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Infestation to Extermination

We used a problem from the Calculus Nspired activity Infestation to Extermination recently during our unit on differential equations:

The rate of increase of bugs is proportional to the number of bugs in a certain area. When t=0, there are 2 bugs and they are increasing at a rate of 3 bugs/day.

What does this mean?

I set the mode to individual and watched as students worked.

Many recognized that the rate of change changes.

Several used the initial condition to write a statement about the rate of change.

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Eventually, we went back to the given information to decipher what it was saying.

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And then we (anti)derived the model for exponential growth, which of course students recognized using in a previous math course.

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So what are the constants for this particular model?

I sent a poll to collect their model honestly having no idea that the bell was going to ring in less than two minutes. A few students correctly answered before the end of class.

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Productive struggle isn’t fast.

I should have paid better attention to the time … I really had no idea it had taken us as long as it did. But students were engaged in “grappling with mathematical ideas and relationships” the entire time. That’s got to be better for their learning than them watching me tell them how to work the problem.

What opportunities are you giving your students to struggle productively? Even if you don’t “cover” as much as you think you should?

And so the #AskDontTell journey continues …

 
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Posted by on March 8, 2015 in Calculus

 

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Area between Curves

Our learning goals for the Applications of Definite Integrals unit in calculus are the following:

I can calculate and use the area between two curves.

I can use the disc and washer methods to calculate and use the volume of a solid.

I can use the shell method to calculate and use the volume of a solid.

I can calculate and use volumes of solids created by known cross sections.

During the lesson focusing on the first goal, we used a scenario from a TImath activity The Area Between to start our conversation.

I rarely send the TNS documents as is to my students or give them a copy of the printed student handout (even though I learn from both in my own planning of how the lesson will play out). This activity gave the following information on the first two pages:

Suppose you are building a concrete pathway. It is to be 1/3 foot deep.

To determine the amount of concrete needed, you will need to:

– calculate area (the integral of the top function minus the bottom function

– calculate volume (area multiplied by depth)

The borders for the pathway can be modeled on the interval -2π ≤ x ≤ 2π by

f(x)=sin(0.5x)+3

g(x)=sin(0.5x)

On the next page, graph the functions. Use the Integral tool to calculate the area under f1 and f2. Then, use the Text and Calculate tools to find the volume of the pathway.

Which takes away any opportunity for students to engage in productive struggle.

I shared this instead:

Suppose you are building a concrete pathway that is to be 1/3 foot deep. The borders for the pathway can be modeled on the interval -2π ≤ x ≤ 2π by f(x)=sin(0.5x)+3 and g(x)=sin(0.5x).

(I’m fully aware that giving them even this much information takes away from the modeling process … but there is always give and take, and for this lesson, the learning goal wasn’t whether they could determine functions for modeling the sidewalk.)

They decided to graph the functions.

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And talked about how they could calculate the area between the curves.

They had never used the Integral tool for graphs, much less the Bounded Area tool, so they oohed and aahed gasped in amazement.

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Sydney asked: Is that the only way to get the area between the curves?

(I knew that she was looking for and making use of structure, composing and decomposing the sidewalk into regions with equal area).

I answered: Is it?

We made Sydney the Live Presenter, and she used the Integral tool to calculate the area between f(x)=sin(0.5x)+3 and the x-axis from -2π to 2π.

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So how can we calculate the amount of concrete needed? The integral and bounded area tools are helpful for visualizing what you’re calculating, but you can’t use those tools on the AP Exam.

And so the students decided to calculate the area between the curves and then multiply by 1/3 to get the volume of the pathway.

Because they were able to tell me what to do, I almost didn’t send a Quick Poll to collect a definite integral that would calculate the volume. I wanted to hurry up and get to a card-matching activity similar to Michael Fenton’s that I knew would be helpful, but instead I eased the hurry syndrome and sent the poll.

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What I saw and heard was well worth the time that it took.

Can you spot the students’ misconception?

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Several students were multiplying the definite integral by and by 1/3, to represent height times base times depth, instead of recognize that the definite integral represented height times base (area), and not just height. (They knew this … we had summed the areas of an infinite number of rectangles for a certain base to calculate area under the curve. But they obviously didn’t know this like they needed to.)

When we calculated their integral, we didn’t get (1/3)*37.699, as expected.

 

Next I purposefully choose a region for which the upper and lower boundaries changed.

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We had a nice look for and make use of structure discussion about different ways to write a definite integral for calculating the area of the region.

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Many of you might notice that there is more opportunity to look for and make use of structure for the concrete pathway. I never asked whether you really need calculus to calculate the volume of the pathway. Nevertheless, I feel like I found two good problems/items/tasks to push and probe student thinking. And there’s always next year, as the journey continues …

 
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Posted by on February 25, 2015 in Calculus

 

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A New Function

One of the NCTM Principles to Actions mathematics Teaching Practices is support productive struggle in learning mathematics. In the executive summary, we read “Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.”

