- Section 1: Graphing quadratic functions
- Section 2: Translating parabolas
- Section 3: Solving equations using square roots
- Section 4: Solving equations using factoring
- Section 5: The quadratic formula
- Section 6: The discriminant and complex numbers
- Section 7: Quadratic systems

Many times at the start of this unit I would show the students how to graph basic parabolas - I would define the key items such as the vertex, line of symmetry, and intercepts, and then graph them. The students would copy what I was doing and then do homework problems essentially replicating this behavior. The students' thinking only went as far as the rote mechanics they were employing when finding the key items.

As I approached this chapter this year, I was thinking about how to organize the lessons to best suit the students. It then led me to the question, 'How can I organize this unit so that I am able to leverage the 8 cultural forces described in 'Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools' by Ron Ritchhart, in a way that helps students the most?'

I have organized this blog post in to several days. Today I will cover day 1, with other days following soon.

On this day, I wanted the students to speak the language of parabolas. Our foreign language teachers tell me that the best way for students to learn about the language is to speak it - so it seemed reasonable that the best way for students to begin to learn about parabolas is to speak about them. I decided to start the unit by using the "Polygraph Parabolas" activity on www.teacher.desmos.com. All the students need a computer and sign in to the activity after receiving a special code that Desmos generates. I decided to sign in as well so I could do the activity with them. When everybody signs in, Desmos pairs everybody up randomly.

The opening screen of the activity looks like this:

Student 1 of the pair chooses their 'favorite' parabola, and then Student 2 of the pair asks questions to try to figure out which parabola that Student 1 picked. Student 2 has to ask yes-no questions, and then Student 1 answers them. Take a look at the sample interactions below. One nice feature of desmos activities is you can anonymize the names so the students' real identities are hidden:

As you can see from the above interactions, the students must use mathematical language in order to communicate with each other. I used this in more than one block, and the activity would start out very quiet as the students communicated via the computer as they typed in and answered their questions (much like they interact when texting each other). However, in every class, as time went on, the students began to interact with each other with live conversations. It was really awesome to see that unfold. I would hear things like "what do you mean by ...", "when you say the graph is positive, what do you mean exactly", and "when you say minimum, do you mean the vertex or another point?" The best part about this computer interaction and live interaction is the students were communicating with each other with mathematical language. At the beginning of the activity, the language was imprecise but as time went on it became more detailed and mathematically correct. The best part of it was when the students used their own language to start with, such as "x-axis crossing points" and "bottom point", and then to see it slowly transform into terms such as "intercepts" and "vertex."

I let the students play this activity for about 30 minutes. After this, there is another desmos activity related to this called "Polygraph parabolas, part II" that I let the students explore. This activity takes some of the imprecise language, such as 'turning point', 'smile', and 'touches the middle line twice' into language such as 'vertex', 'concave up', and 'intercepts.' At the end of this activity, the students are asked to create parabolas with certain characteristics. Here is an example:

I let the students play this activity for about 30 minutes. After this, there is another desmos activity related to this called "Polygraph parabolas, part II" that I let the students explore. This activity takes some of the imprecise language, such as 'turning point', 'smile', and 'touches the middle line twice' into language such as 'vertex', 'concave up', and 'intercepts.' At the end of this activity, the students are asked to create parabolas with certain characteristics. Here is an example:

There are many cultural forces at play during this day in math class. Here is a look at some of them:

1.**Language**: The language of community is very evident as the students are creating the vocabulary that is used to communicate about parabolas. Often times during the course of the polygraph activity, students ask me 'is this the way to say it' or 'is this right?' My response is always 'are you able to communicate with your partner effectively with the words you are choosing? Perhaps you should talk with them and get the vocabulary ironed out. Later in the week we will see what other mathematicians have used to communicate, and we will also talk about why they might use the language they use.' This allows the students to have a sense of community and to see that math is not something that is defined outside of our classroom - rather it is something we develop as a community of mathematicians.

2.**Modeling**: By participating in the activity with the students, I tried to model that I am a learner with them and that I am not the sage of parabola vocabulary. I wanted them to see me as a role model instead of an exemplar. As they worked through questions about parabolas, I noticed the students leaning on each other to make sense of what was going on - they slowly became independent and confident in talking about parabolas. If I would have used transmission of knowledge as the mode of teaching on this day, students would have missed out on this rich activity and would have seen me as an exemplar instead of taking control of their quadratic vocabulary.

3.**Interactions: **It goes without saying that the Interactions were a big part of this day. Most students interacted with over half the class because when one partnership ends, desmos automatically matches the students up with somebody else that is waiting. As I said earlier, what started out as virtual interactions slowly turned into live interactions as the students were trying to sort out their thinking.

4.**Opportunities**: The best part about providing this opportunity for the students was that it was a low threshold opportunity - all students could participate. Some of the students already knew the 'perfect' parabola vocabulary, while those that didn't improvised by using words that made the most sense to them. Some students were using phrases such as 'bowl shaped', or 'smile' to describe a certain parabola. I think that was great. If I would have just *told them* to say 'concave up' instead of 'smile' it would have been my words and not their own.

5.**Environment**: The environment on this day was one of organized chaos. There was structure for sure as the students were at their computers and playing a polygraph game, but the overall atmosphere was one of curiosity, thinking, and laughter as the students tried to communicate with each other about a topic that was new to many of them.

In upcoming posts, I will describe how to leverage Cultures of Thinking with the rest of the quadratics chapter.

Later this week, I will cover what happened on**Day 2: Marbleslides: Mapping the shape of a parabola to the equation that generates it. **

Please visit again!

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In upcoming posts, I will describe how to leverage Cultures of Thinking with the rest of the quadratics chapter.

