Factoring Trinomials with Algebra Tiles Visual Guide

This week, I worked on a visual guide for how to factor quadratic trinomials using algebra tiles. It has 10 examples, and each example includes the number of tiles needed with any zero pairs, notes about the example and a common student misconception about arranging the tiles for that trinomial. 

Free Discriminant Cheat Sheet for your Quadratic Formula Unit

In this post, I want to share a few resources for teaching the discriminant as part of a Quadratic Formula unit. 

This simple discriminant cheat sheet helps students make the connection between the value of the Quadratic Formula's discriminant and the number of real solutions the equation will have. It also shows examples of graphs with 1, 2, and 0 zeros and what the discriminant will look like in each case.

2 Free Factoring Quadratic Trinomials Algebra Cheat Sheets

What's your favorite method for factoring quadratic trinomials? For me, it's the AC method with grouping because there's no guess and check, which frustrated me as a high school student. I remember endlessly erasing my work, and settling with "close enough" on some problems. The box, or area, method has grown on me over the years, especially how it links back to multiplying polynomials.

The structure of the AC method works for me, and is the method I have used most when teaching students to factor. I don't know when I learned it, but I remember thinking that it would have been a helpful tool for me in high school. If making the list of factors is difficult, students can divide AC with 2, 3, 4, ... with a calculator and write down any quotients without decimals, checking which factor pair adds to B.

I made a couple factoring quadratics cheat sheets and posted them on Facebook this week.

We're getting to factoring in a few weeks after imaginary number operations, Quadratic Formula and solving with square roots and we're going to teach the kids the box (area) method, which they may have seen in algebra 1. AC with grouping will be the backup plan for students who aren't clicking with the box method. Multiplication charts will also be available to students who need them.

Box method factoring

AC method with grouping

If you are teaching factoring this year, I wanted to share the two cheat sheets above in case you find them helpful. I also wanted to mention that the cheat sheets I share are never intended to be standalone teaching tools. They always go along with lessons as a way to help students through independent work. This was something that came up this week, too. 

> You can download the two factoring cheat sheets free here.

Factoring Quadratic Trinomials Activities

For practice, there's a free set of solve by factoring task cards here.

self-checking factoring escape room

For self-checking practice, this factoring quadratics escape room covers factoring trinomials when A = 1, when there's a GCF, and when A > 1. The entire escape room is housed in one answer-validated Google Form with no outside links, eliminating your need to grade.

The escape room also comes as a printable PDF with a student answer sheet and key.

Factoring polynomials error analysis

For engaging error analysis, this factoring polynomials 2 truths and a lie activity asks students to find the lie when presented with factored form binomials and trinomials.

 

> download the free factoring sheets

free math resource library

 

Graphing algebraic function transformations with cut paper videos

In this post are a bunch of function transformations videos showing how vertex form functions all transform using the same pattern. The way that functions transform in the coordinate plane can feel pretty abstract to algebra 2 students just learning about nonlinear functions. But every algebraic function in vertex form transforms the exact same way.

Functions can translate vertically and horizontally, and even reflect over the x and y axes just like geometric shapes. The one quirk is that horizontal transformations feel opposite from expected. 

Why are horizontal transformations opposite? It feels backwards for the vertex of y = |x - 3| to translate right 3 units. With all horizontal shifts, we're looking for the value of x that will "zero out" the inside expression. For x - 3 = 0, x would need to be 3. With x gone, we can find the function's lowest or highest y value, i.e. the vertex's y value.

Here is how we can find the horizontal transformation of a quadratic function in vertex form and why the inside shift is opposite. 

function transformations cheat sheet

This function transformations cheat sheet in my Google Drive has graphics from an algebra 2 word wall for absolute value, quadratic and square root graphs. There's also a free set of dancing skeleton functions posters here.

Below are a bunch of function transformation video shorts using cut paper. In each video you'll see familiar nonlinear algebraic functions transformed in the coordinate plane. 

Numberless Quadratics Activity

"But Miss, these word problems don't have any numbers!"

"You're right! They're open-ended numberless word problems!"

Graphing Standard Form Quadratics step-by-step Video and Cheat Sheet

I had stalled a bit on my mission to make videos to go along with all of my math cheat sheets. It feels good to be back at it! The graphing quadratics in standard form quadratics cheat sheet in this post was a teacher request. It's a brand new step-by-step on graphing standard form quadratics.

