Tag Archives: 3-Act Math

Rigor: Trig Ratios

NCTM’s Principles to Actions includes build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. In what ways can technology help our students build procedural fluency from conceptual understanding?

I wrote last year about using technology to develop conceptual understanding of Trig Ratios.

This year, we started the lesson a bit differently. I read a while back about Boat on the River, a 3-Act that Andrew Stadel had published and that Mary Bourassa had used to introduce right triangle trig, but I had never taken the time to look it up.

We watched Act 1 to begin the lesson.

Students submitted what they noticed and wondered.

Then we thought about what information would be useful to know, along with thinking about what information would actually be reasonably attainable.

For example, many bridges are made to a certain standard height or have the clearance height painted on them. This one is no exception … the bridge height is given in the Act 2 information.

Students decided the length of the mast was attainable, too. And the angle at which the boat is leaning. Maybe there was a reading on the control panel.

So we ended up with something like this:

Students’ experience with right triangles to this point had been the Pythagorean Theorem, similar right triangles/altitude drawn to hypotenuse, and special right triangles.

I told them we’d come back to the boat problem by the end of class.

Next, I asked students to draw a right triangle with a 40˚ angle and measure the side lengths. I collected their side lengths, again, telling them that we would use this information later in class. (I wish I had asked them to do this part the night before and submit via Google Form … maybe next year.)

Students practiced I can look for and express regularity in repeated reasoning along with Notice and Note while first watching B move on the Geometry Nspired Trig Ratios activity and then observing what happened as I pressed the up arrow on the slider.

Eventually, we uncovered that the ratio of the opposite side to the hypotenuse of an acute angle in a right triangle is called the sine ratio. We connected that to triangle similarity as our content standard requires.

CCSS G-SRT

1. Define trigonometric ratios and solve problems involving right triangles
2. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

We checked the ratio of the opposite side to the hypotenuse for the right triangle they had drawn and measured. How close were they to 0.643? Students immediately noted that there was a problem with the ratios that were over 1 and talked about why.

We continued practicing I can look for and express regularity in repeated reasoning along with Notice and Note to develop the cosine and tangent ratios.

And then we went back to Boat on the River. What are we trying to find? What ratio could we use? How would we know whether the boat made it?

When I cued up the video for Act 3, the students were thrilled to find out they were actually going to get to see whether the boat made it. And at the end, they spontaneously clapped.

Someone asked me in a workshop recently how long a 3-Act takes. There are plenty on which we spend majority of a class period. Or even more than one class period. This one took less than 10 minutes of our lesson, but the payoff is worth more than our whole unit of Right Triangle Trigonometry. It gave us a way to develop the need for trig ratios that my students have just had to trust we need before. For these students, trig ratios don’t just solve right triangles; trig ratios can help with planning trips down the river. Coupled with the formative practice that students got during the next class, Boat on the River helped us balance rigor, one of the key shifts in mathematics called for by CCSS.

And so the journey towards rigor continues … with thanks to Andrew for creating Boat on the River and Mary for blogging about her students’ experience with it and to my students for their enthusiasm about learning which made evident during this lesson through applause.

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Posted by on January 23, 2016 in Dilations, Geometry, Right Triangles

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 …

Hot Coffee

CCSS say the following about what students should be able to do concerning the volume of a cylinder.

8.G.C.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

HS.G-GMD.A.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.

HS.G-GMD.A.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

The high school standards with an asterisk indicate that the standard is a modeling standard and should be linked to “everyday life, work, and decision-making”.

Our learning targets for the modeling unit are the following:

Level 4: I can use geometry to solve a design problem and make valid conclusions.

Level 3: I can estimate and calculate measures as needed to solve problems.

Level 2: I can decompose geometric shapes into manageable parts.

Level 1: I can create a visual representation of a design problem.

What can learning about the volume of a cylinder look like in a math class using CCSS?

Students made sense of the volume formulas during our Geometric Measure and Dimension unit. For this lesson, we started with a 3-Act lesson by Dan Meyer. You can read more about 3-Acts here if you are interested.

I showed this video and sent a Quick Poll to ask students what we could explore.