In calculus, we started the semester with a unit on Transcendental Functions. On the first day, students figured out everything they could about F(x).

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What is F’(x)?

What is F’’(x)?

What is F(1)?

Where is F(x) increasing, decreasing?

Where is F(x) concave up, concave down?

What is the domain for F(x)? the range?

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Then they sketched a graph of F(x) from what they figured out, and determined that F(x)=ln(x), and F’(ln(x))=1/x.

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(I found the suggestion for students coming up with F(x)=ln(x) by thinking through these questions somewhere else. But I don’t remember where, and I can’t find it anymore.)

 

So the next day, I asked them to differentiate y=log(2x).

I had not given them any “formula” for differentiating logarithmic functions. They had only figured out that the derivative of ln(x) was 1/x.

I sent the question to them as a Quick Poll to watch their progress.

I watched for a long time.

I saw and I heard productive struggle.

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And eventually, their struggle turned into success.

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We can cover so many more examples when we don’t give students time to grapple with mathematical ideas and relationships. But how effective are the examples without the productive struggle?

Ultimately, are my students better off having struggled to think through change of base to get to the derivative of log(2x) using what they already know about the derivative of ln(x)? Or would they have been better off with me giving them the textbook way to calculate the derivative of logb(x)?

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I’m hoping for the former, as the journey continues …

 
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Posted by on February 9, 2015 in Calculus

 

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The Diagonals of an Isosceles Trapezoid

It was the day before the test on Polygons, and so I thought that writing a proof and then giving feedback on another team’s proof might be helpful.

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Students worked alone for a few minutes, thinking about what was given and what could be implied. Then they worked with their team to talk about their ideas and to begin to plan a proof.

Some were off to a good start.

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Some were obviously practicing look for and make use of structure.

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Some were stuck.

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I talked to several groups, listening to their plan, asking a few questions to get them unstuck.

And then I got out colored paper on which to write the team proof.

 

The clock was ticking, but I thought that surely they would be able to trade proofs with another team for feedback within a few minutes.

 

I talked to another group. They were reflecting ∆ABC about line AC.

What will be the image of ∆ABC about line AC?

The answer? ∆ACD.

Of course that is wrong. It seems so obvious that ∆ABC is not congruent to ∆ACD. And I’m also wondering how that helps us prove that AC=BD, since BD isn’t in either of those triangles. But that’s where this team of students is. I now have the opportunity to support their productive struggle, or I can stop productive struggle in its tracks by giving them my explanation.

 

My choice? Scissors. And Paper. And more time.

What happens if you reflect ∆ABD about line AC?

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Oh! The triangles aren’t congruent.

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So are there triangles that are congruent that can get us to the diagonals?

∆ABC is congruent to ∆BAD.

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How do you know?

A reflection.

About what?

This pencil!

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So what is significant about the line that the pencil is making?

It’s a line of symmetry for the trapezoid.

It goes through the midpoints.

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(One of the team members was using dynamic geometry software to reflect ∆ABC in the midst of our conversation, but I don’t have pictures of her work.)

 

So the plan was for team to write their proofs on the colored paper and then trade with other teams for feedback. Great idea, right? So how do you proceed with 15 minutes left? Proceed as planned and let them give feedback with no whole class discussion? Or have a whole class discussion to connect student work? Because as it turned out, no two teams proved the diagonals congruent the same way. I chose the latter.

 

I asked the first team to share their work.

 

Their proof needs work. But they have a good idea.

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They proved ∆AMD≅∆BMC, which makes the corresponding sides congruent, so with substitution and Segment Addition Postulate, we can show that the diagonals are congruent.

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Next I asked the team to share who proved ∆ABC≅∆BAD using a reflection about the line that contains the midpoints of the bases. Their written proof needs work, too. But they had a good idea.

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Another team proved ∆ACD≅∆BDC.

 

Another team constructed the perpendicular bisectors of the bases. Since the bases are parallel, a line perpendicular to one will be perpendicular to the other. I’m not sure they got to a reason that the perpendicular bisectors have to be concurrent. They could have used ∆AZD≅∆BZC to show that. Instead, they used a point Z on both of the perpendicular bisectors (they know that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment) to reason that ∆AZB and ∆DZC are isosceles & then used Segment Addition Postulate and substitution to show that the diagonals are congruent. Not perfect. But a good start.

