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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”.

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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.

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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

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

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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.

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

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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.

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

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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 …

 

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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”.

Screen Shot 2014-06-30 at 6.07.34 PM

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.

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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.

Screen Shot 2014-07-05 at 3.26.13 PM

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.

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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.

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This year’s results:

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Last year’s results:

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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.

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YOU NEED THE AREA OF BASE TIMES HEIGHT OF EACH            1

height hnd radious   1

height and radius     3

radius and height of both    2

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

both radii, height      1

radii and heights of each     1

length and width of the paper        1

height and radius of both    2

radius and height     5

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.

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They answered a second Quick Poll.

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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?

2014-05-08 09.51.37  2014-05-08 09.49.56 2014-05-08 09.49.43  2014-05-08 09.48.49

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.

What’s most significant about this lesson is not the engaging way that students got to learn but what the students did learn. Several students made comments about this lesson on their end-of-unit survey:

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 …

 

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Why I Believe in CCSS

In my geometry class, we talk about the Segment Addition Postulate.

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If we take a segment 2 in. long and put it with a segment 3 in. long, we end up with a segment 5 in. long: the part plus the part equals the whole.

But as I reflect on what CCSS have done for my students, my coworkers, and me, I lean less towards Euclid and more towards Aristotle in my beliefs: the whole is greater than the sum of its parts. We are better together than we are alone. Mississippi educators are better with educators from all over the U.S. than we are alone. Mississippi students are better when their teachers are learning alongside other teachers all over the country than they are when we only share within our own school and state.

What has happened in my classroom over the past two years of implementing CCSS regarding student ownership of the mathematics we are learning is more visible than I would have predicted. The positive changes are not only undeniable; they are important, and they have been transformative for my students and for me.

The CCSS for mathematics are anchored by the Standards for Mathematical Practice, which is how we want students to learn math. The first math practice is to make sense of problems and persevere in solving them. How many times do you have to make sense of problems and persevere in solving them in the work that you do? How did you learn to make sense of problems and persevere in solving them?

Most students that I’ve encountered think that math is about a teacher demonstrating how to work a problem and a student mimicking the teacher. They are surprised on the first day of class to encounter problems that not only can be solved in more than one way but for which they are encouraged to solve in more than one way.

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Another math practice is to construct a viable argument and critique the reasoning others. Most students that I’ve encountered think that math is about getting an answer. They are surprised when I encourage them to talk about how they are getting an answer – and even more surprised when I ask another student whether they agree or disagree with the process. We learn math by talking about how we are doing math. We even learn math when someone discusses a wrong method, and we make sense of the misconception. We celebrate learning from and correcting mistakes.

Another math practice is to look for regularity in repeated reasoning. If you took a high school geometry course, you learned about 45-45-90 and 30-60-90 special right triangles. In how many of your classes were you given the opportunity to think about the triangles formed by cutting a square in half on its diagonal or cutting an equilateral triangle in half on its height?

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Were you given formulas for the relationships between the legs and hypotenuse of the right triangle (as you can see that I used to do from the picture of the transparency from which I used to teach)?

30-60-90

Or were you given an opportunity to make sense of the relationships – and maybe even figure them out yourselves by looking for patterns and talking about what you notice with other students?

45-45-90

Another math practice is to look for and make use of structure. What is the formula for the area of a triangle? Why? Can you calculate the area of a trapezoid? Looking for and making use of structure is about providing students opportunities to make sense of the formulas and equivalent expressions that we have traditionally given them and asked them to memorize without providing them the opportunity to make sense of why.

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I won’t spend our time here talking about how the standards are not curriculum. If you don’t know what this means, maybe it would be helpful to hear that CCSS simply tells us that the Pythagorean theorem is a topic for introduction in grade 8 and then students learn more about it in a high school geometry course, but CCSS doesn’t tell us on what day of the year to teach the Pythagorean theorem or how to teach the Pythagorean theorem. As a teacher, I have full control over what standards I teach when and how I teach them. The standards are “the floor, not the ceiling”. They are the minimum that we expect our students to do in order to graduate as college and career ready. Students will still have the opportunity to take calculus and/or dual enrollment college math courses while they are in high school.

The Math Practices have been transformative for my students and for me. And what’s better is that CCSS says that all students will learn math using the math practices. Not just those who have a good teacher. Not just those who go to a good school. CCSS allows me to learn alongside educators from all over the United States who tweet and blog about their efforts to provide opportunities for their students to make sense of mathematics. I use free sites such as Illustrative Mathematics and the Mathematics Assessment Project that make it easy to search for tasks and lessons by standard.

And so the journey to help my students make sense of mathematics using the CCSS and the Standards for Mathematical Practice continues … And you are invited to come see it in action any day. Just be forewarned, my students will expect you to fully participate in the lesson – no one is just a spectator in our math class.

