I like that she focused not on the practical benefit of student engagement, but rather phrased engagement as what her students deserve. I agree completely. My students generally work quite hard trying to balance school, activities, and teenage life. They deserve a course that is compelling filled with lessons that are compelling.
Of course the big question is how to make that happen, and that is why Nicole examines the question of what makes a lesson compelling which she breaks down by what “type” of lesson it is. I will just add a few thoughts both to the general question of what makes math compelling, and to her specific framework…
I’m sure I’ll be thinking about this more as the year progresses. Thanks Nicole!
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:
(C) Connections with what students have done: We previously looked at the concept of unit (chapter 2) and area and volume (chapter 5).
(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.
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:
(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.
Now that we’ve established the formal definitions of rotations, reflections, and translations, we’re ready to define a congruence as a finite composition of these rigid transformations. That forces us to define the concept of composition and it also leads us to want to develop a more convenient notation for transformations that can handle compositions well, namely parentheses notation. It would be rather upsetting if we had a figure S which was congruent to T, but T wasn’t congruent to S. That is we’d like to be assured that a congruence from S to T necessarily implies the existence of a congruence from T to S, but that we can handle if we introduce the notion of the inverse of a transformation.
We could introduce all of these concepts just motivated by the desire to define congruence, but I have found the notion of symmetry (which for now we consider to be a congruence from a figure to itself) to be more directly engaging. It also helps motivate better why we’d be interested in identifying the specific transformations that compose, instead of just being interested in whether such a sequence exits.
(A) Key Ideas: Students should understand parentheses notation for transformation, what it means to compose two transformations, and what is meant by the inverse of a transformation. Students should understand the formal definition of congruence in terms of rigid transformations. They should also understand the distinction between a congruence (noun) and congruent figures (adjective). Two figures are congruent if there exists a congruence from one to the other. Students should understand that we consider a symmetry to be a congruence from a figure to itself.
(B) Related standards from the common core:
(C) Connections with what students have done: When I did a unit like this last year I was surprised to discover that students had only learned about symmetry in terms of reflectional symmetry. When shown a figure with only rotational symmetry there was some heated discussion about whether it should be considered to possess any symmetry and the consensus reached was it should not because there were no “lines of symmetry”.
(D) Connections with what students will do: The definition of congruence will extend once we introduce dilations to a definition of similarity (chapter 8). These allow us to establish criteria for triangle congruence or similarity (chapter 9). Transformations are nice examples of functions and the notation and concepts used here (such as composition and inverse) will be generalized next year (10th grade). Students are also laying the groundwork here for the concept of groups which they may encounter in college mathematics.
(E) Texts and videos to which students and parents could refer to help understand key ideas: James Tanton has a nice pamphlet on symmetry which explores the question of how symmetry is defined mathematically. By including dilations and hence self-similarity, he looks at an even broader notion of symmetry than just focusing on congruences. He also spends some time proving that the rigid motions are isometries using SAS and properties of parallelograms (as opposed to common core which starts with the assumption that rigid motions are isometries and uses that to prove SAS and properties of parallelograms).
(F) Questions and tasks that can generate discussion to explore key ideas in class: Last year I started by handing out a sheet with several figures on one side (and frieze and wallpaper patterns on the other side) and asking which figures possess symmetry and are some figures “more symmetric” than others. This led to some great discussion, although as I noted above, the students had in their minds that only reflectional symmetry should be considered symmetry. I had to state that mathematicians and artists were often interested in a broader notion of symmetry. (I shared a lot of work of M.C. Escher and other photos of art and architecture using symmetry). I also pointed out some benefit to being able to perform a sequence of symmetries and still consider that to be a symmetry (which would not be the case if we only allowed reflections).
Later we looked at a square and tried to list out all the symmetries and keep track of what happens when combine two symmetries of the square (essentially writing out the multiplication table for the dihedral group of symmetries of the square). This motivated composition which in turn motivated some improved notation (parentheses) as well as the concept of inverse.
