Our First Day Message

What message do your students leave class with on the first day? How do you craft the first day learning episodes to promote that message?

Our students walked into the room with two Which One Doesn’t Belong scenarios.

Which led to a discussion about working on our math flexibility throughout the year.

We wondered what one word students would use to describe their feelings about math.

Algebra 1:

Geometry:

AP Calculus:

(I wasn’t surprised at the negative feelings towards math in Algebra 1 … but I was surprised at some of the responses from geometry and AP Calculus students.)

That led to the Quick Poll that we’ve sent now for a few years from Mindset.

We did a short number talk that I saw on Twitter:

Without counting all of them, how many total shaded triangles are in this picture? How do you know? #mathchat pic.twitter.com/PNDeWdHIli

— David Wees (@davidwees) July 27, 2015

We watched Jo Boaler’s Day 1 Week of Inspirational Math video on mindset and mathematics.

We ran out of time to do our normal opener where students find more than one way to complete a sequence of terms.

We thought about what it looks like when a team is working well together in math class.

TeamWork – Calculus (using Navigator for handhelds)

cohesive         1

everyone is building on each other’s idea 1

synchronization         1

organized discussion            1

clockwork       1

coordinated   1

everyone ends up with the correct answesr and understands   1

everyone ends up understanding the problem    1

communication and listening          1

people help each other, in an oreo 1

everyone gives input            1

everyone talking/ explaining at the same time.    1

clear discussion         1

All ideas and all teammates are listened to.           1

a well-oiled machine.            1

lots of words, pointing, and ideas    1

It looks synchronized            1

Like a well-oiled machine.    1

you are productive and no one gets left out          1

A solid conclusion shared by the entire group is reached based upon the well thought out ideas contributed each individual   1

dividing and conquering      1

efficent           1

We are very focused and productive.         1

lots of high fiving and excitement   1

lots of talking and no one hates each other           1

looks like an oreo      1

Organized discussion            1

explaining thought processes to each other          1

success           1

helping each other and an oreo      1

TeamWork – Geometry (using Navigator for Networked Computers)

People are building off of other ideas and are able to make an educated descision using everyone’s input.           1

When a team is working well together they may not always agree on the answer or how to get to it; they always work to find the right one no matter how difficult or stressful it may be.            1

When a team works well together they are listening to each other and trying to trying all methods (to solve). Everyone listens as much as they talk.     1

When a team is working well together, there will be a lot of communication and listening to others’ ideas and meathods.          1

The team looks uniform and united, and if you fail, then your team can figure out what to do better next time.   1

People bouncing ideas off of each other and creating new ideas or improving old ones. Even if some people are wrong, people correct them in a kind way and tell them how to correctly solve a problem. All team members are putting in effort and carrying their weight, not just leaving others to do all the work. The team finds multiple ways to solve the problem and chooses the best one.    1

All team members working together and solving problems at the same time individually then comparing answers and learning from different views.  1

When a team is working well together, each individual student listens to one another and actually thinks about each teammate’s idea and sees it as a viable solution. A team is working well together when they get along and are respectful to one another.        1

When a team works well together, they work easily and doesn’t argue when someone has a different answer than the other person. And they get the answer right. They should be able to recieve and give feedback from the other.       1

When a team, in math, is working well together, I think of a deep conversation of different ideas. I see different solutions and ways a proplem could be solved coming from all sides of the table… if you know what I mean. 1

When no one argues and everyone considers others solutions to a problem.  1

It looks very good and fluent when your team is working well together. When you have a team to help bounce ideas off of each other and to help each other reach the goal they need it is very useful. Everybody is going to make mistakes, and when your team knows that and will help you to find what you did wrong, you will have a larger success rate.    1

When your team is working well together, every member is sharing their thoughts and ideas, right or wrong. The team members aren’t embarrassed about getting the wrong answers because they know that their other teammates will help them to understand their mistakes and learn from them. Each member comes up with different ideas when the team works together, so that way every person benefits wih a new way of thinking.     1

The team is able to get more done, and do it quicker.      1

When a team is working well the group closer together and they’re all listening to other team members input and are also giving their own input.  1

It looks productive and focused. We are all concentrating on the problem that the team has to solved.      1

It looks like we all know what we are doing. It looks like we have more ideas and know more ways to solve our problem. It makes us look as if we understand the problem more and in most cases we probably do, when we work together well.     1

You are makeadvancements and improve one another and also agree upon an answer.       1

