Tag: Math

Domain and Range

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.

Photo by Erik Mclean on Pexels.com

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:

The one on the left is before the student has clicked on the picture.

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!

  • Domain and Range Challenge
  • Domain and Range Challenge
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Learning Loss in Math

Learning loss is evident this year. I’m sure you have noticed it too. How do you deal with it? We can either complain about it or we can do our best to address it.

When I coached basketball, we practiced fundamentals everyday. The kids could already dribble and shoot layups, but we still practiced these skills to keep them sharp. What would be wrong with doing this same thing in a math classroom? No, don’t bring a basketball into your classroom. All I’m saying is practice those math fundamentals. There will always be learning loss and students will struggle with certain concepts in math. We should plan for it every year! Here are a few ideas on how to deal with it. If you have some other ideas, please let me hear from you.

1) Plan for mini lessons on content that you have a feeling students will struggle with. 
2) Figure out which students are your star students and make them helpers. Let them tutor other students during class time. 
3) Use bell ringers for content students should have learned last year. (I have algebra bell ringers for Geometry students if you are interested.)
4) Add a problem or two to your worksheets with content from the previous year. 
5) Use dry erase boards to have students work problems, then show you quickly if they understand.
6) For fundamental work, do quick timed worksheets. Let’s say students do not know the order of operations. Give them 5 problems each day for a week. Have a set time and do not go beyond that.
7) Announce a tutoring session that covers a basic skill. You could say, “This Tuesday after school, I’m going to focus on one- and two-step equations.” 
8) Use those videos you made last year to reinforce the material this year. You may do an in-person lesson and then post a video so students can watch it if they need it. 
9) Give students flash cards to study. Let’s say some of the students are struggling with operations with integers. Give them some index cards and some problems with solutions. Put the problem on one side and the solution on the other. 
10) Sage and scribe is a great way to get students to work through some problems and see if they know what they are doing. My algebra students are struggling with combining like terms. I could have one person stand (this is the sage) beside the other person’s desk (the scribe) and talk the scribe through simplifying an expression. The scribe can only write and is not allowed to talk at first. This is great since both students are really having to concentrate on what they are doing. The two students switch roles after each problem. 

If I could add a #11, I think I would say to just make learning more fun. Get students excited about your class. Get them more involved. When I feel like I’m being boring, I pull out games. Students want to have fun. Here are a few games that I use in my classroom:

I hope you can take an idea or two and implement in the coming weeks. Let me know what worked and what did not work. This is going to be a tough year on math teachers. Don’t let anyone put too much pressure on you. You can only do so much. Try your best, but remember to take care of yourself. 

Different Forms of Numbers

My daughter recently upgraded one of her resources for distance learning. This resource originally had a set of 24 matching cards where the students match a number to the expanded form and to the word form. This is a great activity to see if students understand expanding numbers with decimals and then to see if they know the place values. The resource is called Matching Different Forms of Numbers.

Now she has made this ready for distance learning. She has transformed it into a Google Slides where students will drag and drop to match the various forms.

I love the last question on the Google Slides which is to create the three forms on your own. We love when students have to create. After working 8 different problems, they will be able to follow the examples and create one on their own. I love that.

The matching cards were originally in her Objective 2 5th Grade Math resource. Now the matching cards are separate and can be purchased independently.

Matching cards are fun and use lots of brain power. If you teach elementary math, go check this activity out to see if it is something you would like to add to your curriculum.

Study Guides for Math Portions of College Entrance Exams and College Readiness Exams

When you think of college entrance exams, I’m sure the SAT, PSAT and ACT come to mind. More high schools are offering these tests during school. Some students will take these tests 2 or 3 times or more. Why? Because they need a certain score to gain entrance into a school or to apply for a scholarship. The PSAT is a nice warm up to help students know where they stand before they take the real thing.

Every October, schools offer the PSAT/NMSQT to their juniors. Although sophomores can also take it , only the junior scores count toward the National Merit Scholarship competition. Schools are also starting to offer the PSAT 8/9 (for 8th and 9th graders) and PSAT 10 (this is the regular PSAT, but does not qualify for the National Merit Scholarship competition).

It’s good that schools are offering these tests to their students so they can see the format of this test and the way questions are asked. The SAT is a very important test for students planning on going to college. Colleges use the SAT (and/or ACT) to make admission decisions which makes these tests very important for students that want to go to a certain school.

The ACT is another college entrance exam. Some schools are offering this test during the school year to their juniors. The ACT is different and has a science section where the SAT does not. Again, this test requires practice and there is no pre-test like the SAT.

There are a couple of more tests worth mentioning. Nowadays, colleges want students to take a college readiness test to see if students have the skills to start taking college courses. The ACCUPLACER is used by many states. Texas has their own college readiness test called the TSI. Both the ACCUPLACER and TSI are similar. Students scores will determine if they are able to start their English and Math courses on level or if they will need to take some remedial classes first. The ACCUPLACER and the TSI can be taken as early as the 9th grade and in some cases, earlier. Early College High Schools have their students take the college ready tests the summer before their 9th grade year to give them plenty of time to retake them until they pass.

I’ve been teaching high school math for 30+ years, and it was not until I became a teacher at an Early College High School that I became fully aware of all the tests students take. I realized that I needed to be the one to help them get to where they need to be. I know how important it is for high school teachers to help incorporate college entrance and college readiness practice into their curriculum, especially math teachers. I’ve spent a lot of time creating many resources to do just this.

Below, you will find links that will take you to my Teachers Pay Teachers Store where I sell these resources. There are many options, but the option that I recommend if you are planning on doing a ton of review is the Math Test Prep Bundle for College Entrance. It contains the SAT Bundle, ACT Bundle and a TSI/ACCUPLACER Bundle that you see below:

College Readiness and College Entrance Exam Bundle
ACT Math Prep

3 Reviews – 86 Questions

SAT/PSAT Math Prep

6 Reviews – Worksheets, Bellringers and a 5 Week PSAT Plan.

TSI/ACCUPLACER Math Prep

7 Reviews!

I’ve recently started a digital version of the TSI/ACCUPLACER College Readiness Bundle (this is the exact same, but for a digital classroom setting). It is incomplete, but will be finished by the start of the next school year.

It is so important that math teachers take the time to prepare students for their future. Start making a plan now on how to meet the needs of your students. We all have different situations, but I’m sure you can find a way to include study material that will increase your students’ chances of success on college entrance exams and college readiness exams. Good Luck!