(G) Questions and tasks that can be used to help assess student understanding of key ideas: One could give the students a figure and ask them to simply list the symmetries. One could also ask students to identify the composition of two symmetries as a single symmetry.
(H) Interesting and challenging ways to extend and apply the key ideas: By looking at only the orientation preserving symmetries of a regular polygon we encounter the cyclic groups which we can then connect with the notion of modular arithmetic (which some students have encountered before). One could also look at classifying all frieze patterns or all wallpaper patterns.
(I) Standard textbook problems related to the key ideas: I’ve seen a standard problem of given a figure draw all “lines of symmetry” We can extend this traditional exercise by also asking for point symmetries and asking what order they are, as well as considering translational symmetry.
(J) Misconceptions and “tricks” to avoid: Parentheses notation often gets confused with multiplication. That can still happen here, although it is not nearly as bad since the elements of the domain aren’t numbers but rather points in the plane.
If you have any questions, comments, and/or resources, please share them. Thanks for your help.
One of the biggest challenges in a high school geometry course is to motivate the formalism. Why do we need precise definitions? Why must we prove things that seem obvious? Even things that don’t seem obvious, like the Pythagorean Theorem, why not just accept them as facts because we’ve been “taught” that they’re true? Why are we often asked to prove things that seem irrelevant? One approach is to give up on the formalism all together. Have students measure the angles of a bunch of triangles with a protractor and “observe” that they all seem to add up to 180 degrees. Another approach is to just follow Euclid (mistakes and all) and painstakingly work ones way up carefully until eventually something interesting is proven. The common core seeks to strike a balance by starting with a few assumptions and definitions that seem reasonable, but which also allow one to quickly prove key geometric facts such as those needed to define the concepts of slope and coordinates.
Even with that compromise, though, there still remains the issue of motivation. In this course I’ve been trying my best to build up some motivation for the formalism instead of starting off on Day 1 with a bunch of definitions and axioms. In chapter one we hopefully came to realize that formalism would be necessary to abstract to study something like higher dimensions where we cannot rely on intuition and that even in lower dimensions patterns don’t always generalize as we might expect them to. In chapter two we got a glimpse of a situation where the Pythagorean Theorem wouldn’t hold and in chapter three a glimpse of a situation in which a triangle’s angles didn’t sum to 180 degrees. These explorations, even if quite brief, will hopefully challenge us to be explicit about our assumptions. Finally we saw when looking at translations in chapter three that some understanding of parallelograms is going to be essential.
So that brings us to this week where we formally define the notion of rotation and reflection, and fill in the necessary details from the previous chapter concerning translation and parallelism. I expect to make great use of Geogebra this week, and perhaps a final act of motivation for formalism can be thinking about how does the computer determine precisely where to put the image of a point under these transformations.
(A) Key Ideas: Students should understand the formal definitions of rotation, reflection, and translation. Parentheses notation, however, can wait until the next chapter when there is more motivation for it. More than just knowing the definitions of these rigid transformations, though, students should also understand how those formal definitions allow us to prove statements.
(B) Related standards from the common core:
(C) Connections with what students have done: Students have often worked with reflecting over an axis using coordinates, but have not thought about how one would formally define reflection over an arbitrary line. In general this week is about providing some formalism to many ideas they have already explored intuitively.
(D) Connections with what students will do: The formal definitions of these rigid transformations will allow us to define congruence and explore symmetry in the next chapter. This is then in turn key to defining similarity (chapter 8) and congruent triangles (chapter 9). The definition of translation and parallelograms is central to our system of coordinates (chapter 18).
(E) Texts and videos to which students and parents could refer to help understand key ideas: I have yet to see a common core geometry text. So for now, the best reference I’ve found is courtesy of Hung-Hsi Wu, but designed for teachers of geometry. His Teaching Geometry in Grade 8 and High School According to the Common Core Standards can serve as a reference. In this week the focus is on defining the rigid motions (pages 96-110), perpendicular bisectors (pages 118-119) and parallelism (pages 127-130). In Wu’s notes the last two sections come after defining congruence, but the proofs depend only on our definitions and assumptions concerning rigid motions.