When the team is working well together it helps other people of the team to increase their knowledge because each person may see things in a different way.            1

Everyone is not arguing. People are using teamwork and getting the right answer while teaching others on the team how to do it differently.            1

Everyone is learning and helping each other when they may not know the answer to something or need help figuring out how to do something. No one makes fun of anyone if they get a wrong answer because we all need to learn and grow.    1

Somebody will suggest something and people will get excited or say “”yes!””. Then another person will suggest something and everybody will enjoy making progress. People that originally disagreed will change their minds because of something another person explained. They will keep working together until the project is finished.      1

The team is working well when they all have corresponding ideas that come together to get a problem correct.   1

When a team is working well together the team everything flows and everyone is participating. Everyone is helping, ideas are exchanged, and people are learning. Everyone is set on one goal and everyone is headed to achive that goal.           1

It is when everyone in that team is listening to and coming up with ideas. The whole group is cooperating and completing the task given. That is when a group is working well together.     1

Everyone understands the objective and is comprehending well. They understand why and how the team got the answer. No one is confused and everyone feels like they are making a contribution to the work     1

We talked about Popham’s four levels of formative assessment.

Our goal is for all students to reach the learning goals and not just for the “smartest” and “fastest” to do so. Which means that we will have to help each other. Which also gets into Wiliam’s Five Key Strategies for formative assessment, in particular, activating students as learning resources for one another.

The bell rang before I could have my students reflect on what they learned and what they will do in geometry this year, so I sent them a Google form to complete outside of class.

Algebra 1 students answered a Quick Poll before leaving about what they learned.

today, I learned that anyone can learn math

This year i am going to improve in math   1

About my teachers to be better at math.   1

I have learned If I practice I can become better.

In math this year, I am going to practice more.     1

be able to learn more about math and I will try harder. 1

I have learned that I have three very cool teacher that will make math fun this year and I also learned that math isnt everybodys fav but evryone a math person

In math this year i am going to get better at it      1

You can’t be a math person everyone is good at math it you pratice, so i am going to pratice.           1

I have learned that all three of my math teachers went to NWRHS and in math class this year i will not get bored in class         1

i have learned to think outside of the box. i am going to learn to thinkoutside the box          1

that we have three teachers.

try to work out everything better. 1

that math will be exciting this year and every one is good at math       1

That everyone is good in math in some way. In math this year, I am going to have an A.       1

I have learned that my teachers are cool

in math im going to inprove.            1

I have learned how to think outter box .

I am going to try to do better than the year before in math.       1

I have learned this year to be more excited to more about math.

In math this year, I am going to try my hardest.   1

That I have 3 math teachers. Also math wont be so bad. 1

pay attention 1

nothing and this year I’m going to start fresh       1

I have learned about my classmates and teachers.

In math this year, I am going to try harder.           1

I learned my other classmate’s names and how to do Four 4’s.

In this year of Math I plan on working harder and to advance in my knowledge of Math.     1

I have learned that you need to thing hard to get some of the questions right.

In math this year, I am going to make my math skills better and i want to work math fast.   1

I learned that everyone can do math. I math this year, I will work really hard and do the best I can.          1

I have learned how everybody can do math even if you are not a math person and your brain grow if you work it out. In math this year i am going to try to learn more and get a better grade in math.   1

I have learned in this class period to hold a more open mindset towards math and the many different answers to a mathmatical situation. Different methods can be accepted, and not every person will see math in the same light. I am going to learn to find an excitement in the subject of math this year, hopefully.           1

i have learned peoples names in my class in math this year, i am going to make good grades           1

I have learned that anybody can learn algebra and be good at it. This year I’m going to payattention and do my best at algebra.         1

I have learned that everyone is smart at math, people just need to embrace it.

In math this year, I am going to become smarter in math.          1

I have learned… that you can get differnet answers just by using 4 4s, also I learned how everyone is well in Math, it’s just kids with more experience are more better in it.

In math this year, I am going to.. learn more about math and hopefully actually start to like it.        1

i have learned everybody is good at math and i am going to try my best to make good grades         1

Everyone can learn about math very well, and no one is a math person. I am goin to learn about everything that i can and try my best at it.I am also gonna try and figure out problems the best of my ability.   1

I have learned that the more you practice and work your brain the more it grows.

In math this year, I am going to practice and work my brain so I can learn and get better.   1

Today I have learned that practicing something, even if you don’t fully understand it, still helps. This year I am going to try my hardest to achieve high grades in math.           1

I have learned today in class, everytime you have to find something out there can always be more than one way to it.

