Making them can sometimes be challenging, so here’s an editable question stack template you can use to save time (it’s a PowerPoint file).Continue reading
It’s that time of year again…you’ve had a bit of a break and now your mind is going wild with ideas for the new school year. To keep your time and efforts focused (and your stress levels down), I’ve created a list of 20 things to do to prepare for the new school year.
If you’re looking to regain some of your essential time, this is a post for you! Many of the daily systems teachers have setup for themselves and students can quickly turn into time-sucks. Now, I’m not talking about the ever-important relationship building part of teaching, but the nitty-gritty paper passing out, finding absent work, and making seating charts side of things. I’ve found a few ways to streamline my routines and classroom practices so that I can stop wasting my own time by being inefficient. Here’s my tips for you:
If you follow me on Twitter, you might have seen the following tweet about a month ago.
You could say I got a bit behind on my semester 1 INB gluing and, as a result, my INB posts have fallen by the wayside. Semester 1 ended the first week of February and I’m just now getting around to catching up on getting it organized, since I’ve had a few snow days in a row (I really thought this would be a snow-day free year, but nope!).
Without any further ado, here are my INB pages for Unit 4 of Algebra 1: Linear Functions. Note: There were activity/quiz/review days built into this unit–the days listed out are for days that note-taking occurred.Continue reading
Unit 3 of Algebra 1 is all about solving equations and their applications. We start off with multi-step equations, because 1-step and 2-step equations were covered in Unit 1: Foundations of Algebra.
In addition to the notes that went into our composition books, students were each given a full-sized flowchart over solving one-variable equations. We did an example as a class, and then I also keep a class set laminated so students can use them with dry-erase markers whenever they like. Students referenced their notes and the laminated flowcharts while working on homework in class.
We then went into a foldable that covers what special solutions are and when they arise.
To get even more practice, students did the following Types of Solutions Sort, which emphasized common student errors and misconceptions I’ve noticed in the past.
From there, we moved into our main set of notes for the day, with an emphasis on marking the text (NOTE: this is the same color-coding we used in Unit 1).
From there, we used the information we’ve gathered to solve absolute value equations a bit more efficiently (without using the modified
cover-up question mark method). Students had the even numbered problems as homework that night.
In addition to the notes that went into the composition books, students were given a flowchart for solving absolute value equations to reference whenever they got stuck. Here’s an example of how they could use it! Just like the others, I keep a class set of these laminated so students can use them with dry erase markers whenever they get stuck. I like to color-code each type of flowchart to make it easy to grab the exact one that they need from that unit.
After these examples, students filled out the other side of the flowchart that they were given on Day 1 with a more difficult example of solving for a variable in a proportion.
Day 7: Percent of Change
Percent of change is a funny topic to cover in Oregon…most of our textbook’s examples are about sales tax, and we have none. If we go to Washington, we just flash our Oregon ID and presto, bingo, bango, no more sales tax (for the little stuff). Anyway, we find other examples to try to make it more meaningful.
After taking notes, we did this Percent of Change Scavenger Hunt. Students worked really hard on it and had a lot of fun. For some of them, it was difficult to remember to put a negative sign on their r-value when it was a percent decrease!
Day 9: Literal Equations, Day 2
We move into more complicated literal equations that require more than one step to solve. After doing a few, students are able to choose which method they wish to solve with (I’m partial to the algebraic method, but some students love the flowchart way).
After notes, we play my favorite Connect 4 game for solving literal equations. We only played until 6 people won, which allowed us to get through about 70% of the problems. From there, students spent the remainder of class working on a festive Carving Pumpkins coloring activity for solving literal equations. This activity was awesome because students were super engaged in the coloring (every last one of them–even the boys! PS: I have 22 boys in this one class…ay, yai, yai), and it was super easy for me to find common trends that I might need to readdress (the eyes for Pumpkin #2 were the most common error). Also, for students, this activity is fairly self-checking, which is a great confidence boost for many of them.
Day 10: Stations Review Activity Day
We did a recap warm-up over solving literal equations and then spend the rest of class doing a stations activity with my solving equations unit task cards.
Day 11: Review Day
Day 12: TEST!
Here are the notes I used this year for the 2nd unit of Algebra 1:
We started off the unit with a classifying variables sort. This was a good way to jog students’ memories about their prior knowledge, and it also served as a jumping point into domain and range!
From there, we started to talk about all of the different ways we could represent a discrete relation, and how we find the domain and range from each representation. We used this foldable, which went over great with the students. They caught on super quickly, and they mentioned that they liked having one example to do together, and one to do on their own for each representation.
We started off with a word problem to review domain and range in a (discrete) relation.
From there, we filled out a Frayer vocabulary model for functions, to make sure that students really understood what they are and aren’t.
