I haven't yet figured out exactly what I might do with this but I definitely see Pizza Sliders in my classroom in the near future...Their slogan is "The circle is the new slice. With 3 different combinations and up to 3
toppings each, there’s never been more ways to enjoy Pizza Hut". hmmmm.

## Wednesday, February 20, 2013

### A simple problem...rich discussion

I love that sometimes a good lesson can come from something simple. Today I handed out copies of this graph (which is from the Annenberg Insights into Algebra site). I asked students to tell me everything they could possibly figure out about the information on the graph. Great discussion about slope and using different scales on the graph...which company is the better deal...what does the point of intersection represent in the problem...What are the equations of the line, and what do the slope and y-intercept represent in this context?

And the best part...all of this was STUDENT GENERATED! Yeah!

This afternoon I got to head to a nearby county to present at their "Math Gathering"...so nice to see so many teachers interested in professional development- loved being able to share a few ideas.

And the best part...all of this was STUDENT GENERATED! Yeah!

This afternoon I got to head to a nearby county to present at their "Math Gathering"...so nice to see so many teachers interested in professional development- loved being able to share a few ideas.

## Tuesday, February 19, 2013

### Systems Exploration and Ditch Diggers

I really enjoyed my Algebra class today...the pieces of the lesson went together well...

First, I passed out pieces of paper with an ordered pair to each student...asked students to determine if their ordered pair was a solution to either equation shown. After students checked, I had all students whose points were solutions to -2x+3y=1 stand up. I used Desmos to create a table and show that the points formed a line (well, some of them, didn't want to show all of the points for time purposes). Then I had students whose points were solutions to -3x+2y=-4 stand, and repeated on Desmos. Several students noticed that one of their classmates got to stand twice, and others pointed out that the lines intersected at that point, and that is must be a common solution. The list of the points I used is here...came from a Teach21 West Virginia website.

Next, we did the exploration activity I described in my last post . I had a random representative from each group come up to share their question and the conclusions their groups came up with. Had some really interesting discussion with the extension that asked students to graph a system that had more than one solution. One group thought that if they graphed three lines in the system they could have three solutions- each where two of the lines intersected. Another came up with the idea of using two parabolas- the actual question just asked for two equations, and didn't specify linear. One did mention that if it was the same line there would be more than one...but they didn't think that would make a system if it was the same. I wrote two equations on the board (equivalent equations...one in standard and one in slope intercept form). I asked a student to choose a random x value, and find the point on the line in slope intercept form. I said "Wouldn't it be great if you managed to pick the PERFECT point from that line and it is also a solution to the other equation?" Students checked it, and IT WAS! :) So we tried it again with another point and IT WAS TOO! So then I asked a student to just give me a random ordered pair, and students were disappointed when it didn't work as a solution to the system. That started a whole conversation about infinity.

After all the groups had presented we moved on...I really wanted to use Dan Meyer's Ditch Diggers 3act problem. I loved the comment someone made about tying it into the "Chunnel" so we started with watching a little clip about how the "Chunnel" was made, and how the engineers weren't 100% sure that the English and French sides would truly meet up as planned. Then I played the Ditch Digger act one...asked my students what they wanted to know, and then what might help them figure it out. Several students requested a grid on top of the image, so they could find some ordered pairs. Enter Act Two. We had to wrap up the lesson after recording the ordered pairs, but will start with it tomorrow.

I like to assign about 5 problems for homework each night...so I thought I'd use this...a little practice and extension at the same time.

First, I passed out pieces of paper with an ordered pair to each student...asked students to determine if their ordered pair was a solution to either equation shown. After students checked, I had all students whose points were solutions to -2x+3y=1 stand up. I used Desmos to create a table and show that the points formed a line (well, some of them, didn't want to show all of the points for time purposes). Then I had students whose points were solutions to -3x+2y=-4 stand, and repeated on Desmos. Several students noticed that one of their classmates got to stand twice, and others pointed out that the lines intersected at that point, and that is must be a common solution. The list of the points I used is here...came from a Teach21 West Virginia website.

