED being able to draw and work on this scale!

I remember doing this myself as a student, and I was so glad for the reminder when I came across the Four 4’s activity at CAS Musings.  In a nutshell, the problem asks students to use four 4’s and any operations they can think of to get to each target number – I asked them to solve for every integer from 1 and 100.  The basic arithmetic operations +, -, ×, ÷ along with exponents, roots, decimals (4.4 or .4), concatenation (44), percentages, repeating decimals (.44…), are all allowed – and some funky ones are necessary (just try to get to 73 or 77!).  Depending on the level you’re teaching, you might include more advanced operations.  This was easily differentiated as well– I previewed the activity and assigned some specific numbers to students who needed quick success or more challenging work.

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There are many ways of getting to each number, and the multiple solutions leave room for kids to create and to DO math. Depending on the needs of your group, you could do this in a competitive or a collaborative way – for my group of 7th and 8th graders last year, collaborative worked better.  I think that my 10th and 11th graders would benefit from the competition this year, so I’ll add some structure and a prize this time – I’m thinking of awarding points for the most complex solutions as well as the most elegant solutions.  I’ll allow students to work together, and make sure to honor and highlight different ways of getting to each solution.

4s DetailLast year, this worked really well for reviewing and cementing proper notation, order of operations, factorial, multiplying exponents, and general number sense.  There were great student conversations.  My notes included snippets like “…wait.  Dividing by .4 is the same as multiplying by 2.5!”  I’ve included the worksheet I used below.

If you’re looking for a beautiful follow up, Fawn has of course upped everyone’s game with Foxy Fives

MATERIALS

MIND YOUR 4s.docx

MIND YOUR 4s.pdf

31 From 25

This is a quick one that I wanted to record here to remember, and to share.  Before we made our playing card Platonic Solids, I asked students to do a little number exercise with their cards, that I adapted from Sarah’s First Day Activity.

Playing Card Array

Their ticket to begin building the solids was the completion of this challenge: Create a 5 x 5 grid of cards, in which every row and column adds to 31.  We decided as a group that J, Q, and K were worth 10, Aces were worth 1, and all of the other cards were worth their face value.  The groups who began with some ideas of symmetry got to the answer much more quickly, but I had one group who just bulldozed their way through until they made it by sheer force of will.  We debriefed as a whole group, shared our different problem solving (…or bulldozing) methods. The kids were totally surprised to find that their solutions were not unique.  A nice extension would be to count the solutions.  How many ways are there to solve this array?

Next time.

Target Number Game

There are lots of iterations of numbers games like this, but this one is especially nice because it’s open-ended and repeatable, and can be extended to your grade level.  Marilyn Burns has published several variations of this game like this one, or many more from her book, About Teaching Mathematics among others.  I’m sure that I’m not alone when I say that Marilyn is my hero.  These games are full of thinking and practical number sense, and require almost no prep.  This one is a perfect back-pocket activity if you have a few idle minutes (though I can’t actually remember that ever happening – we always seem to have too much to do!), or a nice way to regularly open or close class.

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My kids love this game, and ask for it if we haven’t played it for a while.  I heard about it first from Tom at the Dana Hall Math Workshop.  I mentioned how much I loved this conference in an earlier post.

Here’s how the game works

  • Choose 4 numbers between 1 and 10
  • Choose 1 multiple of 10 between 20 and 100
  • And choose one 3-digit number between 200 and 1000 (Your Target Number)
  • Use the first five numbers and any operations to get as close as you can to the 3-digit number.

Example: 4,8,3,2,70      Target 497

First Try: 70 x 8 = 560-4x3x2 = 560-24 = 536

A Better Attempt: (4+3)x70 = 490+8-2 = 496

I’ve opened this up to include exponents or roots, with pretty good success.  Do let me know if you use this, or have any variations up your sleeve.

“First Past The Post” And The Dreaded Disease “Z”

This is a lesson I adapted from a model I first learned about at a math teaching conference that I attended at the Dana Hall School.  This conference was without a doubt, the best professional development workshop I’ve attended.  The faculty was amazing, the workshops were specifically geared by grade level, and the resources were practical and rich.  My teacher for the grade 6-8 workshops was Tom Harding, who is the current math department head at Shady Hill.  He was a wonderful and caring teacher.  I can’t say enough good things about this workshop.

This problem has a fun, if highly unbelievable back-story.  As the kids picked apart the story – and mine spent some time doing this – there was no pretense of the specific situation actually being useful, except perhaps for creative writing, but the ideas are highly transferable.  It is not a real world problem.  Perhaps even better, it falls under the Realistic Math Education (REM) umbrella described by MathEd.net HERE.

I used this as an integrated lesson about forms of government in addition to the math lesson.  I’ll set-up the problem and then explain what I did with it.