In math this year, I am going to try and learn new things that I didnt understand last year.            1

Today I learned that no matter how much i get frustrated that i will always have someone to leaN ON THIS CLASS TO HELP ME WITH MY SITUATION … In math thjis year imgoing to succeed in all things all I do my grades will also be also better.           1

i have learned that anyone can be good at math with good practice.

in math this year im going to learn different things and hopefully get better at math.           1

practice math everyday and that it’s okay to make mistakes because we learn from them.   1

that we can learn things easier by practice so in math this year, i am going to practice if i have troubles until i fully understand it.       1

I have learned that the brain can grow with what you learn.

In math this year, I am going to study more.         1

I have learned that your brain is always changing and to get better at something you have to keep practicing.

In math this year, I am going to practice what I’ve learned in order to get better at it.           1

I have learned that your brain can grow just by learning new things.

I’m going to practice more on math because, if you practice more on something you can get better at it.    1

That math doesnt always have t0 be boring, it cab be reall        1

i am going to try my best to past     1

I have learned that everyone is capable of being in the highest math class there is. There is no such thing as a “”math person””. In math this year, I am going to try my hardest to maintain the highest grade I can, and pay attention in class in order to make good grades.   1

practice and do my best to make a good grade even thooe me and math REALLY dont like eachother. but im going to try to do my best            1

IF you practice you can get better at it       1

I learned that anyone can do math well and in math this year, I am going to try.        1

that your brian can grow the more you practice something , this year i am going to pay more attention.    1

I have learned that anybody can be good at math, you just have to work for it.

In this year of math, I am going to pay more attention than I did last year so I can get better and so I can get good grades.        1

I have learned if I keep practicing and going to different levels I will become better.

This year, I am going to study and practice math on different levels so, I will become better at math.         1

We didn’t really have a lesson, but I have learned about how the brain works with how good you are at something. In math this year, I am going to have no tardys, have straight A’s, and not disrespect the teacher in any manner.    1

that i dont have to dread coming to math. i can grow and learn from it. im going to try to make all a’s        1

You can get better at something if you practice at it.

I am going to try harder. And give up if I don’t understand it.   1

I have learned that as long as I continue to practice math or algebra I will slowly get better at it.

In math this year, I am going to attept to be more optemistic about the work and try harder to get better grades.           1

be able to be good in math by the end of the year. and that i will be able to succeced in anything that i do by just practicing.    1

I learned that when you have a difficult time with a question that your brain is growing at the same time.           1

We went from math is “complicated, hard, frustrating, …” to “I can do this” in 95 minutes. I believe our students left hearing our first day message:

Everyone can learn math.

Our brains are growing when we struggle to solve a problem.

There isn’t just one way to solve a problem.

Learning more than one way to solve a problem grows our mathematical flexibility.

Working with a team is an important part of how we learn mathematics.

And so the journey continues as a new school year begins …

Advertisement

 

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 …

 

Unit 11 – Modeling with Geometry

Student Reflections

I can statements:

Level 1: I can create a visual representation of a design problem. 100% Strongly Agree or Agree

Level 2: I can decompose geometric shapes into manageable parts. 94% Strongly Agree or Agree

Level 3: I can estimate and calculate measures as needed to solve problems. 100% Strongly Agree or Agree

Level 4: I can use geometry to solve a design problem and make valid conclusions. 100% Strongly Agree or Agree

 

Standards:

Modeling with Geometry G-MG

Apply geometric concepts in modeling situations

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

 

Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used make sense of problems and persevere in solving them the most and model with mathematics next.

 

Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

11A Volumes of Compound Objects

11B 2D Representations of 3D Objects

11C A Tank of Water

11D Popcorn Picker

11E Hot Coffee

11F Two Wheels and a Belt

• I didn’t think any of these were repeated. We really looked at geometry and used it to solve hard real world problems, and I’ve never had to do that before.
• I do not feel like this was a repeat. Many of us knew about basic areas and volumes of course, but in this unit we went into depth with them and learned how to get the dimensions of more complex shapes. I think we covered new material, because we have mainly been going over coordinates, trig, and triangles previously and this seemed like a new and fun topic to cover for us.
• I already knew how to find the volumes of compound objects from unit 10.

 

Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit?