I showed my students the first act of the video by Dan Meyer and asked “what question could we explore”.

how big is that cup?  1

how long will it take to fill it?           1

how long would it take to fill this compared to filling a normal cup

how long would it take to drink this           1

how many gallons did it take to fill that cup?        1

how much can the coffee cup hold1

how much coffee can that cup hold?          1

how much coffee could the cup hold          1

how much coffee does it take to fill the mug         1

how much coffee is needed to fill up the giant cup?        1

how much coffee will fill the coffee cup     1

how much coffee will the cup hold?           1

how much paint was used to cover the mug         1

how much tea can go into the giant cup    1

how much time it takes to fill up the container     1

how much volume is the coffee cup itself  1

how would you measure the volume of the handle of the cup   1

the measurements of the cup         1

what is the height of the cup? from the bottom of the inside to the top            1

what is the radius and height of the cup   1

what is the volume of that huge cup?        1

what is the volume of the cup         1

what was the volume of the original block before turned into a cup?   1

why are they filling a giant cup with what looks like coffee        1

I had a few questions this year about the purpose of the giant mug, but I had even more last year, when I simply asked, “What is your question?” While the questions in red certainly aren’t bad questions, they don’t focus on the math that we can explore in the lesson from watching the video. I can see a difference between the prompts.

how much liquid will fill it up          1

How much clay (in pounds) was used to make the giant coffee mug?   1

how long will it take to fill the entire coffee cup?  1

What is the volume of this cup?      1

how much money would that cost?            1

who in the world would need that big of a coffee cup?   1

How many gallons of coffee does it take to fill 3/4 of it?  1

height and diameter of cup?           1

to what height did they fill the mug with coffee?1

What is the volume of the coffee mug?      2

how big of a rush would u get from drinking all of that coffee   1

how much coffee will go in the giant cup?1

how much coffee fills the whole cup          1

how much time will it take to fill the cup to the top?        1

How much coffee does it take to fill the cup?        1

does the enlarged coffee mug to scale with the original?!?!?!???!!!!\$gangsters

swag ultra      1

how wide did the truck used to transport the giant cup have to be      1

Who would waste money on that?  1

how much paint did it take to cover the cup?       1

If this was filled with coffee, how long would that caffine take to crash            1

Is someone going to drink that?!?!?1

How long will it take to fill up the cup?      1

is that starbucks coffee or dunkin donuts coffee?            1

whats wrong with people?

did they use a scale factor?

they are my main coffee mug inspiration?

#SwagSauce   1

How many fluid ounces of coffee can the cup hold?         1

how much coffee goes into the mug….?       1

what is the mug made of?    1

What is the volume of the cup?       1

how much liquid can be held in the cup    1

How many days would it take to drink all of it      1

How much cofffee will the giant mug hold?           1

why are they making giant cups     1

How much creamer would you need to make it taste good?       1

I had recently read a blog post by Michael Pershan where he talks about the difference between asking students “what do you wonder” and “what’s an interesting question we could ask”. I agree even more with Michael now after comparing student questions from this year and last year for the last two lessons that changing the wording a little gets students to think about the math from the beginning.

We selected a few questions to explore – how many gallons will fill the cup, how long will it take to fill the cup, and how many regular-size cups will fit inside the super-size cup of coffee.

Students estimated first and included a guess too low and a guess too high. I won’t collect all of this information through a Quick Poll anymore – it’s too much data to sift through – only the estimate from now on.

300, 500, 700            1

500

1000

900     1

too low-50

too high-25000         1

low: 1

high: 5000

guess: 200000          1

10

1500

325     1

700/2000/1500      1

50-2000-1000          1

High 700 Gallons

Low 520 Gallons

My guess 600            1

too high= 12000

too low=         1

15-201-189   1

High 1000

Low 50

Guess 72π      1

20,000

100

2,000  1

1,000,000–300,000–200   1

100, 750, 125            1

high:1000; low:100; amount:500   1

10, 5000, 650            1

1,000-800-50            1

8000 gallons

500 gallons    1

too high:2500

too low:5

guess:1000    1

Low: 50

High:1000

Guess: 500     1

High = 1,000,000,000

Low = 1

Guess = 3,000            1

10 gallons,5000 gallons, 200 gallons          1

800-5000-8500        1

high-942

low-600

real-700         1

10547888:56:20564           1

low 600

high 1800

actual 1200   1

low 10000   high 500000 guess 50000   1

500, 20,

256     1

7-250-26000            1

500 too high

50 too low

240 my guess            1

Next I asked teams of students to make a list of what information they needed to answer the questions.

I gave each team their requested information. Some teams didn’t ask for enough information, but instead of telling them they were going to need more information, I let them start working and figure out themselves that they needed more information. At some point the class decided about the size of a regular-sized cup of a coffee.

As students began calculating, I used Quick Polls to assess their progress.

One student became the Live Presenter to talk about her calculations for how long it would take to fill the cup.

