Sunday, July 13, 2014

Designing a curriculum: essential questions for Geometry & Algebra 2

I started working at my new school last week and it's been full-speed, non-stop all the way since day 1. I feel like I'm pedaling like mad down one of the steep slopes that are so popular around here and my brakes are broken.

Yup, I live here now

We're in year 2 of a new Upper School and the challenge is to rethink the traditional Math curriculum so that it promotes engagement, interdisciplinary connections, design thinking, and deep content learning without compromising students' ability to do well on the SATs, AP exams, and you know, their future life in a potentially traditional world that may frown upon our hippie ways.

After an amazing day working with Denise Pope, a professor at Stanford's School of Ed and writer of the book "Doing School,"who basically schooled us in the purpose of education, I felt ready to tackle some curriculum writing. She reviewed some of the ideas from backwards design from Wiggins: don't start with activities, but with the desired results. Then, figure out how you will know whether the desired results have been achieved. Then create activities that lead to the desired learning. Her guidelines for selecting enduring understanding goals were:

  • Represents a big idea with enduring value beyond the classroom
  • Resides at the heart of the discipline
  • Geared to students' interests and developmental needs
  • Open-ended, complex, provocative
  • Fits with school standards/expectations

So, I'm putting together my list of "essential questions" and "enduring understandings" for the integrated Geometry/Algebra 2 class that most students at my school will be taking during 9th and 10th grade. Please, please, please give me all of your feedback and criticisms and suggestions for improvement either here or on twitter (@borschtwithanna). I hope that this can be a fruitful springboard for discussion and curriculum framing.

  • How is a system of definitions, postulates, axioms, and theorems created and made cohesive? What does it mean for something to be true?
  • How can objects be transformed? Which of their characteristics stay constant and which change and why? How can we measure and describe what changes? How does this relate to concepts of proportionality and similarity?
  • How are angles on a plane related to each other? How do we know when two lines are parallel or perpendicular to each other?
  • How can we do algebra on a coordinate plane?
  • What is congruency? How do we know when figures are congruent to each other? Why might that be useful to know?
  • How can you determine whether a triangle is isosceles or equilateral? Why might that be useful to know?
  • What are some special relationships for right triangles? How do they relate to polygons? Why might this be useful to know?
  • What can we measure about figures? How do we measure these qualities? How do measurements change when we change dimensions?
  • What relationships are formed when lines intersect circles? 

Algebra 2:
  • What are the underlying principles of solving equations? How do we solve specific kinds of equations: linear, absolute value, quadratic, exponential, radical, and rational?
  • How are equations and inequalities related to each other? How are inequalities represented graphically? How are inequalities solved?
  • What are functions? How can they be represented and what do they represent? How can they be combined or reversed, algebraically and graphically? Why might any of this be useful?
  • What are the key characteristics of the following specific functions: linear, absolute value, quadratic, exponential, logarithmic, radical, and rational? How can we represent situations using specific functions?
  • What are asymptotes? How are they related to graphs and to equations of functions?
  • How are functions transformed? What is the relationship between the equation of a function and its graph?
  • How do functions model data? How do you know if the model is accurate?
  • How can multiple constraints be represented with systems? How can systems be solved?
  • How can expressions be simplified? How can expressions be combined or operated upon? How can expressions be factored? Why might any of this be useful?
  • What are different ways to categorize numbers? How are the different categories of numbers related to each other? How can different categories of numbers be represented graphically? How can we perform operations on different categories of numbers?
  • What are polynomials? What operations can be performed on them and how? How can they be represented graphically? How are their roots related to their graphs and equations? How can their end behavior be described and related to their graphs and equations? Why might any of this be useful?
  • What are properties of exponents and logarithms? Why are they true? How are exponents related to radicals?
  • How can sequences and series be described and evaluated? How are recursive and explicit formulas different from and related to each other? How are arithmetic and geometric sequences different from each other? How do we know whether a series diverges or converges? 
  • What are the measures of variation and how are they computed? Why might they be useful?
  • What are the measures of central tendency and how are they computed? Why might they be useful?
  • How can probabilities of events be determined? How are theoretical and experimental probabilities different from each other? 

* I should add that I haven't forgotten about mathematical practices, habits of mind, etc. We are planning to assess on a common set of these that will be the same across all of the math courses. Just breaking that up into a separate post.


  1. Awesome! I teach honors Alg. 2 so I can relate on how you want to integrate the mathematics with real-life scenarios. I'm there with ya, girl! I will say that, for the last two years since common core, I have not taught solving abs. value, linear, etc. in Alg 2. We solve quadratics but I expect them to know and apply the rest on their own in the stuff we do.

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    I'm so jealous of your beautiful home now! :)

    1. Thanks, Brooke! There are some equation types that students have definitely learned in middle school, but the class is quite heterogeneous, with kids coming from many, many different middle schools, so there will be gaps all over the place. I'm hoping to use richer investigations that will give kids who haven't seen certain things a chance to pick them up, while giving those who have a chance to go deeper. Will report back on how well that goes :)

  2. Anna - these lists are great, and so timely for me, because I am trying to craft a geometry curriculum for this year for a 'Regents-optional' Geometry class. I had a long conversation with @samjshah earlier this week, and we spent a long time talking about defining and classifying in geometry, how important it is for definitions to be accurate, and how we can actually define something more clearly in geometry (and in math?) than we can in the real world. As I'm putting together my curriculum map, I'm using that idea as a recurrent theme; so many things in geometry are related because of common characteristics.
    What a great resource you have created - thanks for sharing!