## Ratios, Rates, and Proportions INB Pages

In Algebra 1, I split the Solving Equations unit into two parts.  The first part was just all of the standard solving equations.  In the second part I did ratios, rates, and proportions.

On this page, I had my students do a review of ratios and rates from Pre-Algebra.  My students are very familiar with these two concepts because our Pre-Algebra teacher spends TONS of time on them.  The unit conversion at the bottom of the page was a little bit of a struggle.  I just typed out problems that I found in their textbook and online.

Next, I had a page for proportions.  I copied the definition from our textbook, and typed a word problem that I found.  I used half of the foldable by Lisa Davenport.  There was another portion to this foldable, but it covered things that aren’t in our Algebra 1 standards.

I put a lot of thought into how I would teach proportions this year.  I asked the MTBoS on Twitter for suggestions.  My students came to me having used cross products in the past.  So, I referenced it at the top of the page.  I ended up showing three different ways to solve algebraic proportions.
- I had my students get a common denominator, then multiply by the common denominator.  This was great…sort of.  They got the idea of what we were doing, but if the x was in the denominator, this method falls apart.  In Algebra 1, they don’t yet have the skills to solve this type of problem.
- I had my students multiply by one denominator, then multiply by the other.  Essentially, it broke the cross products into two separate steps.  This confused my students a lot.
- Last, I had my students use cross products.  They were less confused by this method, because they’ve seen it before.

I liked showing them three different ways to solve a problem. However, after that I let them choose the method they liked the best.  Most of them prefer cross products.  I haven’t graded their tests yet, so we’ll see how it went.

## Equations of Parallel and Perpendicular Lines INB Pages

To finish up our Parallel Lines unit in Geometry, we talked about equations of parallel and perpendicular lines.  You can find the pages for the beginning of the parallel lines unit here.

I always teach a review lesson about slopes of lines and writing equations of lines.  I spend a whole day on review (they need it!).  I used this slope graphic organizer from 4 the Love of Math and I really liked it.  It covered everything I wanted my students to remember.  The graphic organizer is a full page, but I copied it at 85% so that it would fit in the notebook.

Next, I used my Writing Equations of Lines flip book.  My students are still not very comfortable with their graphing calculators, so I included a few calculator tips in the margins.

The last page of the unit was my Equations of Parallel and Perpendicular Lines foldable.  Under each flap is an explanation about the slopes and a couple of examples.  I also included some written examples at the bottom of the notebook page.  The question that I wrote in was something my students couldn’t recognize in class, so I had them add it to their notebooks.  I hadn’t planned on adding that originally.

I think next year I want to add another page of practice with writing equations of parallel and perpendicular lines.  However, I’m flying through notebook pages and I’m getting concerned that I’m going to run out!  I don’t want my students to have a second notebook, so I’m trying to use my pages wisely.

## Parallel Lines INB Pages

I just tested my students over parallel lines in Geometry and they did much better than I expected!  Their proof skills are improving so much.

The first interactive notebook page for this unit was this graphic organizer from All Things Algebra.  I had students color-code the angle pairs and they also marked on their diagrams.

Next, I used my foldable for parallel line angle pair relationships.  Under each flap is a diagram with the definition, the theorem (congruent or supplementary) and an example.  I also had students write the theorems in shorthand under the foldable.  I do not allow students to write the theorem names when writing proofs, but I do allow them to use short hand.  This shorthand that I gave was the minimum amount they could write in a proof.

I created a hamburger book for practice with proofs.  I ended up doing some cut and paste with my proofs task cards to create this.  I want to create a better looking book (that faces the correct direction!) for next year.

I made the students actually write out the parallel lines converse theorems.  I think they needed the muscle memory.  I also allowed them to use the converse of the shorthand they used for the original theorems.  The little mini-book at the bottom has lots of copy and pasted diagrams inside (read: ugly) from their textbook with examples.

I created another proofs practice hamburger book for the proofs with the converse theorems.  I copied and pasted from my task cards.  Again, this needs beautifying for next year.  It also bugs me that they have to turn their books to read it.  I left tons of space on the side of the proof so students could write themselves hints.  They are starting to notice patterns in proofs and I wanted them to use this space to note that in their notebooks.

The next page has the perpendicular lines theorems.  In the past, I have also done proofs with the perpendicular lines theorems.  However, this year I made the executive decision that I wouldn’t include these theorems in their proofs.  It isn’t required in our standards, but I always included it for good measure.  I haven’t yet decided if leaving it out is a good move or a bad move.  The page is to be printed on legal or ledger paper and trimmed down to fit in their notebooks.  A friend has gotten me hooked to using the giant paper.  :)

After this, we did equations of parallel and perpendicular lines.  However, I will leave those pages for another post!

