We talked about properties and end behavior of graphs all year long. We started with linear, transitioned to piecewise through distance-time graphs, then absolute value, then exponential, then quadratic. These are the INB pages for our properties of quadratic graphs.


Since we had talked about end behaviors, average rate of change, intercepts, vertices, positive/negative, etc. before, this was a quick extension of ideas to quadratics.

Each student got two identical graphs. Two because all the color-coding info would be jumbled on one graph. We color coded and created our flash cards all first, then taped everything into the notebook.


When students walked into the classroom on this day, our "First Things First" section of the board told them to pick up two pages. One had graphs and the other is the flashcards and envelope. They also picked up scissors and immediately got to cutting everything out. This is something I often do, give students the directions before class officially starts. I begin on day 1 telling students that they are considered tardy to my class if they haven't  begun the First Things First activity before the tardy bell rings. They learn this very quickly and don't waste time once they enter the classroom. I also promise them that I will give them a 2 minute warning at the end of class so they have time to pack up and clean up the classroom, and therefore leave when the ending bell rings.

Anyway... small tangent there. :)

As students were cutting out their flash cards they were telling each other "hey, we know what these mean!" "I know this!" "Mrs. Hester, this looks easy!" I love when they are confident with their learning!!

Once everything was cut out, I asked students where they wanted to start. They knew we had to talk about everything, but might as well let them choose the order! They would call our words that they felt confident in and provide a definition. Classmates would add to the definition and we would talk through specifics. Once I had guided the discussion to a place I was comfortable with, we wrote down the definition. We then switched to a colored pencil, marked that particular feature on a graph, and wrote the specific answer for that definition in the same color on the card.

First and foremost... this lesson is not orginally mine!!
Remember when I told you how much I loved this blog? When I was first starting INBs last year, I clung to a few blogs for ideas and tips. Sarah's was definitely one of them! HERE is Sarah's original post of this lesson.

I started with the scales just like she recommended. Your copy is here if you want it.

Students worked for a few minutes individually and then discussed ideas with a partner. Of course #3 and #6 really caused some problems... they don't have a solution! I encouraged students that if they got stuck on any particular problem to skip it and come back. Inevidably #3 and #6 were left til the end!

We discussed all six as a class and wrote the solution types. Our notes looked something like this when we were done.

We talked about one solution, no solution, and infinitely many solutions at this point. This was the first time most of my students had heard these ideas so we talked them through quite a bit. And yes...we wrote them algebraically. At first they moaned and groaned, espeically at the all real numbers, but by the end of the year, after coming back to these ideas repeatedly, they really felt acomplished. They would brag about being able to write things like "real mathematicians." I love math nerds!!! :)

Then we made a foldable, including one of each type of solution set.
**From here on the pictures are tiny. GRRRR. If you click they will enlarge. 



We talked about how the algebraic process of solving relates to the scale. For example, in the equation where the only solution is x=3, when we subtracted 2x from each side the resulting 2x=6 could be seen on the scale where two boxes line up to the 6 box. This really helped students grasp what was happening as they solve equations.

 How have you taught this idea? Got any great foldables or activities to share??

Multiple people have asked for it, either through email or comments, so here it is!

The glossary template.
Competely free.
Completely editable.


The blank template in here.
My Algebra 1 completely filled version is here. The font is Berlin Sans FB Demi and comes standard on most computers.
Both are in word so you are free to edit as you like. :)

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Short and sweet with pictures! :)

We used this product by Lisa Tilmon to talk about theoretical and experimental probability. Students included the results in their notebooks on page 17.

Students then completed this simulation investigation made by yours truly. Download your copy for free!

They then completed their own simulations for homework on page 18 in their notebook. In case it's hard to read, I printed thin pieces of paper to be taped in so they didn't have to copy anything down, risk missing a key piece, or waste valuable class time.

The paper says: (1) Describe a real-life event that you are curious about. (2) Create a simulation to model that event. (3) Perform your simulation 15 times and record the data in a table. (4) Describe what your data tells you about your real-life event.





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"How do you retest?"

Alright...here we go.
I work in a district where all math courses 7th-Geometry have retests; it is not the teacher's choice. That being said, I love retesting! My students learn so much, about math and study habits in general, between their first test and their retest.

Here is an overview of how retesting looks in my classroom.
All tests are written as a team and all courses give the same tests throughout the district. This helps to ensure that our students receive an equitable education. As we write the tests, we write them with testing/retesting in mind. Meaning, we write small sections with about one or two topics addressed per section. These are called competences or objectives. Most unit tests have 3-4 competencies of new material. Beginning on the second unit, we also include a review competency. These questions help keep ideas fresh throughout the year. So most tests are 4-5 competencies long. Every once in awhile we have a 6-comp doosey, but we try to avoid that.

