We have the same provider leading these new sessions, so I was a little worried we would end up looking for content to cut rather than incorporate. So far, that hasn't been the case. Obviously, there are certain topics that will require more focus (eg. linear relationships as opposed to exponents) but the goal has been to see how and where these supporting standards fit with the focus standards.
Our team has come up with the following units of study. We haven't reached the point where I can discuss specific activities/tasks, but I'd like some feedback on the pedagogy that motivated the clustering and sequencing.
Transformational Geometry
Use the coordinate plane to discuss transformations, congruence and similarity. Use dilations as an application of the ratios/proportions work done in grades 6 and 7. Use a graph as a tool to describe proportional relationships.
Data Analysis
Use bivariate data to create scatter plots which can then be the jumping off point for informal line of best fit (where the line may have an initial value other than zero) and an introduction for a future defining of function/non-function.
Linear Relationships
Graphing, graphing and more graphing. Take the informal line-of-best-fit and formalize the definition. Allow math to be it's own context. Graph systems of equations and look for common point (read: solution to system).
Equations
Use the work done in graphing systems to motivate more abstract symbolic manipulation required for solving linear equations.
Exponents
This was a tough one. Expressions with integer exponents and scientific notation seemed like an island unto themselves. We are still working on finding a place that fits nicely for these ideas.
Functions
Use the informal introduction to functions and formally define a function. Look at linear and non-linear functions. Compare functions using different representations (ie. graph vs. table vs. equation vs. verbal).
Pythagorean Theorem
I think this one speaks for itself.
3D Geometry
Problem solving involving the volume of cones, cylinders and spheres.
We are trying to move from the concrete/informal to the abstract/formal while allowing students to explore these ideas while creating their own formal definitions. I'm particularly interested in the sequence that runs from Data Analysis to Functions (note: Exponents look to be a unit that can be dropped in and our school calendar lends itself to having that unit kick off the second semester) as it may receive the most push-back from our high school colleagues. Traditional textbooks usually go the route of
Functions-->Equations-->Graphing-->Applications so we're going to have to have solid rationale.
No one pushes back better than you all. I'm counting on that.
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6 comments:
Pythagorean theorem seems pretty specific compared to the rest of the units. As a non-math teacher, what am I missing about that PT that requires such emphasis?
I don't think you're missing anything. In fact, Pythagorean Theorem is a bit like the Exponent unit in that it's an island that can be supported by a couple Number System and Expressions standards.
I guess, the Pythagorean Theorem standards look like they're a bit of a throw-in.
I find this interesting because you have the common core system that groups concepts but then you have the direct insight that teachers like you acquire through seeing what does and doesn't work with your group of students.
Math is a set of rules and processes but there are an infinite number of ways in which concepts can interrelate. I am interested to hear about your thought process when you adapt your lessons?
Why not build applications of the previous units into each new unit.
Exponential dilations that bring back up the first unit.
Pythagorean theorem proved experimentally which brings back the bivariate data unit.
You want more, I'd love to chat over twitter, this is kind of fun.
Maybe I'm missing something obvious (the non-native speaker in me always worries) but may I ask, why is there no "2D Geometry" unit that contains the Pythagorean Theorem and builds into the 3D Geometry unit?
Even if it partly serves a consolidation and review unit (as well as an inquiry-skills-based unit), a "2D geometry" unit could be a significant arena for practice in constructing an authentic understanding of the concepts and skills they will need to advance through 3D geometry.
- Elizabeth (@cheesemonkeysf)
@hilby
The application idea is what motivated the current sequence. What exactly is an "exponential dilation" and how would you experimentally prove the Pythagorean Theorem using bivariate data?
@cheesemonkeysf
I don't think you're missing anything. Our Pythagorean Theorem unit isn't sitting well with me as it is. But, we will definitely use things from the Transformational Geometry unit to prove the Pythagorean Theorem. I'm not exactly sure how the Pythagorean Theorem would fit in with the transformation unit, though, other than to say it's all geometry.
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