This is an initial draft for the sixth 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.
Area and volume are incredibly subtle concepts (see for example the Banach-Tarski paradox). Even if we can’t handle all of the subtleties yet, any course on geometry should spend some time discussing these central concepts and thinking about what properties we’d like them to have. This seems like a great time to have that discussion in this course. One of the central assumptions it would seem we’d want to hold for area is that it is preserved by congruences and we’ve spent the last four chapters defining what that means. Our discussion of area and volume is a reminder of why we’ve been doing that as we returns to our original motivating exploration of dimension. A key idea that should arise in our discussion is Cavalieri’s principle which builds upon the concepts of parallelism and congruence and even introduces a new non-rigid transformation “skew” which preserves distances only in one direction.
Now is also a good time to have this discussion as we are about to explore the concepts of scale and similarity (chapters 7 and 8) in which we’ll want to explore the effects on length, area, volume (and even higher dimensions). Scale will also provide an important way to think about multiplication, but it is worth first considering the area model of multiplication. In both cases we’ll focus especially on the implications of the model for the multiplication of fractions as that’s a topic where students often rely solely on procedure with limited understanding.
(A) Key Ideas: (Wherever I use the word “area” we should also be thinking about volume and even higher-dimensional analogs.) Students should understand that area is a rather subtle concept. There are certain nice properties we want it to have, including: congruent regions in space should have the same area; if we think of a region as being formed by a finite number of non-overlapping subregion; the area of the whole region should be the sum of its parts; a region which is located within another should not have a larger area. The area of a rectangular region should be the product of its dimensions. From this it follows that our procedure for multiplying fractions makes sense as does the commutative and associative properties of multiplication (at least for positive numbers).
With “limiting” arguments we can say that it would therefore be reasonable that if we have a region of a certain area and extend that region to a higher dimensional solid by extending in a new right-angled direction, the resulting solid should have a volume that is the area of the original region multiplied by the length in our new dimension (often called “height”). Similar arguments would imply that it is reasonable that a figure with parallel cross-sections of constant length should have the same area as if all of those cross-sections were lined up nicely in a rectangle. From these follows the formulas for areas of parallelograms, and volumes of solid prisms. With congruence arguments we then get the formulas for areas of triangles and pyramids.
(B) Related standards from the common core:
(The connection between area and multiplication is covered in the common core for whole numbers in 3rd grade and for fractions in 5th grade. As noted earlier it will be some time before I have students who have encountered this common core approach in elementary school. Apart from that, now is a great time to revisit this essential concept in light of our more formal approach to geometry. There are still subtleties involved and students could explore these same concepts in graduate school if they pursue mathematics.)
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
(C) Connections with what students have done: This chapter most directly picks up on some of the key discussions of dimension, definitions, and proof that we introduced in chapter one. Students come into the class having memorized with mixed success many formulas concerning perimeter, area, and volume but often without any sort of reasoning leading them to mix up perimeter and area formulas or area and volume formulas. Multiplication of fractions was also often learned by rote procedure without thinking about it in terms of area.
(D) Connections with what students will do: In the next chapter students will explore how area and volume change when scaling (chapter 7). We will also explore the amazing connection between a circumference of a circle and the area it bounds (and higher dimensional generalizations) (chapter 13). An area model of multiplication will be critical when multiplying polynomials (chapter 14) or solving quadratic equations (chapter 20). It is also key to the binomial theorem (chapter 27). As students do more intensive modeling (10th grade) area and volume concepts will be used frequently and of course the concept of area is central to integral calculus.
(E) Texts and videos to which students and parents could refer to help understand key ideas: Francis Su and his colleagues at Harvey Mudd have a short fun fact on the “Banach-Tarski Paradox.” Gabriel Girdner has a nice under 4 minute video on the “Cavalieri Principle.” In Thomas Banchoff’s Beyond The Third Dimension (pages 16-21) he explores how we might try to generalize the area formula for triangles, and volume formulas for pyramids to higher dimensions.
(F) Questions and tasks that can generate discussion to explore key ideas in class: Asking students whether the situation of the Banach-Tarski paradox is possible, and then telling them that is accepted as proven based on generally accepted assumptions is likely to spark some interesting discussion that you can focus to what assumptions would we want to accept as true about area.
Asking students to think about what should happen to the volume taken up by a deck of cards if we then “skew” the deck could also spark some discussion. One can also ask simply what is meant by the “height” of a triangle.
Take your marker and trace out an open curve. Ask whether the curve has any area. Then scribble the marker back and forth to fill up some space, does the shaded region now have “area”?
(G) Questions and tasks that can be used to help assess student understanding of key ideas: I like to ask students to find the area or volumes of figures for which they’ve never been given a formula, but which can be determined by applying the assumptions about area we’ve made. I also like to ask more standard triangular area, parallelogram area, or pyramid volume problems where the height is not directly given, but these work best after we’ve had a chance to explore the Pythagorean theorem a little more (chapter 10).
(H) Interesting and challenging ways to extend and apply the key ideas: Our main discussion focuses on area and volume, but thinking about higher dimensions can naturally extend these ideas. What would be a higher-dimensional analog of a pyramid? What would it’s volume formula be?
(I) Standard textbook problems related to the key ideas: There are numerous standard textbook problems involving finding the area of some figure or the volume of some solid.
(J) Misconceptions and “tricks” to avoid: The key is to move beyond just memorizing formulas and really trying to understand them better.
If you have any questions, comments, and/or resources, please share them. Thanks for your help.