preparing for calculus?

[daijobu]

If you have a few weeks to prepare a student for calculus, what is your list of key skills they should have mastered?  

 

[Alicia64]

Great question. Following.

[regentrude]

Solid mastery of algebra, manipulation of algebraic expressions. Understanding what a function is; ability to sketch graph of function. 
Contrary to popular belief "precalculus" has nothing to do with actual preparation for calculus.

[Not_a_Number]

6 hours ago, daijobu said:

If you have a few weeks to prepare a student for calculus, what is your list of key skills they should have mastered?  

Oooh, I had to do this with my sister. 

We graphed TONS of functions by hand for months -- I gave her a vast variety, where there wasn't any shortcut other than plugging some points and making observations. We also surprisingly had to remediate her understanding of what speed is -- she'd seen the formulas, but she actually didn't have any kind of intuition for what they meant. We did LOTS of practice problems with Laggy the Snail -- graphing his location on a long straight road and calculating his average speed by hand. 

She got an A in her UChicago calculus class, and she was not a mathy student before we were doing this, so it's safe to say this worked well. 

We actually never quite had the chance to fully remediate her algebra. To me, this was less essential than what we did, although if I had more time, I'd have done that as well. But I think algebra can take a seriously long time to remediate with a kid who isn't solid already. 

[daijobu]

17 hours ago, regentrude said:

Solid mastery of algebra, manipulation of algebraic expressions. Understanding what a function is; ability to sketch graph of function. 
Contrary to popular belief "precalculus" has nothing to do with actual preparation for calculus.

Yes, this is so true, both of me and the general population.  I really dislike things that are labeled in terms of what they are not instead of what they are.  Thank you.  

[daijobu]

16 hours ago, Not_a_Number said:

Oooh, I had to do this with my sister. 

We graphed TONS of functions by hand for months -- I gave her a vast variety, where there wasn't any shortcut other than plugging some points and making observations. We also surprisingly had to remediate her understanding of what speed is -- she'd seen the formulas, but she actually didn't have any kind of intuition for what they meant. We did LOTS of practice problems with Laggy the Snail -- graphing his location on a long straight road and calculating his average speed by hand. 

She got an A in her UChicago calculus class, and she was not a mathy student before we were doing this, so it's safe to say this worked well. 

We actually never quite had the chance to fully remediate her algebra. To me, this was less essential than what we did, although if I had more time, I'd have done that as well. But I think algebra can take a seriously long time to remediate with a kid who isn't solid already. 

Laggy the Snail...I love it.  

Did you graph polynomials?  trig curves?  (r, theta) polar graphs?  What else?  

[UHP]

18 hours ago, regentrude said:

Contrary to popular belief "precalculus" has nothing to do with actual preparation for calculus.

What is precalculus? If a high school offers calculus, will it also typically offer "precalculus" or is it language that is more often used at colleges?

[daijobu]

13 minutes ago, UHP said:

What is precalculus? If a high school offers calculus, will it also typically offer "precalculus" or is it language that is more often used at colleges?

I don't think colleges typically offer precalculus, their math begins with calculus and they expect their students to have completed at least that much math in high school.  

I found "precalculus" a little weird as well because my high school also did not offer the class in the 1980s.  Our sequence was:

9th grade: geometry

10th grade:  advanced algebra and trigonometry

11th grade: functions and analytic geometry

12th grade: calculus

[daijobu]

What about point slope form for lines?  My sense is that students lean to slope intercept form, when really the point slope form is what they should use by default.  Am I wrong to emphasize this form?  

[regentrude]

26 minutes ago, UHP said:

What is precalculus? If a high school offers calculus, will it also typically offer "precalculus" or is it language that is more often used at colleges?

Precalculus typically is a semester of trigonometry and a semester of algebra review (in colleges, you may find this part called "College Algebra")

[Not_a_Number]

1 hour ago, daijobu said:

Laggy the Snail...I love it.  

Did you graph polynomials?  trig curves?  (r, theta) polar graphs?  What else?  

Polynomials and trig functions and lots of varied functions like x^2 sin(x) and such. Also, things that aren't functions like x^2 + y^2 = 1 but also harder stuff. 

But I didn't change the coordinate grid. 

[mathnerd]

If I had limited time, I would check for gaps in important skills. I am assuming that your student has spent a couple of years doing Alg1, Alg2 before getting to calculus and is comfortable  in those areas.

- being rock solid in functions helps a lot. I would focus the review effort on functions (all kinds including polynomial func), manipulation of functions, graphing, periodicity, Domains/Range, exponentials, log functions. These concepts pop up often in Calculus.

- next area of review would be Geo/Trig: focus on Analytic Geo, all the basic trig functions and identities, familiarity with graphs of common trig functions

If I had more time, I would focus on review of sequences and series.

Edited August 10, 2021 by mathnerd

[regentrude]

10 minutes ago, daijobu said:

What about point slope form for lines?  My sense is that students lean to slope intercept form, when really the point slope form is what they should use by default.  Am I wrong to emphasize this form?  

Why???
The standard way of expressing a linear function is f(x)=ax+b, as a special case of the standard way of expressing a polynomial of nth order as f(x)=a_sub_n x^n + a_sub_(n-1) x^(n-1)+.....
The student should be able to express a line in either way and effortlessly move between the forms. However, there is no value in students memorizing "slope-intercept form" or "point slope form"; they need to understand what it is they are doing. Then there's no "emphasizing" one form over the other - a student who understands how a linear function works will easily be able to use either, depending on the information available.

[regentrude]

1 minute ago, mathnerd said:

If I had more time, I would focus on review of sequences and series.

