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

updateHere’s what went into our INBs for the 1st unit of Algebra 1:pic_Page_01

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.  pic_Page_02

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.

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Real Number System Sort from Amazing Mathematics 

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!

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Always/Sometimes/Never Sort for Real Numbers from Amazing Mathematics

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.

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Classifying Real Numbers Flowchart from Math by the Mountain

Day 2:
We started off with a recap warm-up on the real number system, which we covered the day before. pic_Page_05

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

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Words into Math Sort from Amazing Mathematics

From there, we used our key words and started defining what a variable is, and what an expression is. pic_Page_07pic_Page_08

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! pic_page_09.jpg

Day 3:
We began with a recap warm-up over translating expressions.pic_Page_10

From there, we talked about evaluating expressions and also reviewed the order of operations.
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From there, we discussed the properties of real numbers and students made up their own examples for each property.  pic_Page_12

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. IMG_1668

Day 4:
We started with a recap warm-up over evaluating expressions and identifying properties of real numbers. pic_Page_14

Next we took notes on combining like terms and the distributive property, cutesy of Sarah at Math Equals Love.pic_Page_15pic_Page_16pic_page_17.jpg

Day 5:
Recap warm-up over distributing and combining like terms. pic_Page_18

What is a solution?  What does it mean to be a solution?  What does it look like? pic_Page_19

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. pic_Page_20pic_page_21.jpg

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!

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Day 7:
Recap warm-up over solving equations. pic_Page_24

Day 8: Review

Day 9: Test!

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Want the full unit? Get it here!

My No. 1 “Teacher Hack” For Interactive Notebooks

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:printing setup

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:

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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!

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!)

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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.

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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.   Scan0018

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.  Scan0019Scan0020

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The next day, we practice some more with basic factoring when a=1, using the patterns we found from the investigation the day before:IMG_1537IMG_1538

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.

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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:

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Let’s talk through an example:

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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. IMG_1535

Here’s how it works with difference of squares problems. IMG_1536

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.

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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! 

How-To: Synthetic Division

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

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Algebra 1 *Solving Equations* Unit Review Stations/Task Cards Activity

My students are finishing up their 3rd unit which is all about solving equations.  The unit includes:

  • Solving 1-step through multi-step equations.
  • Writing equations from applications and then solving
  • Special solution cases (no solution and infinite solutions)
  • Solving Absolute Value Equations
  • Writing absolute value equations from a graph
  • Writing and solving absolute value equations from a scenario
  • Ratios and proportions
  • Solving proportions
  • Percent of change problems (emphesis on working backwards to find original value or final value)
  • Literal equations

To help them review, I’ve made the following set of task cards (to be done at 11 different stations around the room), using problems from a variety of different resources.  I have my students for 2 periods each day, so we should be able to finish in one class.  If you have only one period per day, this might take you 2 periods.  OR you could give students the choice of picking any 2 problems from each station to complete.

I will have students work in groups of 4 and will give them 8 minutes per station.  If they finish early, I have an additional review assignment for them to work on in the meantime. On the back of each card is the final solution, so students can quickly check if their work is on the right track, or not.  If they’re really off and can’t find where they’ve gone wrong, I’ve also provided the fully worked out solutions for each problem at the given station (but that is only to be used if truly needed).

Click HERE to download the stations/task cards activity.

The fonts Riffic and Arcon are used, throughout.  If you plan on editing the Word Document to fit the needs of your own class, you’ll want to download those two free fonts.  Otherwise, the PDF is good to go!

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I have each station paper-clipped together.  Each station contains 4 problems that are placed inside a white half-sheet of paper that contains the fully worked out solutions.  The  white paper with full solutions are there only in case a full group of students truly get stuck. fullsizerender-16

The front of the cards have the question (and problem number).  The back side has just the answer–no hints as to how that answer was reached.  Students can collaborate together to get the right answer, if their answer didn’t initially match.  If they’re really stuck, they are allowed to use the white solutions paper for the station. fullsizerender-17

Here’s an example of the solution paper for Station 8.  It’s nothing fancy, but it does the job.  It’s meant to get a group “unstuck” if they couldn’t figure something out together.  After all, there’s only one of me and 36 of them, so extra help is sometimes good to provide. fullsizerender-18

Here’s a look at all of the questions, from each station (the problems are to be cut apart, and turn into 3″x5″ rectangles).

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Solving Literal Equations “Connect 4” Activity {Student Approved} FREE DOWNLOAD

Recently, I reached out to the MTBoS looking for fun ideas for practicing solving literal equations.  I had searched pretty thoroughly to find any pre-existing activities on the internet, but there wasn’t a lot available.  On top of that, what was there, required way more pre-existing skills (SO MUCH FACTORING!) than my Algebra 1 students currently had a month and a half into the school year.   Unfortunately, the MTBoS and I were pretty stuck. feed

Farther down in this Twitter conversation, however, it was mentioned that someone recently used BetterLesson’s lesson for teaching literal equations.  At that point I had already taught the lesson and most of my students caught onto solving them quite quickly, but I still was looking for a fun way to get a bit more practice in.  While exploring what BetterLesson had, I found this worksheet  that gave me inspiration for a game I could play with my students.  After a little bit of brain-storming, I created what I’m calling a Connect 4 Activity.  Essentially, it’s BINGO, but 4×4 instead of 5×5.

