Number Lines – Made to measure (patterning)
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
Patterning is the act of analyzing the relationships between elements in a string of repeated occurrances of those elements. š 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.
Circle Counting
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 OutĀ
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.
Number Lines, Made to Measure
How do you feel about number lines? Are they only for the weak kids? Do you use them in your math instruction?
I never used to, until last year when I was discussing them with a colleague who’d been doing some extensive reading on early number routines to use in her grade 1 class. She told me about what she’d read, and how she’d been using these counting models with her students. By the end of the talk, I was sold.
I had a grade 2 class, which is expected to count in various ways up to 200, and I always need to make sure I’m getting as much bang for my buck as possible. So, I took the concept of number lines, tied in the research I’d been doing on patterning instruction, and decided to bring in measurement, for good, ahem, measure.
The project I came up with was to have the students make their own meter measuring tapes. This is a very rich, hands-on task that provides 100 repetitions to reinforce the length of a centimeter. It gives practice printing numbers and skip counting. And it allows for many opportunities for you and the students to assess and problem solve.
Here are the instructions for how we made the measuring tapes, complete with cautions.
In my next post I’ll share ideas for how to use them, and I’ll invite you to add your own suggestions too.
Materials:
Narrow rolls of masking tape
Bristol board or Cash Register Rolls
Fine tip permanent marker/Ball point pen
Meter sticks
1. PrepareĀ strips of paper that are over a meter long.Ā You can do this with lengths of cash register paper, or by cutting strips of bristol board, about 5 cm wide. You’ll need to cut the bristol board sheets width-wise, and tape or glue two short strips end to end to make one strip that’s long enough. The bristol board will be sturdier and less likely to tear when the measuring tapes are being used. And besides, we have about 500 sheets of pink that no one wants to use.
2. Run a line of maskingĀ tape along the edge of a meter stick, leaving a portion of each of the centimeter marks showing. You should do this if expedience and materials are factors. Straight taping is not the goal of the lesson. Be thoughtful about how you’re placing the tape. If a particular student is a bit “rough” with her things, don’t let the tape hang over the edge, or it will be twisted, bent and torn before long.Ā
And, a word to the wise, if the meter stick is also broken down into half centimeters and millimeters, cover these marks entirely with the tape so that only the centimeter lines are showing. Otherwise, some of your more industrious and less attentive students will mark every single dash on the ruler.
3. Have the students use the marker or ball point pen to mark only the centimeter lines on the tape. Demonstrate that they should go only about to the middle of t
he tape with each line. Some guiding questions are “What number do we start the dashes at? (Zero) How many dashes will you have when you’re done? (101) What is the word we use to describe the distance between each dash? (centimeter)”
I promise that some students will skip dashes. Others will do all the millimeters anyway, because they can still sort of be seen through the tape. Whatever the error, each is an opportunity to reinforce the concepts of centimeters and standardized measures. I.e., Dashes that are closer together are smaller than a centimeter. Missed dashes mean that there are some spaces that are more than 1cm long. Not following the dashes exactly means that some “centimeters” will be bigger than others.
If an error happens, it will be better in the long run to pull the tape off and start again. Otherwise you’ll be dealing with those stray, scribbled out lines every time the student tries to use the ruler. But keep note of the mistake for your records.
4. Once you’ve checked all the dashes, peel the tape off the rulerĀ and stick it down the center of a prepared paper strip. Don’t pull the tape too taught, or the whole thing will curl. If the tape tears, just carefully place the two ends together on the paper strip. You’ll never notice.
5. Have theĀ student mark the desired numbers under the appropriate dashes, along the paper.
NOTE, I said along the paper.Ā This is because there is every likelihood that yourĀ students
will miss a number, repeat a number, print the numbers backward or inĀ reverse order, or partition the numbers too close or too far apart. IF they writeĀ the numbers on the tape first, you willĀ have to redo the whole dash process as well as fix any number situations.
This process is an excellent opportunity for you to assess and remediate your students’ number knowledge. You can consider having them use charts or exemplars in the room, or on the actual meter sticks, to find and fix their errors. For example, rather than saying “your fives are all backwards,” you could say “you made one kind of number backwards every time – see if you can figure out what to fix.” If you’ve pre-questioned the child about how many dashes she had to draw, and she’s written “100” five dashes too early, question her about why she still has five dashes left.
6. Transfer the numbers to the tape. After you have confirmed that the numbers were all written properly, in the right order, with the right spacing, it is now just a matter of the student copying the correct numbers onto the corresponding places on the tape, using the correct numbers on the paper as the guide. Note, that one or two students might start from scratch, missing the whole point of writing the numbers on the paper first. Does anyone come to mind?
7. Put each child’s name on the tape itself as they complete the work. Make any notes about their knowledge or learning in your mark book. Then cut the tape away from the paper strip.
You might want to do the cutting yourself, unless you are very confident in your students’ scissor skills.
Here are some points to consider when making these.
Decide what increments you want the students to use. Last year I had them do it by 1s. This was very time consuming, it was not easy for the students with poor fine motor control, and there were a lot more errors. This year I had them work by 5s. This gave me the chance to see who could count by fives (three could not), and I have the added advantage of forcing the students to think in terms of anchors of five and ten when they use the rulers in the future.
Some students will need a lot of hand holding, either due to physical or intellectual issues. Anticipate who these students are, ahead of time, and allot time accordingly.
Make a couple of extras in case a tape gets damaged when you start using them.
Check back for the next installment where I tell how I’ve been using these tools.
A Close Look at Patterning
The Big Ideas in Patterning
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.
This Will Make You Smarter by John Brockman
Oh, if only. Pray that the title isn’t just a clever marketing tactic.
I have always been interested in the study of intelligence. There are so many intense debates about natural ability, and teachability, and testing, and measuring. This book isn’t about any of that. It is about thinking better, and being more aware of who we are, how we think, and what is happening in the world around us.
Every year, John Brockman, via Edge.org, gathers together the thinkers of our era, and asks them a question. This Will Make You Smarter is a collection of answers to this one.
“What scientific concept would improve everybody’s cognitive toolkit?”
In each mini essay, one of today’s intelligentsia describes a concept in layman’s terms, and attempts to connect it to our everyday life, demonstrating how knowledge of this concept, or the practicing of that skill, will improve the way we think. The responses are varied, and some are seemingly contradictory. They have been grouped into overall themes to help the reader make some connections.
Now, just in case you think this might be too “sciency,” Brockman clarifies the question a bit. “Here, the term āscientificā is to be understood in a broad sense – as the most reliable way of gaining knowledge about anything, whether it be human behavior, corporate behavior, the fate of the planet, or the future of the universe.” So, there is lots of philosophy to try to wrap your head around too.
Full disclosure. A few of these ideas are beyond me. So much for my 129 IQ. However, on the whole, the entries are interesting and thought provoking, and justify the effort to understand them.
I think that much of this book will eventually end up in my personal arsenal of intellectual weapons, to reload my pen and tongue, as it were. I will update this section with synopses of some of the entries, and the questions and ideas they provoke, as I work through the book.
But check it out yourself if the state of modern intellectual development is something you’re interested in exploring, or if you just want to get smarter.
Mister Reed's Writing 