• 11F 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.
• The coffee one helped me because it made me talk with others at my table and look for ways to solve the problem.
• Although I enjoyed the popcorn activity, I feel like the most helpful activity was the one where we cut out the cards and thought through the gradual shapes of the water as it emptied the top figure and filled the bottom figure.  It allowed me to work with a team to reason out our arguments and working with my peers in a collective effort enlightened me with thoughts and ideas that I had not previously thought of or would have otherwise ventured to discuss.
• 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.

 

What have you learned during this unit?

• 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.
• How to use the least amount of information to find the need item.
• 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.
• I have learned to attend to precision. Throughout not only the homework lessons but also in class, I learned to be careful and slow down. I often hurried over the question and did not take into account the measurements. For example, I hardly ever noticed when the question gave dimensions in feet but asked for the answer in inches. It’s not a hard concept, but it requires patience and effort that I was trying to shortcut on.
• During this unit, i learned a lot about using what i already knew and combining that to solve difficult problems.
• I’ve learned how to break down 3D shapes into simple 3D shapes so I can get their volume, and how to do actual geometry problems with all of the things I’ve learned this year. I’ve learned what the different cross sections of objects can look like, and I learned that the world’s largest coffee cup help 2015 gallons of coffee.
• I learned that I need to model with mathematics more often.

 

And so the journey continues … figuring out how to provide more opportunities throughout the course for my students to model with mathematics.

 

Unit 10 – Geometric Measure and Dimension

Student Reflections

I can statements:

Level 1: I can use formulas to calculate area and volume. 100% Strongly Agree or Agree

Level 2: I can identify cross-sections of 3-D objects, and I can identify the 3-D objects formed by rotating 2-D objects. 90% Strongly Agree or Agree

Level 3: I can explain the formulas for area and volume. 100% Strongly Agree or Agree

Level 4: I can calculate the area and volume of geometric objects to solve problems. 100% Strongly Agree or Agree

 

Standards:

Geometric Measurement and Dimension G-GMD

Explain volume formulas and use them to solve problems

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

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

3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects

4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

 

Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used look for and make use of structure the most. They used make sense of problems and persevere in solving them and reason abstractly and quantitatively next.

 

Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

10A Length Area

10B Surface Area

10C Volume

10D Enlargements

10E Cross Sections

10F The Best Box

10G Performance Assessment

10H Mastering Arc Length and Sectors

• I already knew length and area. I had been taught the formulas for surface area, but I didn’t actually understand why those formulas worked until this unit. Cross sections were completely new to me.
• I felt like I already knew some of the information for calculating volume and area of 3D forms, but in this unit I learned why and how the formulas work.
• The length and area was a bit of a repeat of what we’ve learned in the past. It was nice to review and be able to bring to memory what we’d already learned as a basis for the new things we were going to learn this unit.
• This Volume lesson was a tiny bit of something I already knew, but I never got as much into it as we did. Before, I just learned the formula and never understood. I also only ever learned the volume of a rectangular prism.
• The very first lesson in which we used the basic formulas was a bit of a repeat from past years. However, in the past we had to memorize the formulas, whereas this year we were given the opportunity to understand the why the formula is what it is. I find that understanding it helps more than simply memorizing it.

 

Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit?

• Yes, Length and Area helped me with Surface Area and Volume by making sense of structure within the problem.
• When you got the orange peels and separated them into four circles to show how the surface area of a sphere worked, it really helped me understand how surface area works. After that I started thinking about how the formulas worked instead of just memorizing them.
• 10G Performance Assessment helped me out the most. It summed up most of the chapter and helped my apply what I learned to many different problems.
• I think the lesson of volume helped the most in understanding the formula. When we talked about the different figures fitting into one another it helped me understand why this certain formula was there. It helped show me that all formulas are taken from a common one.

 

What have you learned during this unit?

• In this unit I have learned how to use surface area, lateral area, volume, and cross sections. I now have a better understanding from where the formulas come from by looking at the figures and in a way putting together the puzzle pieces to come up with the final formulas.
• I learned exactly how area, surface area, and volume are all different.
• I have learned how to calculate lateral area, surface area, and volume of forms and I also learned why the formulas work and not just to plug in numbers to those formulas.
• I have learned how to identify cross sections and explain the formulas for surface area and volume.
• I’ve learned how to use formulas to calculate area and volume and why we use those formulas. I’ve learned how to make sure to convert problems to the same dimensions and same ratios, and I’ve learned what cross sections are and the different ones in different 3D figures.
• I have learned to understand the formulas and not just memorize, but really understand them. I can use these formulas to solve word problems and figure out tough questions. I’ve also learned about cross-sections and how they differ for each 3D shape. I have also understood the surface areas of 3D shapes and how they work with all shapes.
• During this unit, I have come to understand the geometric formulas, and I believe that understanding will help me later on in tests like the ACT if I were to forget the formula that I memorized in 7th grade, I would remember the concepts and understandings of why the formula is what it is.

