Tag Archives: #SlowMath

Marine Ramp, Part 2

After talking about how Marine Ramp played out in the classroom with another geometry teacher, she decided to try it the next time our classes met, and I revisited it.

Our essential learning for the day content-wise was still:

Level 4: I can use trigonometric ratios to solve non-right triangles in applied problems.

Level 3: I can use trigonometric ratios to solve right triangles in applied problems.

Level 2: I can use trigonometric ratios to solve right triangles.

Level 1: I can define trigonometric ratios.

[Since I had recently read Suzanne’s post about #phonespockets and tried it during Part 1, I turned on the voice recording again. When I listened later, I noticed how l o n g some of the Quick Polls took. And I could also hear that students were talking about the math.)

We went back to the Boat Dock Generator to generate a new situation.

I sent the poll, and the responses were perfect for conversation.

About half used the sine ratio, using the requirement that the ramp angle can’t exceed 18˚. The other half used the Pythagorean Theorem, which neither meets the ramp angle requirement:

Nor the floating dock:

We generated one more.

And had great success with the calculation.

And whether we were making any assumptions about the situation.

As expected, they generalized with sin(18˚). And they didn’t really think we were making any assumptions. Except a few about the tides. And that the height was the shorter side in the right triangle.

Which was the perfect setup for the next randomly generated situation.

(I would have kept regenerating until we got a similar situation had the random timing not worked out as perfectly as it did.)

While students worked on calculating the ramp length, I heard lots of evidence I can make sense of problems and preservere in solving them … lots of checking the reasonableness of answers. “No – you can’t do that.” “That won’t work.” “Those sides won’t make a triangle.” And I think it’s telling that no response came in that calculated with the sine ratio.

Here’s what I saw after 2 minutes:

After 4 minutes:

And after 5 minutes:

So were you making any assumptions?

Yes!

And that assumption was … ?

We assumed you could always use sine. But this time, using the sine ratio didn’t work, so we used cosine.

How did you get 26.8?

We did the Pythagorean Theorem and then rounded up to be sure the ramp would meet the floating dock. After looking at the Boat Dock Generator, most of the class decided they might not want to walk on that ramp.

27.9 came from using the cosine ratio. Would you feel more confident on that ramp?

Will the cosine ratio always work?

We ended wondering whether we could generalize what will always work, given the maximum ramp angle, the distance between low and high tides, and the distance from the dock to the floating ramp.

Next year, we will go farther into generalizing, which might look something like this.

We started our Right Triangles Unit with Boat on the River, and we ended with Marine Ramp.  More than any other year, students had the opportunity to actually engage in many of the steps of the modeling cycle – a big change from how I used to “teach” right triangle trigonometry through computation only.

So tag, you’re it now, and I’ll look forward to hearing about your experience with Marine Ramp as the journey continues …

Posted by on February 19, 2016 in Geometry, Right Triangles

Cavalieri’s Principle

Geometric Measure and Dimension G-GMD

Explain volume formulas and use them to solve problems

1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

How do you provide students an opportunity to make sense of volume formulas? I’ve written before about how we use informal limit arguments to make sense of volume formulas for the cylinder and prism and then Power Solids to make sense of volume formulas for the cone and pyramid.

Using a slinky, we briefly discuss Cavalieri’s principle.

Solids: equal height, cross sections for each plane parallel to and including the bases are have equal area.

What are the implications of Cavalieri’s principle here? (the two solids have the same volume)

And here? (none, as the conditions aren’t met)

When we get to the volume of a sphere, I’ve always told my students they’ll have to wait until calculus to make sense of the formula.

(I sneak in this exercise in calculus and wait for someone to notice the result.)

If I ever made sense of the volume of a pyramid or sphere using Cavalieri’s principle while I was in school, I don’t remember. (Surely I’m not the only one.) This year, though, I’m determined to do better. I’ve been saving Pat Mara’s TI-Nspire documents to think this through.

How can you use these images along with Cavalieri’s principle to make sense of the formula for the volume of a square pyramid compared to the volume of a square prism with base and height equal to the pyramid?

