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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 prealgebra 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 Tutorials: on reading, writing and reducing fractions
Math Games, Videos, Worksheets: on fractions, decimals, and percentages
Dining Out: Here are free activities you can use for a coop 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 Antics: Free 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 Snacks: Short 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.



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 gamelike, 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.
MangaHigh: I’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.
TenMarks: This 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:
iPracticeMath: This one is the most workbooklike of the three, with stepbystep text tutorials, and fillintheblank or multiple choice questions. Other than the novelty of doing work online, as opposed to a worksheet, there’s not much of a gamelike atmosphere here, and the only incentives are printable certificates. But, if you want nofrills, this is the one, and it is comprehensive. Here’s one homeschool mom’s review of the site.



The following is a list of full math courses, online and free:
Algebra 1: SAS Curriculum Pathways
Beginning Algebra: Saylor Academy.
Algebra and Geometry: Hippocampus
Algebra 1 Modules: From NY State Education Dept.
Algebra 1 Online: Henrico County Public Schools
Algebra 2 Online: Henrico County Public Schools
Geometry Online: Henrico County Public Schools
Algebra 1, 2, Geometry: Math Planet
Algebra 1 and 2: Open High School of Utah
Calculus 1: Mooculus.
Georgia Virtual Learning: This 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 realworld project.
Geometry: The course is modular, so it can be used to supplement an existing Geometry program, or serve as the main curriculum. It's projectbased, emphasizing realworld 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.
Calculus: Covers Limits & Continuity, Derivatives, Applications of Derivatives, Integrals, and Applications of Integrals. Includes links to workbooks and videos.
Math Video Library: Visit Mathispower4u.com 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).







