# Tag Archives: model with mathematics

## 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

## Systems of Linear Equations

We are working on Systems of Linear Equations in our Algebra 1 class.

The first day, our lesson goals were the following:

Level 4: I can create a system of linear equations to solve a problem.

Level 3: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

We used the Mathematics Assessment Project formative assessment lesson Solving Linear Equations in Two Variables to introduce systems.

Then we asked students to draw a sketch of a system of equations with one solution, a sketch of a system of equations with no solution, and a sketch of a system of equations with many solutions.

We used the Math Nspired activity How Many Solutions to a System to let students explore necessary and sufficient conditions for the equations in a system with one, no, or many solution(s). We used a question from the activity so that students could solidify their thinking about the relationship between the equations for different solutions.

We sent the questions as Quick Polls so that we could formatively assess their thinking and correct misconceptions.

We like using this type of Quick Poll (y=, Include a Graph Preview) while students are learning. The poll graphs what students type so that they can check their results and modify the equation when they do not get the expected result.

(Note: on the second day, we asked again for students to create a second equation to make a system with one, no, and many solution(s), but the given equation is in standard form.)

On Day 2 of Systems, the learning goals are slightly different.

Level 4: I can solve a system of linear equations in more than one way to verify my solution.

Level 3: I can create a system of linear equations to solve a problem.

Level 2: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

We started with a task from Dan Meyer’s Makeover series, Internet Plans. Part of this makeover was a complete overhaul of the context. Students write down a number between 1 and 25. They view this flyer (from Frank Nochese).

And then they decide which gym membership to choose, using the number they wrote down as the number of months they plan to work out.

When I saw the C’s and B’s on the number line on the blog post, I actually thought that the image was beautiful – a clear indication that one plan is better than the other two for a while, and then another plan is better than the other two. This image absolutely leads to the question of which one becomes better when – and thinking about why plan A is never the better deal.

Our students started calculating. Several asked what \$1 down meant before they could get a calculation. Then they shared their work on our class number line.

We were dumbfounded at the results. Really. It was hard to know where to start. We were prepared to help student develop the question with correct calculations. We hadn’t thought about what we would do when students couldn’t figure out which plan was the better deal for a specific, given number of months.

What would you do next if this is how your students answered?

We started with the lone C at 4 months, hoping someone would claim it and share his calculations. He did: I divided \$199 by 12 months and then multiplied by 4 to get the cost for 4 months.

Another student said he couldn’t do that because you had to pay \$199 for 12 months no matter how many months you actually used it.

Another student said that A was the better deal because they were planning to get every other month free by working out more than 24 days each month. We discussed how realistic it is to work out at a gym 24 days a month.

We did eventually look at a graphical representation of each plan. And talked about what assumptions had been made using these graphs.

Can you tell which graph is which plan?

Can you write the equations of the lines?

In the first class we never made it to actually calculating the point(s) of intersection (class was shorter because of a pep rally), but other classes did calculate the point(s) of intersection.

It’s not a waste of time to think about Plan C costing \$199 for 12 months.

But then we really should have created a piecewise function to represent the cost for 13-24 months as well.

The number lines weren’t much better in subsequent classes.

So maybe the students got to practice model with mathematics, just a little, even though we have a long way to go before students will be able to say, “I can create a system of linear equations to solve a problem”.

The teachers were definitely reminded of how much work we have to do this year, as the journey continues …

## Volumes of Compound Objects

We used the Mathematics Assessment Project formative assessment lesson on Calculating Volumes of Compound Objects. MAP also has a Glasses task that includes student work.

Students calculate the volume of liquid that the glasses can hold, and then determine the height of the liquid in Glass 3 when it is half full.

