Divisibility Rules

I often run into students who need to do a complex operation in algebra or even up through calculus, but they struggle because they are missing a skill from a much simpler time. One of the skills that I see students are missing is being able to quickly determine if a number is divisible by a digit between 2 and 9 in order to simplify and reduce fractions, roots, factor equations, and more. One way to fix this is through practice and learning a few tricks. Even when students are in upper level classes. It’s never too late!

To make this easier for students to learn, it helps to have a list to use, like training wheels while they work to memorize how to find possible factors. To that end, I’ve made a great printable in both color and black and white which explains each rule, then gives two examples and a counterexample so students can see at a glance how to check for divisibility. If you print it two pages per sheet, you can have all of them on one side or print them front and back so you have them just a bit bigger!

How can you use this printable to learn the rules? Well, firstly you can use the chart while doing math work. But sometimes you want a fun way to practice the skill first before you need it. To that end, try these ideas:

  • Get 3-5 (or more!) dice and roll them. Line the dice up to make a number and figure out if that number is divisible by anything between 2 and 9. For example, roll 3 dice and get a 2, a 5, and a 3. Use the number 253 (or 235 or 352 and so on) and figure out if it’s divisible using the chart. If they are successful, they get that number of points (2+5+3=8, so 8 points)
  • Using similar rules, take a deck of cards out and make numbers by picking cards at random and lining them up. This works best if you take out the aces and face cards, but leave 10. If you put 10 into a number, you can fit zeros in that you don’t normally get with dice. For example, you pick a 5, an 8, a 10, and a 3, so you have the number 58,103 to test divisibility rules on.
  • For some competition, you can have kids race against each other. If that causes too much friction, do several rounds and have the kids try to beat their own previous best scores. This can be a great 5-10 minute warm-up or break time to make the day more interesting and get the body and brain moving.
  • Math War: Each student gets half of the deck of cards. Using their own stack face-down, they flip over 3 cards to create a 3+ digit number using the rules above. In this case, assign face cards a value of 10 and aces a value of 1. Use the divisibility rules to find the largest number between 2 and 9 that is a factor of the number formed by the cards. Whoever has the largest number gets to keep all the cards and everyone lays out 3 more cards until one person has all the cards.

This skill is really useful from middle school on up. I can’t emphasize this enough. Whether students are trying to find a common denominator, write a prime factorization, or just doing mental math in general, being able to quickly determine if a large number can be broken down more easily is a huge skill for students to master. If a student is still struggling with a basic skill, learning advanced material will be that much harder.

This skill is learned in chapter 4 of my pre-algebra class, but you can get this great free resource today just for signing up for my email list below! Please share this post if you know anyone who might want the file for themselves!

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AP Physics Insights: Non-Content Knowledge

Great tips for all AP students, not just AP Physics!

Val Monticue

AP Physics Insights Caveat: These are my personal thoughts and not in any way condoned or affiliated with ETS or whatever else. If you disagree with any of my points, please shoot me a comment or respond to the post and let me know your interpretations. I am more than willing to re-evaluate my ideas. These are simply my current conclusions based on evidence I’ve gathered, but I’m quite sure there’s more out there I haven’t seen that could lead to even better insights.

Most Important Thing

There will always be things on the AP exam that the taker feels they were never taught. The whole point of the exam is to get them to use their knowledge in scenarios never seen before. It will be a new situation, some twist on the basic ideas learned, and the goal is to figure out how to use that knowledge to make…

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Why are labs in science important?

In the last two posts, I talked about how lab activities benefit your student, but now I’d like to switch gears a bit and discuss how you can help your student get through labs and make a success of those assignments they may be avoiding.  But first, it helps to understand some of the reasons students avoid labs.

  • They’re time consuming.
  • They require planning ahead.
  • They don’t have the materials.
  • They feel overwhelmed or confused.

For the student who does well with bookwork and can get through assignments efficiently, having to stop to complete a lab can seem frustrating or a waste of time.  Since you already know they are important, one way you can help your student is to schedule a regular lab time into your week.  If they know the time is set aside for labs, and possibly means that there won’t be bookwork for that class on that day, they will hopefully see it as less of a time drain and more of a useful tool.

