Forum Replies Created
-
Posts
-
In my thinking, if a student understands how to find the point of intersection with a calculator, the RRT would be inefficient for that student. The PARCC calculator policy is here: http://www.parcconline.org/assessments/administration
I’ve been on a safari looking for that answer, too! I’m thinking rationalizing goes with N.CN.8(+) which one wouldn’t know is in the Algebra 2 standards unless they read Appendix A, pages 8 and 38. I suspect the writers felt it was best not to teach “rationalizing for the sake of rationalizing.” Some of us think radicals in the denominator look cute the way they are, so why…? On the other hand, what are students to do when their answers don’t look like the ones at the back of the book? I am hoping Dr. McCallum or someone else can answer that.
The importance of N.CN.8 was not obvious for me. I now think it is important because students need to connect polynomial identities (specifically difference of squares and perfect square trinomials) from real number quadratics to complex quadratics. Along with that connection comes the idea of “conjugates” that eliminate middle terms when they are multiplied. Of course, that means N.CN.8 connects with rationalizing. However I’m thinking that rationalizing is only done to effect division with two complex numbers. How else can we get an a+bi number from dividing two a+bi numbers? Quotients of complex numbers is only in N.CN.3 which is in the “fourth course” of high school math. So for N.CN.8 I think we only need to connect polynomial identities. We won’t need to rationalize with conjugates since we won’t be dividing complex numbers.
While I was out hunting, I realized that (I think) A.SSE.2 is where we get perfect square trinomials (along with difference of squares).
Please correct me if I’m wrong, but I think I just found the answer to my question on page 39 of Appendix A where the cluster heading “Represent and solve equations and inequalities” is directly linked to specific combinations including absolute value for a Traditional Algebra 2, Unit 1. My confusion stems from A.REI.11, “Explain why the x-coordinates of the points where the graphs of
the equations… I had interpreted that to mean equations only. Indeed, on page 19 of Appendix A, for the traditional Algebra I unit 2, the same cluster and standard seems to limit absolute value to equations.I just found the explanation for “simple” rational and radical functions. It’s a footnote in Appendix A, page 36: http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf
One thing have found helpful is to occasionally take placement tests at colleges (I took two in April). I consistently see traditional cases of arithmetic with rational expressions (including the need to factor), so I continue to teach that in Algebra 2 and above until I hear otherwise. The ACT practice problems include an addition problem where the student must get a common denominator for x and (x + 5). As you pointed out, rational expressions present a challenge for making them seem relevant instead of a “means to an end.” I just finished writing a lesson sequence that bounces off gas laws and capacitors in series. While my disaffected seniors may not have dreams of becoming engineers, it is certainly not a huge stretch for them to consider working in a technical environment, and having a clue how those formulas work has its advantages. I include brief videos of things blowing up when maximums are exceeded and am confident I will at least get their attention. In another lesson, they modify a 8-line calculator program that calculates D=rt to t = D/r to see one is a linear function and the other is rational. Please email me if you want the series: lane.walker@fhsdschools.org But, yes, I am anxious to see if Dr. McCallum has any words of wisdom on this.
Interestingly enough, the f(n) concern turned out to be a paper tiger. The real problem is that lots of people do not realize a sequence is a function. The terminology input-output is not universally understood, and those worried about confusion students might have appear to have been confused themselves. Can I thank you, once again Dr. McCallum, for the beautiful job you have done with these standards?
I was really interested in the link you shared but it doesn’t open to the article. Can you re-post? I’m working on Algebra I curriculum for our district and have really had to look closely at what our students are doing in 8th grade to capitalize on (review) immediately and move on promptly, scaffolding weak spots along the way. I nearly croaked at the idea of teaching 14-year-olds about rational exponents when they struggle so badly with integer exponents, but then I realized there was a shift in emphasis to usefulness with exponential functions instead of 3xy(2xyz)^3…. I am confident, now, they can do it. There are other topics like piece-wise functions that can be integrated along with a review of linear functions, etc. So far I have 20 extra days each semester to work in more project-based activities.
The answer to my questions is no, but I skim too fast. I see that instruction in exponents in the Common Core took a huge shift toward relevance instead of (xy)^2z^3(x^2)y. That makes perfect sense!
I think I just answered my own question: A.CED.a.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
I was just watching your Khan video for multiplying and dividing negative numbers. My remedial Algebra students gain solid traction with multiplying negative numbers like this:
-3 lost 3 friends
2(-3) lost 3 friends twice
-2(-3) the opposite of losing 3 friends twice.The added benefit is that “opposite” connects with -x.
The emotional connections help with retention.
Great question. I asked the same thing! Here’s what Dr. McCallum said:
I would write (x-h)^2 + k Using h and k is pretty much textbook standard for shifts on conics (although students need to know variables are variables). Since h is the horizontal shift from the origin, the line of symmetry runs through x=h
I really like your idea of slicing food like this. I would add a clear statement of the objective at the beginning, something like, “Seventh grade standard G.3 requires students to describe the two-dimensional figures that result from slicing three dimensional figures.” “Vertex” is in the high school standards but I don’t see it in the lower grades, so maybe it would be better to just show we can slice three different ways without using the word “vertex.” You might want to include quick examples like a stack of 3×5 cards. I love the way this standard can lead into calculus and your example of sliced objects does that well.
My examples were ambiguous. I explain here: https://dl.dropboxusercontent.com/u/7405693/public%20responses/crosscancel.avi