If that has been covered in algebra at all. Nowadays, sequences and series are often not covered until calculus 2, since many calc 1 courses no longer introduce the derivative through the epsilon-delta formalism but instead use an intuitive understanding of the concept of limits.

[mathnerd]

3 minutes ago, regentrude said:

If that has been covered in algebra at all. Nowadays, sequences and series are often not covered until calculus 2, since many calc 1 courses no longer introduce the derivative through the epsilon-delta formalism but instead use an intuitive understanding of the concept of limits.

Is "Calculus 2" the equivalent of Calculus BC?

[regentrude]

Just now, mathnerd said:

Is "Calculus 2" the equivalent of Calculus BC?

yes, the portion that's not on AB. High score on AP Calc BC typically gets you college credit for both cals 1 and calc 2

[SoCal_Bear]

18 hours ago, Not_a_Number said:

Oooh, I had to do this with my sister. 

We graphed TONS of functions by hand for months -- I gave her a vast variety, where there wasn't any shortcut other than plugging some points and making observations. We also surprisingly had to remediate her understanding of what speed is -- she'd seen the formulas, but she actually didn't have any kind of intuition for what they meant. We did LOTS of practice problems with Laggy the Snail -- graphing his location on a long straight road and calculating his average speed by hand. 

She got an A in her UChicago calculus class, and she was not a mathy student before we were doing this, so it's safe to say this worked well. 

We actually never quite had the chance to fully remediate her algebra. To me, this was less essential than what we did, although if I had more time, I'd have done that as well. But I think algebra can take a seriously long time to remediate with a kid who isn't solid already. 

This is why I am hesitant to hand over a graphing calculator too early. Trying to decide when to make that available.

[Not_a_Number]

58 minutes ago, calbear said:

This is why I am hesitant to hand over a graphing calculator too early. Trying to decide when to make that available.

It’s a good question. I’d say when kids can both graph well and use graphs and successfully go backwards, too.

[regentrude]

1 hour ago, calbear said:

This is why I am hesitant to hand over a graphing calculator too early. Trying to decide when to make that available.

When the student is planning to take AP calc. Because the test has been constructed so that certain questions cannot be solved without one. (Why, oh why???)

Otherwise, not until they take a college course that specifically requires one. My kids have made it through their physics degrees without ever using a graphing calculator. Btw, many math and physics college classes won't permit one on exams.

Edited August 10, 2021 by regentrude

[NittanyJen]

The unit circle is rather helpful to understand for calculus.

The trig identities can be handy, but those can also be looked up when needed, and then practiced with plenty of problems in context until learned sufficiently.

Basic function graphing. This can be accomplished with or without tech, as long as the student is explaining the major features of what is happening in the graph and what they mean, and can follow up by sketching by hand when needed.

 

 

[Gil]

On 8/10/2021 at 2:06 PM, calbear said:

This is why I am hesitant to hand over a graphing calculator too early. Trying to decide when to make that available.

If you're doing math 100% at home, then never. The Boys made it through Multivariate Calculus (Calculus 3 at university) without using a  calculator. We simply skipped the few questions in each chapter that were designed to be solved with a graphing calculator in the texts.

We got around using a graphing calculator by front-loading the ability to graph fluently.

They can look at most functions and visualize it immediately. They can do a rough sketch of most functions in under 30 seconds and an accurate sketch in under a minute.

[HollyDay]

My math major dd says to prepare for calculus:

solid understanding of  algebra (factoring, FOIL), mastery of fractions, interpretation of decimal approximation and how to convert back and forth from decimal to fraction, strong understanding of how to solve for x and y.

basic trig skills such as unit circle, sin, cos, tan, graphing, phase shifts, how to work with trig functions numerically and graphing including their inverse functions, the differences between degrees and radians and how to convert between the two.

Understanding of rates of change especially how to see these graphically. You will learn more about this in Calculus but it helps to have a familiarity with the concept early.

An understanding of exponential and logarithmic functions.

A basic preview of limits does help as well.

Ability to recognize linear vs non linear equations

Overall though, the importance of a strong algebraic foundation cannot be stressed enough. A lot of problems actually require only a little calculus but a lot of algebra. This is especially true when you get into integration. Sometimes to integrate, you first need to use a lot of algebra to get the equation into a form that can be worked with. The same is true for differentiation at times. Then after using the Calculus skills learned, you often have to use algebra again to get the answer into a form that can be used in application to get the needed answer. 

[Not_a_Number]

1 hour ago, HollyDay said:

My math major dd says to prepare for calculus:

solid understanding of  algebra (factoring, FOIL), mastery of fractions, interpretation of decimal approximation and how to convert back and forth from decimal to fraction, strong understanding of how to solve for x and y.

basic trig skills such as unit circle, sin, cos, tan, graphing, phase shifts, how to work with trig functions numerically and graphing including their inverse functions, the differences between degrees and radians and how to convert between the two.

Understanding of rates of change especially how to see these graphically. You will learn more about this in Calculus but it helps to have a familiarity with the concept early.

An understanding of exponential and logarithmic functions.

A basic preview of limits does help as well.

Ability to recognize linear vs non linear equations

Overall though, the importance of a strong algebraic foundation cannot be stressed enough. A lot of problems actually require only a little calculus but a lot of algebra. This is especially true when you get into integration. Sometimes to integrate, you first need to use a lot of algebra to get the equation into a form that can be worked with. The same is true for differentiation at times. Then after using the Calculus skills learned, you often have to use algebra again to get the answer into a form that can be used in application to get the needed answer. 

The interesting thing about this list is that it's definitely a list of the things you need to do well in a calculus CLASS but is not the list of things you need to understand the IDEAS of calculus. You can absolutely understand calculus ideas without having ever seen a trig function.