How to play: 

  • Before game: print enough game cards so each student has one, and cut apart the 16 problems.  I fold the problems in half (the problem number to the inside) and put them into a plastic bin.  (When printing from your computer, make sure it says “print double sided, flip on long-edge.”)
  • To start off the game, each student gets a game board, on which they randomly place the numbers 1-16.  Students then pull out a piece of scratch paper, where they will be doing their work.
  • The teacher brings the plastic bin containing the 16 equations around the classroom, letting a student volunteer pick a problem at random. (They LOVE getting to pick!)
  • The teacher then places the problem under the document camera (or writes it on the chalk/white-board if you’re at a low-tech school) for students to solve.
  • After all students have solved the problem, discuss the solution as a class.
  • Once all students are silent, the problem number is revealed for students to cross off on their game card. (The excitement levels usually explode at this point, hence the moments of silence in between.)
  • Repeat for as much class time as you have available, or until all 16 problems have been solved.
  • Each time a student gets 4 in a row, they bring up their card and their work for inspection (they showed their work and corrected any mistakes for each problem), and are allowed to choose a small piece of candy (Jolly Rancher, a Starburst, etc.).

Reasons why I LOVE this game:

  1. It is super easy to set up and is so adaptable for other topics.  This has probably been the lowest prep activity I have made for my students, yet it has been one of the most successful.
  2. Students felt much more confident about their skills and were able to get nearly-instant feedback about how they’re doing.
  3. Students LOVED it. The class begged me to continue letting them play the game through passing time.

Download the game here:2-8-literal-equations-connect-4-activity-page-001connect 4 problem cards for blog post picsconnect 4 problem cards for blog post pics2More Literal Equations Activities:
(Updated September 2017)
This year I wanted to find more ways to practice literal equations with my Algebra 1 students.  We teach literal equations the week before Halloween, so I wanted to make something really fun and “Halloween-y.”  I made a Carving Pumpkins activity that’s self-checking and SUPER fun!  I couldn’t wait to try it out, so I gave it to my Algebra 2 students mid-September (patience never was my virtue) since they review literal equations in their first unit.  Students though it was fun, and they also found it really comforting that it’s self-checking.  To quote a group of boys, “this is super dope, we should do this for all of the holidays!”

Students are given 12 literal equations to solve for a specific variable.  Depending on what their answer was, they “carve” color the corresponding pumpkin in a particular way. In the end, each of the pictures should end up looking the same, as far as the color and carvings go.

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I’ll be making more activities, and will update the post!

Percent of Change Scavenger Hunt Activity {Free Download}

This week in Algebra 1 we covered the topic of percent of change, which is one of the many Algebra 1 topics that is covered in middle school but gets revisited in high school.  The concept of percent of change isn’t too challenging, even when working backwards to find an original or final value, but, geesh, it can be boring.  I looked around online and couldn’t find many activities for this topic, and the ones that I could were really geared toward lower middle school grades, so I decided to make my own version that is great for an 8th-9th grade class.

There are 17 problems that are to be posted, alphabetically, around the room (get creative, though!  can you hide any?).  Students work in pairs, each student getting their own work recording sheet.  Each pair of students also gets a path recording sheet, so they can track the order of problems they’ve gone through.  Students can start at any letter, that way you don’t have 30 students starting at the same place (I normally have a 4 person limit per letter).  Whichever letter a pair of students starts at will be the first letter added to their path recording wheel. They will solve the problem at the bottom, and then look around the room at the tops of the other letters until they find the letter with their answer printed on top.  Then, they go to the new letter, record it on their path recording wheel, and solve the problem at the bottom of the page.  The process repeats until the student makes it back to the letter they began at.

You can download the PDF and editable PowerPoint version of the scavenger hunt here.  You’ll need the fonts Wellfleet and HVD Comic Sans if you want to edit the PowerPoint file.  Otherwise, the PDF is good to go!

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My Favorite Resources #MTBoSBLAUGUST #Made4Math

Over the last year or so, I’ve done a lot of work with very low-end students.  Between teaching summer school for two years straight in the inner city, and teaching support classes in my regular semi-rural school, I’ve really been pushed to find other ways to convey information that work for my students.

One thing that I found is that no matter how small and bite-sized of steps I could break a process down to in our notes, many of my ELL students and students with IEPs for processing disabilities just couldn’t follow along and rework through the steps to get themselves “unstuck” on a problem.  Working toward self-sufficiency is really big for me.  I strongly believe that the purpose for high school is to prepare students to be productive once they enter the “real world,” whatever that means for them (school, workforce, military, etc.).  Being self-sufficient and being able to problem-solve on their own is a big part of being able to reach this point.  So, I kept searching and trying new things until I made my first flowchart graphic organizer.  It was a game changer for my class!

