Today's estimation was civil rights leader Jimmy Webb. I love that my 4th period has a few students who grew up internationally so one of two students always estimates heights in centimeters. So, we got to talk about converting Jimmy's height to centimeters by multiplying by 65 inches by 2.54 centimeters.

A student compared their height to Jimmy's, and one estimated that he was 30 centimeters shorter than me (180 cm). |

These students found that these two equations intersected at (1/2,0) |

I complimented this group and wanted to point out how they were all leaning in, listening and invested in finding the connections between their graphs with the arrows. |

With 20 minutes to go I told students to stop and make sure they had their cards and arrows glued down. Then, with 15 minutes to go, they took out their comp books and we started our closure share out where we took notes on our discussion.

We started with examples of two equations with one common solution. I showed them the notation of the "SET" after looking it up just now. I told them that the set symbol, {} is a type of grouping symbol or parentheses for systems of equations. I asked students what they noticed about the first system where both equations were in y=mx+b form. They were able to see that they both had b values of 4, so their point of intersection was (0,4). I asked them what else they noticed and they said one was growing by 2, and the other was decreasing by half. We also discussed how it was harder to analyze the equation when it was in standard form, and the "x =" format threw off some people because of the unfamiliarity with it.

The next discussion headline was equations with no common solution. I asked them how they knew they had no common solution. They said that the lines were parallel. I asked them what they noticed about the equations. Here students observed that they had the same growth rates, but different y intercepts.

Finally, infinitely many solutions was more evident with y=2x+4 and y=2(x+2). In some classes some students distributed incorrectly and it was a great discussion and a place to honor the mistake. We also discussed how if they were the same they had the same growth and same y-intercept.

In accelerated students started right away finishing the Desmos LSRL activity I created. Students needed assistance entering the correct symbols for the regressions line. I wrote on the board y 1 ~ m x 1 + b. That helped a lot of students. Also, they weren't quite ready for the acronym LSRL. Students needed some prodding to click near the vertical line and the line of best fit to see the point of intersection's coordinates would pop up.

We also reviewed this photo as to why a line of best fit is so important: look at the 32 different lines of best fits made by students:

Students saw Steph Curry as an outlier. This student did, but is clearly not a basketball fan: "(1013, 893) is an outlier. That data represents the basketball player known as Stephen Curry. I cannot say if that is an actual basketball player, and I don't want to know. I do not know why anyone's last name would be curry."

But when finding his residual, here is a student J, who clearly understood the concept: "Curry's residual is 341.8. I found this by clicking on the point between the line of best fit and 1013. Then I subtracted 551.2 (the best fit line prediction) from 893 (the real point value) and got 341.8 residual = actual - prediction"

Contrasting this, N did not understand it: "1013-893=151 so 151 is the residual?" But the cool thing about this, is a student who did get it, saw a student's response ahead of him, who did not get it. He mentioned it to me, which allowed me to go talk to N about it.

R understands what a positive residual means in basketball terms at least "A player would be more proud with a positive residual because having a positive residual would mean the player scored more than what was predicted."

This student predicted the points scored for a player with 809 minutes: "425.8 it gives you the point of intersection which is the average that a player should score with the same amount of time." Also noticing that some students who were given the residual that was negative, still calculated a residual that was positive. This was only one student who confused the order.

A lot of students easily interpreted why a negative y-intercept did not make sense, but a smaller group also interpreted what the slope of .6 meant in the context. A understood it as: "The slope shows that for every one minute played, on average, someone scores 0.61481 points. The y-intercept can be interpreted as when there have been 0 minutes played, you have scored -71.6 points, which does not make much sense in real life."

When students saw the residuals plotted, they thought it had negative association. I think they were not used to seeing those plotted.

I like how T and H interpreted the residuals once the scatterplot was turned off:

In reflection, I am definitely going to remove the upper and lower boundary lines. I do like they got a preview of what residual plots look like on the same graph and separated from the graph. I think I am still getting the flow of how much is doable in one period, and then I can have students try Marbleslides out when they finish next time.

We started with examples of two equations with one common solution. I showed them the notation of the "SET" after looking it up just now. I told them that the set symbol, {} is a type of grouping symbol or parentheses for systems of equations. I asked students what they noticed about the first system where both equations were in y=mx+b form. They were able to see that they both had b values of 4, so their point of intersection was (0,4). I asked them what else they noticed and they said one was growing by 2, and the other was decreasing by half. We also discussed how it was harder to analyze the equation when it was in standard form, and the "x =" format threw off some people because of the unfamiliarity with it.

