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Homeschooling the Middle & High School Years

Chaotic Bliss Homeschooling

Fractions, Decimals, Percents, Ratios, Proportions
Category: Math
Tags: multiplying fractions how to multiply fractions multiplying fractions with whole numbers multiplying fractions worksheets

Several years ago, when my son and I were working on multiplying fractions, we were doing some work with multiplying a whole number by a fraction, specifically a proper fraction, where the numerator is less than the denominator. The example was 5/8 x 12, and the focus of the lesson was supposed to be that in order to multiply a fraction and a whole number, you change the whole number into its equivalent improper fraction, in this case, 12/1, then solve. (Cross cancel, multiply the numerators, multiply the denominators, get your answer).

Okay, so he had no problems with the process. What did bother him was when he got the answer and saw that it was less than what he started with.  At first, this didn’t make sense to him because, as he put it, “Doesn’t multiplication mean more?” 

So we talked about it, and with a little more contemplation:  

  • he understood that, yes, when you multiply whole numbers, you get more (except when one of the factors is 1 or 0
  • he remembered that when you multiply by 1 (which is like multiplying by a fraction with the same numerator and denominator) the number stays the same
  • by Jove, he then realized that when you multiply by a fraction that’s less than 1, you’re taking less than 1 of what you started with, so the answer is smaller.
  • he then extended the thinking a bit more by relating that when you multiply by an improper fraction (where the numerator is bigger than the denominator) you’re multiplying by more than 1, so the answer would be bigger

Now, I realize this example demonstrates a rudimentary understanding of numbers, but, to me, this also illustrates a problem I think kids can run into when it comes to learning math: the difference between knowing how to perform the steps of a particular process versus understanding why an answer does or does not make sense. Sometimes, we can learn a process mechanically without thinking about what we're doing in concrete terms. Maybe the pondering comes later – or maybe it never comes at all.

I suspect this may be one reason why kids that do great in basic math and pre-algebra seemingly hit a brick wall when they get to Algebra.  Could it be because when studying Algebra, you may learn to mechanically follow steps to solve complex equations without really comprehending the underlying logic? Or maybe some kids never get a good enough grasp of the basics to sufficiently tackle “higher math?” 

I’m not sure where the breakdown occurs, but I do know that some people have a great facility with numbers (I am, unfortunately, not one of these people), while others may struggle all their lives with math, and others are somewhere in between. Is the facility with numbers from having been taught well, a result of some innate trait, a combination?  How do you determine if your child has a sufficient understanding of math principles? Do you assess with worksheets and tests or use another method? Please leave a comment. And, if you need some free resources for learning fractions, decimals, percents, ratios and proportions, take a look at these:

MathMammoth: videos and worksheets on a variety of fraction topics

Fraction Video Tutorialson reading, writing and reducing fractions

Math Games, Videos, Worksheetson fractions, decimals, and percentages

Dining OutHere are free activities you can use for a co-op or group. Get the kids working with fractions, decimals and percents by figuring tax, tips and discounts when ordering food at restaurants.

Fraction Worksheets and Printables

Math AnticsFree videos with access to some free worsheets.

Yummy Math:  This site stands out from many on the Web, in that it focuses on relating math to real life. There’s more emphasis on concepts and critical thinking than on memorization of steps. For more on the YummyMath philosophy, read this post.

Math SnacksShort animations and games for teaching math concepts in grades 3 - 8. Includes accompanying worksheets (with answer keys).

And for additional math resources, visit our math page.



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Online Math Review: Skills Practice
Category: Math
Tags: online math games online math tutor math facts practice math review

Whether you’re continuing your schooling through the summer, in review mode, or taking a break, these free online math sites offer up math skill drills in game-like, edutainment fashion that your kids can do with minimal input from you (if you are taking that break).


These sites all have similar features that prove useful, namely:

  • they can be used on mobile devices as well as PC
  • they have content for elementary through high school levels, covering a comprehensive set of concepts for each grade
  • you can sign up as a teacher, and assign each of your kids their own “playlist” of exercises that automatically appear when they sign in
  • you can track progress, seeing how well your student is doing, how long they spend on problems, and whatever badges or incentives they’ve earned
  • you can generate and print reports if you want them for your homeschool portfolios

All of these have pay versions, giving access to more features, but the basic free packages offer a wide variety of content.

MangaHighI’ve used this program since it launched in 2010. The kids like it, and I have found it to be an easy way to assign games for specific math skills. The games are adaptive, meaning they adjust in difficulty according to a player’s performance. In some games, students can compete against other players. There's a teaching mode if you want your child to have more instruction on a particular concept, but I think the greatest strength for this one is its games. For a thorough overview of this site, read this article from EdTechReview.
TenMarksThis site also allows you to assign review tasks based on grade level, but is more instructional, with the games used as incentives. The format looks like interactive, multiple choice worksheets, with each question providing hints if the student gets stuck, and embedded video lessons to explain the concept. After successfully completing a given number of concepts, a student can unlock games and badges.
Tenmarks sample questions: 
For an overview of TenMarks that includes a comparison to Khan Academy, read this review from Match Education.
iPracticeMathThis one is the most workbook-like of the three, with step-by-step text tutorials, and fill-in-the-blank or multiple choice questions. Other than the novelty of doing work online, as opposed to a worksheet, there’s not much of a game-like atmosphere here, and the only incentives are printable certificates. But, if you want no-frills, this is the one, and it is comprehensive. Here’s one homeschool mom’s review of the site.
You may also like: Free PDF Math Workbooks 

High School Math Resource Roundup
Category: Math
Tags: geometry online high school geometry geometry online book algebra calculus

The following is a list of full math courses, online and free: 

Algebra 1SAS Curriculum Pathways

Beginning AlgebraSaylor Academy.

Algebra and GeometryHippocampus

Algebra 1 Modules: From NY State Education Dept.

Algebra 1 OnlineHenrico County Public Schools

Algebra 2 OnlineHenrico County Public Schools

Geometry OnlineHenrico County Public Schools

Algebra 1, 2, GeometryMath Planet

Algebra 1 and 2Open High School of Utah

Calculus 1: Mooculus.

Georgia Virtual LearningThis site has a number of online high school math courses that can be accessed for free.

Curriki Courses: 

Algebra 1: This is a full course, broken into modules, so you can use it in its entirety, or just the parts you want. The course includes lesson plans, with links to related Khan Academy videos, worksheets and assessments with their answer keys, and each unit ends with a real-world project.

GeometryThe course is modular, so it can be used to supplement an existing Geometry program, or serve as the main curriculum. It's project-based, emphasizing real-world examples, such as using Geometry to create architectural designs for an apartment or house, or designing a floor plan that maximizes open space and natural light.

CalculusCovers Limits & Continuity, Derivatives, Applications of Derivatives, Integrals, and Applications of Integrals. Includes links to workbooks and videos.

Math Video Library: Visit for video tutorials in Algebra 1 and 2, Geometry, Trigonometry, and Calculus.

(If you know of any other free math courses, please list in the comment section).

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