Parallel and Perpendicular Lines

My Introductory Math students learned about parallel and perpendicular lines this week. My wonderful math intern from Appalachian State University taught my students how to identify lines as parallel, perpendicular, or intersecting from graphs and equations. The following day I was set to teach my students how to write the equation of a line parallel or perpendicular to a specified line going through a specific point, but I wasn't sure how I was going to teach it. I knew my students were struggling with their algebraic skills and I was nervous about how they would learn the material if the algebra was getting in their way, although I didn't know how to teach it without this being an issue.... Cue Tuesday morning and a conversation with a former student.

On Tuesday morning during my Math FLEX session, I had a former student come by to get help in her current math class. She is currently working on transformations of geometric figures and she was struggling with rotations. In her class, they had been using the rules for rotations. I started by reviewing the rules with her, then showed her I keep track the different rules.  I could tell that this wasn't working for her.

I decided to get out some graph paper and show her how to graph the rotations and from there she could come up with the image coordinates. All of a sudden her there was light behind her eyes, where there was once fear, and I could see the joy in her face. She saw how rotations worked and understood what it meant. It wasn't another rule for her to memorize, it just made sense.  I was elated for her and at the same time a little disappointed that I didn't start with the picture first.

How often do our students need to see something concrete, like a graph, before moving onto the abstract, like algebraic rules?   The answer is probably 99% of the time, especially for students have difficultly in math. 

Throughout the day I kept thinking back to this student and how excited I was for her; not only for the knowledge she gained but that there was a simple solution to helping her understand.  All she needed was to SEE what was happening!  

During my planning period on Tuesday, I realized I had to finalize my plans for writing parallel and perpendicular equations. It was Halloween and I was going to spend time with my nieces, so I wouldn't be home until late and I knew I wouldn't want to finalize my lesson plans once I got home.  I thought back to working with the student on rotation and I had an epiphany, what if I taught my students how to graph parallel and perpendicular lines before I taught them how algebraically find the equation?  It made sense in my head but I wasn't so sure it was the right thing to do, or that they would understand and see the connection. 

(My nieces dressed as Descendants Uma and Evie for Halloween.)

My co-teacher wasn't at school, so I couldn't run the idea by her. So I did the next best thing, I went down the hall to visit another teacher and ran my idea by her. She loved it and thought it would work.

Now, I just needed to figure out how to create a foldable for our interactive notebooks to include this new part of my lesson.  In the foldable I wanted to include parallel and perpendicular lines, writing equations by graphing and using algebra, and I wanted it to have an order to it. The teacher I was talking to suggested a flip book and I jumped on the idea. 

I knew that Kathryn had a template for a flipbook because I had used it before so I re-downloaded the template and whipped up a foldable. The foldable came together so quickly and I was excited that I had the time to think about it, talk it over with another teacher, and then create it before leaving for the day. However, I wasn't too sure how it would go over with my students.  I didn't know if it would make sense or if it would be too much at one time.

 

We spent the next two days on writing equations of parallel and perpendicular lines, beginning with graphing, then with algebra. It went amazing, so much better than I expected. This was the first time  that my students could write the equations of parallel and perpendicular lines after only seeing it a couple of times.  While students were practicing, I kept walking by my co-teacher and our interns and would say, "I just can't believe this; the students understand and can do it!" 

I'm happy to report that after two days of working on parallel and perpendicular lines, over 90% of our students had mastered this skill. I don't have the data from previous years but I can say without a doubt, it wasn't that high of a percentage of students nor was it that quickly.

I'm so glad that with the help of a former student and fellow teacher, I was able to remember the power of a picture!

After tweeting about this experience, Tim Brezezinski replied with a link to a geogabra investigation on parallel and perpendicular lines. I plan on using this as one of my stations during our unit review next week!

Here are the links where you can download these files. Here's a pdf copy and an editable Word document. Graphs created with GraphFree.com  (Printing note: You will need to print on both sides of the page, flipping on the short edge.)