Dreaded Z Histogram Image

THE SET-UP

A group of 5 people are boarding a spaceship to Mars. Their health is one of the following:

  • Normal
  • Mono
  • Allergies
  • Common cold

And one has…

  • The Dreaded Disease Z

Students must identify who has The Dreaded Disease Z, as it is highly contagious – and fatal!  If the infected person boards the ship with everyone else, they are all doomed.

THE PROCEDURE

The only way to identify each person’s health is through blood testing; a bag for each person has representative chips for their blood levels (per chart included below).  I used color tiles in brown paper bags for this, and let students take 40 “blood samples” (a sample is taken by blindly taking one chip out of the bag, noting its color, and putting it back in the bag).  With 40 chances, students had to be strategic about which bags to sample from.  Should we take 5 samples from one bag or 1 sample from each bag?  Should we decide on 20 samples and then re-evaluate, or re-evaluate our plan every 5 samples?

Tile Graphic 1

Of course, students need some time to just play with the tiles and make patterns too.

You’ll need to set up at least one set of bags with tiles to match the histogram.  If you want separate groups to choose their own ways of sampling, you’ll need multiple set-ups.  40 samples were just about enough to make students pretty confident in their guesses, but not enough for them to be sure.  For me, the big math idea here is sample size, and given a situation, how many samples do you need.  I had students make individual predictions at 0, 10, 20, and 30 samples, and then come to a group consensus about who to kick off the spaceship.

INTEGRATION

I was working on forms of government as part of an integrated Social Studies unit, and saw a nice opportunity here.  When I did this at the conference with other teachers, there was a certain amount of strife about how we should decide to take our samples.  We each had ideas about how to go about finding the bag representing the Dreaded “Z,” and it was sometimes hard to let someone else decide.  And we were adults in a situation where we shouldn’t have cared about solving the problem, but instead should have been focused on examining the merits of different strategies, and why a student might choose one or another.  I imagined that for kids, this would be much harder than it had been for us – and I couldn’t resist leveraging these potential conflicts.

I did this in a whole group setting.  Each group of 4-5 students got one bag, and I asked them to give it a name.  In naming their bag, they grew attached to it, and were more invested in having their bag not be “infected.”  We later talked about how statisticians remain impartial (…or not).

I divided the 40 allowed samples into eight groups of five, and gave specific instructions for who got to decide which samples were taken.

  1. Samples 1-5: By Consensus:  Students all had to give a thumbs-up (I’m totally behind this method), thumbs-sideways (I can live with this method), or thumbs-down (I can’t live with this method).  They had to keep talking it out until everyone had at least thumbs-sideways, and then we took the samples.  As you might imagine, there were some strong ideas and this took a while.
  2. Samples 6-10: “First Past The Post” Popular Vote:  Anyone could suggest a method, and we voted on which method to use.  The method that received the most votes was the one we used.
  3. Samples 11-15: Alternative Vote: Students suggested methods, and voted for their top three choices in ranked order.  We tallied votes, and eliminated the lowest ranks until we had a vote of over 50%.
  4. Samples 16-20: Students voted for three classmates by secret ballot.  The three classmates got together and decided how to take the samples.
  5. Samples 21-25: I chose one student to make the decisions for the whole class, and didn’t allow any consultation.
  6. Samples 26-30: I kept the single decision maker, but added two advisors to help with the decision.  The final decision was still up to the one student with the power.
  7. Samples 31-35: I chose four students to make the decisions for everyone.
  8. Samples 36-40: I chose names out of a hat, and each student whose name was picked could choose one sample.

After each set of 5, I asked students to name the decision-making method, and reflect on how it felt in writing (The organizer that I used for this is included below).  At the end, I had each group choose who to kick off the spaceship, dumped the tiles for the big reveal, and debriefed the math.  We talked over statistical analysis, probability, and sample size.  This was an excellent big picture statistics discussion, and worked well as a hook for probability exercises to come.

After the math discussion, we debriefed the decision making process.  As you likely figured out, the decision-making methods roughly correspond to different forms of government: Variations of Democracy, Republic, Dictatorship, Socialism, Oligarchy, and Anarchy (OK, Anarchy analogy sort of falls apart).  The analogies only went so far, but the written reflections gave us a great opportunity to talk over the benefits and liabilities of different systems, and the different perspectives of those in charge vs. those not in charge.  “It was awesome! I got to decide for everyone.”  Or… “It was really hard.  There was a lot of pressure and I had to decide for everyone!”  I referred back to both the mathematical and the philosophical content of this discussion several times over the course of the news few weeks’ studies, and the kids are still talking about this and asking to do it again.

 

THE MATERIALS

Organizer for Recording the Decision Making Process

Dreaded Disease Z prediction sheet

Dreaded Disease Z Graph

BONUS: Dreaded Disease X Graph for doing this a second time, or for adding a sixth crew member for more complexity