 

And so the journey continues … still struggling to find a balance between teaching conceptual understanding of mathematics and using mathematics to solve problems … although I do think the students’ language when reflecting on their understanding of geometric measure and dimension is a good sign.

 

Unit 9 – Coordinate Geometry

Now that school is out, I am going back through our last few student surveys to make notes for next year.

I can statements:

Level 1: I can represent and use vertices of a geometric figure in the coordinate plane. 100% Strongly Agree or Agree

Level 2: I can use the equation of a circle in the coordinate plane to solve problems. 96% Strongly Agree or Agree

Level 3: I can use slope, distance, and midpoint along with properties of geometric objects to verify claims about the objects. 96% Strongly Agree or Agree

Level 4: I can partition a segment in a given ratio. 81% Strongly Agree or Agree

 

Standards:

Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Use coordinates to prove simple geometric theorems algebraically

4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

 

Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?

Students answered that they used look for and make use of structure the most. They used make sense of problems and persevere in solving them and model with mathematics next.

 

Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

9A Lines

9B Quadrilaterals

9C Triangles

9D Area & Perimeter

9E Circles

9F Translating Circles

9G Polygons

9H Loci

9I Coordinate Geometry Performance Assessment

9J Mastering Coordinate Geometry – Partitioning a Segment

• Yes and No, for me most of the material was new and different. I learned how to partition, and use loci. I already knew the equations but in the course of Unit 9 I learned how to use equations like the Pythagorean theorem to find the side area and perimeters of a triangle.
• I already knew how to find the area and perimeter of simple geometric shapes but not all the shapes.
• Pretty much none of these were repeats. I’ve heard of bits of these lessons, but I learned so much more during all of the units that I felt like those bits don’t count.
• I already knew how to find the slope and do other things in 9A. I learned it in 8th grade Algebra and it was a nice break from just learning new things everyday and just having to review that day.
• The lesson about the equations of lines and the lesson about area and perimeter were things that I had been taught before, but I was taught them in different ways that weren’t as in depth as what we went to in class.
• Yes, I did feel that content from numerous lessons (namely 9A, 9B, and 9C) was derived from lessons that I have previously covered either in this class or a previous math-related class. For instance, dealing with parallel lines and perpendicular lines, as well as distinguishing between the two by looking at the x-intercept of an equation’s standard form, was a very prominent concept in the algebra I course I took last year. Also, as were the concepts of calculating distance, midpoint, and slope, all of which were key concepts last year. Finally, I felt that observing shapes on a coordinate plane, as was practiced in lessons 9B and 9C with quadrilaterals and triangles, respectively, resembled a lesson that we covered in November/December as part of units 4 and 5, in which we specifically observed each shape and its properties.

 

Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit?

• There was no particular activity that really sticks out in my mind, but simply watching how others worked the problems on the board really helped. Many people brought up ways to do things that I never thought of, and a majority of the time, their way was more effective than mine.
• Yes, I think in unit 9E when we had to write a definition for circle in quick paw helped me to get a better understanding of what a circle is and to understand why how I defined it was wrong.
• 9H was one of the hardest ones for me to do but I think it helped me a lot because it made me think harder than the others.
• Just like in most all the units, the mastering lesson always helps me. This just lets me go back through everything and make sure I understand it.

What have you learned during this unit?

• I learned how to gather measurements for shapes from their coordinates. Also, though, I learned to really persevere in solving problems. This unit was one of the harder ones for me, and a lot of times I wanted to give up. I kept trying, though, and when I would eventually understand a problem, it was worth it.
• I have learned how the coordinate plane and geometry are 2 sides of the same coin and that you can use graphs to calculate geometric properties.
• I learned that you can find the area and perimeter of all shapes not just the simple geometric shapes.
• I learned a lot in this unit. I learned about all the shapes and how to use equations to solve problems involving them. I learned what loci was which was new and kind of confusing. I also learned how to partition a ratio for a given segment which was cool because it took a lot of different ways to look at the segment that i wouldn’t have thought of before learning this. This unit contained a lot of review from past years, but also a lot of new material that i had never seen before.
• If this unit has taught me anything, it is that geometry and algebra can be collaborated so that multiple relationships between and details of certain points, lines, and shapes can be proven, all made possible by borrowing the coordinate plane, distance formula, midpoint formula, and slope formula, all in which we became fluent through two years of algebra.