How to Graph Quadratic Functions in Vertex Form - video, free cheat sheet and free task card activity

My wonderful husband got me a whiteboard for Christmas so that I could start making math teaching videos. After taking a few days to get everything set up, all the while wondering where to start, I decided to work my way through my math cheat sheets and make companion videos, starting with Algebra 2.

In this post is a video example of graphing quadratic functions in vertex form, a link to a free math reference sheet to go along with the video, and a link to a free vertex form quadratics task card activity.

Towards the end of the video is a shortcut for graphing vertex form quadratics with a pattern instead of a table.

Fun Quadratic Formula Activities

Of all Algebra topics, quadratics is by far my favorite. What I love most that students start the unit SO intimidated and by the end are old pros. So much growth happens during this unit. I love seeing my students grow more confident as they learn how to solve quadratics in different ways.

I've written a ton about having fun with quadratics and the activities we do during this part of our curriculum. In this post I want to highlight a few activities just about the Quadratic Formula. 

First, is this Quadratic Formula template that we use in class. It scaffolds the formula with spaces for A, B and C and a "skeleton" for students to use to structure their formula. 


I print a bunch for our classroom warm up basket. Students know to take one on their way into the room and use it to solve the quadratic I shine on the board. 

My friend Kara from Learning Made Radical and I have been working on partner scavenger hunt activities. The one above is a Quadratic Formula partner scavenger hunt. *A GOOGLE Slides version is now included in the download.

Pairs of students get 2 clocks and 2 unique sets of cut-outs. Both partners start with their "START" cut-out. The quadratic functions are different but the answers are the same if both partners solve correctly. 

If student answers are different, they work together to find the error. Once the partners settle on an answer, they look for this answer on the top of another cut-out. They then each solve their next unique problem until their answers match.

Displaying student work has worked really well in my classroom for boosting student confidence. We have "The Fridge" - an area where students can use magnets to hang their graded papers. 

We use those super strong magnets so by the end of the year piles are stacked on top of each other! It's great. On days when a student may not be feeling very confident, it's really nice to be able to point to the wall and say, "yes you CAN do it."

I'm also just a huge fan of math class décor in general. Posters, word walls, anchor charts, lists of prime and square numbers, graph and function examples, anything that works to build student confidence, background knowledge and lower what I like to think of as "math affective filter" (ie: math anxiety). 

This Quadratic Formula Math Pennant combines student work and classroom décor. Students get to color a little too, which is always fun.

My students like this solving quadratics chain activity because I build in extra credit. There are 12 quadratics to solve but I tell students they only need to solve 10 to earn a 100%. Most go on to solve more to get up to a 105%. I know, I'm cheap, but I can't be giving out 120s! :)

Here's my former student Omar holding up his paper chain. Please excuse the hideous word wall in the background... it has since been majorly updated!

Quadratic word problems digital math escape room

I love, love, love teaching quadratic word problems. This new quadratic word problems digital math escape room has students answering questions about rocket launch projectile motion problems. None of the equations are factorable, so students have to either use the Quadratic Formula and the axis of symmetry formula or their graphing calculator to solve. Questions in this escape room ask students to:

• Find the total time the rocket is in the air
• Find the rocket's maximum height
• Find the time it takes the rocket to reach maximum height
• Determine the height of the rocket's launch pad
• Find the height of the rocket at a given time

Quadratic Formula escape room

And I just also made a Quadratic Formula escape room with real and complex solutions. We used the printable version in class.

Quadratic Formula escape room

In this post I wanted to highlight a few fun quadratics activities. I wrote more about the activities we do in class, including the posters and anchor charts that help through this unit, in the post Fun With Quadratics.

>Browse all quadratics activities

Free math resource library

Visualize Complex Solutions in Quadratics Shown Graphically

"So they're called imaginary numbers but they never should have been called that because they're super important for things like explaining the way electricity moves."


"Ummm, hmmm...."


Have you ever had a conversation with your Algebra 2 students that went something like this? I have! 


I recently came across a paper written by Carmen Melliger called How to Graphically Interpret the Complex Roots of a Quadratic Equation that brings alive what complex solutions to quadratics look like when graphed. The image on the second to last page is especially awesome because it magically shows us just what these complex solutions look like and how we arrive at them - all with no words!

I wanted to make a math cheat sheet for students so that they could see what complex solutions in quadratics look like. I used Ms. Melliger's paper as a guide to make a super rough 3-d model so that I could better visualize a drawing.