Later this week, I will cover what happened on

Recently we started a lesson on similar triangles. I thought a lot about what to have the kids do, and a small part of me (very small) wanted to 'direct teach' the lesson to save time - we are behind, after all, and if I just told them what similar triangles were and what to do with them we could move on to the next lesson. In the past, I would typically give them scripted notes that had a set of similar triangles, such as the example in figure 1. I would explain what similar triangles were, and how to find different things about them, such as angle measures and/or side lengths. The students would typically do well with this and then we could say that we had 'learned' similar triangles.

This year I wanted to provide a richer opportunity for them. I put up the following directions:

**Use your math resources to figure out what it means for triangles to be similar.****Once you are comfortable with the notion of similar triangles, take the colored paper in the back and create a set of similar triangles. You can use any other tools as well, such as rulers and/or protractors.****After you create your set, be able to convince somebody that they are similar by using a mathematical argument**

I purposely made the direction vague. Some students asked me what similar triangles were, but I pushed it back to them to use their resources, such as their phone, computer, or textbook, to figure it out. Some students also asked what I meant by a 'set' of similar triangles. I asked them to look up that word as well and then decide what to do. I had to laugh because some students just really needed me to tell them 'how many' triangles they needed to make. I did not want to tell them because I did not want to give them a threshold at which to stop learning.

As time passed, many students were using their protractors to create two triangles that had the same angle measures. Two seemed to be the most popular number of triangles. There were a lot of struggles going on as measurements were being made and sides were being measured. Some students used a lot of paper, while others used a sheet or two.

With about ten minutes to go in the block, two of my students, Filip and Haden, asked me to come and look at their set. I walked over to their desk and this is what I saw:

As time passed, many students were using their protractors to create two triangles that had the same angle measures. Two seemed to be the most popular number of triangles. There were a lot of struggles going on as measurements were being made and sides were being measured. Some students used a lot of paper, while others used a sheet or two.

With about ten minutes to go in the block, two of my students, Filip and Haden, asked me to come and look at their set. I walked over to their desk and this is what I saw:

Needless to say, I was floored! The expectation in my head was to see two triangles that were created using straight edges and rulers. I of course inquired about what had happened and what led them to this result, and I found out what actually happened: they used one sheet of paper and created thirteen (or so) triangles in about 5 minutes. All of the triangles were similar, but the beauty of it was that they did it by folding the paper - rulers and protractors were not needed (one is shown in the picture, but it wasn't used). The best part was that I was going to demo to the whole class how to 'quickly' create two similar triangles (by overlaying them on each other), and instead I had these two students demonstrate a much more powerful demonstration.

I was really thankful afterwards that I didn't give in to the urge to instruct directly because this opportunity would have been completely lost. Instead, it was gained, and was a nice lead in for the next class, as I had Haden and Filip lead the class on how to construct so many similar triangles in such a short time.

]]>I was really thankful afterwards that I didn't give in to the urge to instruct directly because this opportunity would have been completely lost. Instead, it was gained, and was a nice lead in for the next class, as I had Haden and Filip lead the class on how to construct so many similar triangles in such a short time.

Cultures of Thinking is an educational framework developed by the Project Zero team at Harvard University. The framework brings to light the eight cultural forces that are at play in our classrooms: Expectations, Language, Time, Opportunities, Routines, Environment, Interactions, and Modeling. These forces are at play in every classroom; CoT provides tools for the teacher to properly harness these forces to produce 21st century, globally minded learners and thinkers. Teachers can use a variety of methods to create a Culture of Thinking in their classrooms, such as allowing critical time when asking and answering questions, using key language to drive deep thinking, allowing key interactions to occur among the students, to name a few. The tools are varied and many, and since this is a framework, not a program, it isn't a matter of doing a couple of things and suddenly the classroom is transformed - CoT is an iterative, exciting, and invigorating journey.

Many times over the past year I've heard people say, 'Well, I can see implementing a CoT in humanities, but it won't work in the Math classroom.' My response to this has been, 'Wait, there isn't any thinking in Math classrooms?' Hmmmmm. After spending the last year and a half on this framework, and doing an extensive study on the book, "Creating Cultures of Thinking", I can assure you that creating a Culture of Thinking in Math class is possible, re-invigorating, and your students will thank you for it!

How can you begin to transform your class and begin to implement a Culture of Thinking in your math classroom? Well, the good news is that you are probably already doing many things that address some or all of the eight cultural forces.

Here are some ideas to get started:

Many times over the past year I've heard people say, 'Well, I can see implementing a CoT in humanities, but it won't work in the Math classroom.' My response to this has been, 'Wait, there isn't any thinking in Math classrooms?' Hmmmmm. After spending the last year and a half on this framework, and doing an extensive study on the book, "Creating Cultures of Thinking", I can assure you that creating a Culture of Thinking in Math class is possible, re-invigorating, and your students will thank you for it!

How can you begin to transform your class and begin to implement a Culture of Thinking in your math classroom? Well, the good news is that you are probably already doing many things that address some or all of the eight cultural forces.

Here are some ideas to get started:

- Perform a self-assessment of the Culture of Thinking in your classroom. Use the rubric here.
- Watch this video to see if you can pick out any of the cultural forces in action. Some of the forces are 'addressed' in this snapshot, and some are not:

- Pick one of the elements on the wheel above, buy the book 'Cultures of Thinking', and begin to read about it!
- Please look through my other blog posts! There is a lot of information and I would love feedback.
- Take the online course "Creating Cultures of Thinking" through the Project Zero team at Harvard University. I am currently an online instructional coach for that course and I can tell you it is well worth the time and effort to take it! Click here for the link to the course.