And another student became the Live Presenter to share his solution. Since it’s been two months since we had class, I can’t remember what question this answers now.

Students who finished quickly also calculated the amount of paint needed to cover the mug.

NCTM’s Principles to Actions offers eight Mathematics Teaching Practices that need to be part of every mathematics lesson. As I look over that list, I recognize each one in this lesson. One of those is support productive struggle in learning mathematics. How often do we really let this happen? Do our students know that “grappling” with mathematics will cause learning?

Several students discussed this task in their unit reflection survey.

• The coffee one helped me because it made me talk with others at my table and look for ways to solve the problem.
• Hot Coffee was very helpful because it made us find all the different dimensions of a cylinder to find how much coffee the world’s biggest coffee cup could hold and then converted different units of measuring to find the amount of gallons in the cup.
• In unit 11G, the activity we did to calculate the surface area, volume, gallons of coffee needed o fill the cup and time it takes helped me learn how to transfer different units to another and apply it to every day life see whether they make sense or not.
• I really liked the Hot Coffee unit. I understood it well, and it was a good problem to work and figure out. It was also really good for me to make sure to use the right units and convert correctly, which I don’t do sometimes.
• I learned that the world’s largest coffee cup help 2015 gallons of coffee.
• I have learned how to use the least amount of information to find the need item.
• This unit helped me to realize how much I’ve learned this year in geometry and how to do many things like finding volumes and areas of different shapes.
• I learned that I need to model with mathematics more often.

And I have learned that I need to provide my students more opportunities to model with mathematics. And so I will, as the journey continues …

Popcorn Picker

CCSS say the following about what students should be able to do concerning the volume of a cylinder.

8.G.C.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

HS.G-GMD.A.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.

HS.G-GMD.A.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

The high school standards with an asterisk indicate that the standard is a modeling standard and should be linked to “everyday life, work, and decision-making”.

Our learning targets for the modeling unit are the following:

Level 4: I can use geometry to solve a design problem and make valid conclusions. G-MG 3

Level 3: I can estimate and calculate measures as needed to solve problems. G-MG 2, G-MG 3

Level 2: I can decompose geometric shapes into manageable parts. G-MG 2

Level 1: I can create a visual representation of a design problem. G-MG 1

How did you learn about the volume of a cylinder?

Many students have been given the formula for the volume of a cylinder, V=πr2h and then asked to calculate the volume of cylinders given the length of the radius and height. For example, what is the volume of a cylinder with a radius of 5 in. and a height of 4 in.?

What can learning about the volume of a cylinder look like in a math class using CCSS?

Students made sense of the volume formulas during our Geometric Measure and Dimension unit. For this lesson, we started with a 3-Act lesson by Dan Meyer. You can read more about 3-Acts here if you are interested.

The question about which cylinder holds more popcorn isn’t a new question. I’ve used this question for years in geometry. But how the lesson plays out in class when I’m focused on providing my students an opportunity to model with mathematics and I’m paying close attention to the modeling cycle is different than simply posing the question to my students as I had in the past.

I had recently read a blog post by Michael Pershan where he talks about the difference between asking students “what do you wonder” and “what’s an interesting question we could ask”. I showed my students the first act of the video by Dan Meyer and asked “what question could we explore”. I agree with Michael that changing the wording here a little gets students to think about the math from the beginning.

how much popcorn could fill the cylinder1

Is there a difference in volume       1

How much popcorn went into the cylinders?        1

Do the cylinders hold the same amount of whatever he was pouring in them?           1

Are the areas of the 2 cylinedes the same?           1

how much can the cylinders hold   1

whether the two cylinders hold the same amount of popcorn   1

will they hold the same amount of popcorn          1

WHICH CYLINDER HOLDS THE MOST POPCORN?            1

how much cereal can they hold and is it equal or is one greater            1

are the volumes of the two cylinders the same     1

what is the volume of each cylinder?         1

Do both cylinders have the same volume?            1

do the cylinders have the same volume     1

Is the volume the same for both cylinders?           1

Are the volumes of the cylinders equivalent?       1

Are the volumes of both tubes the same?  1

were the volumes of the 2 cylinders equal?          1

Last year, I simply asked, “What is your question?” While the questions in red certainly aren’t bad questions, they don’t focus on the math that we can explore in the lesson from watching the video. I can see a difference between the prompts.

why is he using popcorn to fill the paper  1

What exactly was the purpose of that?      1

Are the volumes of the cylinders the same?          1

were the amonts of popcorn equal1

what are the heights of the cylinders?       1

What is the volume of the space in the cylinder not filled with popcorn?         1