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Students were able to easily follow along.  Using the graphic organizer, they were forced to read and do only one small chunk at a time and they had enough space to do their work right on the flowchart (it’s hard for some students to go back and forth between where the steps are written and where they’re doing a problem on a separate page of paper).  Students were able to use the flowcharts as long as they wanted.  As soon as they felt comfortable enough without it, they stopped using it.  I have also laminated a class set that we used for practice early on.

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I’ve also found that these have been very successful with my older students to jog their memories about a method they haven’t used in a while (such as solving systems by elimination).  For a lot of my seniors, I’m not the only math class that they are taking–many of them are also taking a class called Math Skills that gives them opportunities to take more Work Samples, which are needed for graduation.  Work Samples are an animal of their own and the topics on them can vary widely, so students find themselves needing review on topics that they may have not seen for a couple of years.  I’ve had a lot of these students specifically ask if I had a flowchart for topic _______ that they could look over to remind themselves of the details of how to do ________. 6
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With my younger classes, the first time we learn a method, I have a student working at the document camera as our class’ scribe, and the class (no help from me) discusses their way through the problem.  They determine which path they need to go down (the “yes” path, or the “no” path), and then work in pairs to do that step.  Then, they compare their work for that step as a class, and then move onto the next part of the flowchart and repeat the process.  I love, love, LOVE how student and discussion centered this makes my lessons!  Seriously! LOVE!  It’s almost as if I’m not needed (shh! don’t tell anyone that, because I still want my job).
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From there, we do a few examples that we glue into our INBs, and do some practice with dry-erase pens on the laminated copies of the flowcharts.  I find that starting slow and having them work their way through a problem as a class, without me, helps them remember the ins and outs of the process a bit better, since they had to struggle together as a class.
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Although I don’t have students referring to their notes quite as much as I would like, I have found that they go back to these flowchart examples in their INBs more than anything.  When I ask my students why they like these so much, a lot of what they say comes back to the fact that they have the steps on the paper, and the space to do the work on the paper, and the flowchart really forces them to go one step at a time.  A lot of them know that they have a tendency to rush through steps, and using the flowchart makes that very difficult to do.  Students then self-wean off of the flowcharts at their own pace, which is great in my books!  They are taking accountability for their knowledge.  If they can do their work straight away, they do so.  If they need a bit more help to get through a problem, they don’t just give up–rather, they walk to where I keep extra copies of the flowcharts, grab one, and work through the problem.  This has really helped develop the no opt-out culture in my classroom.  If students want to learn, there are tools to help them learn.  For my classes, the flowchart has been an instrumental tool for their development, both in math skills as well as self-motivation and persistence.

If you like the flowcharts, you can find them at my TPT store!  Today, they are 19% off when you couple your purchase with the 10% discount code OneDay.

Solving Systems of Linear Equations Flowchart BUNDLE 

Solving Multi-Step Equations Flowchart

Thank you so much for reading!

 

Inequality vs. Interval Notation Poster {FREE Download} #MTBoSBlaugust #Made4Math

My school doesn’t cover interval notation in its curriculum.  We focus primarily on inequality notation, although I tend to use the more specific set-builder notation.  Each representation has its merits, so I wanted to include interval notation more this year, as an occasional aside.  I’ve made a poster (8.5×14) that I’m going to hang up in my room to help students see the connections between the inequality symbols, the choice of open/closed points on a number-line, and the choice of soft/hard brackets in the interval notation. I’ve also made a color-coded version where students can ask themselves, “Can I include this point?” Green=”yes, include”, and red=”no, exclude.” Half of my classes this year are geared toward students who had received <40% in their last math class, so I’m hoping that the stop-light colors can make this yes/no, include/exclude concept easier to grasp. [NOTE: Thanks to lovely conversations on Twitter, it’s been noted that the green/red combination could potentially be dangerous if you have any colorblind students! I’m working on another, more color-friendly version that you can use, as well. I will update this post when it’s been made!]

Inequality vs interval notation poster COLOR-page-001Inequality vs interval notation poster-page-001

Before I hang the laminated poster up (I add posters throughout the year as topics arise), I’m going to print another one and cut up the grid into the 36 individual rectangles and hand one piece to each student in my class (if there are fewer students, ask your class “who wants another piece?”–I always seem to have a bunch of volunteers because this means they’ll get to talk to more people!).  Students will then find the two other classmates who have representations equivalent to their own card. Once a triple has been found, students will check their cards with the teacher.  If they are correct, they will move around the class helping the remaining students.  If they are incorrect, they will review which card(s) in their triple didn’t belong as a group of three, and then go back to finding the equivalent representations.

Would you like a copy of the reference poster? Get the color and the black and white versions here! (It’s free!)

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Special Right Triangles Display {FREE Download} #MTBoSBlaugust #Made4Math

A while back I made a display for special right triangles, and realized I never shared the files! You can download the PDF and the editable Publisher files here!  You’ll need to download the free font HVD Comic Serif Pro if you choose to edit the Publisher file yourself.

Here’s a picture of the pre-laminated pieces.  I took a few pieces of the finished product on my walls in the classroom, but each one had a nasty glare from the laminated finish.

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