This is an initial draft for the second 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.
James Tanton introduces his Geometry volumes with the following:
I make a bold and nonstandard claim. Setting technical details aside, there are
essentially only three key (“non-obvious”) components to geometry:
- PYTHAGORAS’S THEOREM
- PARALLEL LINES
- SIMILAR TRIANGLES
I agree with Tanton except I would broaden each of these notions slightly. I would broaden the third concept to similarity in general, the second concept to the notion of direction, and the first concept to distance. I’ll explain my rationale for broadening the other concepts as we get to them, but the reason for extending the first concept is to think a little more about one-dimension. Some of the most common difficulties I see from students (in particular when working with fractions, combining like terms, modeling, or understanding trigonometry) seems to stem from neglecting to think about the importance of units. So I believe there is a need to revisit the concept of unit and the number line.
The challenge of doing this, as I alluded to in the last chapter, is that students may consider it to be “complicating” something they’ve “already learned.” To help combat that I’ve made two adjustments this year to what I’ve done in the past. The first is to use the desire to understand higher dimensions to motivate thinking more carefully about lower dimensions. The second is to spread out the work done on thinking about the meaning of addition on multiplication. So in this chapter we look at the concept of the number line and addition. We wait until the unit on area (chapter 6) and scale (chapter 7) to consider multiplication.
(A) Key Ideas: Students should recognize that the two most important points on the number line are 0 and 1. Where those points go is arbitrary, but once they are determined that sets “the unit” and all other points are determined based on that unit. In general we can’t add two numbers from number lines representing different units (1 hour plus 1 foot is not 2 of anything), but sometimes a number on one number line might represent a different number on another line and so we can use an alternate representation (1 foot plus 1 inch is not 2 of anything notable, but it is the same as 12 inches plus 1 inch). We can abstract that idea to think about a number on one line as being a unit on another. For example, 3 on the number line of 10’s is the same as 30 on the initial number line. This idea is critical to defining fractions. The denominator sets the unit. One-third sets a unit on a number line (the thirds line) so that 3 on that line is 1 on our initial number line. Once that is set, the numerator sets the number on the line with the new unit, so for example, five-thirds is interpreted as 5 on the number line of thirds. From this the key idea of equivalent fractions and adding fractions follows. We extend our number lines to negatives by thinking about an opposite direction and think about addition geometrically in terms of one-dimensional vectors or equivalently translation in one-dimension. The distance between two points is the number of units separating them on the number line and the absolute value of a number is its distance from 0. Finally we can extend our notion of distance in one-dimension to higher dimensions by repeated application of the Pythagorean theorem. We will formally prove the theorem and think about how to apply it more efficiently in chapter 10 after we’ve established similarity, but all students have seen it before in 2-dimensions and it’s worth taking some time now to think about how we could extend it to higher dimensions.
(B) Related standards from the common core:
(Our approach to fractions is that now taken by the common core in 3rd grade, but unfortunately I won’t expect my students to have done common core when they were in 3rd grade until at least 2019. How I approach the topic with a 9th grader will be different than how I would approach it with a 3rd grader, but I believe it should still be done.)
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
(C) Connections with what students have done: Students have done arithmetic with negative numbers and fractions but often procedurally without much understanding which makes them prone to errors and which makes it difficult for them to extend the ideas to new situations. Students also have generally seen the Pythagorean theorem in 2-dimensions but haven’t necessarily made the connection to distance and generally have not extended it to higher dimensions.
(D) Connections with what students will do: In the next unit (chapter 3) students will extend the notion of translation from one-dimension to higher dimensions which also lays the groundwork for thinking about vectors and vector addition (chapter 23). The fundamental assumptions of our geometry is that rotation and reflection (chapter 4) also preserve distance. The latter being defined by reference to a perpendicular bisector which is the line equidistant between two points. The fundamental one-dimensional measurement of length is extended to higher dimensional analogs area and volume in chapter 6 and distance is the key to understanding scale, dilations, and similarity (chapters 7-9). We actually prove and use the Pythagorean theorem more effectively in chapter 10. The concept of fractions is key to understanding rational expressions (chapter 15) and rates (chapter 19) and a circle is defined in terms of equidistance in chapter 22. We also touch upon the definition of parabola (again in terms of equidistance) in chapter 21.
(E) Texts and videos to which students and parents could refer to help understand key ideas: Christopher Danielson has a fantastic lesson built around an animated 4-minute video on this critical concept of unit as well as a related blogpost. Hung-Hsi Wu goes in-depth in his guide to Teaching Fractions According to the Common Core (pages 19-26) and a formal approach to the two-sided number line in his course notes Pre-Algeba (Chapter 2, pages 112-132).
For extending the Pythagorean theorem into three-dimensions this video by James Tanton not only goes through the standard way of making a box and applying the Pythagorean theorem twice, but also gives a cool 18th century approach by de Gua. The second approach is algebra intensive and likely to lose most of my 9th graders at this point, but it might be nice to see now and return to at the end of the year with more algebra skills in hand. It might also motivate a more in-depth look at the Pythagorean Theorem (which we will do in chapter 10) and point to the interconnectedness of algebra and geometry. (One of the most common questions I get at the beginning of the year is are we going to do algebra and then geometry or geometry then algebra. Students see them as two separate subjects.)
If you know of any other references or resources that would be particularly good for 9th graders please let know.
(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: I like to ask a mix of word problems where some could be answered by addition or subtraction, and where for others either a unit conversion needs to be done or for which there is not enough information because we would need to know more about how the units would convert. I have also discovered much by having students evaluate “student work” with fractions and explain whether what has been done is valid or invalid. I make it clear that just because somebody does something differently from what you would do does not in itself make it invalid. I’m in search of more ways to assess student understanding of units and fractions according to the common core.
(H) Interesting and challenging ways to extend and apply the key ideas: Thinking about the taxicab metric is a great way to explore what would would follow if we chose assumptions that didn’t lead to the Pythagorean theorem. (In our case these are the assumptions that are translations, rotations, and reflections exists and preserve distance.) If you know any good resources for this please let me know.
(I) Standard textbook problems related to the key ideas: Any standard problem involving addition or subtraction of rational numbers should be accessible to a student who has understanding of the ideas of this chapter. I’m not concerned about applying the Pythagorean theorem just yet. That will wait until chapter 10.
(J) Misconceptions and “tricks” to avoid: Tina Cardone points out several addition/subtraction “tricks” to avoid in her book, Nix The Tricks. These include keep-change-change (Ch 2.4), two negatives make a positive (Ch 2.5), and the butterfly method (Ch. 3.1).
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 week. Thanks for your help. (Chapter 3)