I need your help. I am putting together a course guide for a yearlong 9th grade course on Algebra & Geometry. I have divided the course into 27 “weeks” plus an appendix. For each week my goal is to have
(A) The key ideas for that week and why we’re focusing on them at that point in the year,
(B) Related standards from the common core,
(C) Connections between the key ideas and mathematics students have done,
(D) Connections between the key ideas and mathematics students will do,
(E) Texts and videos to which students and parents could refer to help understand key ideas,
(F) Questions and tasks that can generate discussion to explore key ideas in class,
(G) Questions and tasks that can be used to help assess student understanding of key ideas, and
(H) Interesting and challenging ways to extend and apply the key ideas.
For some weeks I might also have
(I) Standard textbook problems related to the key ideas, or
(J) Misconceptions and “tricks” to avoid (a la Tina Cardone’s Nix the Tricks).
My plan is to devote a blog post for each week where I will share what I have so far, and then hope that others can share suggestions they may have. I am especially going to be in need of suggestions for (E-G) above (i.e. references, questions, and tasks—I hope that all students will have access to Geogebra in class). My goal is to publish 3-4 posts a week so that the whole course outline in as much detail as I have it now will be available in the next 2-3 months (by the end of June at the latest). In this post, I’ll try to explain a little about the background and motivation for the course. I’ll try to be brief in this post and save the details for the subsequent posts on each week and then revisit the course overview at the end.
The audience for this course is the entire 9th grade student body at our school. We do not track, and this course is designed to fit the needs of a fairly diverse student population. Currently about two-thirds of our students come in having taken an integrated algebra course in 8th grade often scoring fairly well on a standardized state high school level exam. Even those students, though, generally come in not “remembering” much of what they learned. They memorized some procedures to get through an exam and promptly forgot them. Those students that do remember procedures nearly always still lack the conceptual understanding underlying the procedures. The key goal for this course is to provide ALL students with the conceptual understanding that they can build upon going forward. One of the challenges is to convince the students of the need for that understanding and to keep them engaged when approaching some topics that might seem at first glance to be review for them and which are completely new to others in the class.
I see the big ideas for the course as the interrelated concepts of number, similarity, slope, and solution sets. The concept of dimension acts as a hook and common thread that runs throughout the course. I’ll provide details of what I mean by these concepts and what I want students to take away with them in the subsequent focused blog posts. This course is influenced by the ideas of many people, but especially the work of three particular mathematicians whose ideas I draw upon to a great extent. First I make use of Thomas Banchoff’s ideas expressed in his book Beyond the Third Dimension and the pedagogical accompaniment he wrote for that volume which appeared in On The Shoulders of Giants. I also draw upon the ideas of Hung-Hsi Wu (especially those that appear in his course notes Pre-Algebra and Introduction to School Algebra which I believe will soon be made into a book for teachers to follow his Understanding Numbers in Elementary School Mathematics). Finally I have drawn inspiration from and will refer frequently to the work done by James Tanton. I will refer to some amazing online courses he has at gdaymath.com as well as his Thinking Mathematics! (a 10-volume set) and Geometry (a 2 volume set).
So here are the 27 weeks. The course starts by exploring the idea of (1) dimension leading us to think about two of the most important concepts in geometry: (2) distance and (3) direction. Those two weeks both address the concept of translation and this is followed next by (4) rotation and reflection. We are now prepared to define congruence that helps us explore (5) symmetry. After exploring the concepts of (6) area and volume and (7) scale we are prepared to address (8) dilations that allow us to define similarity. We can apply this concept to (9) similar triangles that are critical in both proving and efficiently applying (10) the Pythagorean theorem. A focus on number is not complete without looking at two of the most amazing theorems about integers, (11) the Euclidean algorithm and (12) the fundamental theorem of arithmetic. The first semester concludes with a fourth great theorem that makes great use of similarity and dimension as we look at (13) Archimedes and pi.
The second semester opens with an exploration of the connection between (14) place value and polynomials, followed by applying this understanding to (15) rational expressions. With some algebraic thinking in hand we consider (16) what it means to solve an equation and how this might apply to solving a (17) linear equation in one variable. That exploration included solving equations with several symbols where only one was considered to be a variable, and the others were considered to be constants, but we next look at (18) linear equations in two variables. This introduces the notion of slope that we explore further as we think about (19) rate. As we try to extend our ideas about solving equations to (20) solving quadratic equations we make great use of symmetry. We continue that theme as we (21) graph quadratics. We build upon the ideas we had from lines and quadratics as we next consider (22) circles. As we being to wrap up it’s nice to look back at how we can combine the ideas of dimension, distance, and direction from the first semester with the ideas of coordinates from the second semester leading us to explore (23) vectors and (24) systems of equations. We extend these ideas to inequalities with a look at (25) linear programming. Those last topics allow us to see a great benefit of considering higher dimensions, but we realize there is a great deal to explore even in 0-dimensions with a look at (26) combinatorics and (27) the binomial theorem which we can use to answer some questions we had about higher dimensional cubes back from the start of the year.
The appendix will look at some themes that pervade the course such as the common core mathematical practices, Tanton’s principles of thinking like a school math genius, and the role of proof and abstraction in mathematics.
As I publish the posts on each of the weeks I will add links to the outline above. I thus hope to keep this page open for comments, questions, and suggestions about the course as a whole, all of which I will greatly appreciate. I look forward to your feedback and thank you for your help.