Introduction to Curve Sketching, Part 2

While students were working on this antiderivative from a Desmos Activity called Sketchy Derivatives, I heard several students ask how they know where to place the y-intercept.

We didn’t have time to answer that question during the first lesson of the unit, but we started with it during the second lesson.

Should the cubic start decreasing or increasing? How do you know?

[Yes, I do know that I can anonymize the names. However, by the time I thought of that, these two responses were no longer adjacent. It would be nice to be able to drag the responses to different locations in case you want to compare/contrast several specific responses at the same time. ☺]

Then we looked at these.

They all have the same basic shape. Is one more right than the other?

Students began to think about all of the curves that have y’=2x, and so while they know something about the importance of the constant, one won’t be more right than the other until we learn about area under the curve.

I had the Second Derivative Grapher from Calculus Nspired queued during the previous class, but I decided it would be better for students to “do” instead of “discuss” for the last few minutes of class. We looked at it next to make the relationship between the second derivative and concavity of the original function more clear.

Students put all of this information together to begin to analyze a function given the graph of its derivative. The results of our formative assessments seem to indicate that students have a better understanding of the relationship between f-f’-f” than they have in the past. I’ll know for sure later today, though, after their summative assessment. And so the journey continues …

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Seeing the Definition of Derivative

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

1. relate the concept of a limit to all aspects of calculus
2. Students can connect concepts to their visual representation with and without technology.

 

How do you introduce the definition of a derivative?

We start with the visual of a tangent line at a point and a secant line containing the point.

We don’t need calculus to determine the slope of the secant line. We do need calculus to determine the slope of the tangent line.

How might we use the slope of the secant line to determine the slope of the tangent line?

 

Seeing the Derivative

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

1. Students can connect concepts to their visual representation with and without technology.

 

How do you introduce the concept of a derivative?

We start with visuals of the derivative (from a Getting Started with Calculus activity called Derivative Trace).

Tangent Line Demonstration from Calculus Nspired has some similar ideas.

We then look at the Derivative Grapher to connect the slope of the tangent line to the graph of the derivative, changing the original function as requested by the students.

We begin to develop some common language around derivatives before we formalize what is a derivative, before we formalize the definition of a derivative.

 

Infestation to Extermination

We used a problem from the Calculus Nspired activity Infestation to Extermination recently during our unit on differential equations:

The rate of increase of bugs is proportional to the number of bugs in a certain area. When t=0, there are 2 bugs and they are increasing at a rate of 3 bugs/day.

What does this mean?

I set the mode to individual and watched as students worked.

Many recognized that the rate of change changes.

Several used the initial condition to write a statement about the rate of change.

Eventually, we went back to the given information to decipher what it was saying.

And then we (anti)derived the model for exponential growth, which of course students recognized using in a previous math course.

So what are the constants for this particular model?

I sent a poll to collect their model honestly having no idea that the bell was going to ring in less than two minutes. A few students correctly answered before the end of class.

Productive struggle isn’t fast.

I should have paid better attention to the time … I really had no idea it had taken us as long as it did. But students were engaged in “grappling with mathematical ideas and relationships” the entire time. That’s got to be better for their learning than them watching me tell them how to work the problem.

What opportunities are you giving your students to struggle productively? Even if you don’t “cover” as much as you think you should?

And so the #AskDontTell journey continues …

 

The Fundamental Theorem of Calculus

How do you provide an opportunity for your students to figure out the relationship between differentiation and antidifferentiation?

We have used the Calculus Nspired activities The First Fundamental Theorem of Calculus and The Second Fundamental Theorem of Calculus for several years now to improve our understanding of the relationship between a function and its accumulation function. I actually do print the student handouts for these activities and give students time during class to make sense of the relationship between differentiation and antidifferentiation.

It was time for our whole class discussion.

We defined the accumulation function using a definite integral. What do we know?

Students had figured out earlier that the definite integral of f(x) from a to a would be 0 and concluded that F(a)=0.

Students recognized that the value of the definite integral of f(x) from a to d would be F(d) and that the value of the definite integral of f(x) from a to c would be F(c).

Suppose we want to calculate the definite integral of f(x) from c to d. They could tell area-wise that was equivalent to finding the definite integral of f(x) from a to d and subtracting the definite integral of f(x) from a to c, which of course gives us F(d)-F(c).

And what does F(x) have to do with f(x)?

They could tell from the exploration that F(x) is the antiderivative of f(x).

Really? You mean we don’t have to do the limit-sum-infinite-number-of-rectangles every time? Really. You’ve earned the Fundamental Theorem of Calculus.

We talked for a little while about average value using the Calculus Nspired activity MVT for Integrals, and then checked in on their understanding.

As we moved into the second part of the Fundamental Theorem of Calculus, I posed a question to see how they would answer. (Remember that at this point, they’ve been using the FTOC for about 20 minutes.)

I was excited about a few students getting it right. Without discussing the correct responses with the whole group (I showed their answers but had Show Correct Answer deselected, I sent another question, which unearthed their misconception and revealed my initially bad question.

The students who got the answer correct in the first question had gotten it correct the wrong way, but their mistake wasn’t revealed because sin(-π)=0.

By now we were past the bell, and so we started over the next lesson with the second part of the Fundamental Theorem of Calculus.

It’s always exciting to find both the right questions to ask (the ones that reveal student misconceptions) and the wrong questions to ask (the ones that hide student misconceptions) so that I can continue asking the right ones and discontinue asking the wrong ones. In this lesson, I found at least one of each. And so the journey finding and asking the right questions continues …

 

Properties of Definite Integrals

I love the day in calculus that students figure out properties of a definite integral using the Calculus Nspired activity Definite Integral. When I first started using Math Nspired activities, I often copied the student handout. I’ve used them long enough now that there are only a few that I still copy. Instead, I will show some of the questions from the student handout at the front of the room as a guide for student exploration, or I will just use the questions myself to guide our whole class discussion.

We practice look for and express regularity in repeated reasoning. What changes and you move a and b? What stays the same?

Students explore individually first and write what they notice. While they share with a partner, I monitor their conversation, select what they notice that needs to be brought to the attention of the whole class, and sequence their findings in our whole class discussion.

The functions in the activity are carefully chosen – one is odd, and one is even – so that students can begin to generalize what will be true for definite integrals of odd and even functions as well as all functions.

 

I used to tell my students properties of definite integrals.

Now they tell me.

 

 

 

 

(At least those who notice the word NOT.)

And so the journey continues, providing students opportunities to figure out the mathematics we want them to know without just telling it to them.