(F) Questions and tasks that can generate discussion to explore key ideas in class: Much of the discussion driven intuitive development has already been done in past weeks. One idea I have for the move to formalism is to ask students to try to consider (1) what information we need to provide to Geogebra in order to define a rotation or reflection and (2) how does Geogebra use that information to determine where a point should go.
(G) Questions and tasks that can be used to help assess student understanding of key ideas: One task I like is to give students a triangle with vertices labelled and ask them to find the image of given points under given transformations. NO COORDINATES ARE USED. Even though the placement of image points will need be somewhat imprecise, where students place them can reveal misunderstandings and whether students are actually using definitions. For example, students often place the reflection of a point C across the line AB on the line AC even if the angle CAB is not right. Students often neglect to realize that rotating point C in an angle CAB (oriented appropriately) around point A should place it on the ray AB.
In terms of proofs, I like a task where I ask students to take a point A, reflect it over a line PQ (not passing through A), to obtain an image B. They are then asked to translate that point B by the vector PQ to obtain an image C. Finally the image of C by reflection across the line PQ gives a point D. Students are then asked to prove that ABCD is a rectangle. A proper proof requires knowledge of the precise definitions of rectangle, reflection, and translation. Attempts to prove it without those definitions can bring to light some student misunderstandings.
(H) Interesting and challenging ways to extend and apply the key ideas: Our definition of reflection makes use of the concept of perpendicular bisector which we are then able to show is the set of all points equidistant from two points. A circle is the set of all points equidistant from a single point. It can be interesting to explore this notion of equidistance further. What would happen if we wanted a point (or points?) equidistant from three points or four points. How would any of these figures change if we used taxicab geometry?
(I) Standard textbook problems related to the key ideas: As I noted above I have yet to see a common core geometry text, so I know of no standard textbook problems dealing with these precise definitions.
(J) Misconceptions and “tricks” to avoid: In my experience the main thing to watch out for is students not realizing a need for definitions and thinking it is okay to just rely on an intuitive sense of what is meant by these transformations.
If you have any questions, comments, and/or resources, please share them. I would especially love reference resources or suggestions for how to help lead students from intuition to formalism. Thanks for your help.
Distance is probably the most important concept in geometry, but direction is not far behind. The two concepts complement each other wonderfully. Given two distinct points, A and B, we can think about the distance between the two points as well as the direction which is directed from A to B (which always yields a second opposite from B to A). The direction from A to any point on the ray from A to B is considered to be the same direction as A to B and we assume that translation preserves both distance and direction. Thus in one-dimension we had only two directions: the direction from 0 to 1 defined the positive direction and the direction from 1 to 0 defined its opposite, the negative direction. To extend to higher dimensions we imagine a new point that is not on our number line and thus obtain a new direction. We will still want to be able to use any two points to define a translation in such a way that if the same translation takes A to C and B to D, then the distance and direction from C to D is the same as the distance and direction from A to B and the distance and direction from A to C is the same as the distance and direction from B to D. Central to this discussion will be the notion of parallel lines. This idea of translation now forces us to consider an entire plane of points…our two-dimensional space.
One way of thinking about direction is in this “vector” sense. Two points determine a distance and a direction and define a translation by that distance in that direction. Now let us consider another way that direction will show up throughout high school mathematics. Consider a circle in the plane centered at one point A and passing through another point B. Whereas the ray from A to B considers all the points in the plane that are in the same direction as A to B, but that are at any possible distance. The circle considers all the points the in the plane that are at the same distance from A as B, but that are in any possible direction. So choose a point C on this circle. This defines a direction from A to C, but we can also think about that in terms of a direction relative to the direction from A to B. We do that by defining an angle BAC which is defined as the region between the ray A to B and the ray A to C. We can “measure” that angle in several ways that connect the angle with how much of the circle it intersects.