## How I Teach Intro to Logs

I’m gonna be completely honest.  When I was teaching Algebra 2 Honors for the first time, I was super nervous to teach logarithms.  I just didn’t know how to explain things in a way to make sense, and I was afraid it was going to be an epic disaster.  So, I researched.  I read and read.  The result was that my logs lessons ended up being awesome and I felt way more confident.  So yeah, research if there’s a topic you’re nervous to teach.  Anyway, this is how I taught properties of logarithms.

On Kate Nowak’s blog, she shared a puzzle-type thing she used with her students.  I modified it a little before giving it to my students.  I actually gave my kids fewer examples and made the problems a little more confusing (some of them are similar).

I gave this to my students as their warmup at the beginning of the period.  After a few minutes, I asked if anyone figured any out.  If someone had one correct, I had them write their answer for one of the problems on the board.  I continued that process slowly until a different person had written the answer for each of the problems.  Then, we talked about it.  I explained that they could think of logarithm was another word for exponent.

Then, I passed out notes and we talked about the properties of logs.  For a lot of kids, it only half sunk in.  So, I used symbols and pictures like I do when I teach function notation and factoring.

I usually make up little doodles on the fly for each of the properties of logs.  I have my kids write them down and make up their own.  I follow this lesson up with a few minutes of whiteboard practice.  My students have also really liked my Expanding and Condensing Logarithmic Expressions Stations Maze after this lesson.

Using symbols and doodles to teach the properties of logs always seems to help my students visualize the properties better.  While it’s important to me that they understand logarithms, I feel that the first example helps with that.  Sometimes, kids need to step back from the algebra to see the overall patterns.  Using symbols and doodles helps.

Do you do anything interesting when you teach logs?

## Solving Equations INB Pages

I’m starting to wrap up my solving equations unit in Algebra 1.  My kids are doing well.  This entire unit seemed to be a review from their Pre-Algebra class last year.

First, I created a page for solving two-step equations.  I used the weightlifter as an analogy to keep the sides balanced.  I also added an extra example on the blue post-it at the last minute.  I thought about using the two-step equations page that I designed before, but I didn’t have enough paint samples and I wanted to use the paint samples if I used that page.

On the next page, I had students do practice.

Then, I used a flip book for Solving Multi-Step Equations.  I don’t have a picture of it though.  I taught multi-step equations over four days.  The first day, I only did examples where they had to combine like terms before solving.  The second day, we did problems with variables on both sides.  The third day, we did examples of variables on both sides, including no solution and identity.  Last, we did problems with lots of fractions.  I think this approach worked.  I’m glad that I spent so much time here.  To sum it all up, we did this graphic organizer.  I think next year, I will do a similar layout, but have them create it themselves.  I can't remember where I found this graphic organizer, so if you know, let me know so I can link it!

The next page was practice with word problems.  I just typed out a few words problems that we did together - nothing fancy.

I did a mid-chapter test for this unit.  Up next were proportions!

## Beginning Proofs INB Pages

I spent longer on my Intro to Proofs Unit than I typically do.  As I mentioned in my post about my Logic Unit, I typically combine logic and proofs into one unit.  After meeting my students and seeing the level they were, I decided to break these units into two and spend a little more time on them.  My students have struggled with proofs this year.  Not any more than usual, but it strikes me that so many students are afraid to write something down for fear of it being wrong.  They’re so afraid of being wrong, that they won’t even try.  It’s sad, and it’s something that we’re working on.

At the beginning of this unit, we talked about the Algebraic Properties of Equality and used this foldable.  It went fine and my students had no trouble identifying the properties used - as long as we were working as a class.  Any time I had them try on their own it was an epic failure.  They were so afraid to try!

Then, we did a hamburger book of Algebraic Proofs.  This foldable is nothing special.  I seriously just did a google images search of Algebraic Proofs and used screenshots inside.  After this, we did whiteboard practice with my Algebraic Proofs Task Cards.  My students nailed this!  On their test, I had very few kids miss any of the algebraic proofs.

Then, I included a page of proof tips.  This page is kind of my catch-all for anything I might have missed telling them about proofs and a place for them to write random hints as we stumble on them in class.  I’m also having them keep a list of proof reasons (properties, theorems, postulates, etc) in the back of their notebooks.  I can’t find my digital file, but when I do I will link it.