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NCTM’s Principles to Actions discussion on support productive struggle in learning mathematics says, “Teachers sometimes perceive student frustration or lack of immediate success as indicators that they have somehow failed their students. As a result, they jump in to ‘rescue’ students by breaking down the task and guiding students step by step through the difficulties. Although well intentioned, such ‘rescuing’ undermines the efforts of students, lowers the cognitive demand of the task, and deprives students of opportunities to engage fully in making sense of the mathematics.”

 

So while I didn’t rescue my students, we also never made it to an exemplary proof that the diagonals of an isosceles trapezoid are congruent. Did they learn something about make sense of problems and persevere in solve them? Sure. Is that enough?

 

Would it be helpful to lead off next year’s lesson with this student work? Or does that take away the productive struggle?

 

Is it just that we have to find a balance of productive struggle and what exemplary work looks like, which is easier in some lessons than others? If so, I failed at that balance during this lesson. Even so, the journey continues …

 
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Posted by on December 1, 2014 in Geometry, Polygons, Rigid Motions

 

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Productive Struggle: The Law of Sines

NCTM’s Principles to Actions suggests eight Mathematics Teaching Practices for teachers. One of them is to support productive struggle in learning mathematics. The executive summary states: “Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.”

What does productive struggle look like? What does it sound like?

I saw a glimpse of what productive struggle looks like yesterday. I get to share a room with a teacher (who happens to be a former student of mine), and so I listen with one ear when I’m in the room working at my desk during her Precalculus class. The lesson was on the Law of Sines, but Trisha didn’t tell the students from the beginning that was the learning goal. Instead, the students focused on the math practice make sense of problems and persevere in solving them.

She presented a situation. And the students made assumptions and asked questions.

One I remember hearing was “I guess we can’t just use a measuring tape?”

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Then she asked them to solve the problem.

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And so they did. These students didn’t balk at the task. They all worked. They didn’t even talk very much at first … you could hear them thinking in the silence that encompassed the room. That’s when I looked over and realized that I was seeing productive struggle in action. Productive struggle isn’t always quiet, but it definitely started that way for these students. Eventually, students listened to Ain’t No (River Wide) Enough while they worked.

When solving the non-right triangle without knowing the Law of Sines, the students used another Math Practice – look for and make use of structure – to draw auxiliary lines. Some drew an altitude for the given triangle to decompose it into two right triangles. Some composed the given triangle into a right triangle.

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Trisha collected evidence of what students could do using a Quick Poll.

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So if we are given one side length and two angle measures of a triangle, is there a faster way to get to the other side?

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More productive struggle … the numbers are now gone, students are reasoning abstractly to make a generalization.

And they did.

And they derived the Law of Sines in the meantime.

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How often do we give our students a chance to engage in productive struggle? In how many classrooms is the Law of Sines just given to students to use, devoid of giving students the opportunity to “grapple with mathematical ideas and relationships”?

When I discussed what I saw with Trisha, she noted that last year, only a few of the students in her class successfully solved the triangle prior to learning about the Law of Sines. This year, all of them tried and most of them succeeded. These are the students with whom we started CCSS Geometry year before last. These are the students who have been learning high school math with a focus on the Math Practices. These are students who are becoming the mathematically proficient students that we want them to be. Because we are letting them. As the journey continues, we are learning to leave the front of the classroom behind so that we can support productive struggle in learning mathematics.

 
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Posted by on November 6, 2014 in Geometry, Trigonometry

 

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SMP1: Make Sense of Problems and Persevere #LL2LU

We want every learner in our care to be able to say

I can make sense of problems and persevere in solving them.  (CCSS.MATH.PRACTICE.MP1)

SMP1

But…What if I think I can’t? What if I’m stuck? What if I feel lost, confused, or discouraged?

How might we offer a pathway for success? What if we provide cues to guide learners and inspire interrogative self-talk?

 

Level 4:

I can find a second or third solution and describe how the pathways to these solutions relate.

Level 3:

I can make sense of problems and persevere in solving them.

Level 2:

I can ask questions to clarify the problem, and I can keep working when things aren’t going well and try again.

Level 1:

I can show at least one attempt to investigate or solve the task.

 

In Struggle for Smarts? How Eastern and Western Cultures Tackle Learning, Dr. Jim Stigler, UCLA, talks about a study giving first grade American and Japanese students an impossible math problem to solve. The American students worked on average for less than 30 seconds; the Japanese students had to be stopped from working on the problem after an hour when the session was over.

How may we bridge the difference in our cultures to build persistence to solve problems in our students?

NCTM’s recent publication, Principles to Action, in the Mathematics Teaching Practices, calls us to support productive struggle in learning mathematics. How do we encourage our students to keep struggling when they encounter a challenging task? They are accustomed to giving up when they can’t solve a problem immediately and quickly. How do we change the practice of how our students learn mathematics?

 

[Cross posted on Experiments in Learning by Doing]

 

 
 

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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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