The notion of direction will not only be key in thinking about translation, rotation, reflection, parallel lines, and angle measurements. It will also be key when we think about the slope of a line. We are thinking of a line already as being connected with a certain direction (and its opposite). So if we establish one horizontal line and positive direction on that line, we can think of a line’s direction as being determined by its “angle of elevation” ie one of the smaller angles formed with the horizontal. Once we establish the notion of similar triangles we can say that this in turn defines a fixed ratio, the slope the line. In a later course on trigonometry we give names to the functions (tangent and arctangent) defined by this correspondence.
(A) Key Ideas: Students should understand the definition of angle, degrees, parallel lines, and perpendicular lines. (Note that we define parallel lines so that a line is also parallel to itself.) This week should also build up the intuitive arguments (which will turn into formal proofs next week) for many of the consequences of parallel lines. In particular it should make sense that opposite angles have the same measure, a transversal intersects parallel lines in angles with the same measure, a triangle’s interior angles sum to 180 degrees, and a polygon’s exterior angles sum to 360 degrees. It is nice to see proofs and one of the themes throughout this course is why we do we want to prove things. That being said, the intuitive arguments are more important because they will be more likely to be retained and applied. Students should have an understanding of translation this week that is starting to build more formally on the intuition. That is they should understand why the notion of parallelograms will be key for formalizing our intuitive idea of a translation moving each point. Again hopefully the desire to extend our thinking to higher dimensions can motivate this move to formalism.
(B) Related standards from the common core:
(C) Connections with what students have done: Students have done much of this work intuitively but have not begun to think about them more deeply in a way that can formalize. They’ve often done translations in terms of coordinates, but not without coordinates.
(D) Connections with what students will do: The thinking this week about direction will set us up to formally define rotations and reflections next (chapter 4) and later dilations (chapter 8). It will also be critical as think about lines and slopes (chapter 18). We are also setting up the idea more generally of a vector (chapter 23) and laying the groundwork for a better understanding of the tangent function and its inverse (10th grade) and complex numbers (Precalculus).
(E) Texts and videos to which students and parents could refer to help understand key ideas: In volume 1 of his geometry text (unit 3, pages 48-68), James Tanton uses a “pencil trick” to both explore intuitively why a triangles interior angles should sum to 180 degrees and why the trick is inherently making some hidden assumptions about parallelism. He then returns to the notion of parallelism later (unit 8, pages 156-190). Tanton notes that he is taking a different approach from Euclid, but it is also a somewhat different approach from common core.
I am also hoping to write an essay and video to be used as a reference for these ideas. 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: I like the debate starter: Is a line parallel to itself? This can lead to the idea that WE make choices when we formally define terms, but those choices are made to try to capture some intuition. In particular it can lead to a discussion of what is the important idea we want to capture with paralellism.
Tanton’s “pencil trick” and its hidden assumptions should lead to a nice discussion. I would love to get any other ideas you may have.
(G) Questions and tasks that can be used to help assess student understanding of key ideas: I would consider asking if a student can explain in his or her own words what the problem with the “pencil trick” is. I am going to try to develop some new tasks as well and any ideas you have would be appreciated.
(H) Interesting and challenging ways to extend and apply the key ideas: In the last chapter when thinking about distance we thought about a geometry where the Pythagorean theorem didn’t hold. This week in thinking about direction, it is nice to consider spherical geometry where our assumptions about parallelism doesn’t hold (and thus where the interior of a triangle doesn’t sum to 180 degrees). Tanton does this naturally as part of the “pencil trick” described above.
(I) Standard textbook problems related to the key ideas: There are many problems concerning finding missing angles in a diagram involving parallel lines, transversal, and triangles. Students should be able to answer any such problem.
(J) Misconceptions and “tricks” to avoid: I don’t know of any.
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.
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:
(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:
There are several debate starters focusing on fraction and unit at Math Arguments 180. My favorite (from day 82) comes from this post by Rob McDuff. I would love to get any other ideas you may have.
