16 votes

Why do we teach linear algebra in precalculus classes?

Vector algebra is a standard 3rd-semester calculus topic (e.g., see OpenStax Calculus 3, Ch. 2-3). This includes calculations of the dot product, cross product, and related values. Standard ...
Daniel R. Collins's user avatar
9 votes

Why do we teach linear algebra in precalculus classes?

The College Board made curriculum decisions for their new AP Precalculus course that align with sentiments you express. The course is divided into four units, where unit four is titled Functions ...
user52817's user avatar
  • 10.5k
9 votes

Intuition for order of operations in compound transformations

For teaching this concept, I always come back to tables of values. In Calculus Reform, there was the "Rule of Three" which emphasized that learning about functions should involve tables, ...
user52817's user avatar
  • 10.5k
8 votes

Sources on inequity in precalculus sequence

Here's a hair-raising article I sometimes use as a touchstone: Kenschaft, Patricia Clark. "Racial equity requires teaching elementary school teachers more mathematics." Notices of the AMS 52....
Daniel R. Collins's user avatar
8 votes

Why do we teach linear algebra in precalculus classes?

Because derivatives are linear operators.
Leonardo's user avatar
5 votes

Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

This is kind of a joke answer, but in my favorite math story ever we have the following exchange: Eric pondered a moment. "But... but if it's that simple, why don't my textbooks talk about it?&...
Steven Gubkin's user avatar
4 votes

Chinese and Japanese most important high school textbooks

The only full-length English translations of Chinese or Japanese secondary math textbooks that I'm aware of are the ones listed here, which were produced by the University of Chicago School ...
Justin Hancock's user avatar
4 votes

Intuition for order of operations in compound transformations

One way to build intuition around this is to think about the effects on the intercepts. Vertical Shifts/Stretches: Effect on $y$-Intercept Suppose you want to graph $y=2\sqrt{x}-6$ using ...
Justin Skycak's user avatar
3 votes
Accepted

Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

I think it is just that the related concepts encompassed by little-$o$ and big-$O$ notation are more important than "polynomial asymptotes" and do find many applications. We do teach Landau ...
user52817's user avatar
  • 10.5k
3 votes

Differentiation in integer solutions

I think the scope of your question is too narrow. If you ask more broadly about the use of calculus in number theory (not just differentiation for solving Diophantine equations) then books on analytic ...
KCd's user avatar
  • 3,456
3 votes

f(x+h) in the difference quotient

You start with $f(x)=x^2+x+1$. How about having students determine $f(y)$ where $y=z +1$? First, substitute $y$ for $x$ to obtain $f(y)=y^2+ y +1$. Then, substitute $z+1$ for $y$ to obtain $f(z+1)=(z+...
Dan Christensen's user avatar
3 votes

Intuition for order of operations in compound transformations

I'd say the root cause of your problem is that you're forgetting to include the square root in the sequence of operations transforming $x$ into $\sqrt{2x-6}$: $$ x \overset{\times 2}{\ \to\ } 2x \...
Ilmari Karonen's user avatar
2 votes

Intuition for order of operations in compound transformations

I think that my previous answer addresses this. If we name the intermediate functions, and make the relationships between these functions explicit, things become a bit easier to understand. In your ...
Steven Gubkin's user avatar
2 votes

Advice and Remedial Algebra Resources for Students Committed to Calculus

I know this doesn't directly answer your question which is about helping one student. But I would be wanting to see my institution implement sections of calculus with support. This blog addresses that ...
Sue VanHattum's user avatar
  • 20.2k
1 vote

Intuition for order of operations in compound transformations

Is there a good intuitive explanation for how to think about the order of steps when constructing these compound transformations? Here's a stab at what it would mean to obey the order of operations. ...
Nick C's user avatar
  • 9,406
1 vote

Intuition for order of operations in compound transformations

You can perform the transformations in any order you want as long as you respect this rule: Always transform completely inside, affecting $x$ directly, or completely outside, affecting $y$ directly. ...
Stef's user avatar
  • 546

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