Are the volumes of these cylinders the same?      1

Will the two cylinders hold the same amount of food?    1

is he pouring popcorn?        1

What was he pouring into the paper cylinders?   1

Which way should you hold the paper in order for it to hold the most popcorn?        1

what are the dimensions of the paper       1

are the volumer of the cylinders the same?          1

is the volume the same        2

Do they have the same volume?     1

Is the video showing us the different volumes     1

Is the volume of the cylinders the same?   1

do cylinders hold the same amount of popcorn?  1

what is he trying to do?        1

whats he trying to do?          1

which cylinder can hold the most popcorn?          1

What is the volume of 1 popcorn kernal.  1

how much popcorn could fit in the bowls  1

Do the cylinders hold the same amount of popcorn?       1

Do the cylinders have the same volume    1

do both cylinders hold the same amount of popcorn things.      1

why is he doing this?

which will hold more?

whats wrong with him?        1

how did the paper not fall apart     1

Do the cylinders have the same volume?   1

what is the radius of each cylinder?           1

So we continued, exploring which container will hold more popcorn. Before we started calculating, students made a guess as to which they thought would hold more popcorn. I sent a Quick Poll asking whether container A would hold more, container B would hold more, or they would both hold equal amounts.

This year’s results:

Last year’s results:

So what would normally happen next is that I would give students measurements so that they could do some calculations for which container holds more popcorn. But instead, I asked students what information they needed to explore the question.

YOU NEED THE AREA OF BASE TIMES HEIGHT OF EACH            1

radius and height of both    2

what are the dimentions of the paper, the average volume of each piece of popcorn1

radii and heights of each     1

length and width of the paper        1

height and radius of both    2

is it an average size piece of paper? how much do they overlap?           1

area and height         1

the radii and the height       1

diameter of both

height of both            1

radius and height of both cylinders           3

If it is a regular size of paper (8.5×11)      1

radius, height, size of the paper      1

the dimensions of the paper           1

height and width of paper   1

I want their radii and their heights.           1

And then I gave them the information. For this task, I gave everyone the same information, but on some of our modeling tasks, I gave each team only their requested information. (More about that in future posts.)

Container A is made from an 11-in. x 8.5-in. sheet of paper. Container B is made from an 8.5 in. x 11-in. sheet of paper. Students began to calculate with their teams and construct a viable argument as to which container held more popcorn.

They answered a second Quick Poll.

As the teams finished, they started thinking about another question: Can a rectangular piece of paper give you the same amount of popcorn no matter which way you make the cylinder? Prove your answer.

The 3 who still said that the containers would hold equal amounts showed me their work and ultimately corrected their miscalculation.

Students then watched the video of Act 3 where the conflict was resolved.

Teams then decided which of the following questions they wanted to explore next:

• How many different ways could you design a new cylinder to double your popcorn? Which would require the least extra paper?
• Is there a way to get more popcorn using the exact same amount of paper? How can you get the most popcorn using the same amount of paper?
• How many more pieces of popcorn will the first container hold?

For this part, I provided a bag of popped popcorn. By the end of class, we had a whole class discussion on the plan that each team used to answer their chosen question.

Popcorn Picker was very helpful in helping me learn the targets for this unit. It helped me realize that although two objects may have the same surface area, their volumes may not be the same.

Popcorn Picker definitely helped me understand how exactly different dimensions affect the volume of a cylinder even though the dimensions are nearly the same. Using the piece of paper to compose a cylinder using 8.5 and 11 as two different circumferences as well as the height helped me see that the volumes will be different.

One activity that personally really helped me was whenever we put popcorn into two tubes. One of the was a normal sheet rolled up vertically and the other was horizontally. I learned that even tho they had the same dimensions at first, the one folded horizontally held more pieces of popcorn in the end.

I think the lesson for the popcorn in 11F helped me meet learning targets because it taught me that flipping the dimensions actually changes the volume.

I have learned that two sheets of paper with the same dimensions, but different orientations do not hold the same amount of popcorn.

This student reflection makes me realize how important it is for students to think about what information they need to solve a problem instead of always being given the information from the beginning:

I’ve ;earned how to divide a complex geometric objects into parts and calculate It’s volume. I can find out the necessary information needed to solve this kind of problem and how to use them to solve the problem. I can apply math to every day life and model with mathematics. I can also make visual representation of a design problem.

And so the journey to provide students opportunities to model with mathematics continues, with much gratitude for those who are creating lessons here, here, and here for the rest of us to try with our students …