Over the past couple of weeks I have been posting about the quadratic unit that we are covering in my freshmen math class.  Today I am going to deviate from that a bit to talk about a lesson that occurred today.  The first part of the next unit is using exponent rules to simplify expressions.  There are a lot of mechanics involved, and the practice problems can get quite tedious.  
    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:
"Okay, all of you are in your teams.  Team 1 is responsible for Example 1 in the packet I gave you, Team 2 is responsible for Example 2, and so forth.  Your team needs to think about the problems in your example and talk together to determine how to do them.  Perhaps some of you have seen this before and you can start the group in that direction, or maybe you want to go to www.wolframalpha.com, type in your question, see the answer, and then work backwards.  As always, you can ask me well thought out questions.  In addition, each team gets one "spoon feed" - I will work out ONE example for you if your group requests it.  I won't tell you what I am doing, though, it will just be the work and process written on a small white board.  Okay - enjoy!"

​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.

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:

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:

• "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:

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. 

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.

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​Happy reading!

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

• 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:  

As I contemplated what to have the students do after their 'Polygraph: Parabolas' activity on Day 1 (see Day 1 post here), I needed to take inventory of what the students learned up to this point.  I tried making a list, and then realized that it would be better if the students made the list instead.  This led me to the new thinking routine, the '+1 Routine', under development for the Making Thinking Visible 2.0 book, which is under development by Ron Ritchhart and Mark Church of the Project Zero team at Harvard University.

The routine goes something like this:

1. ​Recall: In 2-3 minutes and working individually, each learner generates a list of key ideas that they remember about a topic.
2. Add (+) 1: Learners pass their papers to the right.  Taking 1-2 minutes, each student reads through the list in front of him/her and adds one new thing to the list.  The addition might be an elaboration (adding a detail), a new point (adding something that was missing), or a connection (adding a relationship between ideas).  REPEAT this process at least two times.
3. Review: Return the papers back to the original owner.  Learners read through and review all the additions that have been made on their sheets.  At the same time, they may add any ideas they have picked up from reading other's sheets that they thought were worthwhile.

The reason I decided to use this routine is because I wanted to see what key ideas the students remembered from the Polygraph activity, and how well they committed them to memory.  I had the students close their laptops and put away their source material.  I gathered the students in groups of five for the routine and I handed each student in the group a different colored pencil.  The routine went something like this:

• Me: 'Okay so each of you take out a piece of paper.  I know you get to work collaboratively a lot in this class, but for right now I want to know only what is in your mind.  So, when I say 'go', I want you to write down any key ideas about parabolas that you can remember.  Remember, this is your own work.  You will have exactly two minutes.  Go...'
• The students wrote their lists for two minutes.  Each student in the group used a different color so they could easily see who was writing each part.  
• After the 2 minutes passed, I said 'Now, take your list and pass it clockwise.  Read the list.'  I paused here and gave them time to read.  Then I said 'please add one new thing to the list.'  After a minute, a student said to me, 'Mr. Watson, I added a new item, but what if I saw something wrong with what is currently there?'  This caught me by surprise because I didn't think about this.  So, I had them pass the papers again in the clockwise direction, and gave a slightly different direction this time.  I said, 'This time, read the list, and then either correct something that is there, or add something new to the list.
• This continued for five iterations before the papers went back to the original owner.  I then had the students read their original paper.  I told the students to chat with their group about what they had read.  Since the writing was in different colors I told them to chat with the person with that color if they had questions or comments.  Here is some of what I overhead the students saying to each other:
  • ​"Wait, this isn't incorrect?  Why did you correct this?"
  • "I'm not sure what you meant by 'middle number.'  Can you explain this?"
  • "Completing the square.  What is that?  Did we do that the other day?"
  • "I still don't understand 'line of symmetry.'  Somebody wrote x equals negative b over 2a.  That didn't make sense the other day either."
• This sparked rich interactions between the students.  I'm glad that we spent the time to do this as it brought their current thoughts about parabolas alive on the paper.  Some students wrote items that they had learned in middle school, but were not yet discussed this year.  In the past, I would tell students 'not' to discuss those things because we hadn't 'covered' them yet.  Now, I encourage all prior knowledge and want the students to share it.

I had one person from each group read their list, and then I said the word 'popcorn.'  This is a signal to students that I am looking for a comment or feedback on what was just read.  The student that did the reading then called on a person of their choice to share their thoughts.
The +1 Routine provided a great recap of the previous day's lesson, and sparked more of the vocabulary that I wanted the students to continue to use.  

Where did we go from here?  The polygraph activity on the previous day involved only graphs of parabolas - the students did not see the corresponding equations that generated those parabolas.  It seemed logical to me that they should spend time associating parabolas with the equations that generate them.  

This led me to the Desmos Marbleslides parabola activity.  As you can see from the sample screen below, in this activity the students have to use the vertex form of a parabola in order to 'catch' stars with marbles rolling along the parabolic curve.  They are given an initial curve, and then have to change the values of a, h, and k in order to get the parabola in a certain shape to catch all the stars.  Once they feel confident that their shape will work, the students hit the "Launch" button which releases the marbles (the first marble is the purple dot) which 'hit' the stars.  Once all the stars are hit, the student gets to move on to the next screen which is a new curve and a new set of stars.   

After the students got a feel for the first couple of slides (teachers are able to pace the activity so students can only work on certain slides at one time), I paused the activity to give a new set of directions.  I had them each take  out a piece of paper and explained that they would be documenting their thinking as they went along.  I felt it was important for them to do some additional thinking about what they were doing with desmos in this activity and to write it down.  I have students record all of their desmos activities so they can go back to them later, but I also wanted to see some documentation of their thinking.  I wrote the general form of the vertex form on the board, and proposed the following questions as they were going through the activity:

• 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?

​
These prompts gave the students some things to think about as they changed the graphs of the quadratic equations.  I gave the them time to complete more slides and think about the above prompts, and then explained that their home thinking due the next class was to finish the marble slides activity, and to answer the above two prompts.

In the Day 3 post, I will discuss how the students converted between vertex form and standard form, and how they used Desmos to 'check' if their conversions were correct.

​Stay tuned!