I recently posted instructions for a hands-on activity wherein the students make their own meter measuring tapes. I promised I’d explain how I’ve used them so far. Below is a description of ways to use the tapes to teach patterning. This will be followed in another post with ideas for using them with number sense and then, of course, measurement.
Patterning is the act of analyzing the relationships between elements in a string of repeated occurrances of those elements. 😛 In this case, we are looking at the relationship between one number and the next after something has been done to that number. We can focus on the individual numbers, or on the change happening to those numbers.
Have you ever noticed that your students can skip count, no problem, rambling on and on until they reach the biggest numbers they know? Have you noticed that they don’t actually know what 10+2 is, despite being able to count by 2s, or what 15 + 5 is, despite knowing how to count by 5s; they don’t actually know how these patterns are constructed or how they work?
This is because they have memorized these patterns the way they memorize the lyrics to the latest Katy Perry song; they know the words, but have no idea what they are actually saying!
To avoid this meaningless regurgitation, it is important to teach the students how the pattern works at the same time as teaching them the pattern. Better still, is teaching them about how changes happen in patterns and letting them figure out the different patterns that can happen because of those changes. Specifically, we can continually add on, or take off a fixed quantity, and say the new number we get, and eventually, we notice the same numbers happening over and over until we see a pattern.
To accomplish this learning, the skip counting must be accompanied by a concrete understanding of the quantity we start with, the quantity being added, and the quantity that we end up with, and the students must understand that it is a recursive process wherein the quantity we end with becomes the quantity we start with.
These measuring tapes (and any other number line example, such as thermometers, hundred charts, etc.) provide both a concrete example of quantity, and a way to figure out, check, and keep track of the pattern until it is memorized with meaning.
With the students sitting in a circle, with their personal measuring tapes in hand, start at zero, and go around the circle, adding the number you’re skip counting by. As you go around, the children mark the place on the number line with their fingers, and they move to each new number in sequence as they hear it called out.
If you followed my advice in the first post, and numbered only the 5s and 10s, then you should start the students practicing this activity with 5s. This will help those who struggle with counting, as the numbers are clearly marked.
Then you can do 10s, noting that you skip the 5s.
Then you can do 1s, easy because they should know the sequence orally, difficult because the numbers aren’t marked on the strip.
The students will lose their places on the tapes. That’s fine. Once they hear the number that came before, they will be able to catch up. Either, they will be forced to go right back to the beginning and count out (because they haven’t yet realized that the number on the tape always represents the same fixed quantity), which is valuable practice, or they will, hopefully, eventually, realize it’s easier to go to the closest anchor of 5 or 10, and work out from there where the next number is.
For example, “The previous number was 34, I can find 34 because it is one down from 35, and now add 2 more, that makes 36.”
Over and above developing a true understanding of the skip counting process, this activity builds estimating skills, builds fluency with number and the distributive property, and sets the ground work for operations with ‘friendly numbers.’
Follow the same activity to count by 2s.
In Ontario, grades 1 and 2 have to count by 1s, 2s, 5s, and 10s, starting from a multiple of those numbers. Grades 3 and higher have to be able to count by those numbers from any starting point, and be able to count by other factors as well. It’s easy to adapt the circle counting activity by changing the factor being added, and/or changing the starting point.
Using the tapes forces the students to figure out what comes next, by adding/subtracting the right amount to/from the previous number. This is better than just hearing the pattern and parroting it back. Also, as patterning becomes algebraic study in the later grades, the students become familiar with the recursive process, or the algorithm of skip counting (n=n+1), so the formula will have meaning when they eventually learn it.
What concrete thing are we counting exactly? Make sure that the students understand we are counting centimeters, “the spaces between the lines.” This might require some groundwork on what centimeters are, in measurement terms.
If centimeters are too abstract, use counters of some kind, small enough to fit between the lines on the tape. Place the correct number of counters, directly on the tape, with each new iteration of the pattern.
In and Out
There is a game played in the Junior grades where the players give a number to put in to the ‘machine’ and the leader gives the number that comes out (input and output). For example, a student says 5, the leader writes 10. A player says 10, the leader writes 15. You can use these tapes to figure out the algorithm for what happens to the input to get the output. By marking on the number line where the input is, and the output is, commonalities will reveal themselves, and a rule will be developed.
The patterning in this case lays in the repeated operation. What is always happening to the number given to get the new number. Incidentally, this is an A pattern. +1, +1, +1, etc. is A, A, A, etc..
Essentially, any number pattern game you can think of can be done with these measuring tapes. If you have games you use to explore the numbers on 100 charts, you might be able to adapt them to these number lines, taking advantage of the linear nature as compared to the array layout of the 100 chart.
If you have any other ideas for ways to use the measuring tapes to build number pattern knowledge, please share them in the comments below.
In Ontario, there are five math curriculum strands that must be covered each year in elementary school. Of the five, I always gave patterning short shrift. I never really understood why having the kids make strings of red, yellow, and blue beads was relevant, and it was usually something I left to the last minute, or something I had the kids do while I assessed students or worked with small groups on “important” math.
Then, last year, I decided to find out more about it, and this past May, I presented my research and exploration of patterning at the Ontario Association of Mathematics Educators conference in Toronto. Unfortunately, I only had about 70 minutes, which, judging by the saucer eyes I saw staring back at me, was only enough time to turn my workshop participants on their heads and send them out the door, walking on their hands.
Therefore, over the next few months, I’d like to use my blog to go into more depth and engage in some discussion about what I’ve learned and the activities I’m developing.
The first thing I want to share is my list of what I consider to be the big ideas that students should learn about patterning. These are compiled and cobbled together from the NCTM standards, readings from Van de Walle and Small, and research by Joanne Mulligan. I have no idea, anymore, which ideas are theirs or mine, so lets just assume it’s all part of a Jungian zeitgeist and carry on.
- patterns are models
- patterns model relationships and structures
- patterns simultaneously represent consistency and change in those relationships and structures
- understanding pattern means being able to recognize, extend, replicate, predict and exploit those relationships and structures
- understanding pattern means being able to identify, diagnose, and perhaps repair breakdowns in those relationships and structures
These are some pretty big ideas indeed, and they seem a bit too grand to be achieved by stringing beads together. They are very likely not something most primary teachers keep in mind when chanting “red, yellow, blue, red, yellow, blue” with their students, during calendar time.
But as I post, I will endeavour to always refer back to these ideas, and we’ll see what other, perhaps more effective ways there are to discover and explore them. More to come.