One of the concepts we have been studying in my freshmen math class is solving systems of inequalities, such as the system seen below.

In the past, the lesson would go something like this:

1) I would model how to draw each curve in the system.

2) I would draw dashed or solid lines and explain why they were dashed or solid.

3) I would shade the appropriate region and explain how to determine this.

4) I would ask if there were any questions and would address them with each student directly.

5) I would give the students a different, but very similar example to do and monitor their progress.

At the end of this lesson, students could claim that they knew how to solve a system of inequalities by simply replicating the steps that I gave them in class. I really wouldn't allow much variation in the procedure - I had the math degree, after all, and I told the students that we all need to be on the same page as far as the 'right' way to do this procedure.

This approach made me think of some of the belief sets that are discussed in the Expectations chapter in "Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools." The belief sets, which are outlined as natural tensions, that came to mind for this lesson are**focusing on the learning vs. the work**, **teaching for understanding vs. knowledge**, and ****encouraging** deep vs. surface learning**. As I reflected on the way that I used to teach this, I imagined that the students had the **knowledge **of how to graph an inequality, that they knew how to complete the **work** required (especially given the fact that I would assign a bunch of these types of problems for homework), and they at least had developed learning on a **surface **level. Obviously, these are not bad things in and of themselves, but how could we develop a **deep** level **understanding **of **learning** that was meaningful? I attempted to do this with the following lesson.

I started class by displaying the problem above. I asked them to draw the solution on their desk (their desks are white boards) using whatever prior knowledge they could remember. There were definitely some cob webs as the students tried to remember what they had learned previously. A lot of discussion ensued: 'how do you draw this parabola?', 'I remember having to use dashed lines - do we do that here?', 'is shading a thing with these problems?', 'I think we shade everything, don't we?'

It was tempting at this point to walk around and quickly check these for the kids and tell them right or wrong. It certainly would have made the lesson go faster, but in trying to foster independence I tried something different. I had the students check in with their group members (the students are in groups of 4) to compare and contrast their graphs and to come to an agreement on what was correct. Then, they had to erase the four graphs and create one that the group could agree on and draw that one rendition. In my classroom there are eight groups of four, so at the end of this there were eight graphs total. Each group collaborated, questioned, erased, shaded, and played until they were happy with their final product. I then had the groups rotate around the classroom. Each group was instructed to carry one marker and to 'annotate' the final product at each group. If they saw something that somebody else had already written, they had to put a star next to it. After a slew of rotations around the class, everybody came back to their home group and were able to see the comments, suggestions, fixes and feedback that were left at their table. I gave them a few minutes to digest everything that they saw.

In the past, the lesson would go something like this:

1) I would model how to draw each curve in the system.

2) I would draw dashed or solid lines and explain why they were dashed or solid.

3) I would shade the appropriate region and explain how to determine this.

4) I would ask if there were any questions and would address them with each student directly.

5) I would give the students a different, but very similar example to do and monitor their progress.

At the end of this lesson, students could claim that they knew how to solve a system of inequalities by simply replicating the steps that I gave them in class. I really wouldn't allow much variation in the procedure - I had the math degree, after all, and I told the students that we all need to be on the same page as far as the 'right' way to do this procedure.

This approach made me think of some of the belief sets that are discussed in the Expectations chapter in "Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools." The belief sets, which are outlined as natural tensions, that came to mind for this lesson are

I started class by displaying the problem above. I asked them to draw the solution on their desk (their desks are white boards) using whatever prior knowledge they could remember. There were definitely some cob webs as the students tried to remember what they had learned previously. A lot of discussion ensued: 'how do you draw this parabola?', 'I remember having to use dashed lines - do we do that here?', 'is shading a thing with these problems?', 'I think we shade everything, don't we?'

It was tempting at this point to walk around and quickly check these for the kids and tell them right or wrong. It certainly would have made the lesson go faster, but in trying to foster independence I tried something different. I had the students check in with their group members (the students are in groups of 4) to compare and contrast their graphs and to come to an agreement on what was correct. Then, they had to erase the four graphs and create one that the group could agree on and draw that one rendition. In my classroom there are eight groups of four, so at the end of this there were eight graphs total. Each group collaborated, questioned, erased, shaded, and played until they were happy with their final product. I then had the groups rotate around the classroom. Each group was instructed to carry one marker and to 'annotate' the final product at each group. If they saw something that somebody else had already written, they had to put a star next to it. After a slew of rotations around the class, everybody came back to their home group and were able to see the comments, suggestions, fixes and feedback that were left at their table. I gave them a few minutes to digest everything that they saw.

I then put up the following drawing using desmos. I asked them to compare what they had with what they saw from desmos and to write down what they were wondering about. Here are some samples:

- is the blue and the red region necessary?

- is the white region part of the answer?

- i still don't understand the dashed lines

- is desmos right or am I right?

I asked the class "Which region is the correct region? Is it the blue, the red, both of these, or the white? Many students chose blue, some chose red, while some chose both. I called their attention to the fact that in the original system, the word "AND" was used, which in math means intersection.

- is the blue and the red region necessary?

- is the white region part of the answer?

- i still don't understand the dashed lines

- is desmos right or am I right?

I asked the class "Which region is the correct region? Is it the blue, the red, both of these, or the white? Many students chose blue, some chose red, while some chose both. I called their attention to the fact that in the original system, the word "AND" was used, which in math means intersection.

I then showed them the same graph in desmos, except now with some points plotted in the different regions. I asked students that were comfortable to go and plot more points in the various regions, so they slowly went up and put more points on the screen. Instead of asking which color regions were the solutions, I asked which points were the solutions to this problem. Many students mentioned the point (0,4), as well as others that were in the blue, but some still called out points (4,2), and (10,-1) in the red region.

