Introduction This activity should take around three days of classtime, and is designed to be used at the beginning of the school year. The students this lesson is designed for are nervous and scared about taking calculus. The goals of this lesson are three-fold:

1) To remind students of some prior knowledge from the previous year (e.g. basic functions they should be familiar with like trigonometric functions and lines)

2) To have students understand that the biggest idea of calculus (differentiation) is not actually very challenging to understand on a conceptual level (and for a teacher, is even easier for them to think about without having formalism and symbols to get in the way)

3) To draw an analogy between a function, which takes an input number and expels an output number, and the derivative operator, which takes an input graph and expels an output graph.

This activity uses a lot of pattern recognition as well as reasoning articulation.

Part I: Students Recall Prior Knowledge about Functions

Students are given a packet with a strange scenario involving little cute monsters. Yes, the setup is contrived. But the monsters are so cute it'll work! They are asked to figure out the pattern of various functions. Through this, hopefully they will be recalling things like trigonometric functions, inverse trigonometric functions, exponential functions, and quadratics. The point is just to have them talking about these ideas and using mathematical language to bring up all this stuff that has been dormant for the summer.

[packet in .docx form]

1) To remind students of some prior knowledge from the previous year (e.g. basic functions they should be familiar with like trigonometric functions and lines)

2) To have students understand that the biggest idea of calculus (differentiation) is not actually very challenging to understand on a conceptual level (and for a teacher, is even easier for them to think about without having formalism and symbols to get in the way)

3) To draw an analogy between a function, which takes an input number and expels an output number, and the derivative operator, which takes an input graph and expels an output graph.

This activity uses a lot of pattern recognition as well as reasoning articulation.

Part I: Students Recall Prior Knowledge about Functions

Students are given a packet with a strange scenario involving little cute monsters. Yes, the setup is contrived. But the monsters are so cute it'll work! They are asked to figure out the pattern of various functions. Through this, hopefully they will be recalling things like trigonometric functions, inverse trigonometric functions, exponential functions, and quadratics. The point is just to have them talking about these ideas and using mathematical language to bring up all this stuff that has been dormant for the summer.

[packet in .docx form]

The solutions to the packet:

(1) x+3 (2) 2x-3 (3) 4^x (4) 0.5cos(x)

(5) 4tan(x)-4 (6) 5arcsin(x) (7) 3x^2-x

Some notes about the packet: You can and should change the functions to suit your needs. You may want to make one of the trig functions include radians to recall that prior knowledge. The very last one (a quadratic) is pretty tricky. To help them, encourage them to plot the points. Once they see it may be quadratic, encourage them to determine what they can from the point (0,0). (So they will hopefully come to the conclusion that f(x)=ax^2+bx+0.) Finally, help them plug in any two other points from the table to come up with a system of equations to find a and b. This is a challenge, and we don't intend everyone (or even anyone) to get it.

Part II: Blurpo and Breakfast Graphs

Each group will now be given two giant whiteboards -- one for "noticings/observations" and one for "wonderings/questions." They then are given scenario two, with the cute monster Blurpo.

[page in .docx form]

(1) x+3 (2) 2x-3 (3) 4^x (4) 0.5cos(x)

(5) 4tan(x)-4 (6) 5arcsin(x) (7) 3x^2-x

Some notes about the packet: You can and should change the functions to suit your needs. You may want to make one of the trig functions include radians to recall that prior knowledge. The very last one (a quadratic) is pretty tricky. To help them, encourage them to plot the points. Once they see it may be quadratic, encourage them to determine what they can from the point (0,0). (So they will hopefully come to the conclusion that f(x)=ax^2+bx+0.) Finally, help them plug in any two other points from the table to come up with a system of equations to find a and b. This is a challenge, and we don't intend everyone (or even anyone) to get it.

Part II: Blurpo and Breakfast Graphs

Each group will now be given two giant whiteboards -- one for "noticings/observations" and one for "wonderings/questions." They then are given scenario two, with the cute monster Blurpo.

[page in .docx form]

After having a student read the scenario out loud (hopefully in a dramatic fashion!), groups are given 8 sets of graphs (input and output graphs) and asked to write down noticings and wonderings. Download the graphs here (this download will include the graphs for the entire activity). The teacher knows these are original graphs and derivative graphs, but students don't have any clues about derivatives. They will be lost at first. The teacher is not to answer questions from groups during this part of the activity, but just encourage them and prompt them when they are stuck. Encourage them to try to spread out their graphs and re-organize them on the table!