Then a rough drawing.

And finally after a lot of trial and error, this image of the solutions to y = x^2 + 4x + 5 shown graphed.

I added in a starting image of what the parabola looks like when graphed in the real plane and the solutions we get from the Quadratic Formula.

The first unit in this set of Algebra 2 worksheets covers imaginary numbers and is based on the Common Core standards. 

You can find the complex solutions in quadratics cheat sheet here.

Update! An instagrammer sent me a link about Math teacher Philip Lloyd and his research on phantom graphs. So cool! 

MORE FREE MATH CHEAT SHEETS

Difference of Squares through Pictures

Our unit on Quadratics is probably my favorite unit to teach. There is so much growth! One of the topics we cover in our Quadratics unit is factoring differences of squares. 

I love looking at concepts through pictures (Have you seen the Pythagorean Theorem proof with water or this one with sand?). When math concepts can be shown through pictures, they seem to stick with kids longer. 

Quadratics Posters

For the last few years, we've been using this quadratic keyword poster during our quadratic word problems unit and it has been so helpful. 

Linking Quadratics to Rectangle Area

Disclaimer: This is not my idea. If it is yours, I would LOVE to meet you, chat, virtually shake your hand and pat you on the back!

Factoring Quadratics in Special Ed

This week, my Algebra 2 students started factoring. We started with a Let's Factor! Powerpoint with no trinomials in sight- just a bunch of thinking questions like "What 2 numbers multiply to 6 and add to 7?". My students responded really well to this introduction and they were so much more open to the idea of factoring quadratics after getting this confidence boost.

Hands-on Factoring Quadratics Using Cereal

Hands-on factoring quadratics with cereal? Yes!

My students have always had a hard time factoring trinomials during our quadratics unit, probably because of the multiplication and backwards thinking. This year I started our unit with a whole lot of thinking and a whole lot of hands-on - with cereal! 

Intro to Factoring

"What 2 numbers multiply to 12 and add to 13?" My students loved that one! It was a class favorite because it was so obvious...AFTER learning the answer:) 

We started factoring with no trinomials in sight, just these types of number puzzles in this Let's Factor! PowerPoint.  Once the thinking was there, it made moving on to trinomials a whole lot easier and less intimidating. 

Fun with Quadratics

I love teaching quadratics! It's the unit in Algebra 2 where I see the most student growth and where students seem to be the most proud of what they learn to do. It's just the best unit! 

In our classes, we first learn to factor. My students have all seen factoring in Algebra 1 and Geometry, so it's not totally new to them. Still, factoring can feel completely new to some students, so I always start from the very beginning.

In this post are lots of activities and ideas for teaching quadratic functions in algebra class. 

Over the years, I've made a ton of math cheat sheets to support student learning in my classes. I hand them out at the beginning of a unit and practice referencing it with students so that they can use it on their own. I have found it's one thing to give students references and a whole other to teach them how to use them, so I make sure to set the example. Here is a video and free graphing quadratic functions cheat sheet.


And a shorter video for graphing a vertex form quadratic where a=2:

We start without any variables with this Let's Factor! PowerPoint Game that gets students thinking the way they'll have to think when factoring, just without the x's.

Let's Factor! PowerPoint Game

On our Algebra 2 word wall there are factoring reminder posters that we reference throughout the unit. We also refer to our nonlinear graph references all year. Here is the one we use during our quadratics unit.

We start by factoring binomials and trinomials where a=1. A couple days in we go through this factoring Prezi as a way for students to self-assess and for me to see who knows (and doesn't yet know) how to factor when a is 1. I give them a simple answer sheet and award classwork participation credit.

(zooming factoring Prezi)

Once everyone has a good grasp on factoring trinomials where a=1, we complete this simple factoring activity before moving on to more difficult factoring.

In this activity, students find the lengths and widths of rectangles given quadratic "areas". It's a nice throwback to Geometry and gives factoring a little context. Students can also color their factoring activities.

To meet the needs of classrooms with technology, I have been making digital math escape rooms that run in Google Forms. Here is puzzle #1 of a factoring quadratics digital math escape room:

The trinomials are presented as areas of rectangles to give some context to what students are trying to find. And here is Puzzle #2 of a quadratic word problems digital math escape room covering projectile motion through rocket launch problems:

This "2 truths and a lie" error analysis activity (below) shows students a graph and its equation, along with 3 statements about the zeros, y-intercept, axis of symmetry and vertex. Students need to figure out which of the statements is false (the "lie") and correct the lie on their answer sheets. There are more "2 truth and a lie" math error analysis activities in this post.

