Unit 5 Scatter Plots, Correlation, and Line of Best Fit

Right after mid term exams we cover this mini-unit. The lessons take about a week (remember we meet every day for 82 minute blocks) then we work on our performance task. We use one class block to get started on this and then students get about a week to complete it.

Students worked with scatter plots in eighth grade, so they know about creating scatter plots and correlation. They might not have done much with lines of best fit and probably didn't learn how to do it on the graphing calculator yet. We do them both "by hand" and using the graphing calculator (TI-84 or TI-83 also works).

PreAssessment

Here's a "preassessment" kind of thing that works well in determining understanding and having students think about what we will be doing next. I started doing these in a multiple choice format so I could "plicker" to quickly assess how students are doing. If you have checked out https://www.plickers.com/ it's great!  You have to have a smart phone of some type (android or iphone) and the students hold up cards to display their answers and you read the answers by scanning across the room with your phone. It works really well and gives you a quick real time summary of what students know.

The Lessons

Correlation:

We start with what they understand correlation to mean. Then we fill in our left hand learning page about correlation.

Then students sort cards that indicate possible relationships. They decide if they have positive, negative or no correlation. Here are the "cards", caution students can take a long time cutting things out, you can hand this out the night before and have them cut them out for class (use envelope on inside back cover to hold the cut out cards).

I have them glue in a few cards of each. I think next year I'll make this into a foldable - just fold an 8.5x11 sheet of paper in half and create a little booklet. Front cover could be positive correlation, inside negative correlation and no correlation. It will give us more space. At bottom of each can have "sample correlation sentence" corresponding to one of the examples on that page. 

Scatter Plots by Hand:

Then we do work with making scatter plots by hand. Here are some materials I use with students: NBA players and Full Analysis example. And a foldable to summarize the steps. 

 We also talk about interpolation & extrapolation. Here's a foldable to illustrate this:

We fill in definitions of interpolation and extrapolation (apparently I didn't do that, this was before I started using a document camera). Next year I'll add a place to also write a correlation sentence for this example plus what type of correlation is shown.

Correlation and Causation:

The tricky thing about correlation is that most people assume the correlation implies causation, wrong!  This is a common misconception and something that warrants a bit of discussion. Here is an introduction worksheet that I have students work together on. *updated in 2016 I used a powerpoint to lead the discussion on this topic. All materials in the link at the at of this post.

Then we summarize with an ISN foldable. (but I put the examples from the foldable on the board before handing it out for some discussion first).

with cute little saying I found on pinterest:

Common misconception - students think a relationship is either a correlation or a causation. They don't see that something is either a correlation OR a correlation and causation. It's a bit tricky and abstract for them. 

Outliers & Influential Observations:

Next topic is Outliers and Influential Observations. I've started to do both because not all "distant data points" are outliers. If they follow the general trend line they are influential observations. I didn't actually know that and I think it's important to know. Here's a classwork discussion worksheet. And here is my ISN foldable for summarizing. *again in 2016 I used a powerpoint to lead the discussion on this. files all available at the end of the post.

Scatter Plots on the Graphing Calculator:

Now we bring in some technology. Students learn how to create scatter plots on their calculators. Not all my students have the same operating system so my foldable has instructions for both. I like to have students use the [VARS] menu to insert the linear regression equation directly into Y1 and calculating the linear regression looks a bit different depending upon the operating system. Here is their instruction foldable. I expect advanced students to learn this and know how to do it without instructions on assessments. (they need to get very fluent with graphing calculator).

I also like the "diagnostics" to be "on" so they can examine the regression coefficient to evaluate the strength of the correlation. Here is the foldable for that.

Data Sets:

Here are some data sets that I then have students work with and here is the answer key. 

Performance Task:

Our summative assessment for this unit is a performance task that has students compare fat and carbohydrate content of snack foods. It's a fun activity with some good analysis. *previously I had a question that introduced an item that would be an outlier. After correcting my 2016 performance tasks I tweaked my write up a bit more (in addition to fixing typos on the rubric) so that we have a weird item in our data to start (steak!) and let students discuss what that does in the write up. All update files below.