Then, using the definition for function we just wrote down on the Frayer model, we made a cheat sheet to refer back to that tells us all of the different ways a relation (discrete or continuous) would NOT be a function.
We practiced classifying functions using a card sort from Amazing Mathematics. Instead of cutting and pasting, we decided to color-code instead! Love it! (In the words of one of my students, this is the page that has “fourteen thousand graphs.”)
We then filled out another cheat sheet, this time for domain and range of continuous functions. Students reasoned together through the inequalities and we talked about what a bound actually means (we used a lot of basketball references).
We practiced finding the domain and range for continuous relations (as well as determining whether or not they were a function), using the following set of notes. PS: It took me a LONG time to figure out how to make a parabola or a trigonometric wave using Microsoft’s shape tools. I feel overly proud of this set of notes! You can download them here!
We began with a recap warm-up on domain and range for continuous relations.
To make sure that students didn’t forget about discrete relations, we went back and did more practice with determining their domain and range, and also stating whether or not the relations were functions.
Inside this set of notes, we really emphasized interpreting what we were being given in a problem (input or output value) and what the problem was actually asking us to find (input or output value), before starting the problem. This helped students from making a lot of careless mistakes. After we practiced function notation in both directions (evaluating a function, and solving for an input given the function’s output), we mixed up the problems and even threw a few variables and function compositions in there!
Recap warm-up on function notation. Problems 5 and 6 both spurred amazing conversations about order of operations.
After doing this recap warm-up, we did my function notation mystery sum activity, which was a blast. It encourages students to collaborate together and it’s really high engagement each time.
From there, we continued talking about function notation, but now in terms of a graph. Interpreting what the function notation was telling us was such a huge part of the previous day’s lesson, that I wanted to see how they could do when we attached a context to the problem.
Inside, we worked on graphing functions, and using the graph to find an x-value. Some students preferred solving for x, but others were impressed by my tracing over on the graph method. To each their own–that’s the beauty of math, in my opinion.
Recap warm-up over function notation with graphs, and then we reviewed for the test.
Day 7: Test!
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.
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?!).
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!
We began with a recap warm-up over translating expressions.
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.
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.
Recap warm-up over distributing and combining like terms.
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.
We filled out a foldable for solving 2-step equations. Those pesky fractions are going to be our friends by the end of today!
Recap warm-up over solving equations.
Day 8: Review
Day 9: Test!
Want the full unit? Get it here!
Making things for interactive notebooks can be tedious, at times. If you’re like me, you use a composition notebook so students will (hopefully) resist the urge to tear out pages for scratch paper. The issue with composition notebooks, however, is their sizing. A full sheet of paper is much too large to fit, but a half sheet makes the page feel a bit empty.
Also, unless you want to make everything from scratch to perfectly fit in your interactive notebook, you’re a bit stuck on what to do to get full-sized materials you may have used in the past to fit.
My hack: print any normal sized paper at 80-85% the size and, after cutting out the paper, it will fit PERFECTLY into a composition interactive notebook. Use this hack to make the world your oyster.
Here’s how to do it:
1. Make sure your document has been saved as a PDF.
2. When you go to print, select the following setting:
Rule of Thumb:
If the margins on the original paper are 1″, print at 85%.
If the original margins on the paper are .5″, print at 80%.
If the original margins on the paper are at .25″, print at 75% (not common).
Here’s the difference it makes:
This has saved me a TON of time making interactive notebook pages, and also allows the writing space to be much larger for students. Sometimes a half-sheet can be cramped. Hopefully this teaching hack can help save you a ton of time, like it does for me!
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, A⋅B.
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!
During my Algebra 2 unit on polynomials, I had asked my (support) class if they would like to stick to just using polynomial long division, which works for every single problem, or if they would like to also learn another method (synthetic) that, while far quicker, only works in certain situations. It was almost unanimous that they favored sticking to polynomial long division, which was fairly surprising to me. I almost figured they would want a quicker method, but their rationale was sound. They thought that having another method would just trip them up, and they didn’t really see a point if it could only be used for linear binomials.
However, a few weeks after our unit on polynomials, we had a bit of down time so I introduced synthetic just for fun. The students caught on quickly, but still preferred long division since it made more sense to them. (I agree that Synthetic is harder to wrap one’s head around. It feels a bit more “magic.”) Unfortunately, most of the class was gone that day due to an optional viewing of the school play being offered for students during the first four periods of the day.
As we start moving toward reviewing for finals, I figured I’d make a slideshow for students to view on their phones if they wanted to get a refresher on synthetic division. Here it is! I like it because it has a quiz-yourself and work-at-your-own-pace feel to it.
Do you cover both synthetic and long division for polynomials? Which does your class seem to prefer?
Download a PDF of the slideshow here: synthetic-division-how-to