## Algebra 1 – Unit 1 INB Pages | The Foundations of Algebra

Here’s what went into our INBs for the 1st unit of Algebra 1:

Day 1:
We glued in a reference sheet for the real number system. Our textbook uses I for the set of irrational numbers.  I went with the same notation this year, but I think I’m going to go with R-Q for next year, since I is used for imaginary numbers, later on.

To practice working with these definitions, we did a real number system sort, which I found from Amazing Mathematics! My students enjoyed doing it, and it spawned many great conversations about the difference (however subtle they may be), between the sets of real numbers.

For homework, students did this Always/Sometimes/Never sort, which is also from Amazing Mathematics. They were given about 20 minutes in class to begin their assignment, and then had whatever was left as their take-home assignment for the night.  This one was even better than the last card sort, in terms of spurring student conversations.  Students were justifying with counterexamples and providing fully flushed out reasons for where each card should get placed.  It was awesome!

As a note, we also keep a binder for the class which holds extra handouts, like additional reference sheets and homework assignments that don’t go in the INB. My favorite reference sheet that didn’t go into the INB was this real numbers flowchart that I made.  The day of teaching my lesson on real numbers, I noticed that using the “Venn diagram” approach wasn’t meshing well with some of my students.  That afternoon, I went home and made a flowchart handout that they could refer to, in addition to their INB pages.  Next year, I think I’ll just use this flowcharts in a mini-book format for notes, instead!  I found that students started making more connections about the sets each number belongs to (i.e. not only is a number natural, but it’s a whole number, and an integer, and a rational number), and students were able to remember the questions they need to ask themselves when determining the best classification for a real number.

Day 2:
We started off with a recap warm-up on the real number system, which we covered the day before.

From there, we did a translating expressions sort, also from Amazing Mathematics.  (Can you tell I love her sorts?!).

From there, we used our key words and started defining what a variable is, and what an expression is.

For homework, students did the following problems.  They had about 15 minutes of class time to get started.  We color-coded “turn-around words” in pink, “parentheses-words” in green, and “equals words” in blue.  Students marked the page in highlighter before beginning to translate the expressions.  They mentioned that this made the process much easier for them!

Day 3:
We began with a recap warm-up over translating expressions.

From there, we talked about evaluating expressions and also reviewed the order of operations.

From there, we discussed the properties of real numbers and students made up their own examples for each property.

For in-class practice, students did the a properties of real numbers puzzle from Lisa Davenport.  A student volunteered to glue it into my notebook.  Notice the lack of glue?  Notice the crooked edges?  It was a very sweet offer, but I’m I don’t think it’s one I’ll be taking again any time soon.

Day 4:
We started with a recap warm-up over evaluating expressions and identifying properties of real numbers.

Next we took notes on combining like terms and the distributive property, cutesy of Sarah at Math Equals Love.

Day 5:
Recap warm-up over distributing and combining like terms.

What is a solution?  What does it mean to be a solution?  What does it look like?

Up next, we focused on solving and verifying solutions to 1-step and 2-step equations.  I’ve found that verifying a solution is a skill that students struggle with more than solving (at least in Algebra 1), so I wanted to make sure it got emphasized.

Day 6:
We filled out a foldable for solving 2-step equations.  Those pesky fractions are going to be our friends by the end of today!

Day 7:
Recap warm-up over solving equations.

Day 8: Review

Day 9: Test!

Want the full unit? Get it here!

## How I Teach Factoring Trinomials

When I teach the unit on polynomials and factoring in Algebra 1, I start off my first lesson on factoring trinomials with a discussion on which has fewer options: multiplying to a number, or adding to the same number?  Students take a couple minutes to list out all pairs of numbers they can think of and then share out to the class.  After doing this twice, quite a few students start catching on to the fact that there’s an infinite amount of ways to add to any given number, but only a handful of ways to multiply to the same number.  Multiplication gives us fewer options, which will allow us to do less work.  This will be really important to what we do in just a moment. (NOTE #1: I provide students with a factor pair chart as an aid to help with identifying factors later on. NOTE #2: We originally began using only whole numbers as a starting point, but students then wanted to branch out further.  Could we extend this question to include integers?! Yes-and we did!)