All my files for this unit can be found HERE. I did update some of my ISN inserts in 2016. They are not hugely different and the window for blogging about them is gone so I'll probably blog about them in 2017! But all the documents can be found in my box file posted above.

Linear Inequalities in 2 variables

What was interesting when we first started working with the common core curriculum in our state workshops four summers ago is that linear inequalities in 2 variables was NOT in the topic list but systems of linear inequalities was included!  So we realized pretty quickly that we had to add this topic if the students were going to have a successful experience with systems.

Here are my ISN pages. I realize that I need to have one of my examples be a negative slope. So I'll change one of my examples for next year. Here is the document with both pages.

Here is a homework worksheet I used for graphing and a nice enrichment worksheet for those who catch on to this topic quickly and need a challenge (given graph, write linear inequality).

I'm a fan of having students compare & contrast things in math class and Venn Diagrams are a great way to do that. These make a good warm up activity for students to fill in with their groups and then we have a giant one on the board to have students come up and fill in. Here is my worksheet.

My advanced students are very good at learning procedures and working through problems. But I want them to be able to articulate what the solutions are for each of these. (we talk about solutions quite a bit all year - haven't mentioned it in the blog so far - but a lot of what we are doing in algebra one is working with solutions.....and a solution isn't necessary a single number). This is a nice critical thinking worksheet getting them to think about the solution sets for equations and inequalities. 

We also did applications of linear inequalities in two variables - but I didn't get to make an ISN foldable for these. At this point in our school year it was the end of semester one and we were wrapping up so we could have a unit test and start to prepare for our midterm exam. But here are some materials I used with my students.

Application Introduction.

Applications HW

Piecewise Functions

We saw piecewise functions earlier in unit 3. We worked with them quite informally as descriptive functions. Then we saw them as a separate function type in the "non-linear" category. Even though the pieces are often linear since there is not a constant rate of change throughout the whole function we're going to classify them as non-linear.

This is a nice introduction worksheet to use when getting students to remember what we did before and what we can do with it now that we've covered linear functions.

Now we look more closely at piecewise functions. Students should be able to graph a piecewise function given the function equations and also to be able to write the function equations for a piecewise graph and finally to evaluate a piecewise function.

We do this two-page spread. I give them the large graph and the function equation slip. We examine the function equation (seeing how each part is defined for a certain domain) and I ask them what those little inequalities might be telling us. Then we use colored pencil or crayon to give each piece a color (highlighting the equation for that piece). And we lightly make vertical lines separating the various parts of the domain. Finally we graph the pieces lightly with pencil (we may need to work "outside of the domain" but then once we get the line we only use the part within the appropriate domain).

ISN large coordinate plane and function equation

Then we make some summary statements about the domain & range. And we go over how to evaluate a piecewise function for given values (notice the color coding). 

This was my first year doing this with students. I gathered a bunch of practice worksheets from on the internet but want to go through those more carefully and create a good packet that students can work on. But here are the sheets I found and mostly used. WS1    WS2    WS3    WS4    exit slip

Parallel and Perpendicular Lines

https://www.youtube.com/watch?v=vnnwfcDcNlY

This is a quick topic with my advanced students. Usually I have them do an exploration packet that they take once they finish a quiz on our previous topic (linear equation applications). Once everyone finishes the quiz students work together on completing this packet. I want to come up with some enrichment option for those who finish early.

Then we summarize everything in our ISN. Usually the right hand reflect ends up being the warm up activity for the next class.

Here's a nice little homework worksheet I've used. And a very cool sorting activity. 

Linear Equations - All Forms & Applications

We cover all three forms of linear equations in my Algebra One class. Students learned about and used slope intercept form last year (8th grade common core) so they have had some experience working with y = mx + b. My goals for them now is to graph a linear equation in any form, write a linear equation in any form given information about the line, convert between the three forms and use all three forms in application situations. This takes a number of blocks but I will post all the ISN materials in one post.

We go through each form - discuss, work in groups to explore & practice. We summarize in our ISN. All the bright yellow "left hand learning" are found here (in large 11 by 17 document - I got a great deal on this paper, but you can change size to 8.5x11).

Two page spread for Slope Intercept Form:

Right hand reflect for each are examples that students work through. 

