Sunday, October 19, 2014

The Bestest Things in the World

    One of the things I love about being an elementary school math specialist is getting to work in primary grade classrooms.   They have lots of cool stuff down there!


    With Mr. Harris as my inspiration...

... I thought I might try them with the first graders to explore visual representations of complements of 10.  I am fortunate to work with some amazing first grade teachers who allowed me to experiment.
     First we let the kids play around:

They went wild.  "These are the bestest things in the world," exclaimed one girl.  A boy remarked, "This is better than having an i-pod!"  They made necklaces, bracelets, key chains, swings, snakes, and handcuffs.  They linked them together in patterns, measured them, and sorted.  All without any teacher direction.

After about ten minutes, I called them all together.  I showed them a chain that I had made and asked:

    I asked them to turn and talk to a partner, and when I called on a student, I asked him to share what his partner noticed.  They had quite a bit to say, including observations about the colors, the total number of links (10), and their arrangement.  Some children saw 2 red, 6 yellow, and 2 red, others 4 red and 6 yellow.  Others saw 5 and 5: 2 red and 3 yellow and 2 red and 3 yellow.   I explained that they were now going to make chains of 10 links, but that they could use no more than 2 colors.

This was a popular arrangement: alternating colors 5 and 5.

Everyone was able to make at least two chains.  I added another stipulation after reviewing the results from the first class: once you had used a combination (5 of one color and 5 of another, for example), you could not repeat.  This was because, out of 44 chains, 35 were 5 and 5 combinations, 5 were 10 and 0, and 4 were 6 and 4.   I wanted representations of all the complements of 10.

The next step was to record the chain on a piece of paper.  The first class drew only the links, while the second class recorded the number of each color used.

     This took about an hour.  I collected their work and came back several days later with...

...strips of paper to make paper chains!

The kids had to look at their original picture and create a paper chain.  We asked them to attach an index card with a number model describing how they made 10.

The completed chains made for a colorful display, and a reminder of the complements of 10.

Turns out the project had many benefits:

  • Linking and gluing promoted fine motor skill development.
  • Students had practice sorting as well as creating patterns.
  • The process of turning their pictorial representations of the chains they had created with the plastic links into paper chains required an attention to detail.
  Now that the students have the procedure down, they can begin to compose and decompose other sums.  I know there are more links hiding in closets and collecting dust on shelves.  Let's put them to use!


  1. Love seeing you hang out with the little people Joe! And the fact that you found a sweet use for the old linking chains is even better. I‘ll be dusting them off for sure.

    If you were to revisit this task I wonder if you could pull out the standards for mathematical practice a bit more. I’m sure you’re hitting most of them through your questioning throughout the lesson but I’m thinking particularly focusing on SMP #7 (look for and makes use of structure) and #8 (look for and express regularity and repeated reasoning). Just trying to promote as much love for SMP 7&8 as the first 6 get.

    These 2 SMPs don't get nearly enough play in the primary grades. James Kaput (the Godfather of algebraic thinking in the elementary grades) would definitely agree.

    Here's my thinking:

    Give the students 2 colors like you did in your post and have partners find all the combinations they can to make a chain with 2 links- 0 red & 2 green; 1 red & 1 green; 2 red & 0 green (3 combinations)

    Another group builds chain with 3 links- 0 red & 3 green; 1 red & 2 green; 2 red & 1 green; 3 red & 0 green (4 combinations)

    Another group builds chains with 4 links- (5 combinations)

    …and so on all the way up to 10.

    Students could use this repeated reasoning to ensure that they had found all of the compliments for a particular number. Be great to see if your students could find the number of combinations for a chain with “x” links without ever having to build it (x+1).

    I’m sure Rich would appreciate it by the time they got to him in 5th grade…just sayin!

    1. Wow! That's an activity worthy of the Georgia Frameworks (unless it's already in there). I like how you concretize the SMPs. That's not always easy, especially in the primary grades. I'm anxious to see how the first graders respond. Just need to clear up the confusion about combinations relating to number, not color pattern/arrangement (see Chris's comment below).

  2. Hey Joe,

    Only 35 out of 44 were 5 & 5? My bet would have been 40. The AB pattern is sooo popular with the 5/6-year-old set.

    Me: Can you tell me about your pattern?
    Kid: Blue, green, blue, green, blue, green, …
    Me: Can you make a different pattern?
    Kid: Sure! Green, blue, green, blue, …

    I picked up on a question that the teacher asked in a classroom I was teaching in last year: "Can you make a pattern that has more blue cars than green cars?" (The context here was the picture book Beep Beep, Vroom Vroom! The "!" is there b/c it's part of the title, not b/c I'm super excited as I wrote it.) This would likely give you more combinations, like 6 and 4.

    Also, I agree. As a former HS teacher, best part of my current gig is working with young children. Definitely gives you a new perspective.

    Thanks for sharing this. Love it!


    1. Thanks for your comments Chris. Even in the second class, there was confusion about what "different combination" meant. One kid made a chain of 10 red, then 10 yellow, then 10 blue! The intersection of language and math plays itself out in fascinating ways in the primary grades.

  3. Joe,
    I love how you've made the whips, wallet chains, tails, and all sorts of other interesting but not particularly mathy link creations from my kindergarten days of yore into powerful math tools of discovery. Makes me wistful for that time and for a do-over with the links. Thanks, as always, for being such a thoughtful teacher to all of us. Your view of the world expands mine exponentially.

    1. Thanks Turtle. Your words are much appreciated. It's funny, after the activity one of the first grade teachers said to me, "I feel so bad! I've had all these links sitting in my closet (left from the teacher whose classroom she took over) for all these years, and never knew what to do with them!" Like the links were sitting around just waiting to get work again.

  4. LOVE the story, the lesson, and the ensuing discussion. I have thought a LOT about how to help kids make patterns and then try again to make other patterns different from the first (especially in the math/dance context). I've also noticed a lot of ABAB thinking in little kids when I have transferred the Math in Your Feet making process to basic making supplies of paper, glue and beads. Here's a post I wrote about some summer math & making I did with primary kids:

    The essence of my approach is that making our own patterns using *more than two attributes* can open up a whole new world of exploring sameness and difference AND can help kids conceptualize how to move beyond red-blue-red-blue or circle-square-circle square.

  5. Thanks for your insight. The work you do connecting movement and math is fascinating. I hope to meet you one day and experience it first-hand.