 

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The Protractor Tool

Let’s look for and make use of structure. Which of these is not like the other?

A. SMART Notebook software protractor tool

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B. Promethean ActivInspire software protractor tool

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C. PARCC digital tool

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D. Smarter Balanced digital tool

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If you were trying to attend to precision, would you prefer one over the other?

 

Question 26 on the PARCC EOY Grade 4 Practice Test asks which angle measures 65˚, which I can do without using the protractor.

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But I tried using the protractor tool, just to see how it worked, and I had a difficult time using it.

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It struck me that I don’t think using the protractor tool should be tricky. We are already asking students to measure while negotiating rotating the computer protractor with a mouse or trackpad. Do we expect them to guess where the rays would extend as well? On paper, I would use my pencil and a straightedge to extend the rays.

I felt confident about placing an initial ray of an angle at 0 using the other protractor tools.

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I’d be more confident about my precision using the PARCC protractor if there were a horizontal line connecting the 0/180 measurements on both sides.  But maybe there is a trick to using the PARCC protractor tool that I don’t know? Or maybe PARCC will consider improving their tool?

And so the journey continues, wanting the best tools available for our students so that they can show they know how to use appropriate tools strategically

 
3 Comments

Posted by on June 25, 2014 in Angles & Triangles, Geometry

 

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Ratios of Areas Performance Tasks

On the performance task day for our unit on Geometric Measure and Dimension, we worked on Circles and Squares most of the block. With about 20 minutes remaining, we moved to Dan Meyer’s Three-Act Math Task Some Really Obscure Geometry Problem.

Students watched the first act and responded to a Quick Poll with their guesses about the percents.

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Then they began to work.

I loved watching students use appropriate tools strategically. I purposefully had not given them a pre-made diagram. Some students chose to build the diagram to help make sense of the relationships between the regions, but not every student. Some students used their handheld as a calculating device for work they were doing on paper, but not every student.

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I loved watching students look for and make use of structure.

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The bell rang before we finished. I told students they could continue their work on the task for Problem Solving Points. No one did.

 

There was a comment on my post about Circles and Squares suggesting that the scaffolding provided by the questions from the Mathematics Assessment Project “shoehorned” students into one solution. It made me wonder whether I should have structured the class period differently.

If we had removed the questions from Circles and Squares, it would have taken longer. Some students might have gotten to the obscure geometry problem, but not all of them would have, and we would not have had the class time together looking at student estimates; the task would have played out differently had students just gone straight to calculating the ratios of the areas.

Last year, the reverse happened. We started with the obscure geometry problem, and that took the majority of the period, so we spent only a little time on the Circles and Squares task.

So what’s best in this situation? Providing a little guidance so that students can see both tasks? Or providing less guidance on one task? Or introducing both tasks at the beginning and giving teams a choice in which task they pursue first, culminating in class discussion about solutions to both? Or even discussing solutions to both the next day so that students have the option to spend some time on the tasks outside of class.

I don’t have answers … only questions. But maybe next year as the journey continues I’ll try introducing both tasks and giving students a choice …

 
 

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Use Appropriate Tools Strategically – The Student

One of the Standards for Mathematical Practice is use appropriate tools strategically, and one of my calculus students sent me the following reflection about this practice.

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KH has reached a point in mathematics where the functions that we use have become part of what she thinks of as available tools to use when solving a problem. She was referring to problems where logarithmic differentiation was helpful. She was solving differential equations, and after integrating it was helpful to make both sides of the equation the power of e. Her comments struck me as something I want to remember as the journey continues …

 

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Use Appropriate Tools Strategically – The Teacher

I have sent students questions like the following many times.

Screen Shot 2014-06-10 at 10.36.19 AM

And my students almost always get this question correct.

A few years ago, when I tested a similar question on a no-calculator, no-multiple choice part of their test, I was surprised to find several students with the following answer:

QP1

The next day in class, without mentioning their responses, I sent my students the following Quick Poll.

Screen Shot 2014-06-10 at 10.37.10 AM

As students start typing their response into the Quick Poll, it graphs the equation.

Screen Shot 2014-06-10 at 10.37.58 AM Screen Shot 2014-06-10 at 10.38.06 AM Screen Shot 2014-06-10 at 10.38.38 AM

Several of them tried to enter the equation the way they had written it on their test – and then wondered why the function was being translated down 1 instead of to the right 3 and up 2.

They learned the lesson I needed them to learn by correcting their own work.

Screen Shot 2014-06-10 at 10.39.25 AM

And I am learning to use appropriate tools strategically along with formative assessment better every day. The y= question with the “Include a Graph preview” option gives my students the opportunity to attend to precision while they are learning – and then they are better able to attend to precision with whatever type of function I might give them on the no calculator part of a summative assessment.

And so the journey to continue learning the appropriate use of technology continues …

 
 

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