During the unit students complete qualifiers. These are particular activities/investigations/homework assignments/projects that will qualify them for specific sections of the retest later on. In my classroom, we do lots of activities and assignments that are not qualifiers as well. I specifically tell students which assignments are qualifiers and which are not; I'm all about transparency! After I grade a qualifying assignment, I give it back to students and they record the grade on their grade sheet for the appropriate competency. You can see a copy of my grade sheet in this post. Like I said, we do other things that are scored, they are just not recorded on the grade sheet. I require that every qualifying assignment have a grade of at least an 80%. If a student receives less than an 80% they must correct the incorrect work and resubmit their work. I give full credit for resubmitted work. This also relieves a lot of the stress students feel and insecurity about having one shot to complete their math homework. They go home, they try, we discuss it again the next day, they submit their work, I offer feedback, they tweak and resubmit. I really like the system.

Anyway...moving on.
After an entire unit is complete, students do a review assignment and take their test. Each competency of their test receives a separate score out of 100 points. I hand back their tests and they record their competency scores. Any competencies that they scored at least an 80 on are considered "mastered" meaning they do not have to retest. They are welcome to try for a higher score, but they do not have to. Any competency where they scored lower than an 80, they are required to retest.

I really like that my grade sheet shows homework and test scores for the same material stacked vertically. It is a very often occurrence that when a student makes a low test score, such as a 40, they also have a relatively low homework score. They can see their hard work (or lack thereof) really paying off.

I give students about a week between tests. During this time we go over common errors and misconceptions and I hold tutoring times during seminar (a 40 minutes block built into our day for tutoring, homework, and assemblies). I also remind students repeatedly throughout this week to check their grade sheets and make sure they are qualified. I will often create a second review-type assignment before the retest that also counts as a qualifier.

Once it is retest day, students turn in any qualifying assignments. These are often homework assignments they have been correcting to get their 80%. This also includes any test corrections that I assign for a particular unit. As students take their test, I quickly check the assignments that have been turned in, update their homework grades, and make sure every student that is taking a test is qualified for their test.

After their retest, we go over final errors and students record their retest scores. The sample grade sheet would look something like this now.

 Notice the student fixed their 70 in comp 4 and was allowed to retest that section. They did not retest comp 1 since they scored a 100 the first time. They tried to improve their comp 2-4 scores and succeeded in comps 3 and 4. I always keep the highest grades that students make in each section. So in this case, I would keep the highlighted 80 for comp 2 since their retest score was lower. Students usually like to calculate their overall grade, in this case 360/400, or 90%. I keep each competency score separate in my grade book.

Since this style is fairly repetitive, I also give a problem-based task for each unit that forces students to look at the material in a new way. Those tests do not have a retest opportunity and are worth half the points of the mastery test. In this case, since the mastery tests were worth 400 points the task would be worth 200 points. All of those scores are recorded at the bottom of the grade sheet in their own section.

I hope that answers some of your questions! Feel free to comment with more questions and I'll do my best to answer them.

Also...
Get your copy of my grade sheet here! I created the original file in publisher, email me if you want a copy of that one.

One of my favorite pages from this year was our Solving Systems of Equations page. In my opinion, this page was truly interactive. Students were sorting, classifying, and studying with these notes. YAY!!

This page took a few days to complete. First, we discussed solving systems graphically. After some exploration and activities, we completed THESE notes (new file link that the bottom). They were still connected as one piece of colored paper. Students then put those notes away in their handy-dandy pocket. We came back the next day and completed notes over solving by substitution and solving by elimination (new file link that the bottom). Students then put those in their pocket as well.

After making sure students were comfortable with solving systems, we started talking about one/none/infinitely-many solutions. Students were already thinking about these ideas because each colored set (graphically, substitution, elimination) had one system with each type of solution set. Here's where the fun part happened!

On page 103 of their notebook we took basic boring notes about what a system of equations is, the notation for a system, and what a solution point means. On page 104 we created a flipable pocket. The pocket is one full sheet and one half sheet of blank paper. Fold the full sheet in half width wise (hamburger style if you know what I mean) and then put the matching sized half sheet inside the larger, now folded paper. I know... it sounds confusing... it's really simple! Tape up the sides and fold the whole thing in half. You have four pocket areas. We only used three for our purposes. Do these pictures help at all??

 

 Anyway...

Students cut their colored papers apart and sorted into three piles: one solution, no solutions, and infinitely-many solutions. They looked for themes and commonalities within these new groups. We then had a class discussion about their observations and added general notes to the fronts of each pocket. We visually represented how one/none/infinitely-many looks on a graph and the type of solution when solved algebraically.


 ifinitely??? yeah... I dunno either...



The sorting and processing part really made things click for students. I also saw them studying with these notes later. They were able to take out all notes of the same color to study how to solve with a given method, or take our all notes within a pocket to study solution types.
Overall...success!

How have you done systems of equations in the past? Any great ideas to share??

New links that should *hopefully* work!