(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)
The rubric seems to expect a student to use the table to determine the average weekday consumption rate of the various size popcorn containers. It then expects one to use that and the given information to extrapolate the number of containers of each that would be consumed on the weekend (Fri-Sat). Then one would use a given conversion rate to convert the consumed popcorn to an amount of popcorn seed. One would also scale the 4 given weekdays usage of popcorn seed to get how much would be used over the 5 weekdays (ie account for Sunday) and finally throw in some extra popcorn seed to bring us up from our dangerously low level of seed into the desired comfortable range.
This sample solution (provided in the rubric) assumes that on the weekend no popcorn is popped unless it goes out in a box, but the data indicates that at least during the week quite a bit of extra popcorn seed is used. How would graders handle quite different solutions with alternate (probably more reasonable) assumptions? Stay with me past the jump for my solution when I did the problem and some other alternates.
I first figured that about 320 cups of seed were used over 4 weekdays. We are told that weekends (Fri-Sat) are busier. For example we sell twice as many small and medium boxes. So my initial model assumes we can treat weekends as 2 weekdays. This leads to 9 weekdays in our week. So scaling by 9/4 we would need 720 cups of seed plus another 80 cups since we’re currently too low. But we are also told that we sell 200 to 300 large boxes per day on the weekend compared with the 180 we sold over 4 weekdays. Our model would have predicted the same 180 over the 2 weekend days. In other words, instead of selling twice as many large boxes as we did for small and medium, we expect to sell more than 5 times as many. That’s quite the discrepancy and I would wonder what the reason for it was. In any event, there are many different reasonable assumptions that could be made at this point leading to vastly different estimates. I chose to just try to account for the seed needed for the extra 220 to 420 large boxes sold over the weekend. Figuring 5 cups of seed makes enough popcorn for 6 large boxes, I settle on a convenient estimate of 330 extra large boxes and hence 55 x 5 or 275 extra cups of seed. So I settle on a total estimate of 1075 cups of seed and essentially 50 more cups over Friday and Saturday than that estimated by the rubric solution. Would I get full credit for this solution? It doesn’t seem so. The rubric says students should recognize how many small and medium boxes are sold per day and that they MUST first estimate how many cups of popcorn are sold on Friday on Saturday. I did neither. The rubric specifically awards 1 point for “an adequate estimation strategy for two sizes of boxes for both days.” I neglected to do so. In fact I completely ignored all the data in the table concerning small and medium boxes.
The rubric also awards 1 point for accurate use of popcorn seed to popcorn. I happened to do that, but what if somebody made an assumption that didn’t require them to do so. For example, after discovering that our model was way off on the number of large boxes I might have chosen another way to reconcile the discrepancy. I might argue that the since we sell twice as many small and medium boxes, that it is reasonable to assume we should also be selling twice as many large boxes. So maybe the inventory indicates an atypically slow week and shouldn’t be used as a reliable indicator of typical daily boxes sold, but only as an indicator of boxes sold to seed actually used (which is more relevant than the given ratio of seed used to popped seed). How much credit would that strategy get if well argued?
Or what if a student determined that the ratio of large boxes to medium boxes to small boxes sold was roughly 3:1:2 (slightly overestimating small boxes). The ratio of popped corn between the boxes is roughly 5:4:3 (again slightly overestimating small boxes) and thus the ratio of popped corn used is roughly 15:4:6 or 3:2 in terms of cups used in large boxes to smaller boxes. So if we scale the number of large boxes sold by 5 to 6 and the number of smaller boxes sold by 2 maybe it’s reasonable to assume a weighted average of about 4 times as much popcorn needed on a Friday or Saturday compared to the weekdays. Thus we can consider a week as having the equivalent of 13 weekdays and scale our 320 cups of seed by 13/4 for a total of 1040 cups of seed needed next week plus an extra 80 cups or so because we’re currently low for a total of about 1120. This is probably the most reasonable estimate so far and it did not make use of the 1/3 cup of seeds to 8 cups of popped corn ratio at all. What would this solution score?
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:
(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)
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.