Many labs that you complete will require planning ahead.  If your student doesn’t look at the lab ahead of time, they may not know that the steps will take several hours or days and won’t plan accordingly.  In addition to what they might learn in the lab, this is a great skill to learn for life.  However, bear in mind that most teens don’t have the ability to do this on their own.  The part of their brain that controls executive function, which includes planning, isn’t fully developed yet.  They may need you to model for them how to plan out the activity, and in the beginning, the planning may fall entirely to you.  If the activity will take several days, help them by getting them started each day and showing them what needs to be done.  Write out the steps and which days they need to be completed on and follow up by checking that those steps have been done so that Friday doesn’t roll in and no progress has been made.

Sometimes the planning process will include gathering materials.  Just as the planning may initially fall to you, so might the materials-gathering process.  If the book is broken into units or chapters with multiple labs, try gathering the materials to a single box or cabinet (if it is safe to store the materials together).  Demonstrate how to organize the materials by activity and soon your student will be able to follow your model and gather future materials themselves.

And the final reason many students avoid lab activities is that they feel overwhelmed.  This is often a combination of the first three obstacles and by following the steps above, your student can often avoid this issue.  If labs also overwhelm you, try starting by breaking the labs for a course into groups, then follow the steps above.  Get out a calendar and plan for when each activity must start and how long it will take, giving yourself lots of buffer room for life and mistakes.  Plan to do a little each day or set aside an entire day for labs, whichever works best for your family.  Then gather the materials for the first few labs.  Separate and label the material so you can see what you have and what still needs to be found.  If you see the lab as small steps it won’t be so overwhelming and is much more likely to be done. Think of it like cooking a meal, you My Post Copy (4)don’t do all the steps at once and it’s much less overwhelming if you have all the pieces together before you start.  Planning is your best weapon when it comes to labs. In the end, it’s ok if you don’t do every lab, but make an effort to do a good chunk of them.

And finally, my biggest piece of advice is to let students do the labs themselves and that struggling is alright, even good, for your student.  It’s the struggle during learning that makes the connections in our brains and doing the lab for your student won’t help them in the long run but working by their side and showing that you are learning right along with them will do wonders for their self-confidence and interest in the material they are studying.  Even if you are familiar with the experiment, showing your interest and enthusiasm in the subject will help ignite a fire of interest in your student, even if the topic at hand isn’t their favorite subject.  Lab activities are important, and more importantly, totally doable for both the scientist and non-scientist alike!  If one activity doesn’t work, chalk it up to experience and move on.  Use what you’ve learned to make the next activity a success!

Find a great FREE lab activity here that you can practice with!  It’s fun and it doesn’t count toward your grade, so try it out with your student and see if you both don’t have fun and learn at the same time!  Comment below with how it turned out!

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Why are labs in science important?

Last post we talked about two great reasons why having your labs complete the lab component of a course is integral to success and the benefits it produces that may not be obvious.  This post will cover the other two reasons I feel that labs are truly important to the study of science, and even math.  Any one of these reasons should be enough for you to overcome any reasons you might have to skip the lab portion of a class, but these last two are really important when it comes to upper-level science classes in high school.  Whether your student plans to pursue college or not, these benefits will help them both inside and outside the classroom.

First, a real, but not always obvious, gain of having students complete labs is that they get the opportunity to learn about aspects that are difficult to grasp from just reading about them.  These aspects include, but are not limited to, sources of error, the interconnectedness of concepts, and the physical limitations of lab experiments.  When students have to recreate a famous experiment themselves, it brings to life the realities that those highly-regarded scientists dealt with.  It also shows students why it can be very difficult to prove an idea using a physical experiment and why it often takes so many trials and years of hard work to prove an idea and turn it into an accepted theory.  Labs also demonstrate to students how interconnected concepts they learn are; like when they try to prove that there is no horizontal acceleration when learning about motion in physics, they have to disregard the effects of friction and air resistance.  This also gives them an opportunity to brainstorm their own ideas of how to minimize these effects, which gives them a chance to exercise logic and abstract thinking.  Or when they perform a chemistry titration and forget to put the indicator, phenolphthalein, in and the test tube never changes color.  This never happened to me, but I’m sure some poor soul kept adding acid and the pH never seemed to change.

