## Using Technology Alongside #SlowMath to Promote Productive Struggle

Using technology alongside #SlowMath to promote productive struggle
2017 T³™ International Conference
Sunday, March 12, 8:30 – 10 a.m.
Columbus AB, East Tower, Ballroom Level
Jennifer Wilson
Jill Gough

One of the Mathematics Teaching Practices from the National Council of Teachers of Mathematics’ (NCTM) “Principles to Actions” is to support productive struggle in learning mathematics.

• How does technology promote productive struggle?
• How might we provide #SlowMath opportunities for all students to notice and question?
• How do activities that provide for visualization and conceptual development of mathematics help students think deeply about mathematical ideas and relationships? [Cross posted at Experiments in Learning by Doing]

Posted by on March 12, 2017 in Professional Learning & Pedagogy

## A Heuristic Approach to Angles in Circles

I am taking a qualitative research class right now, and my mind is full of lots of new-to-me words (many of which my spell checker doesn’t know, either): hermeneutics, phenomenology, ethnography, ethnomethodology, interpretivism, postpositivism, etc. One that has struck me is heuristic, the definition of which I can actually remember because I try to teach heuristically. (The word does not yet roll off of my tongue, but the definition, I get.) On Monday, our content was G-C.A Understand and apply theorems about circles

1. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. We started with a Quick Poll. I asked students for their best guess for the angle measure. I showed the results without displaying the correct answer, noting the lowest and highest guesses. Students moved to the technology. What happens to the angle measures as you move the points on the circle? They moved to the next page, which revealed more information. What happens to the angle measures as you move the points on the circle?  I sent the poll again. There was one team who hadn’t answered yet, so I made a brief stop by their table. Last semester, I remember reading something about how a certain example might give students the eyes to see what you’re trying to get them to see. So we moved the points around to look something like this. If you have 49 and 43, how can you get 46?

Changing the numbers purposefully helped them see. I sent one more poll before we talked about why. So we gave our best guess, and then we used technology to explore. Students practiced MP8 I can look for and express regularity in repeated reasoning as they noticed what stayed the same and what changed with an angle whose vertex is in the center of the circle. They generalized the result. But we hadn’t yet discussed why that happens.

Students practice MP7 I can look for and make use of structure. By now they know our mantra for MP7: What can you make visible that isn’t yet pictured? I saw a line constructed parallel to the given line, which made alternate interior angles visible. I saw a chord drawn that made a triangle visible. I asked students to write down everything they knew about the angles in this diagram. They made suggestions about what we know. They didn’t say the relationships exactly like I would. I wrote them down anyway. They didn’t recognize the exterior angle of the triangle and so ending up proving the Exterior Angle Theorem again off to the side. I wrote it down anyway. And so the journey continues, always trying to enable my students to discover or learn something for themselves (and sometimes succeeding) …

Posted by on February 9, 2017 in Circles, Geometry

## MP6 – Defining Terms How do you provide your students the opportunity to attend to precision? Writing sound definitions is a good practice for students, making all of us pay close attention to what something is and is not.

I’ve learned from Jessica Murk about Bongard Problems, which give students practice creating sound definitions based on what something is and is not. What can you say about every figure on the left of the page that is not true about every figure on the right side of the page? (Bongard Problem #16)

Last year when I asked students to define circle, I found it hard to select and sequence the responses that would best contribute to a whole class discussion without taking too much class time.   I remember reading Dylan Wiliam’s suggestion in Embedding Formative Assessment (chapter 6, page 147) to have students give feedback to student responses that aren’t from their own class. I think it’s still helpful for students to spend time writing their own definition, and possibly trying to break a partner’s definition, but I wonder whether using some of last year’s responses to drive a whole class discussion this year might be helpful.

• a shape with no corners
• A circle is a shape that is equal distance from the center.
• a round shape whose angles add up to 360 degrees
• A circle is a two-dimensional shape, that has an infinite amount of lines of symmetry, and a total of 360 degrees.
• A 2-d figure where all the points from the center to the circumference are equidistant.  We recently discussed trapezoids.

Based on the diagram, how would you define trapezoid? Does how you define trapezoid depend on how you construct it?