Another fun way to assess student understanding is with a math pennants. Here is a set of factoring pennants that focuses on trinomials where a=1.

And this set for graphing quadratics. 

For support factoring quadratic trinomials, I made a few new factoring quadratics cheat sheets that are free to download. One supports the AC grouping method. There are two versions for factoring with the area (box) method.

My students always seem to struggle with factoring out greatest common factors. This new factoring trinomials math game called Voyage to the Treasure! includes trinomials with GCFs to factor. After the GCFs are factored out, all A values = 1.

Negative signs are also always tricky, but GCFs are the biggest offender. I made a set of factoring task cards with GCFs for my students that all involve factoring out a GCF as the first step. I find they help remind students to look for GCFs.

After we factor, we move onto the Quadratic Formula. Later when we solve quadratic word problems, my students can choose to solve by factoring or with the Quadratic Formula. 

My students have trouble with the structure of the formula and all of the variables, so this Quadratic Formula template really helps. We use it a lot for warm ups. It also works great for quick checks for understanding and as an exit ticket in the last few minutes of class.

I print a stack of the template to have ready to go at the beginning of class. The routine is great for student confidence and knowing that my warm up is ready to go is great for my warm up sanity! 

Before templates, I'd always stress about having a meaningful warm up that was long enough or that gave the right sort of practice. To get to know different solution types, I like to slip in this discriminant sorting activity within the first couple days. It gives students more practice evaluating and shows them the pattern for when the Quadratic Formula will yield either 1, 2 or no real solutions.

I find that my students need a lot of practice with the Quadratic Formula and they work through a lot of frustration throughout the process. So to summarize and celebrate their learning we do this fun quadratics chain activity. 

My students liked the hands-on aspect of getting scissors and cutting the strips. They also really like being able to see their progress right away. There are 12 links and I always give a 100% after 10 are completed correctly. 

All of the equations are factorable but my students usually chose to use the Quadratic Formula. When a student completes a problem, I am able to quickly scan the QR code with my QR scanning app. 


Once an answer is correct, it is ready to be made into a chain link. I award 10 points for each correct link and extra credit points if anyone goes beyond 10, which a few students always do.

Here's one of my students holding up his chain in front of one of our math word walls.

I have gotten a lot of great feedback about this quadratics puzzle that links the forms of quadratics to their graphs.

Another fun activity to assess student understanding of the Quadratic Formula is this Quadratic Formula pennant activity. Students solve the quadratic on each of the pennants, color and hang them up in the classroom to display their hard work.

My friend Kara at Learning Made Radical and I have started collaborating on these partner scavenger hunts. 

Each partner gets a clock and their own unique set of problems that yield the same answers. This encourages collaboration as students know their answers should match before moving on to the next problem. 


After The Quadratic Formula we move on to quadratic word problems. There are visual supports everywhere in our classroom for this part of our quadratics unit. 

This quadratic keywords poster hangs in our room. 


I also made this Quadratic Word Problem Keywords Flipbook for student notebooks.

After introducing some of the terminology associated with quadratic word problems, we do this quick quadratic keywords sorting activity to get everyone tuned into how the keywords will present themselves in the problems.

At first, my students complain that there are no numbers on the cards. But then things start to click. They use the keywords poster and think about what each problem is asking them to find. 


I find that taking the time to do this sorting activity helps my students feel less intimidated by the word problems once we get to the equations. 

I'm not a huge fan of tests, so I use a lot of task cards to assess student understanding. We use 2 sets of task cards with quadratic word problems - quadratic trinomial word problems and another set where the equations are all binomials. These now include a link to a GOOGLE Form version.

Some years I use both sets, other years I use one set or the other. I find that factoring out GCFs can sometimes be hardest for students, so some years we only use this set of quadratic binomial word problem task cards. These also now include a link to a GOOGLE Form version.

This year my students used the task cards with their graphing calculators. Because the "left bound", "right bound" on the graphing calculator is always tough, this poster hangs in our classroom:

Activities in this post can be found in this Quadratics bundle.

Quadratic Functions Bundle

I love this unit because of how much my students learn in such a short amount of time. Within a few weeks, they go from having worked primarily with linear equations to finding the amount of time a thrown object is in the air given an initial velocity. It's just so cool!