And, finally, the last way that I want to mention why labs are important for science classes is that it, if done right, can give students the chance to discover a scientific principle on their own.  For example, in my AP Physics classes, students are asked to complete some experiments before the unit starts.   This gives them the chance My Post Copy (3).jpgto think through what they see happening the same way countless scientists have before them, sometimes even deriving the equations that will be used in the unit from their own experimental data.  Giving students that opportunity can do wonders for their confidence and ensures that they understand why the abstract ideas presented are true and how the equations are developed.

I have always been more interested in theory than applied math and science.  Labs weren’t always my favorite part of class.  But having taken many of both types of classes and having performed numerous labs in my time, I can confidently say that labs are an integral part of science (and activities in math often have the same benefits).  I encourage my students to not skip over the lab activities and I would encourage you, as parents, to help your student work through the labs even if you both find them challenging.  The benefits you’ll reap from completing them will far outweigh the trouble you’ll go to getting them done.  Here’s one lab activity you can try with your student and see the benefits yourself, even outside of a course!

The final part of this lab series will cover how you can help your student do well on labs.  It will contain real practical tips that you can use right away.  What barriers have you found that keep your student from tackling lab activities?  I’d love to hear what you think below!

Why are labs in science important?

From the first lab science I took in ninth grade, biology, I had a terrible time understanding what the importance of labs was.  I figured I could learn anything that needed to be learned from a book or the teacher’s lecture.  Labs were just a waste of time.  As a student that lived inside my own head and often spaced out during lab time, the importance of labs was lost to me until I began teaching students myself and I realized that labs have the power to:

  • make concepts more personal and memorable
  • motivate kinesthetic learners who need a break from the book work
  • demonstrate to students the realities of lab work like sources of error and interconnectedness of ideas
  • allow students to discover concepts on their own before they learn about them in a more abstract sense.

This post will cover why the first two reasons are so integral to learning, the next post will cover the last two reasons, and then the third post will give you some ideas of My Post (28).jpghow you can help your own student be successful, so be sure to come back for the other two parts of this series.

If one type of assignment is going to be skipped in my science classes, it’s generally the labs.  Sometimes students tell me they just don’t have the time or they need parental help so they don’t always get to the assigned lab experiments and investigations.  However, I continue to assign labs in my classes because I find that students often need that physical experience to really cement an idea in their head.  Sure, you can read about how much trash you make in a day, a week, or a year; but having students collect their trash and sort and weigh it makes much more of an impact and makes the topic much more personal for them.  When it comes to assessments later in the year, a student is more likely to remember the experience a lab provides versus something they read in a book.  As proof, I submit the fact that I still remember, 20 years later, that both plant and animal cells contain and endoplasmic reticulum, which I left off of my own plant cell model in ninth grade.

Another great aspect of labs is that it helps students who are kinesthetic learners, especially boys, get a chance to physically learn about a concept.  When I student taught, all the students at the school were required to take physics, but there were several levels to choose from.  The least “academic” version was generally chosen by the students who also attended technical school, mostly because of scheduling.  This version was catered to their need for a hands-on learning experience, as each unit was taught with a project as the focus.  The students built hot air balloons to learn about density, bridges to learn about forces, and k’nex cars to learn about kinematics.  Those students who, in other classes, often performed poorly on written assignments soared when given the chance to create and demonstrate their knowledge in a different way.  They still had homework and tests, but they also had the project to let their grade reflect their knowledge rather than punish them for not being reading/writing learners.

Come back next week to learn more about the importance of labs and how you can help your student when it comes time for a lab activity, but for now please post below with the successes you’ve seen with your own students with regards to labs.  Or if you’ve struggled, please share that too!

Monday Review: HMHCO Environmental Science

by Kristen Lauria

Note: I purchased this curriculum myself and have received no compensation for this review.

Today I am going to tell you about a great book I found for teaching Environmental Science.  The book is published by Holt McDougal and is available here.  This link goes to the homeschool package, which includes both the student and teacher materials.  Another alternative is to buy the online access here and then a student text only.  The online access gives you access to the teacher’s edition plus lots of activities (like labs and such) that really make the course interesting.