Next, we did a flap book about the different Angle Pair Theorems and Postulates.  The theorems are written out under the flaps.  My students always have a hard time with the Congruent Complements Theorem and the Congruent Supplements Theorem, so we added a little bit more about those on the bottom.

The last page for the unit was another proofs practice hamburger book.  I didn’t take a picture of it because it’s super-ugly.  It was the same idea as the other one with lots of screenshots though.  I want to make up my own proofs and type them out nicely, but I just didn’t have time this year.  Summer project!

## Logic INB Pages

This year in Geometry, I taught Logic as it’s own mini-unit.  Before, I’ve always lumped it in together with beginning proofs.  However, this year I decided to spend an extra day or two on it and then have a separate test for this unit.  It worked out pretty well and most of my students did well on the test.  We spent a week and a half on this unit.

First, I used my foldable for conditional statements.  I used white paper because I wanted them to color-code the hypothesis and conclusion.  My students used two highlighters and used the same colors for the hypothesis and conclusion on the entire page.

On the next page, they did three examples.

I made a page dedicated to counterexamples.  I’m not really sure why I did this.  Half of this information is included in the biconditional statements hamburger book on the next page.  I might not include this page next year.  If I do, I might include a few more examples or something.  This page was included in the lesson with biconditional statements.

The information in this biconditional statements foldable is good.  Overall, I liked it.  However, it bugs me that I made the font so big.  I think it's size 20.  My students aren’t blind!  I will make it smaller for next year and include a few more examples since I will have extra space.

Next, came the Law of Detachment and the Law of Syllogism.  Students always have such a hard time telling these apart, and I’m not sure why.  In class, I felt like I was beating a dead horse with how many times we practiced this and how many different ways I explained it.  They still missed it on the test.  You can find this Law of Detachment and Law of Syllogism hamburger book here.

I don’t do symbolic logic or symbolic proofs.  I know some teachers do, but I after these four lessons, I move on.  Beginning proofs is next!

## Tips for Teaching How to Identify Functions

Today, my friend Amanda is going to share some tips about teaching students how to identify functions.  She has experience teaching middle school math and has lots of great tips for Pre-Algebra and Algebra 1.  Amanda blogs at Free to Discover and also has a TpT store under the same name.

Thank you, Karrie, for giving me the opportunity to share my ideas today!
In eighth grade, we learn about identifying functions pretty early in the year.  To help my students remember the rule I have a few tricks up my sleeve, and today I’m sharing them with you.

### We sing.

Oh yes.  I sing.   They sing.  I love the look on their faces during this experience.  After our Problem of the Day and homework discussions, I play That’s Amore by Dean Martin.  They all look at me like I’m nuts.  Some will recognize the song.  Then I sing (or I try to sing):
“If the x that you try goes to only one y, that’s a function!”
I sing it twice then have them join in for a few rounds.  In my experience, this has been a fun way to get the definition stuck in their head.  Months later when we are studying for the midterm many students still remember the song!

### We show.

At this point, I’ve hooked them in.  They don’t understand what they just sang yet, but we fix that by taking some quick notes.  We make two columns in our notes.  The left we label Function and the right we label Not a Function.  I start with a set of coordinates, then tables, then mappings.  I also make sure I include weird cases like when there are two y values that share the same x value.  I can see that many begin to see the pattern and understand.  The light bulb expressions go off like popcorn.

When we get to graphs, we pause and talk about the Vertical Line Test.  We all hold our pencils and talk about the rule.  It is so important here to connect it back to the original rule, too.  We look at specific coordinates on the graphs, and discuss how the Vertical Line Test is just a visual way of checking the same rule.

### We practice.

First, concrete rule-checking is in order.  I have an awesome Algebra with Pizzazz riddle that I use here.  Students are given a dozen relations represented as sets of ordered pairs, tables, mappings, and graphs.  They work quickly through the worksheet to place a check or x based on whether the relation is a function or not.  This is a great self-checking exercise and we quickly check together as a group as well.

Then, we take out our classroom set of mini-whiteboards and markers for a bit of challenge.  I present students with some higher-order thinking practice by having them work backwards.  I will say something like, “Make me a table that is NOT a function.”  Everyone’s will be different but I can do a quick scan to make sure I see a table with matching x values and different y values.  We go through several rounds of this type of practice.

These activities hit on a bunch of learning styles and interests and have worked well in my classes.  I hope you can take away something new to try!  Thanks for reading!

I hope you found Amanda's post helpful!  Lessons are always more fun when you can incorporate songs or short video clips!