This summer I'm spending my time looking for rich contexts or problems to engage my Algebra 2 students. Generally, Algebra 1 is a piece of cake whereas Alg 2 gets a much more booky class. I intend to change that.
I'm taking a no-holds-barred approach: surfing the internet for good contexts, activities begged borrowed and stolen, and--this is a first for me--using the projects section out of the back of the textbook I use. Surprisingly, I like some of these. What a dolt I am for never having really looked before.
The project on the left is rich enough that I think I can use it to replace two sections 2-8, 2-9 I currently teach. (sorry about the picture--I used an iPad 2) Sections 2-7, 2-8, and 2-9 deal with data modeling and combined variation. The combined variation aspect is really neat because you can investigate each variable involved in a situation separately and then put them together for a combined variation function. For example, in this situation, the Weight (M) a board can hold varies directly as the width of the board (w), directly as the square of the thickness (t) and inversely as the distance of the supports underneath the board (d). The variation constant (k) changes with each particular situation. When you put them all together you get: M=(k*w*t^2)/d.
The difficulty with this project is finding the materials to make it work. Finding strips of Balsa wood was doable but I couldn’t find enough weights to make a multi-group experience of the project. Here’s how I plan to modify it: Instead of Balsa wood, each group uses spaghetti noodles. Suspended paper cups from the spaghetti using paper clips. Then place metal washers or marbles in the cups as weights. They’re not standard weights, but that could even lead to a discussion of whether or not we could just weigh one washer and scale up or whether we need to weigh the whole stack when predicting how much weight it will take to break a "board" of this many noodles.
Here’s a quick write up of what I’d give to students. If you have any suggestions, I'd love to hear from you.
Directions
We will do 3 different experiments to explore how different variables change the amount of weight a board can hold: width, thickness, and the distance between supports. Because wood is much harder to break and I don’t have lots of weights laying around, we’ll make pasta boards by taping together spaghetti noodles. Please do at least 5 trials in each of the 3 categories. As a group, you may either choose to use washers or marbles as your weights for the hanging cups.
Spaghetti Width
Investigate how the width of a board changes the amount of weight it can hold without breaking. Be sure to keep all other variables the same and only change the width of the spaghetti board.
Spaghetti Thickness
Investigate how the thickness of a board changes the amount of weight it can hold without breaking. This one is a bit more on the tricky side of things. I suggest using tape and to help force your spaghetti to stand vertical. I also would use a width of more than 1.
Support Distance
Keeping all other variables the same, change the support distances to gather information about how the distance between supports affects the amount of weight a board can hold without breaking. Spaghetti noodles are generally 25.5 cm please use metric units.
Questions to Ask:
After collecting your data, analyze it as a group. (I expect you to be able to share your analysis with the class using math.)
What relationships do you see?
Create models describing each of these experiments.
Make a prediction using your data and test it with your spaghetti and weights.
Make a mathematical model that takes into account all three variables and how they relate to the weight a board can hold.
How can we test our model?
I'm taking a no-holds-barred approach: surfing the internet for good contexts, activities begged borrowed and stolen, and--this is a first for me--using the projects section out of the back of the textbook I use. Surprisingly, I like some of these. What a dolt I am for never having really looked before.
The project on the left is rich enough that I think I can use it to replace two sections 2-8, 2-9 I currently teach. (sorry about the picture--I used an iPad 2) Sections 2-7, 2-8, and 2-9 deal with data modeling and combined variation. The combined variation aspect is really neat because you can investigate each variable involved in a situation separately and then put them together for a combined variation function. For example, in this situation, the Weight (M) a board can hold varies directly as the width of the board (w), directly as the square of the thickness (t) and inversely as the distance of the supports underneath the board (d). The variation constant (k) changes with each particular situation. When you put them all together you get: M=(k*w*t^2)/d.
The difficulty with this project is finding the materials to make it work. Finding strips of Balsa wood was doable but I couldn’t find enough weights to make a multi-group experience of the project. Here’s how I plan to modify it: Instead of Balsa wood, each group uses spaghetti noodles. Suspended paper cups from the spaghetti using paper clips. Then place metal washers or marbles in the cups as weights. They’re not standard weights, but that could even lead to a discussion of whether or not we could just weigh one washer and scale up or whether we need to weigh the whole stack when predicting how much weight it will take to break a "board" of this many noodles.
Here’s a quick write up of what I’d give to students. If you have any suggestions, I'd love to hear from you.
Directions
We will do 3 different experiments to explore how different variables change the amount of weight a board can hold: width, thickness, and the distance between supports. Because wood is much harder to break and I don’t have lots of weights laying around, we’ll make pasta boards by taping together spaghetti noodles. Please do at least 5 trials in each of the 3 categories. As a group, you may either choose to use washers or marbles as your weights for the hanging cups.
Spaghetti Width
Investigate how the width of a board changes the amount of weight it can hold without breaking. Be sure to keep all other variables the same and only change the width of the spaghetti board.
Spaghetti Thickness
Investigate how the thickness of a board changes the amount of weight it can hold without breaking. This one is a bit more on the tricky side of things. I suggest using tape and to help force your spaghetti to stand vertical. I also would use a width of more than 1.
Support Distance
Keeping all other variables the same, change the support distances to gather information about how the distance between supports affects the amount of weight a board can hold without breaking. Spaghetti noodles are generally 25.5 cm please use metric units.
Questions to Ask:
After collecting your data, analyze it as a group. (I expect you to be able to share your analysis with the class using math.)
What relationships do you see?
Create models describing each of these experiments.
Make a prediction using your data and test it with your spaghetti and weights.
Make a mathematical model that takes into account all three variables and how they relate to the weight a board can hold.
How can we test our model?
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