Is it weird that I love prime factorization??
Every year of teaching math, I have come to appreciate prime factorization more and more! Maybe it’s because when I was a student (forever ago!), I didn’t learn how to use prime factorization to find greatest common factors, least common multiples or to reduce fractions. (I will admit to the possibility that I learned and forgot….but I truly think I didn't learn it!). In addition to missing this information as a student, I didn't find it in math teacher manuals until I'd been teaching for more than 20 years.
I'll share why I love it so much by explaining three ways to use prime factorization: to find GCF, LCM, and lowest terms for fractions.
How to Find Greatest Common Factor With Prime Factorization
1) First, find the prime factorization of each number.
Using the example in the image:
Why do I like this method? I like using prime factorization to find greatest common factor because when my students use the 'listing method, they often miss factors of some numbers; and when they miss factors, I swear they always end up missing the GCF.
Using prime factorization, they DON'T miss these factors, so they're more successful in identifying the GCF.
How to Find Lowest Terms Using Prime Factorization:
In past years, when students reduced fractions, they often chose ANY factor to divide by, (unless they were forced to find the GCF). Then, they would reduce and reduce again, and sometimes they still didn't reach lowest terms.
For example, some students would take 54/72 and divide by 2 to get 27/36. Then they might divide by 3 to reach 9/12. Some might stop here and never reach 3/4 as the lowest terms.
To find lowest terms using prime factorization:
1) First, find the prime factorization of each number.
2) Next, cross out the factors that are in common.
How to find Least Common Multiple with Prime Factorization
Using prime factorization to find the least common multiple is fantastic! Listing multiples can be pretty tedious (though it does reinforce multiplication facts), and although finding the prime factorization might be difficult for students to begin with, it will eventually be quicker than listing multiples.
Once I started talked about prime factorization a lot in math class, and thought about it aloud so the students could hear my thought process in breaking down numbers, my students started to find prime factorizations much more quickly than students in the past.
To find LCM with prime factorization:
1) First, find the prime factorization of each number.
2) Next, identify the different factors of each number.
A 2nd Example: Find the LCM of 6, 7, and 14
1) Find the prime factorization of each number.
The factors 2, 3, and 7 occur once at most, so they are each multiplied once to find the LCM of 42.
Other Prime Factorization Benefits
Besides helping students to find GCF and LCM and reduce fractions more quickly, I love the fact that using prime factorization for these concepts helps students develop a better understanding of relationships between numbers....I see and hear this awareness developing.
Here's a great benefit - students like it! While some students are comfortable with ways they've learned in previous years and are hesitant to use prime factorization, other students have actually come to me during our study period to double check how to use prime factorization in these ways, because they LIKE it and think it's cool!
I've created a note sheet for my students to keep in their notebooks so they can refer to it throughout the year. Feel free to download and use it!
Use the Ladder Method to Find Prime Factorizations
Of course, I also love how the ladder method can be used to find prime factorizations, as well as GCF, LCM, and lowest terms fractions. You can read about that in this post!
I hope you enjoy prime factorization! Are there any other ways you use prime factorization with your students?
Other Resources for GCF, LCM, and Prime Factorization Instruction and Practice
If you're looking for other resources to help your students practice with GCF, LCM or prime factorization, I have several in my TeachersPayTeachers shop - some print and some digital:
To Read Next:
Six jobs to Help you Supplement or Replace Your Teaching Income
Do you love your teaching job, but need extra cash? Maybe something to do on the side or during the summer?
Do you love teaching, but your circumstances require you to find an alternate career?
Or, are you hoping to leave the classroom, because teaching isn't what you thought it would be and you want a change, but aren't sure where to start?
I've been where you may be! I loved teaching and taught elementary and middle school for 24 years, but throughout those years, there were times that I wanted to be home with my own children, and that made me want to leave the classroom. And sometimes we needed extra money. AND I wanted to work for myself! So, from as long ago as my second year of teaching, I was always seeking other ways to earn money - especially ways that would let me work from home.
I tried several ways to earn extra money over the years, and I'll share the most successful of those here (yes, there were some unsuccessful ones too:-).