When I got out the play dough to make more sense of the dissection of the cube, my coworker joined me. Our solid isn’t beautiful, but we get why the three square pyramids have the same volume and why one square pyramid will have a volume that is one-third of the square prism with base and height equal to the pyramid.

I like the visual image of seeing cross sections that aren’t congruent but have equal area.

Now for the sphere.

What do you see?

On the left, a hemisphere.

On the right, a cone cut out of a cylinder.

What’s the same about the solids?

The sphere and the cylinder have equal radii and equal heights. Since the “height” of the sphere is its radius, the cylinder has height equal to radius.

What are the horizontal cross sections?

On the left, a circle. The radius decreases as the cross section slices go from bottom to top.

On the right, a “washer” (or officially, an annulus), where the outer radius is always the radius of the cylinder (constant) and the inner radius is equal to the height of the smaller cone formed with the inner circle of the slice and the center of the base (shown by similar triangles).

Dynamic geometry software shows us that the cross sections have the same area. Convince yourself that they do.

I convinced myself here:

And then I looked at the next page, which allowed me to move the cross sections and see the similar triangles change.

So what does this tell us about the volume of a hemisphere?

According to Cavalieri’s Principle, it has the same volume as the solid on the right.

How can you calculate the volume of the solid on the right?

Subtract the volume of the cone from the volume of the cylinder.

And then what about the height of a sphere for which this hemisphere is half?

The Illustrative Mathematics task Use Cavalieri’s Principle to Compare Aquarium Volumes could be helpful for exploring Cavalieri’s Principle. I’ve had it tagged for several years now. Maybe this will be the year we take time to try it.

And so the journey as both learner (student) and Learner (teacher) continues, with gratitude for those who share their work and those who are willing to pause their work long enough to learn alongside me …

Experiencing Dandy Candies as learner and Learner

Back in April, I had the pleasure of attending a CPAM Leadership Seminar with Dan Meyer on mathematical modeling, where he lead us through Dandy Candies. Dan wrote about this 3-Act recently here. I’ve used several 3-Acts with my students, but this was my first time to participate in one from a “lower-case l” learner’s perspective. I’ve read about “purposeful practice” and “patient problem solving” for several years now, and I know that I have some understanding of what they mean, but seeing them in action from the learner’s perspective is powerful.

A few things struck me during the seminar. We don’t do 3-Acts just for fun. (I knew this, but Dan made it very clear that this isn’t just about engaging students in doing something; it’s about engaging students in doing math. I’m not sure I’ve made that as clear to other teachers with whom I’ve discussed 3-Acts.) As Smith & Stein point out in 5 Practices for Orchestrating Productive Mathematics Discussions, there is pre-work for the teacher: identifying the math content learning goal for the lesson and then selecting a task that is going to provide students the opportunity to engage in that math content. Even with a 3-Act, where we let our students’ curiosity develop the question, we do have an underlying question that will engage students in the math content we want them to know. What they ask might not be worded exactly the same, and it might extend the mathematical thinking in which we want students to engage, but the math is there.

I have said before that I use technology to give every student a voice – from the loudest to the quietest, from the fastest to the slowest. When Dan solicited questions we could explore from the group, I was never going to volunteer mine for the list. (I am not criticizing Dan’s move here … just noting that I find it challenging, both as Learner and learner, to establish trust in a short session with participants that I’m likely not going to see again.) I *might* have participated had I been asked to submit my question somewhere anonymously.

And finally, I really like the opportunity that we had before each question to answer before performing any calculations. I’ve been working on providing this opportunity for my students, but it still isn’t automatic. I have to remind myself to ask students to use their intuition first. As I heard from Magdalene Lampert, “Contemplate then Calculate”.

We were working on Modeling with Geometry (G-MG) when I returned to class after the seminar last year, and so I tried Dandy Candies with my students.

are the heights the same     1

what are the surface areas of the boxes    1

the similarity between how thevolume stays the same and te cross sections change1

Do all the solids have the same volume?    1

are the surface area and volume the same throughout the same changes?     1

do all the boxes have the same volume     1

how many cubes       1

what shapes could be made            1

how does the surface area change 1

same surface area?   1

could the volume make an equal ratio       1

whats the volume of each cube that makes each shape  1

how many different shapes can be made with those boxes        1

Do they all have the shme volume  1

Is the area of any gift formed by the candies the same? 1

What do you *think*? Which package(s) use the least cardboard?