Another two page spread for Point Slope Form:

And another two page spread for Standard Form.

At this point students were asking, "how do we know which form we are working with?". So we brainstormed characteristics and made a page in our ISN for helpful hints on identifying.

Graphing Practice:

And here are some "graphing packets" for each form that I had students work on for practice:

slope intercept packet

point slope packet

standard form packet

 Working between the three forms (convert, write, etc!) 

 ISN "Writing all 3 forms given a graph"  and  the three examples

....with some documents I had students use to work between the forms:
Class Practice
More Class Practice
Quiz

 Applications

I use the ISN to go over examples for each and how you identify which form is used. This only takes up three pages. I need to make a fourth page that is a "you choose" application. Not sure what I'll do with that yet, feel free to send me suggestions via comments below.

After introducing we do a bunch of practice - in their small groups and for homework. Here are some documents that I have used;

 Helping Others
Homework
More with Standard Form

 Which Form Do You Use?
Windchill Exit Slip
More Writing & Using
You Choose Water Packet

Horizontal & Vertical LInes

This is another topic that students have seen before in 8th grade. But I find that students really don't understand how the simple equation of y = -2 or x = 4 result in a vertical or horizontal line. They are used to seeing those simple equations as solutions of equations in a single variable. So I have to remind them that we are now working on a coordinate plane and these are equations in two variables.

My advanced students are quick with this concept and remember it well from last year. All that we need to do is review with some ISN inserts. I LOVE the HOYVUX inserts I found via Sarah Hagan at Math=Love.

Since this is pretty quick I finish up class by giving them an investigation to get ready to talk more formally about linear functions. We buy this great red graph paper from GEYER and I give them a sheet for this investigation. It comes in 6 per side (front & back) or 3 per side (front & back). disclaimer: I do not profit from mentioning this paper, I've been using it for years and it's really wonderful. Being printed in red ink allows the student work to show up easily. 

                         

Rate of Change & Slope

My Algebra One students learned about rate of change and slope last year in 8th grade. And they are an honors level class so I find they are quite good with this material, so this is mostly review.

We start with "rate of change" and we recall that a linear function has a constant rate of change. I use a graphic organizer to illustrate the many ways to consider what rate of change is on our "left learning page".

 On the "right reflect" we do some rate of change examples.

At this point I like to include a short little performance task called "Homecoming Balloons".

 Parts of it can be solved with an equation, but I want them to use rate of change so purposely do this before we review the different linear equations (they learned at least slope-intercept form last year in 8th grade). If you want the teacher "answer key" with rubric for scoring contact me and I'll email it to you.  

I have this "warm up" worksheet to wrap up "rate of change" the next class and lead us into slope of a line. 

Next we look more formally at slope. First we consider the 4 types of slope using the very popular slope dude youtube video. I've used this for years!  Well before we had our interactive student notebooks. 

and we support these ideas with an ISN page:

I added the letters WWSDS to this page (what would slope dude say) and have a large poster of these letters on both my front and back walls. Thank you Sarah Hagan of Math=Love for this great idea!

Students do a little "right hand reflect" on the bottom of this left hand page with some roller coaster slope examples:

And for additional practice, the real right hand page we do this "4 Types of Slope" foldable that I found via Mrs. Math at teachers-pay-teachers. This is a free downloadable. (my copy linked above includes any edits I made).

All of the previous material is essentially review for my students so moves rather quickly. Here is a nice "cool down" for students, it can be collected as an exit slip to see what students are thinking and how well they can explain. It just looks at using the formula. 

A great warm up on the second day is to do a slope treasure hunt. I'll have to do a blog entry going over this really cool "move around the room" activity that can be adapted to any math concept. But it's not too difficult to interpret and students have a lot of fun with it. 

And I love doing "You Be the Teacher" problems, where students examine worked through examples and explain what's wrong with them. Here is a worksheet of "you be the teacher slope formula". 

Then we do some additional examples involving slope as seen in this foldable. (sometimes I start with an investigation that the students work on in their groups). 

With a Right hand reflect page of examples that I use as a warm up the next class:

Or even assign those problems for homework.

There is a ton of material out there on the web on teaching slope, the above is just what I've pulled together for my class.