At this point, I displayed the following Venn diagram on the board. I asked them to draw this Venn Diagram on their desk and to place the coordinate points in the appropriate places in the Venn diagram. They had been exposed to Venn diagrams in the past, and for the most part knew that the AND part was the portion in the middle of the diagram. They slowly started placing points in various spots on the diagram, and were trying to figure exactly where the coordinates would fit.

However, the most fun occurred when I asked each group to create their own systems. They could include any functions that they were familiar with as long as they had general knowledge of how to draw them. This was fun to see develop as some groups created a system with a line and quadratic (just like mine!) and I told them to go back to the drawing board and make it a bit more interesting. One group brainstormed a list of the functions they were familiar with before choosing. Still another group picked three functions instead of two. Regardless, the kids came up with much better examples and insight than I could have, and it certainly was much more rich than me just handing them a system to solve!

The eight new systems served as a springboard for more practice. I gave the students a choice on which systems they wanted to tackle, and when they wanted to tackle them, whether it be at home, during class, or after school.

]]>The eight new systems served as a springboard for more practice. I gave the students a choice on which systems they wanted to tackle, and when they wanted to tackle them, whether it be at home, during class, or after school.

Day 1: The students used Desmos polygraph to discuss the vocabulary surrounding the graphs of quadratics.

Day 2: We used the +1 Routine to generate all of our thinking about parabolas and quadratics.

This led to the Desmos marble slides activity where the students wrote equations of quadratic functions in vertex form in order to generate graphs that 'catch' stars.

Day 3: The students continued the marble slides activity by plotting the vertex on each page of the activity, converting the equations from vertex form to standard form, and looking at the 'wideness' and 'narrowness' of parabolas.

For their home thinking that was due today, I had asked the students to manually find the vertex for each of the parabolas in the Desmos marble slides activity and to check their answers as they went along. Each student logged in to their Demos marble slides activity and had it showing on their computer screens, with the home thinking that they created sitting out as well. I used the Team Shake app to mix the class up in to pairs and gave them five minutes to discuss the results of their thinking. I walked around and answered various questions about finding the vertex, looking at the math work as I went. Some students were still having trouble simplifying correctly, while other students had mastered the concept. I then had a discussion with the class about doing more practice with this concept. It went something like this:

<The class and I all laugh at this comment>

At this point I present the class with the following pictures of two different calculator screens:

I then pose this question to them: "Take a look at the calculator screens that I have put up on the overhead. Take a minute and process what you are seeing here. Think about what is happening on each calculator screen and be ready to discuss." The students thought for several minutes. I realized after we did this that it would have been a perfect moment to have the students do a 'See-Think-Wonder' routine; this is essentially what we did, though, but with different words. I asked the students for some input and gave appropriate wait time. The students chimed in and realized that what they were seeing was the calculation of the y coordinate of the vertex, done two different ways. I then asked them which solution was 'better.' This part surprised me, as most of the students chose the picture on the left. Here were some of the comments:

**"I would pick the one on the left because it makes more sense to me"**

"The one on the left makes you think about what you are doing"

"The left one is easier to see what is going on"

"The one on the left gives you the steps, but the one on the right is quicker. So, if you know what you are doing, do the one on the right; if you are still learning it do the one on the left."

For years, I didn't give the students a choice in the matter. I made them calculate the y-coordinate using the screen on the right because I thought it was easier and cleaner; I didn't consider their viewpoint in that the steps made more sense to them if they entered it like the screen on the left. I left this activity pleasantly surprised because their response to this was not at all what I had expected. A student then asked, "Well, which one do you want us to do on the test?" My reply was, "Well, can we ask that question but leave off the phrase 'on the test'?" The student replied, "Well which one do you want us to do?" I said, "Which one makes better sense to you?" She said, "The one on the left for now, but maybe the one on the right." My response was "well, do the one that make the most sense to you. As mathematicians, there are many ways to do the same thing, and we sometimes need to do what we are comfortable with."

"The one on the left makes you think about what you are doing"

"The left one is easier to see what is going on"

"The one on the left gives you the steps, but the one on the right is quicker. So, if you know what you are doing, do the one on the right; if you are still learning it do the one on the left."

For years, I didn't give the students a choice in the matter. I made them calculate the y-coordinate using the screen on the right because I thought it was easier and cleaner; I didn't consider their viewpoint in that the steps made more sense to them if they entered it like the screen on the left. I left this activity pleasantly surprised because their response to this was not at all what I had expected. A student then asked, "Well, which one do you want us to do on the test?" My reply was, "Well, can we ask that question but leave off the phrase 'on the test'?" The student replied, "Well which one do you want us to do?" I said, "Which one makes better sense to you?" She said, "The one on the left for now, but maybe the one on the right." My response was "well, do the one that make the most sense to you. As mathematicians, there are many ways to do the same thing, and we sometimes need to do what we are comfortable with."

I then showed the following slide to the class. I gave them a minute to read it and then asked if they had questions on what they were expected to do. One question that was asked was: "What do we put in each box in the chart?" I said, "Each box is a process in and of itself. Each box will most likely take up a good portion of a sheet of paper or a white board. Feel free to use chart paper, loose leaf paper, a small white board, or one of the large white boards on the wall. It is time to dig deep! One recommendation: start with the vertex form to standard form cell, and then go from there."

The next block of time, over 30 minutes, was one of organized chaos! Students were all over the place in the classroom: some chose to work with pencil and paper, some used small white boards, some stood at the classroom white board, others were using the computer to look up items on the internet. At one point, a group of students was standing around a laptop and when I went over they were watching a Khan academy video about completing the square. I asked them what they were watching and they said that they had read that to convert from standard form to vertex form, they needed to complete the square, and they wanted to see what it was. Now, I knew that completing the square was not a focus of this unit and it would be something that they would do in later years, but for now I did not want to extinguish their energy and excitement.