The jobs I share here are the ones that allowed me to make enough money to replace my teaching income and leave the classroom.
1) Be a curriculum writer
This is the first teaching-related job I took that seriously started me on the path to leaving the classroom. While I was still teaching, I became a math curriculum writer for an online education company. They hired me to write lessons for upper elementary and middle school math, and I was able to do this work from home.
I wrote the content for the lessons; designed the activities that a developer would turn into online, interactive activities; wrote practice problems; and wrote assessments. I worked for this company for several years. After a couple years, the company was bought out, and I continued to working for the new company. Then the manager I worked with left the new company and began her own online learning company (Accelerate Education). I continued to do the same type of work with them for several years. I worked in math, but curriculum writers were hired for all content areas.
There were a lot of late nights during these years, but it was definitely worth it - it was a great experience that taught me a lot.
2) Teach online
I did just a little of this with Accelerate, back when the company was new, while I was writing math curriculum. I have to admit teaching online wasn't my favorite, and it wasn't as lucrative as writing, but it was a little extra cash and, again, I was able to do it from home:-) You may find that you really enjoy teaching online!
3) Write for elearning companies
This type of writing doesn't have to be K-12 education-based. There are elearning companies creating courses for people/businesses in ALL areas!
After my curriculum writing (and while I was still teaching), I started writing storyboards for an elearning company where a friend was a project manager (she's a former teacher, as were four or five other employees in this company). As teachers, we know how to set objectives for learning and how to organize and teach information. Writing storyboards for a course that teaches something falls right into our wheelhouse.
As a writer, you're given the content the company needs to teach, and your job is to turn their information into an engaging learning experience. In your storyboard writing, you include presentation of information, activities to practice the information, and questions to assess the information. And this can be done as creatively (or not) as the company wishes it to be!
(This job was a remote one too!)
4) Be a proofreader or copyeditor
In addition to writing for elearning, I've spent a bit of time copyediting storyboards others have written. As a teacher, proofreading and editing may be a real area of strength for you. You can find many proofreading and editing jobs listed on sites like Upwork and Fiverr.
I've been working in the elearning industry for six or seven years now (still am), and there are many tasks that teachers can do. You may even be interested in being a course developer! (I've done some of that too - it's how I learned to make the digital activities on my site:-)
(It was actually the work in elearning that gave me the security to leave the classroom. When I left the classroom, I took a job working in elearning 20-30 hours/week and worked on creating teaching resources about 15-20 hours/week.)
5) Become a Teacher-Author
If you love creating your teaching resources and want to share them with teachers and students around the world, you can become a teacher-author on TeachersPayTeachers! There is so much guidance out there to help you get started:-) I started on TPT when I was still in the classroom and working at some of these other side gigs.
Writing curriculum, working in elearning, and being a TPT teacher-author have all complemented each other. I've learned so much in each area that I was able to apply in the others.
6) Be a Virtual Assistant
MANY Teacher-Authors hire virtual assistants to help with the numerous tasks they need to get done. And outside the TPT world, there are a plethora of businesses looking for assistance. As teachers, we amass a wealth of knowledge and abilities - in content areas, technology, planning, data analysis, time management, organization...the list goes on and on.....we know how to do so much!
If you know teacher-authors, you can get in touch with them to learn more about Virtual Assistant work, and Virtual Assistant jobs can also be found on Upwork and Fiverr.
This is the one job on the list that I haven't done, but I know it's a great job. You can take on as much work as you want and build your business at your own pace.
I hope these suggestions are helpful! There are many more possible avenues of income, but I wanted to share the ones I've tried and had success with. Even if these suggestions aren't quite right for you, I hope they can jump start some brainstorming of additional job ideas to help you reach your goals.
When I started my math curriculum writing adventure, I never thought it would lead me into the elearning and TPT worlds! It's been so exciting to keep learning new things, meeting new people, and experiencing new roles. I can't wait to see what comes next!
To read next:
Emergency sub plans! Do you have them?
Will you need them?