From there, we go back to something students have just learned a few sections earlier: multiplying two linear binomials of the form (x+A)(x+B).  We do this a few times and then look at a bunch of already expanded examples and I ask students what they notice.  It doesn’t take long before students start realizing that the middle term, the coefficient of the x, always comes from adding the two numbers A+B, and the end term, the constant, always comes from multiplying the two numbers, AB.

Then, I switch the question around.  How can we figure out what someone expanded to create a quadratic expression?  Is there any easy way to figure this out?  Students start to volunteer info that they know: the middle term comes from the addition (A) of the two factors, and the constant term comes from the multiplication of these two numbers (M).

So, then the question is, which number do we look at?  The addition number, or the multiplication number? Technically, it doesn’t matter, BUT mathematicians love to be lazy efficient, so we’ll look at the multiplication number.  Students justify looking at the the multiplication number first because, just a few questions prior,  they determined that there’ll be fewer options with multiplication than for addition.

From there, I ask students to make further generalizations and predictions about the signs of the terms and the signs of the factors and use that information to work both forward (expanding) and backward (factoring) using some diamond puzzles.

The next day, we practice some more with basic factoring when a=1, using the patterns we found from the investigation the day before:

Then, we kick it up a notch.  How the heck do we do this factoring thing when there’s more than one x-squared?!  No problem! GCF to the rescue.

After that, we look at what do we do if a GCF alone isn’t enough to get rid of the a-value, or, even worse, there’s no GCF at all?

This brings us to my FAVORITE part of factoring quadratic trinomials: Slide, Divide, Bottoms Up!  If you are unfamiliar with this method, let me start off by telling you that it’s awesome.  It’s firmly rooted in the same concepts we’ve been using for the last three sections of factoring, and it just makes sense.  Another benefit to the Slide, Divide, Bottoms Up method is that it is efficient.  Doing guess and check (or the box method) can become very frustrating for students when the a-value is larger than, say, 4 or 5.  There’s just too many options and it ends up taking forever, even with a decent intuition about which numbers to test out as factors.  Also, this method even works for special factoring cases like difference of squares!  Students can certainly utilize the factoring shortcut for difference of squares, but, if they forget, Slide, Divide,  Bottoms Up still has their back.

Here’s how Slide, Divide, Bottoms Up works:

Let’s talk through an example:

Like all factoring problems, we check if there is a GCF, first.  If we’re lucky, that will remove the a-value and we will be good to do what we normally do.  However, in this example, we weren’t that lucky.  No GCF, so what to do with the 6?  We certainly don’t want more than 1 n-squared, so we’re going to temporarily transfer it to the constant term by multiplication (we “SLIDE” it over).  At this point, we discuss what “temporarily” means.  It means, “only for a while,” so that tells us that, at some point, we’re going to have to undo it. This should be perfectly “legal” because if we do something but then undo it later, that just cancels out to what we started with.  It might also be worth noting that we transfer the a-value through multiplication because we are factoring, which literally means returning an expression back into a product (multiplication) of two factors.

Now that we’ve gotten rid of the a-value of 6 for a moment, we’re left with a standard trinomial that students know how to factor in their sleep with their eyes shut, at this point.  The only thing they have to remember after factoring it is that our factored form is for our temporary expression, not the one we started with.  So, how to undo what we did?

Well, if we multiplied the a-value into the coefficient, it stands to reason that we should do the inverse operation and just divide it back out (DIVIDE)!  Since we’re dividing, make sure to reduce the fractions!

Lastly, we didn’t start out with any fractions.  Actually, we started out with a number that was a coefficient (our a-value of 6).  To get rid of any fraction(s) that we introduced, we bring the denominator(s) back up in front of the variable to be a coefficient, once again (bring the “BOTTOMS UP“).

Here’s some more examples.  Note example 5 where there’s a GCF but we’re still left with an a-value of 4.

Here’s how it works with difference of squares problems.

After using Slide, Divide, Bottoms Up for the past 3 years, I can’t see myself doing factoring any different.  I’m pretty smitten with this method, and, hopefully, it’s easy to see why.

After doing all of the different factoring, I give students one last reference sheet to use in their notebooks, which can be used at any time to refresh their memory on how to solve ANY quadratic trinomial.

If you are interested in this flowchart, it is available in three different sizes here.

Let me know how you teach factoring quadratic trinomials in the comments below!