As I kept listening to the class, I heard things like 'foil', 'simplify', 'completing the square', and 'factoring.' I was excited to hear these key words and phrases. One pair of students were 'arguing' over one of the conversions because one of them entered it in wolframalpha.com and the other on symbolab.com and they got different answers.

One student came up to me and our conversations went something like this:

**Student: ** "Mr. Watson, I think I know how to convert from vertex form to standard form, but I'm not sure what to call it."

**Me: **"well, what is it that you did to do this?"

**Student: **"I used foil and then made it simpler."

**Me: **"well, that sounds like a good way to describe it - let's go with that and we can see what the class thinks later."

**Student: **"We haven't spent any time on this, have we?"

**Me: **"Remember when you did the conversions in desmos last week? And you checked them?"

**Student:** "OHHH!!!! That's what we were doing when we were doing that. It makes sense now!"

This made me laugh as this particular student finally made the connection of something we were doing a while back. This type of thing used to irritate me before, but now I accept that learning is a process and different students process and master things at different rates.

Class was nearing the end so I asked the students to pick two conversions that they felt the most comfortable with at this point in time. I asked them to find two quadratic equations and convert them accordingly for our next class.

Please stay tuned for Quadratics Day 6, where we will fill in the conversion chart, which will naturally lead us to foil'ing as well as factoring!

]]>As I kept listening to the class, I heard things like 'foil', 'simplify', 'completing the square', and 'factoring.' I was excited to hear these key words and phrases. One pair of students were 'arguing' over one of the conversions because one of them entered it in wolframalpha.com and the other on symbolab.com and they got different answers.

One student came up to me and our conversations went something like this:

This made me laugh as this particular student finally made the connection of something we were doing a while back. This type of thing used to irritate me before, but now I accept that learning is a process and different students process and master things at different rates.

Class was nearing the end so I asked the students to pick two conversions that they felt the most comfortable with at this point in time. I asked them to find two quadratic equations and convert them accordingly for our next class.

Please stay tuned for Quadratics Day 6, where we will fill in the conversion chart, which will naturally lead us to foil'ing as well as factoring!

Lately I have been telling my freshmen that teachers are like personal trainers. I gave them this scenario: imagine you go to the gym with your personal trainer. For an hour you watch your trainer run 5 miles, then afterwards you watch them lift weights. At the end, you leave the gym and say that you "worked out." Ridiculous right? The personal trainer should show you some exercises and guide you to best practices, but at some point you need to perform those exercises and routines yourself, and go through the sweat and tears. Teaching and learning is much like that. The teacher can guide and show certain things, but the students must perform the exercises and go through the sweat and tears much the same way as in the gym.

I thought to myself: 'How can I guide my students through these exponent rules while giving them the opportunity to go through the sweat and tears necessary to deepen their learning?'

First, I created examples for them and handed them out. Above is one of the examples that I gave the students. You can download the full worksheets here. As you can see, the worksheet isn't anything special - it is something that I've more or less handed out every year. I then organized the class in to six groups using the Team Shake app. You can see the organization of the teams on the right. After the class re-organized themselves in teams, I said the following:

"

I left the team arrangements showing on the screen. The students went off in their teams and began their problem set. Here are the things I observed:

- Some of the teams would call me over for their one spoon-feed problem, but others in the team would say, "wait, don't use it yet, I can explain it to you!"
- Many students would ask me 'is this right?' I would turn it back on them and ask them to type it into www.wolframalpha.com. It turns out, though, that wolfram alpha didn't always have the answers in the perfect form. Below is one example. On the left is the original problem; on the lower right is Wolfram alpha's simplification of the problem. It turns out that it is correct, but not quite as simplified as one would wish. This led us to a discussion of helpful math websites.

- We talked about using www.symboab.com, www.mathpapa.com, and desmos.com. It turns out that symboab.com and mathpapa.com didn't simplify the original problem down very nicely either. We used desmos.com to help us. We entered the original expression as well as the proposed solution into desmos (see the picture at right). It turns out that desmos wouldn't process it because it got confused over the x and y being in the same expression. So, we changed the x and y to a and b, and then added sliders. As you can see, the answer boxes are the same for both expressions, even if you change the 'a' and the 'b.' The bottom line is that this gave the students and I the opportunity to discuss different ways to check answers using online tools.
- As the groups finished their assigned problem, they began heading to other groups to get help with the next problem set. At one point, a student was walking towards me and I thought he was going to ask me a question, and he walked right by me to head to another group so he could ask them a question!
- I commended the students after about a half hour because of their effort and for asking each other so many thoughtful questions. I opened it up for about 5 minutes for them to ask me anything about any of the problems or rules that they encountered. There were a few questions, and I was able to offer some 'helpful tips' that I have used after years of experience which they appreciated.
- It was interesting that not one of the teams ended up using their 'free spoon feed' ticket that I offered them at the beginning of class. Many teams didn't want to 'waste it', but by the end of class they didn't use it anyway!
- I reflected on the cultural forces that were leveraged during this class:

**Expectations**- my expectations for the students on this day was for them to be less dependent on a teacher's answer key, and more dependent on the math tools that exist online such as wolframalpha.com or symbolab.com.**Time -**giving the students time to struggle through exponent rules and exercises in class. It would have been very easy for me to demonstrate all the rules at the front of the classroom with lack of time as an excuse for doing so.**Routine**- one routine that is in place in my classroom is putting students in groups my number, assigning a problem to that group, and then having that group be 'responsible' for that problem. Students know that other students in the class will be coming to them for help at some point during the block.**Interactions**- the interactions between the students on this day were invaluable as they leaned on each other. I believe that students in math should 'answer questions' and should 'question answers' and there was a multitude of this going on between students.