One amazing suggestion that we got, but didn't have time to implement, was to have all input and output graphs not have a scale on the y-axis, nor gridlines. We highly recommend doing this, because it will force students to focus on x-values, and look for general upwards-downwards trends without getting hung up on precise values.

After students have had enough time and you see the're slowing down, you can either (a) do a gallery walk with the noticings/wonderings or (b) have each group present one of their noticings and wonderings each to the class.

Part III: Lunch Graphs

In this part of the activity, students are given 8 more graphs (all the graphs are included in the download above), but this time the "output" graph is hidden (covered with some colored paper). Students are initially asked to pick two of the 8 graphs and do one of two things: (a) with what they know now about how Blurpo's digestive system works, they try to draw a sketch of the output graph, or (b) they write down in words any features they think will appear on the output graph.

Then students remove the colored paper and they now have 8 more pairs of graphs to use (along with their breakfast graphs). In other words, they have 16 input-output pairs. They can continue marking off new things they notice and wonder, and should be encouraged to use colored pencils to mark up the graphs. They should be heavily encouraged to group graphs in ways that make sense to them.

This is where the teacher starts getting a little "leading-questioning"-y. You want students to start to recognize that the output graph tells you some information about the shape of the original graph. The big things you want them to notice: (i) where the output graph is zero often leads to places where there are humps or valleys (or if none, "flatness") on the input graph, (ii) where the output graph is positive corresponds to places where the input graph is going up, and (iii) where the output graph is negative corresponds to places where the input graph is going down.

You can do this is multiple ways. If one group happens to stumble upon one of the three ideas, they can be asked to share that insight with others. This is more organic. However if no group stumbles, you can start being more leading.

You can ask:

Hint Question 1: Look at where input graphs have “peaks” and “valleys.” Draw a red dot at those points. Now look at the corresponding places on the output graphs. Draw a red dot at those points. What do you notice? [The output graphs hit the x-axis]

Hint Question 2: Look at the places where the output graph is positive/negative [shade over the positive values in one color, the negative values in another]. Does that tell you anything about the input graph [shade again, using the same colors as before]? [yes! if the output graph is positive, the input graph is increasing; if the output graph is negative, the input graph is decreasing]

Hint Question 3: If the graph looks like an easily identifiable polynomial, can you determine the degree? Is there a relationship between the degree of the input and the degree of the output?

When students are done, and groups have come to the key points, you can do some sort of sharing out activity. (We haven't quite figured this out yet.)

Part IV: Dinner

Students all work together -- not in groups but as a class -- to figure out what's happening. You post four input graphs throughout the room, and have poster paper below them. Students circulate and write down what they know about the output graph, based on what they've seen. They can sketch one, they can write something like "The graph will have a peak at x=3," and they can write things like "I agree with this!" or "This feels off to me... I believe..." You can have this be a silent activity if you want.

When this is done, you pick one poster paper and input graph and you read off and show what the class has produced. Then on the big projector, you make a big dramatic reveal of the real graph. Hopefully the kids will erupt in "YESSS!" or "AWWWW!" You can continue with the others.

Give students, when this is done, an opportunity to ask questions to the class about things they don't understand. Have other students answer their questions.

Part V: Debrief

This was the hardest part for us to design, and we aren't quite sure the right way to do this. Recall, one of the impetuses (impetii?) of the activity was to make kids not scared of what calculus is. They have through this activity figured out one of the big ideas: the derivative graph can tell you something about the original graph.

You should tell them that at this point. That they've figured out one of the biggest parts of calculus. And also have them understand why they might care (optimization, understanding the precise behavior of a function, etc.). It's an abstract idea, but it is analogous to functions -- there is an input and an output and a rule that gets you from the input to the output.

At this point, you have a choice. You can stop here, and say: we'll keep returning to these monsters later (and start your course however you want), or you can "give" the big reveal about what the rule is that gets you from input to output (the slope-iness of the input graph at a point gives you the output graph).

We suggest giving the big reveal. Although it might seem like it's "giving away the surprise of calculus," we ask you to think: how did you introduce derivatives before? When push came to shove, did you give away the idea or have students discover it?

If you do decide to talk explicitly about the rule, we don't suggest saying the words "slope of the tangent line." We suggest pulling up a graph of a function on geogebra (or winplot), picking a point on the top of a hump or bottom of a valley, and zooming in a lot. Students will see that the graph will start to look like a straight line with slope zero. Then do it again for another point (maybe one on an increasing part of a curve). It will end up looking like a point with a positive slope. Do this a few times. And then have them look at the many graphs they have in front of them. If they zoomed in on any point a lot (with their minds) can they see how it might look like a straight line? And would the slope sort of match the y-value of the output graph at that point?