these posts are hopefully less traditional ways of introducing or exploring calculus

## LIMITS

Often limits of nice or piecewise or rational functions functions are given a lot of time, but very little time is given to very uncommonly weird-o functions. Explore the limit as x goes to 0 of y=sin(1/x). (here)

Don't lose sight of the conceptual idea undergirding limits: they help you get at the very big and the very tiny. Have a little fun playing around with the meaning of these things. (here)

Do you want a simple check to see if your students understand the idea of a limit? Here's a single powerful question that reveals misconceptions about what information is "embedded" in a limit and what information is not. (here)

Do you want some (possibly) new ideas on how to teach limits? Here are some thoughts as well as a Pictionary Graph Game to get students discussing how limits are tied to continuity. (here)

How well do your students understand the Intermediate Value Theorem? Give them these "real-life functions" as a formative assessment. (here)

Don't lose sight of the conceptual idea undergirding limits: they help you get at the very big and the very tiny. Have a little fun playing around with the meaning of these things. (here)

Do you want a simple check to see if your students understand the idea of a limit? Here's a single powerful question that reveals misconceptions about what information is "embedded" in a limit and what information is not. (here)

Do you want some (possibly) new ideas on how to teach limits? Here are some thoughts as well as a Pictionary Graph Game to get students discussing how limits are tied to continuity. (here)

How well do your students understand the Intermediate Value Theorem? Give them these "real-life functions" as a formative assessment. (here)

## derivatives

An graphical understanding of derivatives as slopes of tangent lines can help students understand why the derivative of 5x^9 is 5*9x^8. In other words, why does the 5 "remain" there. This can be shown algebraically, but there is also a simple graphical reason too. (here)

Use a video of a liquid being poured into a conical cup (martini glass) and Logger Pro to understand the conceptual undergirdings of related rates. (here)

Here is a warm up, an introductory conceptual activity, and a numerical activity that helps students understand related rates concretely -- using balloons and a rocket ship in geogebra. (here)

Introduce the idea of inflection points and concavity organically by having students simulate a horrible infectious disease. They model a logistic curve, and from that, see the real world meaning of the shape of a curve, concavity, and inflection point. (here)

Before diving right into the algebra involved in optimization, here is a set of activities that lets students "tinker" with optimization questions without having them get all math-y with it. (here)

By tackling the simple question "is it possible for things in cans give you more stuff in them if they were created with different dimensions (but use the same amount of materials)?" students analyze multiple types of cans and present the information they have discovered -- both algebraically and graphically. In other words, they are solving an optimization problem that arises in the real world. (here)

Implicit differentiation is often taught with only equations, and without a visual understanding of the derivative. Delve deeper with this activity which gets kids drawing connections between a relation, the derivative of the relation, and the graph of the derivative. (here)

How are f, f', and f'' related? Kids need to know the ins and outs of this question. (here)

Students sometimes have a hard time remembering that they need to use chain rule. Here's a way to help them discover the chain rule using product and quotient rules. (here)

Do you struggle with finding the balance between proving each derivative rule and just giving it to your students? How about using a matching game for trig functions (here)? Or a Desmos lab for exponential and log functions (here)?

Use a video of a liquid being poured into a conical cup (martini glass) and Logger Pro to understand the conceptual undergirdings of related rates. (here)

Here is a warm up, an introductory conceptual activity, and a numerical activity that helps students understand related rates concretely -- using balloons and a rocket ship in geogebra. (here)

Introduce the idea of inflection points and concavity organically by having students simulate a horrible infectious disease. They model a logistic curve, and from that, see the real world meaning of the shape of a curve, concavity, and inflection point. (here)

Before diving right into the algebra involved in optimization, here is a set of activities that lets students "tinker" with optimization questions without having them get all math-y with it. (here)

By tackling the simple question "is it possible for things in cans give you more stuff in them if they were created with different dimensions (but use the same amount of materials)?" students analyze multiple types of cans and present the information they have discovered -- both algebraically and graphically. In other words, they are solving an optimization problem that arises in the real world. (here)

Implicit differentiation is often taught with only equations, and without a visual understanding of the derivative. Delve deeper with this activity which gets kids drawing connections between a relation, the derivative of the relation, and the graph of the derivative. (here)

How are f, f', and f'' related? Kids need to know the ins and outs of this question. (here)

Students sometimes have a hard time remembering that they need to use chain rule. Here's a way to help them discover the chain rule using product and quotient rules. (here)

Do you struggle with finding the balance between proving each derivative rule and just giving it to your students? How about using a matching game for trig functions (here)? Or a Desmos lab for exponential and log functions (here)?

## integrals

Students will learn about wealth inequality, the Gini coefficient, and work with real world data. This activity brings up the calculus idea of the area between two curves and using a Riemann Sum to approximate this area. (here)

Instead of asking the forwards question of "find the integral of x^2 from x=-1 to x=3" take a moment to ask the backwards question "draw a function where the integral from x=-1 to x=3 has value 8." This gets at if students have a solid understanding of the integral as signed area. (here)

Do you love geogebra applets that create 2D representations of the 3D shapes we study in calculus? Would you like to take it one step further? Try making these solids (volumes with known cross sections and disk method) for your classroom. (here)

Instead of asking the forwards question of "find the integral of x^2 from x=-1 to x=3" take a moment to ask the backwards question "draw a function where the integral from x=-1 to x=3 has value 8." This gets at if students have a solid understanding of the integral as signed area. (here)

Do you love geogebra applets that create 2D representations of the 3D shapes we study in calculus? Would you like to take it one step further? Try making these solids (volumes with known cross sections and disk method) for your classroom. (here)

## differential eqns, series, and other stuff

Get kids up and moving with this quick slope field activity. (here)

There are several theorems we need to teach in calculus. Here's a way to introduce them that may be less intimidating for kids. (here)

Motion is an idea central to calculus. Test your students' knowledge with this matching activity from Lin McMullin. (here) {Note: solutions may need revision.}

There are several theorems we need to teach in calculus. Here's a way to introduce them that may be less intimidating for kids. (here)

Motion is an idea central to calculus. Test your students' knowledge with this matching activity from Lin McMullin. (here) {Note: solutions may need revision.}