 

We have struggled to know whether it is best to have a separate coordinate geometry unit or whether we should be using coordinate geometry all year. To me, it is kind of like wondering in a calculus class whether you do early or late transcendentals. I like late transcendentals, because students go back through differentiation and integration towards the end of the course, and they really have to think about which one they are using and why. It turns out to be a nice culminating review that wouldn’t happen if we had been differentiating exponential and logarithmic functions along the way.

 

And so the journey continues … always trying to figure out what works best for student learning.

 

Great Tasks: Infinity Pizza

Connie Schrock and I recently presented a webinar on Great Tasks:

What is a “great task”? How is a great task used effectively in the classroom? Published by the National Council of Supervisors of Mathematics (NCSM) with support from Texas Instruments, NCSM Great Tasks for Mathematics 6-12 provides opportunities for students to explore meaningful mathematics by challenging their thinking about essential concepts and ideas. Explore tasks that are aligned to the Common Core content standards, with special attention given to the Common Core Mathematical Practices. Learn how tasks can be enhanced with the TI-84 Plus family and TI-Nspire™ technology to promote mathematical reasoning and active engagement for students of the 21st century.

I shared a story about using one of the Great Tasks with my students, which I have included in this post.

My geometry students are in a unit on Geometric Measure and Dimension. Last week, I gave them the first part of the Infinity Pizza task. Students are asked to create a fair method for cutting any triangular pizza into 3 equal-sized pieces of pizzas. I asked students to work alone for a few minutes before they started sharing what they were doing with others on their team. I walked around and watched. Many students started on paper.

I like seeing the progression of ideas from this student.

Of course at least one student had to decorate her pizza with pepperoni.

My students and I use TI-Nspire Navigator to help manage our classroom. I took a Class Capture to see what students were doing on their handhelds. Last year I read Smith & Stein’s Five Practices for Orchestrating Productive Mathematical Discussions.

The practices for teachers are to anticipate how students will respond to a task, monitor while students are working on a task to select which student work should come out in a whole class discussion and sequence that student work so that students will ultimately be able to connect the mathematics in the different solutions.

Let’s look at the second class capture I took – I can update the Class Capture as often as I want manually by pressing the Refresh button, or I can set it to update automatically every 30 seconds, which I often do. Whose work might you choose to discuss first if you were leading the class discussion?

I chose Alex first. Alex proceeded to tell us that he joined midpoints together and ended up creating a pizza with four equal slices instead of three. Alex answered a different question, but he gave us some good mathematics to remember about the triangle formed by the midsegments.

Next we moved to Kristen. Kristen was in the process of trying to trisect the angle. She hadn’t made it to whether the triangular pieces were equal in size.

Then we moved to Reagan. I made Reagan the Live Presenter, which meant that her calculator and interactions with the calculator were showing at the front of the room. Reagan told us that she created the medians of the triangle to cut her pizza. You can see from the measurements that she took that the pieces are equal in size. Reagan had hidden the lines that contained the medians to make her pizza look better, but it is by seeing the six triangles formed by the medians that she was able to explain why her three triangular pieces of pizzas have equal area.

I called on one other student to share his solution – but I have to show two other pictures that I had taken of students working on paper.

Flynt had thought about using the centroid to divide the triangle into three equal areas, and he had also noted that he could partition the base into three equal parts. One of the CCSS Standards for geometry talks about partitioning a directed line segment into a given ratio. We worked on that standard in our last unit, and it really excites me for my students to attend to precision with the language that they are using during class. I’ve been teaching geometry for 20 years, and my students have not used that language in the past.

Benny also talked about partitioning. He suggested that we needed to partition the base of the triangle into a 1:1:1 ratio.

I asked Benny to share what he had done with the rest of the class. He did, and one of the students asked whether it mattered which side of the triangle he used as the base. Benny said no, but he wanted the image rotated (which is quick and easy thanks to our interactive whiteboard) so that the second side he showed as the base would still be horizontal, and on the bottom. That’s the image you see on the bottom right.

You can see that Katlyn was using the same method as Benny with technology; however, if we look at her file, we can tell that she had not yet figured out how to trisect the base. We always have more to learn.