(No one answered more than one.)

What do you *think*? Which package(s) use the least ribbon?

(No one answered more than one.)

What do you *think* are the dimensions for each box?

I enjoyed watching students use appropriate tools strategically while they were working.

And then I sent the polls again.

Which package(s) use the least cardboard?

Which package(s) use the least ribbon?

Some mistook “better” for “best”, and others are apparently going to cut the candies in halves.

Modeling with Geometry G-MG

Apply geometric concepts in modeling situations

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Content-wise, students had the opportunity to learn more about modeling with geometry. And they were able to engage more steps of the modeling cycle than just computation.

I am not *yet* writing 3-Acts, but as the journey continues, I am grateful for those who do and share …

Hot Coffee + Show Your Work

I’ve written about this lesson before, but I wanted to write again because of several observations from last spring.

I know that I need to find a way to provide students the opportunity to engage in math modeling more often and earlier in my geometry course. I’m having a hard time finding a way to do that. (Ideas for providing students the opportunity to engage in math modeling while proving theorems about congruence and similarity are welcome!) For now, we focus on modeling during the last unit of the course.

I decided last year to show students the modeling cycle from CCSS at the beginning of each lesson so that students would recognize what I am asking them to do differently and why I’m not giving them all of the information they need up front.

Our learning goals: I can model with mathematics, and I can show my work (leveled learning progression from Jill Gough).

Once we decided what questions to answer after watching Act 1 of Dan’s World’s Largest Hot Coffee Three-Act, students estimated responses.

And then teams made a list of the information they needed. I gave them information only as they requested it. Most teams realized later rather than sooner that they would need some type of conversion for cubic feet into gallons.

When they decided they needed to know how much coffee a regular cup would hold, two of the girls remembered that the teacher with whom I share the classroom always had a cup of tea. They asked to borrow her cup so that they could come up with an agreed upon amount for a regular cup of coffee.

At one point, a student asked whether getting the right answer mattered. I asked why. She and her teammate didn’t have the exact same calculation.

It struck me that what we were really working on today was identifying a problem, determining what was essential to know, and creating a model to answer the problem. It’s not that the calculations aren’t important, but for this lesson, the questions were more important. By the time I got back around to that team, they had resolved their computational issue because of a conversion error. Even so, I’m glad I was asked whether it mattered that everyone got the same answer, as it helped shape how I launched our remaining modeling lessons.

And so the journey to provide students the opportunity to engage in all steps of the Modeling Cycle continues …

Carrying a Figure Onto Itself + #ShowYourWork

G-CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Our content learning goal for the day: I can map a figure onto itself using transformations.

Our practice learning goal: I can attend to precision.

Combining those, we were working on: I can show my work.

Jill Gough has written a transformative leveled learning progression for showing your work. This was our first day in geometry this year to focus on it.

Level 4: I can show more than one way to find a solution to the problem.

Level 3: I can describe or illustrate how I arrived at a solution in a way that the reader understands without talking to me.

Level 2: I can find a correct solution to the problem.

Level 1: I can ask questions to help me work toward a solution to the problem.

For this task, our focus was on describing clearly the transformations that would carry a rectangle or equilateral triangle onto itself so that a partner could follow the steps.

Which of the following is clear?

Reflect ABCD about a line through the middle of the rectangle.

Rotate ∆ABC 60˚.

Reflect ABCD about the perpendicular bisector of segment AB.

Rotate ∆ABC 180˚ about point A.

Students set to work individually, paying attention to their language. I walked around to see what they were writing.

I noticed MR’s first, which said, Translate ∆ABC using vector AA. As I looked more closely, I realized that she was mapping the triangle on the left side of the page onto the triangle on the right side of the page, but even so, she had come up with a remarkably trivial solution, had she been mapping the triangle onto itself.

The next student that I saw had rotated ∆ABC 360˚ about point A.

And then the next student that I saw had dilated ∆ABC about point A using a scale factor of 1.