I have used this curriculum for the last year to teach an online environmental course and plan to use it this fall for both regular and AP level environmental science (sign up here).  It is adequate for both levels with a little extra work put into making it an AP level course.  In the next few weeks I’ll be posting an article about how to get a course approved through College Board so you can put AP on your student’s transcript.

But back to the textbook.  One thing that I really like about this text is the use of color, graphics, and current events/case studies to break up all of the material.  This book is meaty and has lots of content to cover, but it is really user friendly and has so much information broken into bite-sized chunks.  There are 6 units, each of which contains 3-4 chapters, with each chapter having 2-3 sections.  There is lots of review available at the end of the chapter and if you get the teacher materials you can have access to pre-made tests available in either multiple choice or open ended or a combination.

In the future I hope to offer a guide to using this textbook for your own student including a recommended schedule, activities, and labs.

The Good:  Very thorough and includes all topics I would want covered in an environmental science course.  The online component has links to lots of relevant articles, course materials, and easy to do at home labs.

The Bad:  The HMHCO website is kind of a bear to navigate and they limit certain products to only schools for purchase, so you have to be sure to have the right item in your cart or you won’t be able to proceed.

Recommended For:  Students who are looking for a math-light science program or who are in middle school or 9th grade and are not ready for biology or chemistry.  This is also a great class for students who have finished their high school requirements and want a less intense science class for senior year or need another AP for college applications.

Website of publisher:  https/www.hmhco.com/

Where to buy: I’d recommend either buying a used copy from Amazon here (aff) or getting the online materials, which are only available at the publisher’s website.

Has your student considered taking environmental science?  Tell me how it went!

Freebie Friday – OpenStax Textbooks

by Kristen Lauria

For my first Freebie Friday post, I’d like to tell you all about an awesome group of textbooks available online (or downloadable in pdf or in iBooks) for absolutely NOTHING!  These books are written by the same people who write most textbooks, PhDs and the like, who really know what they are writing and how to explain difficult subject matter.  Plus the textbooks are peer-reviewed, which means they’ve had lots of eyes on them who have the knowledge and experience to look for errors and fix them before they get to a student.

I got started using OpenStax textbooks last year when I started teaching physics and AP physics online (see here to look at the course description and sign up!).  I wanted a textbook that was thorough and would provide rigorous questions for my students to practice with, but without the usual high price tag of a high school or college level textbook.  My answer was found with OpenStax.

OpenStax offers more than 20 textbooks, not just physics, and they are all available to thepublic at no charge.  There are often even other materials available to students that will help them use the textbooks in the best ways possible, including an answer key that has some, but not all, of the solutions including the work that went along with solving it.  If you register as a teacher (as a parent), you can access even more materials.

Screen Shot 2018-03-09 at 2.17.21 PM

Image courtesy of OpenStax.org Press Kit. Updated September 15, 2017.

If you are looking for books to use with an AP course (which you can register to teach as a homeschooling parent.  Look for a blog post on this coming soon!), there are many choices for AP courses through OpenStax that have AP style questions right inside the book in each section and in the problems at the end of each chapter.  Currently available are Physics, Economics, and Biology.

If you’re wondering about OpenStax themselves, they are a non-profit affiliated with Rice University.

Please comment below and let me know if you’ve used any of the textbooks that OpenStax offers.  Or do you know of another great textbook resource?

Why take calculus?

by Kristen Lauria

I often get asked by parents whether their child needs to take advanced math.  Maybe their kid is just not into math, or they have a learning disability, or they already know what they plan to do after high school and just don’t see the point.  I am here to suggest that you do push your child to take higher math beyond geometry and algebra 2.  It may be difficult, but the rewards are worth your time.

To explain my reasoning, I want you to think about sports.  When a student is a star football player, do they just practice football all the time?  When it’s football season, are all their practices on the field just running plays?

Of course not!  They spend time in the off-season practicing or playing other sports to rest their “football muscles” and keep other muscles in shape.  Or keep their cardio up.  During the regular season, they also probably spend part of the week working on skills like running for cardio or weight-lifting to increase strength and muscle mass.  Doing these other exercises doesn’t practice running plays, but it does directly impact their ability to do so.