Can you construct a dynamic quadrilateral with exactly one pair of parallel sides? And so the #AskDontTell journey continues …

Posted by on November 14, 2016 in Circles, Geometry, Polygons

## 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 …

## Introduction to Curve Sketching, Part 1

Learning Intentions:

Level 4:

I can use the graph of the derivative to sketch a graph of the original function.

Level 3:

I can use the graph of the original function to deduce information about the first and second derivatives.

I can use the graph of the derivative to deduce information about the second derivative and the original function.

I can use the graph of the second derivative to deduce information about the first derivative and the original function.

Level 2:

I can determine when a function is concave up or concave down and where it has points of inflection.

Level 1:

I can determine when a function is increasing or decreasing and where it has maxima and minima.

We were on the first day of a new unit. I included two questions on the opener to ensure students know what we mean by increasing/decreasing and concave up/concave down intervals. As expected students were familiar with increasing/decreasing and not so familiar with concave up/concave down.

Based on the results, we discussed what it means to be concave up and concave down. Someone asked how we would be able to tell for sure where the graph changes concavity, which we get to learn during the unit.

We started the lesson with a few Quick Polls for students to determine which graph was the derivative, given the graphs of a function and its derivative. The polls were based on Graphical Derivatives from Calculus Nspired. I sent the poll, asked students to answer individually, stopped the poll, asked students to explain their thinking to a partner. If needed, I sent the poll again to see whether they wanted to change their response after talking with their partner. I had 6 polls prepared. I sent 3.

I listened while students shared their thinking. I selected three conversations for the whole class.

1. A student who knew which was which based on the power rule, which she learned during the last unit.
2. A student who knew that the slope of the tangent line at the minimum of the parabola should be zero, which is the value of the line at z=0.
3. A student who noticed that the line (derivative) was negative (below the x-axis) when the parabola was decreasing and positive (above the x-axis) when the parabola was increasing.

Again, as I listened to the pairs talking, I selected a few students to share their thinking with the whole class.

1. The first student who shared used the maximum and minima to determine which had to be the derivative, since the derivative is zero at those x-values.
2. The second student thought about what the slope of the tangent line would be at certain x-values and whether the y-values of the other function complied.
3. A third student volunteered a fourth student to discuss her thinking: she noted that the graph of the function (b) changed concavity at the max/min of the derivative (b). After students talked, I sent the poll again to see if anyone was convinced otherwise. Two students briefly discussed how they used increasing/decreasing and concavity to determine the derivative. Next we began to solidify what increasing/decreasing and concave up/concave down intervals look like using Derivative Analysis from Calculus Nspired.

I asked students to notice and note.

Where is the function increasing? Where is it decreasing?
What is the relationship between the slope of the tangent line and where the function is increasing and decreasing?  Where is the function concave up? Where is it concave down? What does the tangent line have to do with where the function is concave up and concave down? Can you look at a graph and estimate intervals of concavity? I was able to see what students were noting on paper and hear what they were noting in our conversation, but I didn’t send any polls during this part of the lesson.

Next we looked at Derivative Grapher from Calculus Nspired. We changed the graph to f(x)=cos(x). We already know the derivative is f’(x)=sin(x). What if we were only given the graph of the derivative? How could we use that graph to determine information about the original function?  I had more for us to discuss as a whole class, but I wanted to know what they had learned before the class ended. I used a Desmos Activity called Sketchy Derivatives to see what students had learned – given a function, sketch its derivative; and given a derivative, sketch an antiderivative. The original activity was from Michael Fenton. I modified it to go back and forth between sketching the derivative and antiderivative instead of doing all derivatives first and all antiderivatives second, and I added a few questions so that students could begin to clarify their thinking using words.

We spent the last minutes of class looking at an overlay of some of their sketches.

Could you figure out exactly where to sketch the horizontal line? Most students have the vertex of the parabola near the right x-coordinate. Should the antiderivative be concave up or concave down? Most students have the derivative crossing the x-axis near the correct location. The bell rang. Another #lessonclose failure. But thankfully, there are do-overs as the journey continues …

1 Comment

Posted by on October 17, 2016 in Applications of Differentiation, Calculus

## MP5 – The Center of the Circle

How do you give students the opportunity to practice “I can use appropriate tools strategically”? How would your students find the center of a circle?