This was in one of the first classes. I'm noticing some misconceptions in this poster. Many students made diagonal connections as well (not pictured). |

The next discussion headline was equations with no common solution. I asked them how they knew they had no common solution. They said that the lines were parallel. I asked them what they noticed about the equations. Here students observed that they had the same growth rates, but different y intercepts.

Finally, infinitely many solutions was more evident with y=2x+4 and y=2(x+2). In some classes some students distributed incorrectly and it was a great discussion and a place to honor the mistake. We also discussed how if they were the same they had the same growth and same y-intercept.

Great summary of the rich FAL activity. |

In accelerated students started right away finishing the Desmos LSRL activity I created. Students needed assistance entering the correct symbols for the regressions line. I wrote on the board y 1 ~ m x 1 + b. That helped a lot of students. Also, they weren't quite ready for the acronym LSRL. Students needed some prodding to click near the vertical line and the line of best fit to see the point of intersection's coordinates would pop up.

We also reviewed this photo as to why a line of best fit is so important: look at the 32 different lines of best fits made by students:

Students saw Steph Curry as an outlier. This student did, but is clearly not a basketball fan: "(1013, 893) is an outlier. That data represents the basketball player known as Stephen Curry. I cannot say if that is an actual basketball player, and I don't want to know. I do not know why anyone's last name would be curry."

But when finding his residual, here is a student J, who clearly understood the concept: "Curry's residual is 341.8. I found this by clicking on the point between the line of best fit and 1013. Then I subtracted 551.2 (the best fit line prediction) from 893 (the real point value) and got 341.8 residual = actual - prediction"

Contrasting this, N did not understand it: "1013-893=151 so 151 is the residual?" But the cool thing about this, is a student who did get it, saw a student's response ahead of him, who did not get it. He mentioned it to me, which allowed me to go talk to N about it.

R understands what a positive residual means in basketball terms at least "A player would be more proud with a positive residual because having a positive residual would mean the player scored more than what was predicted."

This student predicted the points scored for a player with 809 minutes: "425.8 it gives you the point of intersection which is the average that a player should score with the same amount of time." Also noticing that some students who were given the residual that was negative, still calculated a residual that was positive. This was only one student who confused the order.

A lot of students easily interpreted why a negative y-intercept did not make sense, but a smaller group also interpreted what the slope of .6 meant in the context. A understood it as: "The slope shows that for every one minute played, on average, someone scores 0.61481 points. The y-intercept can be interpreted as when there have been 0 minutes played, you have scored -71.6 points, which does not make much sense in real life."

When students saw the residuals plotted, they thought it had negative association. I think they were not used to seeing those plotted.

I like how T and H interpreted the residuals once the scatterplot was turned off:

TIFFANY

Residuals tell us that linear lines aren't always really accurate. You want your residuals to be closer to 0.

HARRISON

The further apart and greater the residuals, the less linear the data is. A closer residual to 0 means a more linear line.

When Curry's data was removed, students responded:

# What happened to your line of best fit?

Also, what does this say about Stephen Curry's value to his team?

NURI

Stephen Curry is the league MVP for a reason. The warriors would still survive if Curry got injured they would still be ok but Curry takes the to the next level.

DINOSAR DAVIN

This means that is stephen curry stopped scoring, then the whole average would drop.

MARKOS W

steph curry is valuable. Weak curry, loses to bucks. No curry, loses to Dallas. MVP!MVP!MVP!MVP!

ANNABEL S

The line of best fit went downwards considerably. This shows that Stephen Curry is very valuable to the team to help them get points.

NICOLAS

It got less steepe

ROXANNE

The line of best fit got a much lower slope. Stephen Curry is important to the team

MOREEN

It means he is an important player

SARAH

Stephen Curry

ALEX NOT DAVID

the line of best fit went down in the right side and it probably makes curry useless if he isn't making the team worse

ALEX D (REESE'S PUFF)

It became more horizontal. HE TOP TIER YO

JEFFREY LIU

My line of best fit flattened. Stephen Curry appears to be very important.

MARINA W

This says that he's bad at basketball

In reflection, I am definitely going to remove the upper and lower boundary lines. I do like they got a preview of what residual plots look like on the same graph and separated from the graph. I think I am still getting the flow of how much is doable in one period, and then I can have students try Marbleslides out when they finish next time.

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