It’s the truth! Factoring is a major topic and somehow, we have to make sure students can do it. Factoring is needed for all math classes after Algebra and for all college entrance exams (SAT, PSAT and ACT) and placement exams (ACCUPLACER and TSI). Algebra teachers have enough on their plate without this pressure, but it’s our job to teach it and hopefully it will be reinforced in future math classes.

About ten years ago, one of my coworkers showed me a cool calculator method that I use with struggling students. Some students have a hard time with their multiplication facts which will make factoring a nightmare for them.

I hate most calculator tricks, but this one is actually a great tool. Let’s say a student needs to know all the factors of 135. Have them go to the graph of the calculator and type 135/x (135 divided by x). Next have the student look at the table. In the table, they will look for whole number values. For instance, across from an x of 1, is a y of 135. That of course means that 1 and 135 are factors of 135. The next set of whole number values are x = 3 and y = 45. When the list of numbers starts repeating, all of the factors have been found.

Look at the sample factoring problem below this paragraph. I have my students multiply the 9x^2 and the -15. The answer is -135x^2. To the right of the problem, they draw a large X . On the top, they write the -135x^2 and on the bottom of the X, they write the middle term: 22x. Next, they start making a list of all of the factors of 135. I tell them not to think about the negative at first…just make a list of factors. If they are not able to do that, then use the calculator to make the list. Once the list is made, then the students decide which factors will multiply to get -135 and subtract to get 22. The answer would be 27 and -5. Those two numbers are written on the left and right side of the X. Next, the original trinomial is turned into a polynomial with four terms. The second step below was 9x^2 + 27x – 5x – 15, before I started the grouping process. The problem is grouped and the factors are found. (Yes, I teach grouping. It helps with this type of problem and it helps with factoring out a GCF. Don’t skip grouping. If you’d like to see more about how I teach factoring go to this Factoring Blogpost.)

Here’s a quick video explaining the same problem:

All students can factor! Believe it, teach it and recycle it!

Real-life examples in math are super important, but it takes time to think of examples and to prepare a lesson using your examples. A quick way to make a lesson interesting and tied to a real-life situation is to take a picture then pose a question. This gets students to analyze details of a situation.

In the next few weeks, I will be talking to my Algebra students about arithmetic sequences and direct variation. I have a great blog post titled, “Examples of Real-Life Arithmetic Sequences.” Check it out if you’d like. I love all the pictures in that post, but I thought I’d take a new picture that I could pose a question to see what the students would say. Below is the question and picture. Feel free to use it yourself if you like it.

(Yep, that’s my dishwasher in the background😂)

I’d give students a little time to think and jot their thoughts down. Next, I’d ask for feedback. Finally, my plan is to let them create a table using height of cups vs. number of cups for each situation. We will create equations and graphs and talk about the similarities and differences. Students will pay more attention to the details and take part in this activity. All the students will need is some grid paper and the picture which I will post on the board and in Canvas for them.

As a side note, I took this picture with my iPhone, then I used a free app called Layout from Instagram to create the collage. I used another free app called Typorama to add the question. Very simple and easy once you’ve done it a couple of times. I save all my photos in Google Photos which is easy to get to via phone or computer.

How is it going in your classroom? If it seems that your students are not paying attention and just not getting the concepts you are delivering, could it be that you are not engaging them? When school really gets going and you are super busy, it seems like we go into survival mode. The way we survive is lecturing because we really don’t have time to plan and be creative. I’m going to give you some ideas that turn a dull boring lesson into an engaging lesson without much prep.

Here are 5 Easy Ideas:

1) Get the dry erase boards out and dust them off! Kids love to draw on the boards, so give them equations to solve, equations to graph or shapes to draw. Maybe you had a worksheet planned. Don’t do it the traditional way, instead call out the problems and let them work them on the board then raise the board up to show you. You can make corrections and help kids that are struggling. You can have students show their partner and talk about which person may or may not be correct. Dry erase boards are a savior for me. I get them out anytime I feel like I have a boring lesson and it really spruces it up. 2) Find a related Desmos lesson. Desmos is easy to use and can be something quick to search and find quick lessons or activities for your students. If you are teaching exponential functions soon, I have a good activity from Desmos that I created. I would say to do this with Algebra 2 rather than Algebra 1. It’s called “The Towers“. I love the Tower of Hanoi and I use it in my Exponential Functions Stations. 3) Another quick way to gain interest in note taking is make the notes colorful or turn it into a graphic organizer. If you have 4 things the students need to know, then create a paper folding graphic where students write on the outside 4 flaps and they open to reveal answers, definitions or a diagram. Here’s two examples of using colored pencils or using a foldable:

4) Let the students partner up and go to a spot on the board or use poster paper. Ask them to write everything they know about a topic. I recently did this and the students did not realize how much they actually knew. I kept adding stuff and reminding them of a few things along the way. Before they knew it, they had a ton of concepts on the board. 5) Turn the lecture into a game. One way is to make it a Bingo Game. Create a list of things you know you will be saying that day and put it on the board. The students will be given a blank bingo card and can write the words randomly into the boxes. As they hear you say the phrase or word, they cross off that box. If they bingo, you will take off a couple of problems on the homework to shorten the assignment.

If you look up from a lecture and you have kids falling asleep or looking at their phones, you know you’ve got to do something to change the dynamics of the class. Try implementing one or two of these ideas in the next few weeks and let me know how it goes!

Why is domain and range so tough for students? I’ve really worked on slowing down and teaching domain and range in detail the past few years. The idea that students are asked to find all the x’s and y’s that belong to a graph or situation is overwhelming. I believe part of the problem is that the concept is abstract. It’s not physically possible to name all the points that exist on a line. Students are not digging deep and realizing what the line or curve is really displaying (or saying). We’ve got to get them to see the details on the graph. Where does it start? Where does it end? What is happening between? I’ve created some online items that have worked well recently, but I have some older resources that I like to use too.

It’s important that students distinguish between finite and infinite values and which type make more sense for the problem. Give students real-life situations and make them think about these situations and what the graphs would look like in detail. A great example of a continuous function is when someone is getting gas out of a gas pump. Here’s a video you can use as an opener. There are tons of questions you could ask students about what is happening.

The concept of a vending machine is a great example of input and output (which would be a finite situation). If you push a certain button, then you get a certain item.

Last year, I created the lesson below. It is a Google Docs with an embedded Google Drawings. Students can click on a Google Drawings and move pieces around or type. I have found this to be very interactive and useful.

Here’s a sample:

In the same activity, before I start having student plug into a table, I have them work on a function machine:

Function Machines in Google Drawings

This activity has 3 parts:

Relations and Functions

Domain and Range

Infinite vs. Finite Domain and Range

Below is another useful activity that I think works well for students each year. This lesson introduces reasonable Domain and Range and we practice which variable is independent and which one is dependent. The worksheet seen below is a tiny look at this activity that contains 10 graphs where the students are asked to find the domain and range. There are 5 linear graphs, 3 quadratic graphs, one circle and a graph with only points. Another worksheet asks students to create graphs with a given domain and range.

I love Boom Cards, so I’ve got a couple of Domain and Range activities that are great for extra practice. The problem below is a sample question in this activity. The students do not know what a parabola is at this point, but I still give them all types of functions. We will come back to domain and range again when we get to quadratics and exponentials. If you click on the picture, you will be taken to a chance to get this FREEBIE.

Finally an escape challenge is the ultimate fun activity to end Domain and Range. If students can get through this tough challenge, then not only do they understand domain and range as well as reasonable domain and range, but they are good at following directions, reading carefully, figuring out combinations and they have that finishing spirit!