Just like football players train using other skills than playing football, our minds need exercise of all kinds as well.  If your student struggles in math, then practicing those skills they’ve learned in earlier math and continuing their studies can really be beneficial.  The same is true if they have a learning disability.  This doesn’t mean they have to take a class within typical time constraints.  Maybe take two years to finish what is typically done in a year, but don’t quit math all together just because you have checked off the state’s requirements for math.

If your child knows what they want to do after graduating high school and those plans don’t require math beyond geometry or algebra 2 it may seem like a no-brainer to stop there.  However, keep in mind that plans can change!  My high school chemistry teacher still likes to tease me that I declared in 10th grade that I’d never do anything in my adult life that had to do with math or science.  Wise 15 year old me never saw the joy I would find when I studied physics the following year (neither did 16 year old me who griped about actually having to do homework to be able to understand the material).  In fact, I was 25 before I decided to go back to college and get a degree in physics and found my true passion, math, during the course of my undergraduate degree.

The reality is, plans change.  It’s easy to write off higher math, but wouldn’t it be better for your student to have seen the math before having to take it in college at a much faster pace and with a professor who probably won’t be as patient with your child as you can be?  Even if they don’t understand everything this time through, they will be much better prepared to learn the rest if they’ve seen the material before.  Even if they don’t end up going further in math as an adult, the problem solving skills and determination to make it through a difficult subject will help them be successful no matter where they end up!

Have you pushed your own kids to keep going in math?  Have you seen the results pay off?  Have you had other results?  Please let me know your thoughts below.

Monday Review: Math Mammoth

by Kristen Lauria

Note: I purchased this curriculum myself and have received no compensation for this review.

This is my first ever post for the blog and I wanted to start off by answering a question I get frequently: What’s a great curriculum for teaching math in the elementary years?  For my children, it’s been Math Mammoth by Maria Miller.

I stumbled across Math Mammoth when I was looking for a mastery-based program for my own kids.  I’d heard of many of the big names in math like Saxon and Singapore, but both of those were spiral-based programs.  I could tell from even an early age that my son wouldn’t be interested in small bites and constant repetition.  He has an intuitive nature when it comes to math and often knew the concepts before I had an opportunity to teach him.

Math Mammoth is a series of worktexts, which are consumable combination workbook/textbook, that the student both learns the material and practices it all in one.  Each year is covered by a book level of the same number (so grade 1 is book 1A/1B) in the Light Blue Series and is split into 2 volumes so that it’s not so overwhelming to the student.  The nice part about just one book is that the student learns and practices together and they can flip back to refer to the instruction when they get confused.  Another nice part is that absentminded kids (like DS) don’t have 2 books to keep track of.

Both my children have worked through this series, with DS in grade 5 now and DD in grade 4.  In the beginning, my DS couldn’t read, so I needed to work through the lesson with him.  By second or third grade he was reading well enough he could work through the program relatively independently, only coming to me when he was struggling to understand.  My DD progressed about the same and they both now work on their own, only occasionally needing more explanation.

For perspective, DD is most decidedly not a math person and has still had success with this program.  The only thing that I think it lacks to some degree is practice for the operations (+ – x ÷, but that is easily outsourced to apps and other programs to practice.  The program also comes with links to an online worksheet generator, so if there is a skill that your child needs more practice on, you can always make custom worksheets (with answer keys!) to give your child that extra reinforcement.

I’ve been exceedingly pleased with this program and am disappointed that there’s not a companion curriculum for middle/high school, but I’ve found other great resources to fill that gap that I’ll review in coming weeks.

The Good:  Very thorough and includes all topics I would want covered in elementary, examples and instruction teach right to the student, colorful and engaging, inexpensive.

The Bad:  Not a ton of review of concepts since it’s mastery based.

Recommended For:  All children, including those children who are naturally adept at math and those that struggle.

Website of publisher:  https/www.mathmammoth.com/

Where to buy: I’d recommend Rainbow Resource Center if you want a printed version or Homeschool Buyer’s Co-op (aff) if you want digital.

Please comment below with what elementary math curriculum has worked for you!