Before we embark on day 4, here is a quick summary of the first three days of the unit:

Day 1: The students used Desmos polygraph to discuss the vocabulary surrounding the graphs of quadratics.

Day 2: We used the +1 Routine to generate all of our thinking about parabolas and quadratics.

This led to the Desmos marble slides activity where the students wrote equations of quadratic functions in vertex form in order to generate graphs that 'catch' stars.

Day 3: The students continued the marble slides activity by plotting the vertex on each page of the activity, converting the equations from vertex form to standard form, and looking at the 'wideness' and 'narrowness' of parabolas.

I started the class with the following prompts:

Day 1: The students used Desmos polygraph to discuss the vocabulary surrounding the graphs of quadratics.

Day 2: We used the +1 Routine to generate all of our thinking about parabolas and quadratics.

This led to the Desmos marble slides activity where the students wrote equations of quadratic functions in vertex form in order to generate graphs that 'catch' stars.

Day 3: The students continued the marble slides activity by plotting the vertex on each page of the activity, converting the equations from vertex form to standard form, and looking at the 'wideness' and 'narrowness' of parabolas.

I started the class with the following prompts:

I had the students read and think about the above prompt quietly for ten minutes. I then had them have a discussion at their tables. I picked up some of the things the students were discussing:

As a teacher, It's always hard not to step right in when you hear incorrect information, but I resisted, choosing to let the 'incorrect' stuff to hang out there for a bit. After about 10 minutes, I then announced: "Ok, in a moment you are going to have Watson time - a ten minute period where you can ask me anything you want. So, take a moment at your tables and come up with the burning questions that you just can't seem to answer amongst yourselves and jot them down. Take a moment...."

This always sparks rich discussion as the students always want to use Watson time wisely. The funny thing is that they can ask me any questions at any point in time during class, at lunch, after school, or by email. However, when I set aside a block of time in class, they always use it as if it is an exclusive time to ask questions! Here are some of the questions that they came up with:

**Question 1 comes from Luke**

**Luke:** "Does intercept form mean x-intercepts or y-intercept?"

**Me:** "What happened when you graphed it in Desmos?"

**Luke:** "I didn't."

**Me:** "Take a minute to graph it, jot down the intercepts, and then we can talk some more."

<A couple minutes go by as I answer another question from a different student>

**Luke:** "I graphed it. The y-intercept is 6. The x-intercept...I think there are two.....1 and 3?"

**Lizzy:** "Yeah, there are two. Just take the opposite of the numbers in the intercept form and they are the x-intercepts."

**Me:** "So, what do you think about your original question?"

**Luke:**: "I forgot what it was.....ok, wait, I remember 'does intercept form mean x-intercepts or y-intercepts'. It should be called x-intercept form, it means x-intercept form."

**Saif:** "Why do they call it that?"

**Me:** "Who?"

**Saif:** "The people that named it."

**Me:** "Well, I got this name from a textbook and a website that talked about quadratics. They are mathematicians just like you - they chose a convention and it stuck."

**Question 2 comes from Angela**

**Angela****:** "I like the vertex form because you don't have to show work to get the vertex. It's just right there. Do we have to show work on the test if we get vertex form?"

**Me:** "Can you ask the question again but leave out 'on the test'?

**Angela:** "Do we have to show work if we get vertex form?"

**Me:** "Did everybody hear that question?"

<a few students shake their heads no>

**Me:** I reiterate something that I have mentioned time and time again. "Remember, this isn't just a discussion between me and the person asking the question. It's a discussion between all of us. Please repeat the question."

**Angela:** "Do we have to show work if we get vertex form?"

**Me:** "What do all of you think about that question?"

**Will:** "Wait, show work when we are doing what with vertex form?"

**Angela:** "If we have to find the vertex, and we get a quadratic in vertex form, do we have to show work to get it?"

**Me:** "Can we have some opinions on this?"

<Hands start to go up, and I give wait time>

**Will:** <This student looks at me, but I redirect him to talk to the person that asked the question, so he does and they make eye contact with each other> "I don't think you need to show math work, the vertex is right there and that is the cool thing about that form - you can look right at it and get the vertex."

**Me:** "How does that answer sound?"

**Angela:** "I think it makes sense."

**Me:** "Can we get some other thoughts on this?"

**Haden: **"I would do the work anyway. FOIL it out and find it using the formula."

**Richard:** "What formula?"

**Danielle: **"Yeah, I remember a formula too. It's -b/(2a)."

**Me: **"Okay, Danielle recommends a formula. Let's try it out. Open up your Desmos marble slides activity. You already plotted the vertex by looking at the graph and by looking at vertex form. Haden said we should use the standard form and then use the formula that Danielle is proposing. Haden, is the formula that Danielle remembers the same one that you remember?"

**Haden: **"Yeah, I think that's it."

**Me: **Okay, on slides 2, 4, 5, 6, 8, & 10, try the formula x = -b/(2a) and see if you get the x-coordinate of the vertex. After you try a few, go over to Haden and Danielle and see if what you are doing matches what they are doing."

At this point, the class is headed in a new direction. I realize I broke my promise of "ten minutes" to ask me questions, but I think this is ok because the discussion led us this way and important points and concepts were brought up. The students work on finding the x-coordinate of the vertex for a while and many of them do not get the same value they currently see on their marble slides screen. Look at the sample marble slides screen below.

