This is an initial draft for the first chapter of a course guide for a year-long 9th grade Algebra & Geometry course. I am requesting your assistance. For more information on the course and this project, please see this post.
I used to start previous versions of this course with factoring of positive integers. My thought was that it made sense to start with one of the more basic “skills” in the course (arithmetic with natural numbers) and then gradually get more complicated. I have come to believe that this is the wrong approach. The first problem with starting with the basic is that it is not terribly engaging. Just as Dan Meyer bases much of his work on the philosophy that a mathematical task should tell a story starting with a visceral first act that draws one in and motivates the learning, I have come to believe that the same should hold for a mathematical course. The second problem with starting with any material that students already “know” is that they often fail to see a need to try to understand the underlying concept. They were capable of learning “that topic” once (even if they don’t now remember “details”) and they did so not by any productive struggle, but rather by being shown a procedure and practicing it. Learning it again seems to be a waste of time as “that topic” is already checked off their list, and furthermore I’m just making it unnecessarily difficult.
My hope is that starting the year by considering higher dimensions will help with both of these issues. I used this idea in the middle of the semester last year and the students were fascinated with the idea. (End-of-semester surveys generally had it as one of their favorite topics and one they wanted to learn more about). Furthermore many of the students bought into the idea that in order to explore an arena like higher dimensions, where we lacked a great deal of intuition, it would be necessary to think more carefully and try to formalize some of our intuition from lower dimensions. (This incidentally is what I think of when I think about mathematical rigor.) The concept of dimension makes for a great first act for this course. It is both a hook that draws students in setting up a conflict they want to resolve, and a thread that runs throughout the course (see part (D) below). Throughout the course we will answer some of our initial questions about higher dimension, but raise new ones. The course should end not with every question answered, but with enough resolution about our initial questions that we leave with a sense of accomplishment.
(A) Key Ideas: The concept of dimension will be explored throughout the course so many of the most important ideas concerning dimension (like how it relates to scaling and coordinates) can wait until later. In this first chapter we just consider an intuitive idea of dimension. Students should develop an intuitive sense of what it means when we say an object is 0-dimensional, 1-dimensional, 2-dimensional, and 3-dimensional. They should understand that we can slice an n-dimensional object into (n-1) dimensional cross-sections and this might be one way we can get insight into higher dimensions. Students should see 2-dimensional space as being of particular interest for us to study because it is the lowest dimension where we can really think about a variety of directions and where something surprising (the Pythagorean Theorem) occurs with regards to distance. In this chapter we also set the agenda for what we will need to do formalize our intuitive concept of congruence in 2-dimensions so that we can later extend the idea to higher dimensions. Students should also see an example (such as dots on a circle) where a seemingly obvious pattern turns out to breakdown hence motivating a need for proof.
(B) Related standards from the common core:
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
- CCSS.MATH.CONTENT.HSG.CO (Particular standards in this strand will develop later, but the general idea of thinking about congruence in terms of superposition starts in this first chapter)
(C) Connections with what students have done: Students have heard the word dimension a great deal but generally have not thought about what it means. I’ve noticed there is a great deal of misconception concerning the dimension of an object and its ambient space. So, for example, the student might think a point is 2-dimensional if has two coordinates. Most students come in thinking congruence means “same size and shape” (only one last year was able to define it in terms of superposition). Almost all students have experience with translation and reflection but almost exclusively from the standpoint of coordinates in the plane and procedures.
(D) Connections with what students will do: We will explore the connection between length, area and volume in chapter 6 and the effect of scale on those measurements in chapter 7. Dimension will play a key role in circular measurements (circles, disks, spheres, etc.) in chapter 13 (the end of our first semester–kind of like the fall finale in shows). Ideally thinking about dimension should prevent mixing up geometric formulas for different dimensions. More generally the notion of dimension is critical for geometric modeling. Dimension also places a critical role in our second semester thinking about exponents place value, and polynomials in chapter 14. It is important when thinking about spaces of solutions (for equations in chapter 18 and inequalities in chapter 25). Dimension is also key for thinking about vectors in chapter 23 and systems of equations in chapter 24. We will soon realize that counting (chapters 26 and 27) is a form of 0-dimensional measurement that can help us answer many questions about higher dimensions. At the same time an understanding of higher dimensions can help us answer many counting questions.
The notion of congruence will be explored with precision in chapters 2-4 and that in turn leads to the notion of similarity which is critical for understanding slope, graphing equations, transformations of functions, and trigonometry just to name a few areas.
(E) Texts and videos to which students and parents could refer to help understand key ideas: Thomas Banchoff is a great source for thinking about dimension. His talk last fall at the Museum of Math on Encountering Salvador Dali in the Fourth Dimension is a great resource. I’d also consider the opening chapter of his Beyond The Third Dimension which I believe he is making into an updated online text. Martin Gardner has a chapter “Hypercubes” with similar themes (and Dali references) which appears in Mathematical Carnival (pp. 41-54). Alex Rosenthal and George Zaidan have a TED lesson using the premise of Flatland to explore higher dimensions.
Above I also link to a video by James Tanton on “Dots on a Circle” that I would most likely do in class, but may assign for students absent that day.
(F) Questions and tasks that can generate discussion to explore key ideas in class: The discussion I had last semester was prompted by some visuals of hypercubes including Dali’s 1954 painting Crucifixion (Corpus Hypercubus). As mentioned above, I would spend a day on the “Dots on a Circle” task. I’m still searching for more tasks and discussion starters.
(G) Questions and tasks that can be used to help assess student understanding of key ideas: I’m also still searching for ways to assess understanding here (although much of this chapter is just laying groundwork for ideas to be developed and assessed later). I could ask questions about what one would get if one sliced a certain solid in a certain way or ask something about surfaces and solids of revolution, but again for now I’m more interested in that they’re starting to think. In some sense I’m more interested in whether the students are starting to ask questions.
(H) Interesting and challenging ways to extend and apply the key ideas: One can explore the idea of what the “basic isometries” should look like in other dimensions (one-dimension, three-dimensions, four-dimensions?)
(I) Standard textbook problems related to the key ideas: I don’t know of any.
(J) Misconceptions and “tricks” to avoid: Watch out for the mix-up of the object and its ambient space as discussed above.
If you have any questions, comments, and/or resources, please share them. As you can see I’m especially in search of tasks and discussion starters for this chapter . Thanks for your help. (Chapter 2)