Minutes before I gave students their handout with the glasses, I decided to cover up all of the measurements on the glasses and ask the students what information would be sufficient for calculating the volume.

circumfrence, height 1

Height and radius of the cylinder and the volume of the solid holding up the cylinder          1

surface area uf cup part

height of cup part     1

dimensions, shape    1

area of circle and height      1

Just the radius and the height, but either if the volume is given.           1

height and base        1

radius of circle, height of glass        1

radius and height of the cylinder   1

height of cylinder and radius          1

radius of base, height of cylinder    1

diameter, hieght, to get volume.      1

height and radius of the cylinder   1

Every thing you need to know for volume of a cylinder  1

height of cylinder, height of hfmishpere, and radius       1

height of the straight side and radius        1

radius and height of the cylinder   2

radius and straight side length       1

Asking what information students needed gave a different spin to mindless calculations with volume formulas, even for volumes of compound objects.

Once students had the needed information, they calculated. I sent a Quick Poll with all four questions at once so that students could work at their own rate without feeling the pressure of a timed poll.

I monitored student responses as they came in, talking directly with teams who needed to revise their work.

Until we got to the question about the glass being half full.

From the student responses, it seemed that the glass was half empty instead.

What do you do when no one has the correct response?

There are times that I have given the correct response and asked students to determine how to get it. I’m not sure how effective giving a decimal approximation of the correct height would have been here. Instead, we looked at the picture. Can you draw where the height will be when the glass is half full? What do you see?

Similar triangles are often elusive for my geometry students. We need more practice.

And so the journey continues, trying to find a balance between giving too much information and too little information …

## 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 …

## Teacher Appreciation Week

Several bloggers have reflected on their teachers this week, which has made me do the same.

Those who have come before me in my own family and those who have taught me have had the most influence in helping me to discern my own calling to teach. My grandfather taught high school mathematics for over 53 years. Several years into my own career as a mathematics teacher, he passed along his box of calculus teaching materials, along with the admonishment to take good care of it in case he needed it back some day. My grandfather celebrates his 99th birthday tomorrow. I’m not sure he would still say at this point that he will want his teaching materials back, but he is still active in his community. He attended a retired teachers’ lunch meeting earlier this week and will attend a distinguished alumnae luncheon tomorrow for his school district. My family and I reserved a place on his calendar Saturday afternoon to celebrate his birthday.

My grandmother, aunt, mother, sister-in-law, and mother-in-law have also been teachers, running the gamut of subjects between 2nd grade, special needs, music, and middle school English. My family encouraged me to enter the teaching profession, offering both their emotional and financial support, rather than discouraging me from teaching, as many of my friends’ parents did, because of teachers’ salaries. My husband and daughters continue to support me in this calling.

My love of school and my desire to teach was evident early on as my sister and I used to play school, lining up our dolls for lessons, using carbon paper to create multiple copies of tests that our “students” worked so that we could perform the ultimate teacher duty of grading papers. Of course I know now that grading papers is not the ultimate duty of my profession, but I am grateful that I knew early in my life that I wanted to teach, because that helped me pay more attention to my teachers.

As my students and I enter the practice of learning each day, I recognize parts of all of my teachers: Mrs. Berry helped me make connections between algebra and geometry to greatly increase my own understanding of mathematics, and she taught me how to clearly communicate that understanding to others. I often hear myself say, “I’ll buy that” to an explanation like I heard Mrs. Berry say to us. And because of our focus on the Standards for Mathematical Practice, I now hear myself asking my students “Do you buy that?” after someone has given an explanation to the class.

Dr. McMath taught me that each method we learn for solving problems is another “tool” to add to our bag of mathematical tools. My bag is overflowing because of all of the different types of problems he introduced to me. Hopefully, my students’ bags are at least a little more full after they consider the problems that I assign to them.

Dr. Travis taught me to sit down and let the students teach each other, “passing the pen” from student to student, as we have conversations together about important concepts and applications. Dr. Travis taught me not to be concerned with an answer as much as the process for getting an answer. He helped me develop good, logical arguments for the mathematics that we studied; I try to do the same with my students.

Dr. Floyd gave me a strong foundation in the history of mathematics, teaching me to incorporate math history into my lessons, getting students interested just enough to learn more about a mathematician on their own outside of class. Her own travels overseas made the stories that she told come alive … you could see it in her eyes and hear it in her voice.