There is a part 2 for the Infinity Pizza task that I haven’t had time to try with my students yet. Every year, though, they come knowing more mathematics than the year before. Maybe next year we will get to Part 2!

My goal for my students is not just to provide opportunities for them to learn the content standards and use the math practices while they are working on a task – but for them to begin to recognize when they are using the practices. I gave my students a list of the practices along with a short explanation of each practice at the beginning of the year. I have that same list posted in the front of my room. This language has become part of how we talk to each other about what we are doing in class – and what our goals are for learning in each unit. Which math practice would you expect to see your students demonstrate with the Infinity Pizza task?

I asked my students which practice they think they used the most during this task. They completed their responses on a Google Form.

AK: “In this challenge, we used many of the Math Practices, but the one that sticks out to me is construct viable arguments and critique the reasoning of others. I think this was most prominent because of the way each student was given the opportunity to share their solution and the class in turn was given the opportunity to compare and support that student’s method or argue for a different approach. Either way it provoked in-depth thoughts to look beyond the simple “find the solution and move on to the next problem”.”

Katie noted, “I chose [reason abstractly and quantitatively] because when you have a problem like this it’s easier to look at numbers and general terms. I first had to think about the question. … It was hard trying to think of where to start, so I thought about cutting the pizza with the lines connecting the midpoints together. This would have worked had the quadrilateral these lines made been a square, but it wasn’t because I was using a right triangle. So I went to look at the lines of medians. Well, the area for each triangle then was not equal. This was trying my patience, but I kept going. I then looked at the circumcenter and the lines connecting that, but it ended up just like the first one, I had three triangles and a quadrilateral. Which was four pieces that definitely did not have the same area. Then I just thought for a few minutes. It then dawned on me that we were dealing with area and the formula for the area of a triangle is 1/2bh(where b=base and h=height). So, I thought how can I get three triangles from one where the base and height never change. I then used 21 for my base of the triangle because 3 will go into 21, 7 times. I had three triangles with a base of 7 and the height couldn’t possibly different because they all came from the same triangle.”

Chandler: “I felt that I mostly modeled with mathematics throughout this activity because the entire goal of Infinity Pizza was to utilize a certain math practice (or more than one) in order to determine how to evenly split a pizza into three perfectly even pieces. … Since I knew that the centroid would give me the point of central mass, I searched for the centroid by drawing a line from the perfect center of each angle to the exact median on the opposite side of the triangle. … Note that this was conducted on an equilateral triangle. Next, to elaborate, I tried the same procedure on an isosceles triangle and a scalene triangle. The results were conclusive — my method was feasible for any triangle despite any side length differences, even if the pieces’ sides were of different lengths. …”

Kelsey: In this activity you had to construct viable arguments and critique the reasoning of others the most. The reason I say this is because you had to come up with a way to split the pizza into 3 equal slices and you had to persuade the class that your reason worked everytime. Also you had to listen to other peoples reasons why and critique if their answer was wrong or write.

Roselynn chose make sense of problems and persevere in solving them: “In order for us to solve this problem, we had to have perseverance. This wasn’t a problem where the answer took a minute or two to solve. It took us a good portion of the block to construct our argument. It also took many approaches. We didn’t get a good answer the first try. It took a while to make sense of the problem, and we had to persevere in solving it.”

And so the journey continues, trying to provide students great tasks that help them make sense of mathematics …

 

Unit 7: Right Triangles – Student Reflections

I have just read through student reflections for our unit on Right Triangles.

Unit 7 – Right Triangles

Level 1: I can use the Pythagorean Theorem to solve right triangles in applied problems. G-SRT 8

Level 2: I can solve special right triangles. G-SRT 6

Level 3: I can use trigonometric ratios to solve right triangles. G-SRT 6, G-SRT 7

Level 4: I can use trigonometric ratios and special right triangle ratios to solve right triangles in applied problems. G-SRT 8

Similarity: G-SRT

Define trigonometric ratios and solve problems involving right triangles

G-SRT 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

G-SRT 7. Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Lessons:

7A – 45-45-90 Triangles

7B – 30-60-90 Triangles

7C – Trigonometric Ratios

7D – Solving Right Triangles

7E – Performance Assessment (Hopewell Triangles)

7F – Mastering Right Triangles

7G – Assessing Right Triangles

Most students felt like the Math Practices they used most in this unit were Construct viable arguments and critique the reasoning of others and Attend to precision.

Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

• Trigonometry is a completely new subject for me, so no, none of the lessons were repeats to me. I had no idea about anything except for the Pythagorean Theorem honestly.
• A student from Iran: yes, I leaned part 7A,7B and 7D in middle school, but the difference was that in middle school we just taught how to use Pythagorean theorem and that the side of triangle opposite to 30 angle is half of h, … which were formula but i didn’t know about sin and cos.
• The only part of this lesson that I knew how to do before was the Pythagorean Theorem.

Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit? If so, how?

• Trigonometric ratios and solving right triangles were the most helpful lessons because we learned the most new information in these lessons. What we learned in these lessons also applies to all triangles, not just a special type of right triangle.
• The self-check bell ringer that we did a few days ago was very helpful to me. When I know that one of my answers are wrong, I persevere in solving it correctly.
• When we finally put it all together and we were solving right triangles really helped me to finally grasp the whole idea. I had slowly learned and built a solid base and the solving right triangles lesson put it all together for me.
• I believe that the Trigonometric Ratios helped me the most because they allow me to solve a triangle with a different set of information.

Our Trigonometric Ratios lesson came from Geometry Nspired. I’ll write a post about it some day.

What have you learned during this unit?

• I have learned how to use sine, cosine, and tangent, and also the purpose of these formulas. Before I was taught what the formulas were used for, I was always curious as to what they meant on the calculator. Now not only do I know, but I also know how to work problems using sine, cosine, and tangent with accurate results.
• I have learned that there are “special” right triangles, and that there are quicker ways to solve these triangles. I also learned about trigonometry, which is something completely new to me. It was definitely new, and a little hard since it was the first time I saw it.
• I have learned how to solve right triangles, how to do trigonometry, and how to think about the problem before I try to draw it out.

What I have learned during this unit?

I have learned that the “I can” statements are really important for students. We reviewed them daily so that they could see where we were and where we were going. We are slowly making progress towards having students recognize what they are learning (I can …) and how they are learning it (Standards for Mathematical Practice). And so the journey continues …

 

Student Reflections for Unit on Angles & Triangles

I’ve been asking my students to reflect on each unit as we finish so that I can have some feedback on what changes to make for next year. I collect their responses through an assignment in Canvas.

1. I can use inductive and deductive reasoning to make conclusions about statements, converses, inverses, and contrapositives.
2. I can use and prove theorems about special pairs of angles.
3. I can solve problems using parallel lines.
4. I can prove theorems about parallel lines.

 

Which Standard of Mathematical Practice did you use most often in this unit?

The top 3 responses:

Make sense of problems and persevere in solving them.

Construct viable arguments and critique the reasoning of others.

Look for and make use of structure.

 

Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?

Many students recognized vertical angles, parallel lines, and the triangle sum theorem as something they had used before. However, they also recognized that we were learning why those relationships worked – and not just that they worked.

One student replied: None of these were repeats so to speak. I already knew what parallel lines and the different types of angles were, but I had no idea why or how or how you use them so much in math. I never sat there and thought, “Oh, we learned this last year, I don’t have to pay attention.”

 

Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit? If so, how?

• I really like 3A Logic for this lesson. It allowed me to think through things and use “logic” to solve problems.
• Interactive activities were particularly helpful as they visually showed the justification of postulates and theorems such as the Angle Addition Postulate, the Definition of Vertical Angles, etc
• Logic in the beginning helped out with the whole unit because that’s mainly what everything linked to.
• All of the lessons were helpful, but I found that the logic lesson was particularly helpful. It helped me to understand what a proof is and get introduced to them. Proofs were very important to meeting the learning targets for this unit.
• The symbolic logic was helpful in the case that if I could figure out one part, it could lead me to unlocking other parts, kind of like a puzzle.

 

What have you learned in this unit?

• I have learned how to prove things that I already knew were true.
• I have learned all the objectives. I am also better at solving proofs and proving theorems. I learned about triangles and theorems about those as well as the way I can draw in Auxiliary Lines to find angles in problems.
• Do not procrastinate on homework.
• I’ve learned that math isn’t always numbers and equations; it can be proving things through logic and reasoning and postulates and many other things that I really didn’t know existed before this. I learned WHY parallel lines are parallel and WHY the alternate exterior and alternate interior and corresponding angles are the same and WHY all of the angles of a triangle added together equal 180 degrees. I also learned how to apply this knowledge in geometry and how it can help us in real life.
• I have learned that I cannot give up when problems are complicated. I also learned that my way isn’t always the only way.

 

I think these reflections are good news as the journey continues ….