Home Forums Teacher’s Notebook Page Constraints

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5 replies, 2 voices Last updated by  Miss Lacy 5 months, 2 weeks ago
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• #1034

Mrs. Post
Keymaster
@Mrs. Post

In class, someone asked if there were a list of constraints somewhere. There wasn’t until now. I thought I’d make a general list of constraints that almost always apply and then post images of a structure that can be modified by changing the constraints. Feel free to add your own constraints.

Why do we use constraints? Constraints allow us to explore mathematics. Real-world problems come with constraints, for instance, if I want to know how much carpet to purchase, it is measured in square yards. That is a constraint. If however I only want to measure the perimeter, that would also be a constraint.

If you are using the notebooking page, constraints can fall under “what can be changed” and “what can be compared, contrasted or constrained?” It’s not important what category heading you place your work under. What is important is that you are playing with the math ideas with your kids.

When using the rods, you’ll find these common constraints:

1. Measuring length. For instance, using a rod of a single color.
2. Number of rods. For instance: making only two car, three car, or 4 car trains.
3. Permutations. A permutation is a way, especially one of several possible variations, in which a set or number of things can be ordered or arranged. Let’s say I have the rods red, red, white, white. Our constraint could be that any variation of 2 red rods and 2 white rods counts. We could also count those differently and ask the student to find all possible variations or permutations of 2 red rods and 2 white rods.
4. Use rods/numbers larger or smaller than a certain rod or number.
5. Count or write statements using only a single operator such as addition, subtraction, multiplication or fractions.
6. Combine 2 or more constraints. For instance, maybe we can find all 2-rod trains (number constraint) of a single color (measuring length constraint) that are equal to the dark green rod.

One of the things we like to do is apply a constraint – notice and wonder, apply a different but similar constraint and compare and contrast. Maybe we want to find all two-car trains of a single color for all the rods. Then we might find all the possible 3 car trains of a single color for all the rods. Which rods have 2 car trains, which do not? Which have three-car trains, which do not? Is there a pattern that we can find? Is there a way to find the next rods that will have both a 2 car train and a 3 car train? Do we have enough information? Do we need to extend the rods past the orange to make a prediction?

#1035

Mrs. Post
Keymaster
@Mrs. Post

Below is your standard staircase starting with a white rod. What constraints could we add to this structure?

1. Change the length between the steps. Maybe we want to change it from white to red.
2. If your students are older, maybe you want to distance between the steps to grow by one. Distance from 1st step to second step is white. Second step to third step is red, third and fourth the distance is light green and so one.
3. Maybe you want to compare staircases starting with a white that have a common difference of white with a staircase that starts with a white and a common difference of a red. What does this remind you of? What is the same about the staircases? What is different? What if you now add a constraint and say that you must start with a white and the common difference is dark green?
4. This is not a constraint but renaming: what if we rename the white and make it 1/2? What if we rename the white and make it 3? What will all the other rods equal?
5. We could also make a staircase starting with the first even number and the common difference is odd. What would that look like? How does the staircase change when you change the common difference?

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#1057

Mrs. Post
Keymaster
@Mrs. Post

Some constraints for trains:

Find all two-cars trains for a single rod.
Find all three-car trains for a single rod.
Find all three-car trains for all rods larger than a light green. Compare the number of trains. Is there a pattern? Can you predict the number of trains?
Find all the permutations of all trains for the blue rod. Permutations are variations: w,w,w,w,w,r,r is a permutation of r,r,w,w,w,w,w.
Find the smallest two-car train and the largest two-car train (this is good for preschooler and kindy children).
Find all two car trains for each rod, one rod must be a white.
Find all two car trains for each rod, one must be a red.
Find all two car trains for each rod, one must be a light green. etc.
Find rods that can be made with a train made of red rods.
Find all rods that can be made with a train of light green rods, what about purple, and yellow? What rods cannot be made into 2 car trains for a single rod?

Do you have any you want to add @miss-lacy.

#1076

Miss Lacy
Keymaster
@Miss Lacy

I tackled constraints for staircases in my staircase task cards. There is a lot you can do but I can expand on what I didn’t do.

-Create a staircase of odd numbers that is plus 2. Is there another plus staircase that is made of only odd numbers? Plus 3? Plus 4? Plus 5? Why or why not?

-Create an even staircase that is plus 2. Is there another plus staircase that is made of only even numbers? Plus 3? Plus 4? etc.

-Create a staircase that is plus 4 that starts with white. Create a plus 4 staircase that starts with 2. How do they differ? How are they the same.

Once you start down this path, the road is endless and you won’t need any assistance in coming up with constraints. In fact, your student should take over at some point with their own explorations.

#1078

Mrs. Post
Keymaster
@Mrs. Post

Lacy, I agree, once you go down the path of constraining things, it is endless and the kids have their own ideas and want to know what happens if…

They begin to be driven by the internal motivation to know vs. the external motivation of the threat of mother.

#1084

Miss Lacy
Keymaster
@Miss Lacy

They begin to be driven by the internal motivation to know vs. the external motivation of the threat of mother.

This is so true. I found that notice and wonder was the gateway into developing that internal motivation to know. For my DD, it is where she began to blossom in math. We don’t dive into figuring out everything she wonders either. It is more important for her to just wonder.

But constraints are wonderful in that they extend students to experience more so that they wonder more. It is gentle leading to get them to want to know more. It’s a breadcrumb that leads them to the wonder of math.

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