War: Comparing Slopes & Y-intercepts

I haven't been blogging lately because life has been busy and I never seem to find the time. However yesterday I posted a picture on Twitter of an activity I did with my Introductory Math students and  large number of people asked for files of an activity. I figured the easiest way would be to blog about it and share the files.

There have been a lot of changes since I last blogged in May 2016. I have switched jobs and started teaching in different school district,  where I began teaching a new course for me, AP Statistics, and had three preps. My new school also has a different schedule that I had to get accustomed to: my planning period alternates every other day, half of my classes are on a block schedule, the other half are on an A/B modified block schedule.  I really love where I'm teaching, what I'm teaching, and my schedule, but it was a lot of changes in a short amount of time.

On top of that, my husband and I sold our previous home and bought a new house.  Now I don't have a lot of time because I'm working on completing my National Board Certification and decided to start doing Standards Based Grading with my students. Which is A LOT of work the first year!

That's enough of a tangent, I'll actually get to my post about comparing slopes and y-intercepts war.
......

After my students played Sarah Carter's @mathequalslove, Evaluating Functions War, they insisted that we play War again. We ended up having a two hour delay on Tuesday because of flooding, so I had time to create a card set for my students to play War again.

We are currently working with linear functions and students needed continued practice with identifying the slope and y-intercept from various representations. So I figured I'd create a set of cards for students to play War and compare slopes and y-intercepts from multiple representations. I  ended up created 18 different cards with linear functions.

I printed these pages should that every student had one of the two sets of functions. I printed these on different colors of paper to make it easier to sort out the cards in the beginning of the game. I went ahead and grouped 4 sets of cards together into bags so that it would be easier to distribute the groups.

I then went over the directions for playing War, if needed, see Sarah's general explanation. Here's how I adapted it for these cards and for my students.

1. Distribute the cards to each player. Each player should get 9 cards of a matching color. Each player should shuffle their deck several times.

2. At the beginning of each round and every 3-4 minutes of the game I'd call out "greatest slope wins" or "smallest y-intercept wins." After all, I needed students to practice finding slope and y-intercept, then comparing their values.

3. Each player turns over their card and determines the slope/y-intercept.

4. The player with the highest or lowest (depending on the rules stated out loud), wins every other players' cards. These cards will go on the bottom of their stack.

5. Repeat #3 and #4 until one player runs out of cards. Once a player has run out cards, the remaining players count the number of cards they have and the player with the most cards wins. Another round is played if time allows. (This allowed all students to be engaged the entire time we were practicing.)

Note: If two or more players had identical slopes/y-intercepts, each of those players turned over a new card to act as a tie breaker.

It worked really well and all of my students, even those that are reluctant to do anything in class, enjoyed this activity.

Here are the links where you can download these files. Here's a pdf copy and an editable Publisher version. Now that I've blogged again, I'm going to try keep and and blog on a more regular basis. 

Linear Functions (part 2) - Interactive Notebook Pages

If you missed the first part of my linear functions unit, you can find it here.

The second half of my Linear Functions unit covered linear regression, residuals & correlation coefficient, literal equations, and domain & range.

This booklet has been the most referred to page in my students interactive notebooks. It is a set of calculator screen shots to help students use their TI-84 calculator. There are pages on entering data, graphing data, finding the linear regression equation, and calculating the correlation coefficient. In my school system, Math 1 is the first time students have used this particular calculator. As a result these type pages really help students self-help when they have questions on how to use the TI-84.

After spending a few days on linear regression models we moved on to residuals. I had students use a spaghetti noodle to approximate the line of best fit and then approximate the residual. I then gave students a graph with the actual line of best fit and students had to compare their line and residual with the true line of best fit.

 For correlation coefficient, I found an idea I liked on Teachers Pay Teachers for correlation coefficients, but I don't like to buy my resources. Once I saw the idea, a coworker and I decided to create similar foldable using Publisher and taking screenshots of data using our calculator. 