- "I think we learned how to convert from vertex to standard form, didn't we?"
- "Standard form tells you which way the graph points"
- "All of the forms tell you which way it points, not just standard form"
- "Isn't the quadratic formula another form, so there are four forms?"
- "It looks like the intercept form gives you the y-intercepts"
- "Having multiple ways to write a quadratic is sort of like having different languages to communicate with each other"

As a teacher, It's always hard not to step right in when you hear incorrect information, but I resisted, choosing to let the 'incorrect' stuff to hang out there for a bit. After about 10 minutes, I then announced: "Ok, in a moment you are going to have Watson time - a ten minute period where you can ask me anything you want. So, take a moment at your tables and come up with the burning questions that you just can't seem to answer amongst yourselves and jot them down. Take a moment...."

This always sparks rich discussion as the students always want to use Watson time wisely. The funny thing is that they can ask me any questions at any point in time during class, at lunch, after school, or by email. However, when I set aside a block of time in class, they always use it as if it is an exclusive time to ask questions! Here are some of the questions that they came up with:

<A couple minutes go by as I answer another question from a different student>

<a few students shake their heads no>

<Hands start to go up, and I give wait time>

At this point, the class is headed in a new direction. I realize I broke my promise of "ten minutes" to ask me questions, but I think this is ok because the discussion led us this way and important points and concepts were brought up. The students work on finding the x-coordinate of the vertex for a while and many of them do not get the same value they currently see on their marble slides screen. Look at the sample marble slides screen below.

Many students plugged in 3.2 instead of -3.2, which caused their value to be incorrect. They were able to self check, though, because the vertex is plotted for them on the screen. I surveyed the class and asked them how many were able to get the correct x value of the vertex using the formula. A group raised their hands, so I asked them to go to other students to check in with them to see why their values were not matching.

After a while, I ask how many of them had seen the x=-b/(2a) formula before that Haden and Danielle proposed. All but two students raised their hands. Twenty-six out of twenty eight had done this before, but there were a good number that were not successful in getting the correct x-coordinate. I then asked if any students knew how to find the y coordinate of the vertex. Several students raised their hands, and I ask them to form a group and talk over their methods, and to put any and all methods on the side board.

What is interesting is that some students just plugged in the x value to the original equation to get the y-coordinate of the vertex, while other students used the formula y=-b^2/(4a) + c. At this point we have two 'competing' methods on the side board (unfortunately I didn't take pictures of this math work).

I ask the class for their attention at this point:

"Okay, please put your pencils down and close the laptops just so we can focus on something else. Everybody look at the side board.....can we put our pencils down and close the laptops? Let's look at the side board. Now read through the methods you see and first seek to understand what is going on."

At this point the students read the side board for several minutes. I then hear chatter as they try to understand what is happening. As students start asking questions, I direct them to the students that wrote the work on the board. Those students head to the board and explain their processes.

I then say, "I think it might be a good idea if we all go to slide 5 and try to get the x and y coordinate of the vertex using the methods we discussed. Why don't you try both methods to get the y coordinate and see which method you like better?" This takes a good amount of time as some students struggle through the mechanics of finding the vertex. Several students finish quickly, so I ask them to wander and help others.

At this point in time, class time is about over. I realize this is not at all how I planned for things to go (I planned on having them convert between the different forms of quadratic functions), but the quality discussions and learning they were doing made it too valuable to get upset about keeping to my schedule. This is material that we were going to cover anyway, it was just moved forward based on the needs of the class.

I then ended the class. "Wow, we definitely accomplished a lot today. This is not exactly what I had planned today, but I think this is what we needed at this point in time. So, for home thinking for next time, find the vertex of the equations on the rest of the slides by using the formulas proposed today. Let me know how it goes."

Please check back soon for Day 5! I will be discussing the results of finding the vertex, and then move on to converting between the different forms of a quadratic.

If you would like to subscribe to this blog, sign up on the upper right of this page.

Happy reading!

]]>After a while, I ask how many of them had seen the x=-b/(2a) formula before that Haden and Danielle proposed. All but two students raised their hands. Twenty-six out of twenty eight had done this before, but there were a good number that were not successful in getting the correct x-coordinate. I then asked if any students knew how to find the y coordinate of the vertex. Several students raised their hands, and I ask them to form a group and talk over their methods, and to put any and all methods on the side board.

What is interesting is that some students just plugged in the x value to the original equation to get the y-coordinate of the vertex, while other students used the formula y=-b^2/(4a) + c. At this point we have two 'competing' methods on the side board (unfortunately I didn't take pictures of this math work).

I ask the class for their attention at this point:

"Okay, please put your pencils down and close the laptops just so we can focus on something else. Everybody look at the side board.....can we put our pencils down and close the laptops? Let's look at the side board. Now read through the methods you see and first seek to understand what is going on."

At this point the students read the side board for several minutes. I then hear chatter as they try to understand what is happening. As students start asking questions, I direct them to the students that wrote the work on the board. Those students head to the board and explain their processes.

I then say, "I think it might be a good idea if we all go to slide 5 and try to get the x and y coordinate of the vertex using the methods we discussed. Why don't you try both methods to get the y coordinate and see which method you like better?" This takes a good amount of time as some students struggle through the mechanics of finding the vertex. Several students finish quickly, so I ask them to wander and help others.

At this point in time, class time is about over. I realize this is not at all how I planned for things to go (I planned on having them convert between the different forms of quadratic functions), but the quality discussions and learning they were doing made it too valuable to get upset about keeping to my schedule. This is material that we were going to cover anyway, it was just moved forward based on the needs of the class.