Next, we did the exploration activity I described in my last post . I had a random representative from each group come up to share their question and the conclusions their groups came up with. Had some really interesting discussion with the extension that asked students to graph a system that had more than one solution. One group thought that if they graphed three lines in the system they could have three solutions- each where two of the lines intersected. Another came up with the idea of using two parabolas- the actual question just asked for two equations, and didn't specify linear. One did mention that if it was the same line there would be more than one...but they didn't think that would make a system if it was the same. I wrote two equations on the board (equivalent equations...one in standard and one in slope intercept form). I asked a student to choose a random x value, and find the point on the line in slope intercept form. I said "Wouldn't it be great if you managed to pick the PERFECT point from that line and it is also a solution to the other equation?" Students checked it, and IT WAS! :) So we tried it again with another point and IT WAS TOO! So then I asked a student to just give me a random ordered pair, and students were disappointed when it didn't work as a solution to the system. That started a whole conversation about infinity.

After all the groups had presented we moved on...I really wanted to use Dan Meyer's Ditch Diggers 3act problem. I loved the comment someone made about tying it into the "Chunnel" so we started with watching a little clip about how the "Chunnel" was made, and how the engineers weren't 100% sure that the English and French sides would truly meet up as planned. Then I played the Ditch Digger act one...asked my students what they wanted to know, and then what might help them figure it out. Several students requested a grid on top of the image, so they could find some ordered pairs. Enter Act Two. We had to wrap up the lesson after recording the ordered pairs, but will start with it tomorrow.

I like to assign about 5 problems for homework each night...so I thought I'd use this...a little practice and extension at the same time.

## Friday, February 15, 2013

### Direct Instruction vs. Inquiry

This is my 18th year teaching middle school math- all in Maryland, but across several different counties and schools. I've been at my current school for the last 10 years. My focus in my classroom is helping students to build their understanding of concepts through a mix of inquiry based learning and direct instruction. Growing up, direct instruction was "lecture and take notes". I truly believe that students don't gain true conceptual understanding unless they have the opportunity to "discover" it. When I plan lessons, I really try to think about how I can get students to figure it out...what questions can I ask, what problems will allow them to "happen" upon some aspect that will bridge what they know to what they need to learn.

I love the "3Acts" type of lesson and have really felt true student engagement during those activities. Next year our district is going back to 47 minute periods after many years of a block schedule at the middle school. I am so disappointed because I'm concerned that many teachers who were willing to dabble in inquiry will go back to lecture/notes because students can "handle" that in a short period.

I guess my question is this...does it have to be all one or the other? Or can lessons be engaging and yet still directed in some ways. Is it still direct instruction when groups are exploring different aspects of a problem. For example, on Tuesday I'll be trying out the following lessons with my Algebra students...

Graphing Systems Lesson

Each group will get the same initial problem...graph the system and identify the solution. We haven't talked about systems at all...students know how to graph an equation so I'm thinking that they will very quickly figure out that the solution is the point where the lines intersect.

Next each group goes in a different direction...one focuses on justifying the solution, another explores what a system might look like that has no solutions, etc. My plan is that after groups have had time to explore their concept they will share what they have discovered with the rest of the group.

The end goal is that students will be able to graph a system of linear equations, identify a solution (if there is one, and justify that it is a solution to the system.

So is this direct instruction? Inquiry? Somewhere in between?

I love the "3Acts" type of lesson and have really felt true student engagement during those activities. Next year our district is going back to 47 minute periods after many years of a block schedule at the middle school. I am so disappointed because I'm concerned that many teachers who were willing to dabble in inquiry will go back to lecture/notes because students can "handle" that in a short period.

I guess my question is this...does it have to be all one or the other? Or can lessons be engaging and yet still directed in some ways. Is it still direct instruction when groups are exploring different aspects of a problem. For example, on Tuesday I'll be trying out the following lessons with my Algebra students...

Graphing Systems Lesson

Each group will get the same initial problem...graph the system and identify the solution. We haven't talked about systems at all...students know how to graph an equation so I'm thinking that they will very quickly figure out that the solution is the point where the lines intersect.

Next each group goes in a different direction...one focuses on justifying the solution, another explores what a system might look like that has no solutions, etc. My plan is that after groups have had time to explore their concept they will share what they have discovered with the rest of the group.

The end goal is that students will be able to graph a system of linear equations, identify a solution (if there is one, and justify that it is a solution to the system.

So is this direct instruction? Inquiry? Somewhere in between?

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