I decided at this point that perhaps a class discussion was in order to limit additional trivial solutions to this task. So we talked about transformations that will, of course, map the figure onto itself, such as rotating the image about one of its vertices 0˚ or 360˚, and also, really, are simple and not very interesting.

And then I let them work some more. The idea was for them to write a transformation or sequence of transformations and have their partner try it, following their directions exactly. The partner helped revise the directions as needed if the directions didn’t work the first time.

Instead of selecting particular students to share their work with the whole class, I asked students to write at least one set of their successful mappings in a shared Google Doc so that they could see multiple solutions to both the rectangle and the triangle.

Thanks to the leveled learning progression, I think we are off to a good start practicing “show your work”, as the journey continues …

Posted by on September 7, 2015 in Geometry, Rigid Motions

The Last Day of Class

Another school year has finished (two months ago), and a new one is about to begin.

Our teachers did a lot to promote growth mindset this last year.

Many of us sent our students a poll with statements from Carol Dweck’s book, Mindset, on the first day of class

and again on the last day of class.

You can see some change in the way that students responded.

I have wondered whether talking about mindset and promoting growth mindset makes a difference in what and how students learn. I know plenty of teachers are skeptical. I’m convinced that it matters.

Many students participated in Jo Boaler’s How to Learn Math: For Students online course through Stanford University. I heard students talking about making mistakes, their brain growing, and synapses firing on many occasions throughout the year.

What has convinced me more than student responses throughout the class, though, are the voluntary reflections that my students have offered. I received a handwritten letter at the beginning of May from a former student who no longer attends our school. An excerpt follows:

“When you taught my geometry class last year you polled us at the beginning and the end of the year to see if our opinions on innate/static intelligence vs. one’s ability to improve intelligence had changed. I just want to say that though I was doubtful at the time, this idea of an evolving and increasing intelligence through questioning and learning through wrong answers has stuck with me and served me well. I was once pretty insecure in my academic abilities: yes, I made good grades without much trouble, but there’s always someone faster or more confident or more eloquent, and so much of my identity was wrapped up in being a ‘smart’ kid that I was often afraid to speak up and make mistakes. Now, though my grades and academic integrity are still very important to me, I don’t see successes and failures quite so black and white. Rather, I try to see it all as a learning moment, and I thank you for introducing me to some of the ideas of growth mindsets and ‘GRIT’.” – CM

This thoughtful reflection a year after the class ended is coupled with a thoughtful reflection from another student who wrote as the class ended last year. You can see his reflection in this post.

We will start another school year on August 6 … our students are going to hear the message not only that they can be successful in mathematics but that we, their teachers, want them to be successful in mathematics … our students are going to be greeted with open-ended problems that are accessible to all (many of which will come from youcubed’s Week of Inspirational Math) – problems that allow them to realize from the beginning that we don’t all think the same way and that making our thinking visible to others is a good and important learning opportunity for all … our students are going to set norms for how the class will learn together throughout the year … our students are going to hear from Carol Dweck on the power of “Yet” and they might even hear from Sesame Street, too.

What message will your students hear on the first day of class? What will they say about your class when asked how they think classes are going to go this year?

I look forward to school starting again, as the journey continues …

Posted by on July 25, 2015 in Student Reflection

Tags: ,

I was invited to write a few posts for NCTM’s Mathematics Teacher Blog: Joy and Inspiration in the Mathematics Classroom.

Ask, Don’t Tell (Part 4): The Equation of a Circle

Ask, Don’t Tell (Part 3): Special Right Triangles

Ask, Don’t Tell (Part 2): Pythagorean Relationships

Ask, Don’t Tell (Part 1): Special Segments in Triangles

While you’re there, be sure to catch up on any other posts you haven’t read. There are some great ones by Matt Enlow, Chris Harrow, and Kathy Erickson.

“Ask Don’t Tell” learning opportunities allow the mathematics that we study to unfold through questions, conjectures, and exploration. “Ask Don’t Tell” learning opportunities begin to activate students as owners of their learning.

I haven’t always provided “Ask Don’t Tell” learning opportunities for my students. My coworkers and I spend our common planning time thinking through questions that we can ask to bring out the mathematics. We plan learning episodes so that students can learn to ask questions as well. (Have you read Make Just One Change: Teach Students to Ask Their Own Questions?)