Then since we had Saturday School the next day (Snow days aren't very fun when you make them up on Saturday) I created a set of scatter plots using GraphFree based on the data from Lesson 8.5 of the Mathematics Vision Project - Modeling Data unit.  I had students draw a number line from -1 to 1 and then have arrange the scatterplots in order by correlation coefficient, then they were to approximate the correlation coefficient. Afterwards, I gave each group a data set for one of the graphs and they found the correlation coefficient. This activity prompted many great discussions among students. 

 Next up was interpreting slope and y-intercept in the context of a problem. We reviewed this while doing linear regression, but student still needed more practice.

Then we focused on solving literal equations. The reasoning for solving literal equations in this unit was to refresh their skills. The next unit we would be covering would be systems of equations, so I wanted to remind my students how to solve an equation for a variable.

The last topic in this was was domain and range. The DIXI ROYD organizer was taken from Sarah Carter at MathEqualsLove and the domain and range foldable from JournalWizard.

If you would like the files to any of the documents I've created I'll be glad to share. :) 

Linear Functions (part 1) - Interactive Notebook Pages

The second unit I covered in Math 1 focused on linear functions, which was suppose to last for four weeks. In actuality it lasted for over a month and a half due to many days off and early releases/delays because of snow! In the future I need to make this two units because it got super long; a unit of linear functions and a unit of linear regression. Today's post will have pictures and descriptions from the first half of the Linear Functions unit. I will try to post on the second half tomorrow.

Students have covered linear functions in 8th grade however it is a key part of the Math 1 curriculum. In Math 1 students are responsible for interpreting linear functions and comparing linear functions to quadratic & exponential functions. Even though it was thoroughly covered last year, students still need quite an extensive review.

I started this unit with a review of writing and solving equations with word problems, since my goal was to incorporate them through the semester, instead of having one unit of review. We just practice solving equations as they come up naturally in the curriculum but I did want my students to have a point of reference in their notebooks. These files were adapted from Sarah Carter at MathEqualsLove. You will notice I get a lot of my ideas for my interactive notebooks from Sarah. She does an excellent job of blogging about her class. In fact, without her resources, I don't think I would have been able to incorporate interactive notebooks into my classroom.

We then jumped into sequences, as students in Math 1 must be familiar with arithmetic and geometric sequences and writing equations in Now-Next form and function notation. I decided to cover both types at one time simply because we had already discussed (briefly in unit 1) exponential functions and it flowed well together.

We started with the green cards and students had to fill in the next three terms and come up with a pattern; we then matched the sequences to a graphs. Afterwards, we formally went over the recursive and explicit formulas and they wrote the corresponding equations for each sequence. The equation notes were also adapted from Sarah Carter at MathEqualsLove. 

After sequences, we jumped into formalizing linear functions by discussing slope.

The types of slope foldable is from Jessie Hester at Mrs .Hester's Classroom.  On the inside we included a visual of Slope Dude's path.

 I adapted the finding slope from different representations from Julie Reulbach at ispeakmath.

Then we discussed y-intercept and putting the slope and y-intercept together to write the linear equation in function notation. We used color with a purpose on these notes so that students could refer back to the slope and the y-intercept individually. 

 Then we discussed various forms of linear functions including: horizontal & vertical lines, slope-intercept form, and standard form. This foldable was inspired by Sarah Carter at MathEqualsLove.

We then went over multiple representations of linear functions. Students were given four problems each with a given representation then they had to create the other representations. This semester my page only has the blank problems as I spent my time working with groups on these problems and I didn't go back and fill in the page. However, I have pictures of this page from last semester.

To end the first half of the linear unit, we covered parallel & perpendicular lines and their equations. We started by discussing where parallel and perpendicular lines appear in real life (and the most creative response was rewarded with a Starburst) then we moved onto the algebra.

If you would like the files to any of the documents I've created I'll be glad to share. :)