Dr. Leavelle shared with us interesting tasks from his visit to Japan that I still pose to my students. His wealth of practical applications for modeling with mathematics are now mine to use with my own students as we seek to understand how engineers and physicians and developers really use the algorithms that we are learning in our classroom.

Dr. Gann’s energy and enthusiasm for mathematics cannot help but show as my students and I develop formulas and prove theorems to add to our deductive system. Dr. Gann taught us the importance of saying what we really mean mathematically, and when she was doing so, she always lifted her hand close to her mouth for emphasis, a habit which is apparently now one of mine … my AP Calculus students asked me to remind them to attend to precision one last time (and lift my hand to my mouth when I did) before I sent them to take their AP exam on Wednesday.

At this point, I am no longer attending class with the professors from whom I have learned so much. But I am still attending class with great teachers … they just all happen to be younger than I.

And so thankfully, the lifelong journey of learning continues …

Posted by on May 8, 2014 in Student Reflection

## Running around a Track I

I have been at the T3 International Conference in Las Vegas this week. This morning, I had the opportunity to present a Power Session on CCSS: Bringing content and practice standards together. My next few posts will be a few of the stories from this session.

I gave a class of students Running around a Track I recently.

I started by showing them a picture of the start of the 2012 Olympic Women’s 400 M race. What’s your question? (I took a picture while watching the video.)

The students had good questions to get us started in our exploration.

who won        1

What is the circumfrence of each lane?       1

how far is th last runnr to the circle     1

Why are the not in a line       1

how far away is the first runner from the last runner?          1

which track is longest 1

what are we trying to find      1

what is the ratio from lane 1 to lane 9          1

What’s the circumference of the track?        1

I know everything?   1

why are they not in a line?    1

who gonna win?        1

how far is the outer lane from the inner lane            1

does contestant number 9 have an advantage over contestant number 1        1

What is the ratio of the inner-most lane to the outer-most lane?  1

how much shorter distance does the inner lane runner have to run than the outer runner?         1

how long is the circumference of each circle            1

What are we looking for?     1

Are they all going the same distance.         1

Who won?     1

What is the measure of the intercepted arc?          1

is this the winter olympics     1

Next, I sent out a Quick Poll of a question from an ACT practice test that students should be able answer as a result of the lesson.

Dylan Wiliam talks in his book Embedded Formative Assessment about teachers needing time for collaboration. Our administrators take this seriously, and so our geometry team has the same planning block. One of the other geometry teachers had the idea to include the ACT question in with this lesson.

Who already knows how to solve this? Who knows how to solve it quickly, since the ACT is timed? I removed the choices, just to see how students answered without them. We didn’t talk about the results. I told them we would revisit the question at the end of the lesson. Just over half of the students had it correct initially: 14/26.

We talked about a few of the student questions from the picture of the Olympic race. And then I passed out the student handout for Running around a Track I. I let the students work in groups to answer the questions, and I sent Quick Polls every once in a while to be sure that they were working productively. I sent a QP for question a) and noticed that 20/26 students had it correct. I was able to use the Navigator to determine who had missed it & provide help for them at their desk while the rest of the class worked ahead.

I didn’t send a poll for question b). Instead I provided students the correct answer so that they could determine on their own whether they needed assistance or were ready to move on to part c).

Most students got a bit stuck on part c. I knew this because I sent a poll to find out student responses. No one had it correct.

We used this as an opportunity to figure out incorrect thinking. One student came up to share his work, and the others critiqued his argument, figuring out where their own thinking had gone wrong, and correcting their work.

To close the lesson, I sent the Quick Poll from the start of class:

75% of the students answered it correctly with no choices (up from 50% at the start of the lesson), and 96% answered it correctly with choices.

I think it is interesting to ask students which practices they used when working on a task.

And which was the most used practice:

At this point in the session, we asked the participants to process this part of the story using the protocol “I like …” “I wish …” “I wonder …” for the discussion. What worked? What would you have done differently had these been your students? What would you do differently if you were going to use this task with a group of students in the future? What if you were going to give students Running around a Track II? How would you intend for the lesson to play out?