Every year, I am amazed at the connections students make between properties of circles that we have explored and what the center of the circle has to do with those properties.

We started on paper.

Some students moved their thoughts to technology.

Whose work would you select for an individual and/or whole class discussion?

Could we use the tangents to a circle from a point to find the center of the circle? Could we use the intersection of the angle bisectors of an equilateral triangle inscribed in a circle to find the center of the circle? Could we use the perpendicular bisector of a chord of a circle to find the center of the circle? Could we use the intersection of the perpendicular bisectors of a pentagon circumscribed about a circle to find the center of the circle? Could we use the intersection of the perpendicular bisectors of several chords of a circle to find the center of the circle? Could we use a right triangle inscribed in a circle to find the center of the circle? And so the journey continues …

Posted by on September 15, 2016 in Circles, Geometry

## Derivative Rules

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.

Learning Objective 2.1C: Calculate Derivatives

Essential Knowledge 2.1C2: Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.

Essential Knowledge 2.1C3: Sums, differences, products, and quotients of functions can be differentiated using derivative rules.

Mathematical Practice for AP Calculus (MPAC) 1: Reasoning with definitions and theorems

Students can: develop conjectures based on exploration with technology.

How do you provide students the opportunity to develop conjectures?

After determining the derivative of a few quadratic functions using the definition, we use our TI-Nspire Computer Algebra System (CAS) software to explore derivatives. We use the power rule to make conjectures about the product rule. (I think that I saw this suggestion in a Mathematics Teacher magazine in the early 90s, but I can’t find the reference now.) We know what the derivative should be, because we know the derivative of x^5. How could we use f, f ‘, g, and g ‘ to get to what we know is the derivative from the power rule?

Once students made conjectures about the product rule, we formalized the rule.

I asked students to predict the derivative of f(x)=sin(3x). As expected, many thought that it would be f ’(x)=cos(3x). When we looked at the graph of the derivative of f(x), students realized that f ‘(x)=3cos(3x). We used CAS to explore the chain rule (power and composite) in more detail.

Students practiced “Notice and Note”. Several generalized the chain power rule before I asked.

Once students knew the chain rule, we used the chain rule to derive the quotient rule.

And so the journey providing opportunities for students to make sense of rules instead of just telling them rules continues …

Posted by on September 14, 2016 in Calculus, Derivatives

## 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? 1 Comment

Posted by on September 9, 2016 in Calculus, Derivatives

## Assessing the Centroid of a Triangle

The centroid of a triangle is often called the balancing point of the triangle. It is the point at which the medians of the triangle intersect.

Students used technology to explore the relationship between the vertices of a triangle in the coordinate plane and the vertices of the centroid.

If your students knew the relationship between the vertices of a triangle and the vertices of the centroid, how would you expect them to answer the following question? (I included this question on an end of unit assessment.)

The vertices of a triangle are (a,b–c), (b,c–a), and (c,a–b). Prove that its centroid lies on the x-axis.

A few of my student responses are below.

What learning opportunities could I have provided in class to better prepare my students for this question without just giving them a similar problem?

And so the journey to provide meaningful learning episodes that prepare students to answer questions they haven’t seen before continues …

1 Comment

Posted by on August 22, 2016 in Angles & Triangles, Geometry

## MP8: The Centroid of a Triangle

We had been working on a unit on Coordinate Geometry.

How do you give students the opportunity to practice “I can look for and express regularity in repeated reasoning”? When we have a new type of problem to think about, I am learning to have students estimate the answer first.

I asked them to “drop a point” at the centroid of the triangle. We looked at the responses on the graph first and then as a list of ordered pairs. What is significant about the coordinates of the centroid? Students then interacted with dynamic geometry software. What changes? What stays the same?

Do you see a pattern?
What conjecture can you make about the relationship between the coordinates of the vertices of a triangle and the coordinates of its centroid?

Some students needed to interact on a different grid setup to see a relationship. After a few minutes, I sent another poll to find out what they figured out. And then we confirmed student conjectures as a whole class.

And so the journey to make the Math Practices our habitual practice in learning mathematics continues …