This is an initial draft for the seventh 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.
Keith Devlin had a series of columns in 2008 (June, July-August, September) arguing that multiplication is not repeated addition, but rather scaling. But, unfortunately, far too many of my students come in thinking multiplication is repeated addition (or perhaps just a symbol on a calculator). It is vital that we spend some time discussing the all important concept of scale.
When to do so, though, is a tough question. I think it would be helpful to have the discussion before getting into formal definitions of dilation and similarity (chapter 8), but I also wanted to hold off on asking students to revisit their thinking in this area for as long as possible. Thus while we certainly touched upon scale when looking at unit (chapter 2) and area (chapter 6), it is only now that we’ll closely examine multiplication as scaling.
(A) Key Ideas: Just as combining is a fundamentally additive process, scaling is a fundamentally multiplicative process. Whereas when we add we shift the start (zero) but keep the length of the unit fixed, when we multiply we stretch the unit (one), but keep the start fixed. Scaling by a factor of r has the effect of scaling n-dimensional measurements by r to the n.
(B) Related standards from the common core:
(This standard which ask students to “solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.” is the closest standard I could find to saying students should understand the concept of “scale” although certainly the concept comes up quite a bit with dilations and similarity which I address in the next chapter.)
Interpret multiplication as scaling (resizing)…(The standard then splits into sub-standards that look at consequences of that interpretation. Again we have a 5th grade standard but it so important and so often misunderstood that I want to make sure all of my students have an understanding of this concept.)
(D) Connections with what students will do: Understanding scale will be key to understanding dilation and similarity (chapter 8). Taking advantage of scale can often make geometric computations easier especially when using the Pythagorean theorem (chapter 10) or determining areas and volumes (chapter 13). The concept of scale is also key to understanding rates (chapter 19) and a fundamental operation involving vectors (chapter 23). In future grades scale is also key to understanding trigonometry, transformations of functions (both in the abstract and in context), and geometric growth.
(E) Texts and videos to which students and parents could refer to help understand key ideas: In addition to the Devlin articles linked above, in Thomas Banchoff’s Beyond The Third Dimension (pages 16-21) he discusses the connection between effects of scaling and dimension.
(F) Questions and tasks that can generate discussion to explore key ideas in class: I might consider asking students what does it mean to multiply two numbers? Or why is a negative times a negative a positive? Dan Meyer’s “Incredible Shrinking Dollar” has led to interesting discussion of what does it mean when we set the photocopier at 75%. A response of “The image will be 75% the size of what it was” highlights a need for MP 6 (attend to precision ), for what is meant by “size” in that response.
(G) Questions and tasks that can be used to help assess student understanding of key ideas: Much of this concept will be assessed in the next chapter (dilation and similarity), but one can ask students to multiply fractions and give to explanations for why their method of multiplying worked.
(H) Interesting and challenging ways to extend and apply the key ideas: When one considers dimension as being determined by the effect of scaling on lengths, one may be led to the question is it possible to have fractional dimensions. A look at Sierpinski’s gasket can highlight an amazing result. Actual computation of the dimension (in this sense) requires logarithms, but the idea that scaling the lengths by a factor of 2 leads to 3 copies of the original is quite a contrast compared to a segment where’d we expect 2 copies, a square where’d expect 4 copies, or a cube where we’d expect 8 copies. It also highlights some of the dangers inherent in infinite processes (which we will not heed when computing the area of a disk in chapter 13).
If you have any questions, comments, and/or resources, please share them. Thanks for your help.