I then ended the class. "Wow, we definitely accomplished a lot today. This is not exactly what I had planned today, but I think this is what we needed at this point in time. So, for home thinking for next time, find the vertex of the equations on the rest of the slides by using the formulas proposed today. Let me know how it goes."

Please check back soon for Day 5! I will be discussing the results of finding the vertex, and then move on to converting between the different forms of a quadratic.

If you would like to subscribe to this blog, sign up on the upper right of this page.

Happy reading!

The home thinking that the students had to do at the end of day 2 was the following:

The students turned in their documentation of their thinking as they walked in to class. I looked at it briefly, checked that they completed it, and then handed it back to them. I used the "Team Shake" app on my phone to generate groups of 3. I then asked them to run through the MicroLab protocol to discuss their thoughts on the values of a, h, & k (If you are not familiar with the MicroLab protocol Visible Thinking routine, see the image below). I displayed the slide to the right in class and did not have to explain too much because we had used it before. I just reminded them that as each person is talking, the other two needed to listen only. We ran through the protocol, and then I asked the groups of three to have an open discussion. After this I had the students go back to their home tables and to verify their thinking about the values a, h, & k, and then we had a class discussion about these values.

As a final check for understanding, I had the students close their laptops and answer the following questions:

**What effect do the values of a, h, & k have on the graph of y = a(x-h)^2+k?****Identify the vertex on slides 2,4,5,6,8,10 in the Marbleslides desmos activity. Plot the vertex in desmos. How does the vertex relate to the vertex form of the equation on each slide?**

The students turned in their documentation of their thinking as they walked in to class. I looked at it briefly, checked that they completed it, and then handed it back to them. I used the "Team Shake" app on my phone to generate groups of 3. I then asked them to run through the MicroLab protocol to discuss their thoughts on the values of a, h, & k (If you are not familiar with the MicroLab protocol Visible Thinking routine, see the image below). I displayed the slide to the right in class and did not have to explain too much because we had used it before. I just reminded them that as each person is talking, the other two needed to listen only. We ran through the protocol, and then I asked the groups of three to have an open discussion. After this I had the students go back to their home tables and to verify their thinking about the values a, h, & k, and then we had a class discussion about these values.

As a final check for understanding, I had the students close their laptops and answer the following questions:

I circulated around the class and looked over their shoulders at their responses. After about ten minutes I told them to open up Desmos.com to check their answers. I circulated and the students entered the 5 equations in Desmos to check the 'width' of the graphs to verify if they had them in the right order. Some of them had questions, which I addressed and overall the problem went well.

The second question, though, was a different story. I thought it was going to go quickly. I thought to myself, "If you take 3x^2 and shift it left 4 units, you obviously get 3(x+4)^2 and you go on your way." The students had real trouble with this. Many of them wanted to add an x term and a constant term to get the graph to shift. Most students shifted the graph to the left, but not by 4 units. Some of them shifted it left 4, but then it moved up or down as well. I was puzzled as I thought since they had played with the equation y=a(x-h)^2+k in desmos, and tried different values, they would understand the connection. As I questioned more students and tried to get the big picture it was clear to me that they did not see y=3x^2 as possibly in vertex form. So, I wrote the equation on the board as y = 3(x-0)^2+0. Bingo! Lights on! They caught it immediately and went on their way. One of them said to me, "But I thought y = 3x^2 was in standard form?" Another responded to her saying, "It looks like we have an example that sort of fits both forms." Still another said, "What is standard form?" I laughed out loud, not at the kids of course. I explained to him that this was one lesson that did not go as I had scripted, but it was necessary because that is what the class needed.

After we got this squared away, I had them open the Marbleslides activity (they can log in so it is saved - see the sample screen below). I walked around and checked if they had plotted the vertex on their parabolas and they had, and asked them how they found the vertex. One student said, "I just looked at the graph and knew where the vertex was at and made a best guess on what the coordinates would be." I asked how many had did it this way, I about ten of them raised their hands. I said, "How did the rest of you do it?" One student chimed in, "I just looked at the vertex form, took the h and the opposite of k, and knew that that was my vertex. Then I plotted it."

I said, "It sounds like we have two methods: one is looking at the graph, the other is to look at the equation. Which is better?" A student said, "I like looking at the graph and making a best guess. I'm visual" while another said that using the equation to get the vertex was more exact. I said that sometimes in math we have several methods to do things. However, I did mention that if they didn't have the graph in front of them to look at, that they needed to be able to use the equation exclusively to find the vertex.

At this point I projected the marble slide screen shown above for the whole class to see. I then typed in a different form of the equation that was already there, but this time I put it in standard form (see picture below). I asked the kids what was happening here, and I made sure to give proper wait time. Here are a few responses:

- "There isn't any difference, they overlap each other."
- "Is desmos working right, I don't see anything different?"
- "It's right, you just FOIL'ed to get the new equation"
- "What's FOIL - did we do that last year?"
- "FOILs easy - we don't have to do it this year"

I asked the class how many of them had familiarity with FOIL. Most of them raised their hands but there were some that did not. I wrote the conversion on a sheet of paper and when I was finished, I projected it on the document camera. I then asked them to take a couple minutes at their tables to read what I had written and to discuss it. I purposely put a mistake just to see if they would catch it. The discussions were rich as the students that remembered FOIL were explaining it the students that either never saw it before or forgot about it. The one comment that I took away from this was from a student that said, "I knew how to do FOIL, but I never knew why I was doing it."

I then gave them their thinking for the rest of the class period and for home thinking for that evening:

Please check back soon for Day 4! On that day, the students will learn a third form for quadratic equations and will investigate converting between all three forms. Also, if you would like to subscribe to get automatic updates for this blog, please subscribe using the button in the upper right of the page. |