This fall especially (2020), you may need to have those emergency sub plans ready. Perhaps your school district is requiring you to have a week, or even two weeks, of plans. Or perhaps you just want to be extra-prepared. What's the best way to approach emergency sub plans in middle school math?
If you're like me, you don't want your absence to cause your math students to 'get behind' in the curriculum...you don't want things to be put on hold or to stop in the middle of a unit. But, you don't always know who will be taking over your math class....will it be a math teacher or someone who has never taught math and isn't comfortable with math?
If you must have the plans set at the beginning of the year, you can't really include the specific content you'll be teaching when those plans are needed.
So what's the best approach for sub plans?
First off - if you do have to be out and there's a test (or some other really critical item) scheduled for your first or second day of absence, I'd keep that on the schedule. Do whatever you can to be sure students get to take the test as scheduled. Make sure you're prepped for the test a day or two before....copies made or links prepared, so no one has to scramble to take care of it that morning (or has to postpone it). After the test, your sub can move on to the emergency plans. (I only mention this because I was guilty of last-minute 'morning of' prepping too often. Eventually, I started prepping my classroom for the next day before I left in the afternoon....writing the agenda on the board, being sure all copies were done, etc.)
OK, on to some ideas...
Problem solving is always an area that students can practice!
I typically include problem solving sheets requiring students to show all their math work AND do some writing, to explain how they arrived at their answers.
Some problem solving suggestions include:
Color by Number Activities
These are my favorite:-) Once I started creating these, they were always part of my sub plans! If I was out for just a day, I'd include one for the math topic we were covering. However, in the emergency plans, I included 'mixed practice' color by numbers.
Members of my email community receive a free color by number every month, so you might have several of these saved or still waiting in your inbox!
I also loved including Faceing Math activities. These incorporate drawing as well as coloring...so much fun and so creative! My own children brought these home as homework and got me hooked on them:-)
Many students love number puzzles, and depending on the type, they can be excellent to help students problem solve, work on math fluency, develop persistence, and look for patterns.
If you have time, creating review lessons focused on areas of student need could be very beneficial. At the beginning of the year, you might do a pre-assessment of math skills needed for your grade level. Review lessons would then focus on those skills students seemed to have the most difficulty with. In 6th grade, students have always needed extra fraction and decimal practice, so I'd include those in review lessons.
If this is a possibility, I'd recommend having students work on whatever program your school may use...students can complete a lesson every day or two that you're out. Fluency lessons and review lessons are great options.
If your district doesn't have a particular program, Kahn Academy is always great!
There are also several free activities on my math activity site.
Do Your Plans Need to be Computer-Based?
If you're currently virtual, your sub plans may need to be more computer-based. If it's possible for students to print some of the math activities and interact on paper, I think that would be great. However, students can still complete many of these suggested math sub plan activities on the computer.
Do you have any other sub plan ideas to share? Feel free to add suggestions in the comments!
To Read Next:
Do you teach divisibility rules in your math class?
I've always enjoyed teaching divisibility rules and my 6th grade math students have always seemed to have fun using them!
I've read different opinions about whether or not teaching divisibility rules should be a focus in math class, because they may be viewed as 'tricks.'
However, I think understanding and using them in middle school helps students develop number sense and number fluency.
Rather than being taught as a 'unit,' I think divisibility rules should be introduced and then referred to again and again in any applicable situation throughout the year. To make the continuous revisiting easier for students, I've always liked to have a resource for them to refer to throughout the year. We used to create fold-it-ups, but then I moved to using a Doodle Notes resource or a Math Wheel.
Any reference sheet is helpful so that when you ask a divisibility question, students can grab it (or look at it on a wall) to quickly refresh their memories, if needed.
Where can students apply divisibility rules?
There are several math concepts where students can apply the divisibility rules:
1) Working with division facts that are beyond the 'basics.' Like 51 divided by 3, for example. Students so often believe 51 is prime, but if they take a second to test the rule for 3, they can quickly see that 51 is in fact a composite number.
2) Prime factorization. When determining the prime factorization of a number like 51, 57 or 87, using the divisibility rules can be very helpful!
3) GCF. To find GCF, students need to determine what each number can be divided by, so the divisibility rules are quite helpful here.
