Category Archives: Mathematics

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.

measuring tapeThe 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

Photo 12-10-2013, 4 32 37 PM

Photo 12-10-2013, 4 34 13 PM

1. Prepare strips of paper that are over a meter long. 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.

Photo 12-10-2013, 4 35 36 PM2. Run a line of masking 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. 

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.

 

mark all the lines3. 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)”

Photo 12-10-2013, 4 36 51 PM

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.

tape to paper

4. Once you’ve checked all the dashes, peel the tape off the ruler 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 the student mark the desired numbers under the appropriate dashes, along the paper.

numbers on paper

numbers on paper

messy numbers

NOTE, I said along the paper. This is because there is every likelihood that your students
will miss a number, repeat a number, print the numbers backward or in
 reverse order, or partition the numbers too close or too far apart. IF they write the numbers on the tape first, you will have to redo the whole dash process as well as fix any number situations.

transferred wrongly

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.

transfer the numberstransfer the numbers6. 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.

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Throwing Junk (just tossing some numbers around)


stock image
I’m big on cross-curricular integration. The Ontario curriculum is huge, with more than I feel is possible to cover in any meaningful way. And I hate gym. So, if I can kill cover as many birds as possible, yay for me.

Here is a game I call Don’t Throw Your Junk in My Backyard (after the song), and some ways to integrate math and physical activity.

First, here’s the game.

Have the class divide itself into 2 or 4 groups, each team getting a half or a quarter of the gym. Or you can divide them, unevenly, and see how fast they figure out about equal division and same sized fractions.

Put out “Junk,” which is any piece of equipment that can be lobbed from one side of the gym to the other, safely, such as bean bags, soft balls, foam frisbees, etc. Have a fixed amount that can be shared equally among the groups.

Blow the whistle and have the students pick up and lob or slide (whatever you feel is safest – unless you enjoy filling out insurance forms) the “Junk” from their side to the other team’s side.

Give a time limit and blow the whistle again when the time is up.

Have each team count the “Junk” in its yard. Get ideas for the fastest way to count these, so they can GET ON WITH THE GAME ALREADY!

Determine which team has the least “Junk” And how much less (many pieces fewer). Working with a friendly number, like a ten, or even 100 can build number sense.

Discuss the winning team’s strategies for clearing their junk quickly, and challenge the other team to incorporate those strategies next round.

A Close Look at Patterning


The Big Ideas in Patterningimage

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.

  1. patterns are models
  2. patterns model relationships and structures
  3. patterns simultaneously represent consistency and change in those relationships and structures
  4. understanding pattern means being able to recognize, extend, replicate, predict and exploit those relationships and structures
  5. 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.

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