After the Special Right Triangles post, someone commented on NCTM’s fb page something like the following: “Really? You told students the relationships without any explanation?”

I have always used the Pythagorean Theorem to show why the relationship between the legs and hypotenuse in a 45˚-45˚-90˚ is what it is. But I think that’s different from “Ask Don’t Tell”.

I have been teaching high school for over 20 years. And yes. I really used to tell my geometry students the equation of the circle. I told them definitions for special segments in triangles along with drawing a diagram. I told them how to determine whether a triangle was right, acute, or obtuse. And I told them the relationships between the legs and hypotenuse for 45˚-45˚-90˚ and 30˚-60˚-90˚ triangles.

I’ve also been in meetings with teachers who have not thought about decomposing a square into 45˚-45˚-90˚ triangles or an equilateral triangle into 30˚-60˚-90˚ triangles to make sense of the relationships between side lengths.

You can see on the transparency from which I used to teach that I actually did go through an example where an equilateral triangle was decomposed into 30˚-60˚-90˚ triangles; even so, I failed to provide students the opportunity to look for and make use of structure.

Purposefully creating a learning opportunity so that the mathematics unfolds for students through questions, conjectures, and exploration is different from telling students the mathematics, even with an explanation for why.

As you reflect on your previous school year and plan for your upcoming school year, what #AskDontTell opportunities do and can you provide?

Connecting Factors and Zeros

NCTM’s Principles to Actions suggests Mathematics Teaching Practices for teachers. Two of those are the following.

MTP 1 Establish mathematics goals to focus learning

MTP 6 Build procedural fluency from conceptual understanding

If the goal for students is to use the factors of a quadratic function to determine its zeros, what concepts must students understand to meet that learning goal?

Our team wrote this leveled learning progression for our lesson.

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros.

Level 2: I can expand the product of two binomials.

Level 1: I can solve an equation in one variable.

Level 1: I can determine the zero(s) of a function from the graph of a function.

We decided to first ensure that students know what a zero is, and we checked this is more than one way on the opener for the day. (See this source for similar Level 1 problems.)

Students had to place a point at the zero of the function.

Almost all students were able to note that the point of interest is where the graph intersects the x-axis.

Students had to name the coordinates of the zero of the function, which about half could do.

And then students had to answer a question about a zero in context. A few more than half could do this.

We decided that students also need to be able to solve an equation in one variable.

Which they could easily do.

And we also decided that if students are going to meet the learning goal, they are also going to have to be able to multiply binomials. Which you can tell from the results that they could not easily do (Q8 and Q9).

In the lesson, we started with the zeros of a linear function.

What do you notice?

If I give you a similar equation, can you tell me the zero?

If I give you a similar equation, can you tell me the zero?

We checked in with students using some Quick Polls.

(I noticed that not all are x-intercepts.)

Students showed some improvement as we continued.

How can we tell that (-6,0) is the correct choice using the equation?

We spent a long time on linear functions. Some might think we spent too long.

Then we looked at a quadratic function.

And we related the linear factors to the quadratic visually.

This is part of a Math Nspired activity called Zeros of a Quadratic Function, where there is a lot more flexibility in changing the factors.

Our leveled learning progression for the second lesson changed a little:

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros, and I can use the zeros of a quadratic function to determine its factors.

Level 2: I can rewrite a quadratic function given in factored form to standard form.

Level 1: I can determine the zero(s) of a quadratic function from the graph of a function.

When we checked for student understanding during the opener of the second lesson, we saw that students were able to determine the zero(s) of a quadratic function from the graph of a function.

Lots of students were at Level 1, determine the zeros when given the graph and the equation.

Not as many were at the target – but definitely more than had reached it the day before.

We have worked to build procedural knowledge from conceptual knowledge in our unit on Zeros and Factors. Our standards say that we want students to “Factor a quadratic expression to reveal the zeros of the function it defines”. The standards don’t say that we want students to factor a quadratic expression just for the sake of factoring.

What opportunities are you providing your students to concentrate on relationships rather than just results?