4) FACTORING. If you teach divisibility rules in elementary school, you might not be thinking of this eventual application. However, the more frequently math students work with divisibility rules, the more their number fluency improves, and the easier factoring will be for them.
Finding the sum for checking 3 and 9 sometimes led some students to start adding the digits to check for every number. I found the use of visuals with the Doodle Notes or the Math Wheel reduced this tendency, as compared to when we used the fold-it ups.
What's your favorite way to teach (or use) divisibility rules?
To Read Next
You walk around the classroom, looking at students' math work, listening to students' conversations, chatting with students about their math work, correcting possible misunderstandings, and reinforcing correct thinking. This is a big part of your 'normal' teaching day, right?
But this probably won't be the scenario for your fall 2020 math class. Whether you're totally online, in a hybrid model, or face-to-face but need to social distance, providing feedback will look different.
Providing feedback in distance learning
Feedback is so important. When a student completes a task - a practice problem, responding to a reading passage, reading aloud, writing an essay - whatever they're learning - they need to know if they did it right or they need to know what to do differently. Then they can repeat and practice correctly.
When you're with your students, giving feedback is relatively easy. You can nod, give a thumbs up, give verbal feedback, etc. But when the learning is virtual, or you need to social distance, it's not so easy! You can't walk among the students, look at their work, and take the quick moment to correct a little misunderstanding. So providing feedback becomes even more critical with distance learning, when you can't see students' body language, facial expressions, etc, that indicate they don't fully understand the concept. Even on a Zoom call or Google hangout, it's tough to just see everyone, let alone notice everyone's cues!
How can you provide feedback to students in this situation? Or in a situation where you're in the classroom, but can't be close enough to students to check their math work in the same way? Self-correcting digital activities are one great way to provide that feedback.
Benefits of Self-checking Digital Resources
How are self-checking resources beneficial? Obviously they aren't the same as being next to your students, checking their work yourself, and giving them verbal feedback. But, they're so much more beneficial than assigning an activity that gives students no indication of whether they 'get it' or not. The self-checking element is a step toward avoiding a lot of incorrect practice.
1) Self-checking resources provide immediate feedback
One benefit of self-checking math activities is that they provide immediate feedback - research has shown that feedback is most effective when it's given immediately. While you may not be able to give very detailed feedback all the time in a distance learning situation, it's helpful for students to at least know if they're correct or incorrect when they're practicing math concepts.
2) Self-checking activities give students more independence
Self-checking, digital math activities, especially non-graded activities, allow students to feel independent and more responsible for their work. The benefits here are that students may take more time to retry a question they answered incorrectly, or retry the entire activity because they aren't being monitored. Since they aren't being watched as they're practicing, they can try again without feeling self-conscious about it.
I remember doing math homework back in high school and loving when the answers were in the back of the math book - not because I wanted to cheat, but because i wanted to check myself! If my answer was wrong, I'd redo the problem until I got it right.
Three types of self-checking digital activities
So far, I've created three types of self-checking activities.
1) My favorite are the Truth or Dare games:-)
These are in Google Slides (and on my digital math activities site) and do require a good amount of navigation. This is a quick overview of the steps (video demo below).
2) The next favorite are the self-checking task cards in Google slides (video on the left, below). These are multiple choice questions. Selecting an answer takes students to a 'correct' or 'incorrect' answer slide. If the answer was incorrect, there's a Back button they can select so they can try again.
3) I also love the pixel art color by answer activities in Google Sheets (right, below). These offer more 'fun' feedback because students see the colors appear if they get the answer correct. They can try again on these as well, by deleting the incorrect answer and entering a new one.
I'll be continuing to brainstorm to find other activities that can provide feedback, instructional when possible.
What other ways do you use to provide feedback during distance learning or social distancing?
To Read Next
Do your math students love using technology? Playing math games?
Do you love math resources that are easy to use? AND great quality?
The digital math activities on my site are perfect for those who answered yes to these questions!
As more and more teachers and students are using the digital games on my site, I've received a few commonly asked questions, so I'm taking some time to answer them here, for anyone who has the same questions, but hasn't asked:-)
Evolution of the Digital Math Activities:
The digital math activities page of this site has been active since August 2019. It started out with 14 Truth or Dare math games for only members to use. Over the past year, I've added seven free activities for anyone to play, as well as several new member activities (member activities are up to 28 and I'm adding more as quickly as I can).
I've migrated the activities to a stand-alone math activities site, called cognitivecardiomiddleschoolmathdigitalactivities.com (I'm still adding/updating new activities there.) On that site, I added 'upgraded' versions of the free activities to the member section - I added question banks and more exercises to them, so students don't get the same questions each time. So, while you may see the 'same' activity on both the Free and Member pages, the activities aren't quite the same.
On to the FAQs...
Q: How do I know what content is in the activities?
A: This document gives an overview of the current activities - content covered, types of questions, number of questions in the question banks. (Plus, there's a free trial week July 22 - 28 so you can check out all the activities - you can sign up at the bottom of this post.)
Question: Do students get the same math questions every time they do an activity?
For the Truth or Dare games: yes. The logic in designing the Truth or Dares was pretty intense and can't be quickly changed to offer different questions each time; however, I will explore making versions that will allow for that.
For all other member activities: NO. All the other member activities will pull from question banks, so while a student may get one or two repeat questions when they play multiple times, the entire set of questions they see will be a bit different each time.
For free activities: yes
Question: How many students can log in at once?
A: As many students as you want. Students use a password to access each activity and there's no limit to the number of students who can access at the same time. None of the teachers who've used the site have reported issues with using the password/logging in.
Q: Can I assign an activity in Google Classroom?
A: Yes! Each activity has its own link, so you can assign an activity by adding the link to your assignment in Google Classroom. In the left image below, I circled the instructions I might write for students; and I circled the 'Link' section under 'Add.' This is where you'll add the link to the activity you want to assign. In the right image, you can see the way the link appears for students (since the page is password protected, that does show in the assignment.)
Q: How do I see students' results?
A: All of the math activities show results in some form. For example, the Truth or Dare games have an "Answer Sheet" (shown below). Many of the other activities show students' points on the main screen; some show the point totals on an ending results screen. Others show results by color, like the absolute value activity shown below (green for correct, red for incorrect).
Regardless of where the results are shown in the activity, students need to take a screenshot of their results page to send to you (or upload into Google Classroom, or another LMS you may be using.) Unfortunately, the software I use can't be programmed to report scores to all the different LMSs classes are using.
The image below shows the student side of Google Classroom, with the screenshot of the results page.
* The Truth or Dare games do have a Print button on the Answer Sheet, so if you're in the classroom and students can access a printer, they could print their results.
Q: How are teachers using the digital math activities?
A: Teachers are using the activities for:
Q: How are parents and tutors using the activities?
A: Both parents and tutors have been using the digital math activities to provide extra practice/reinforcement for students at home.
Q: Where can I find pricing/membership information?
A: Membership Information is listed on Teachers Pay Teachers.
If you have any questions, please feel free to get in touch - email@example.com
To Read Next:
Color by number math activities and distance learning? You bet!
Digital color by numbers? Nope, not for me:-) I like the paper and pencil!
A Little Distance Learning History
As we know, distance learning and digital learning are not synonymous. Distance learning has existed for a looonng time. Early on, it was called 'correspondence education' or 'correspondence learning.' Students received assignments in the mail, completed them, and mailed their work back to their educational institution.
A few examples of early distance learning include:
Check out this infographic for more detail about distance learning history.
Distance Learning Doesn't Have to Mean Digital Learning
Why consider this history? With schools moving to distance learning in light of the COVID-19 pandemic, I think it's important to remember that even though we have the technology to provide digital learning experiences, distance learning doesn't have to mean just digital learning. Paper and pencil activities have their place in distance learning; in some cases, such activities can provide greater benefits than digital activities.
Color by Number Activities Help Children (and Adults)
Why are color by number activities an important part of the current distance learning needs? Because of the many benefits coloring provides!
During this difficult time, the benefits of coloring are so helpful to our students and our own children (and to us). Here are just a few: