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:
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 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 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 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.
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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:
All of these have pay versions, giving access to more features, but the basic free packages offer a wide variety of content.
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 real-world 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 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.
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).
]]>Here are some free personal finance courses and resources. Scroll to the bottom of this post for some free, downloadable readers about money.
High School Financial Planning Program: Designed by the National Endowment for Financial Education, these materials for middle and high school age, come in six units that cover Money Management, Borrowing, Earning Power, Investing, Financial Services, and Insurance. Register at the site and you can download the booklets or order print copies.
Next Gen Personal Finance: Offers a free set of online lessons covering personal finance basics. The curriculum, published by the non-profit Center for Financial Capability, is presented in a series of units that include text and video resources, and activities.
Practical Money Skills: Features teachers guides and activities on money management for grades K - 12. Lessons for the earliest learners introduce money (what it is and how to earn it), then moves on to such topics as savings, loans, credit, budgeting and investing. There are 22 lessons for high school.
CompareCards.com: This credit card comparison site has free lessons for middle school and high school students on the basics of credit, credit cards and investing.
Banzai: This site teaches key concepts in personal finance by presenting students with a number of real life scenarios. Tasks to be completed include paying rent, understanding your paycheck (taxes withheld, direct deposit, gross and net income), budgeting for gas, groceries and other expenses.
Money Wise Teen, a series of financial literacy video lessons created by the California Council on Economic Education. Designed for middle- and high-school students, the animated vignettes cover basic economics topics such as scarcity, budgeting, credit, compound interest, and paying for college. The 10 video lessons are available on the Money Wise Teen YouTube channel; there's also an accompanying 84-page PDF Teachers Guide.
Financial Football: Players gain yards and score touchdowns by correctly answering financial questions. There are three age levels to choose from: Rookie (ages 11–14), Pro (ages 14–18) or Hall of Fame (ages 18+). For each age group, there are downloadable lesson guides with discussion questions, activities, quizzes and other exercises. Combined with the game, this seems like something that might work well in a group setting or homeschool co-op.
Live "Stock" Adventure: A savings and investing game to download, played with a standard deck of cards.
Saving and Investing for Students - U.S. Securities and Exchange Commission PDF e-book
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Looking for non-traditional methods and materials for math instruction? These free sites offer creative math exercises and problem-solving activities that can help your children learn math concepts -- minus the dry textbook questions.
Relating Math to the Real World
RealWorldMath: This site enables students to take the math concepts they’re learning and apply them to a variety of tasks using information, images and tools in Google Earth. After downloading the Google Earth application, download lessons at the RealWorldMath site, and you’ll be able to open the files in Google Earth. One lesson gives students practice solving volume problems using some famous geometric places (the Great Pyramid of Giza; the Leaning Tower of Pisa; the World Trade Center site; and others). When you open the file, you can click on and fly to different locations and you’re prompted through the exercises.
Other activities inlude: calculating time zone differences; using historical data and solving cipher messages to find hidden U-boat longitude and latitude positions; finding area and circumference of circles after flying to various “crop circle” locations; estimating and measuring distances between landmarks; exploring mazes and labyrinths around the world. The site’s author also provides suggestions for using the activities in other subject areas, and additional resource links. There is a password-protected “Teacher” portion of the website where you can access lesson overviews, worksheets, and some answer keys. Fill out the form under the “Contact” tab and the author will send you the password.
FoodMaster: The hands-on activities at this site use food and cooking to teach math (and science) concepts. You can download individual chapters and answer keys for grades 3 - 5, and labs for high school. Grades 6 - 8 materials are currently being evaluated, according to the site. Explore measurement and conversions, ratios and percentages, comparisons, etc. using a variety of foods. Many activities involve graphing and experimentation.
Problem Solving
MathCounts: Solve a variety of word problems: compare the value of two summer jobs; perform calculations with gift cards; explore percents and probabilities with ice cream, and more. There's an archive, where you can view past problems and answers. The exercises can be searched by topic (Moon Math; Pizza Problems) or concept (measurement, percents and fractions), with new problems presented each week. Also at the site are MathCounts Minis, videos teaching math concepts, with downloadable worksheets and answer sheets.
Math Maven Mysteries: This Scholastic site presents short story mysteries to solve, and in the process, you're reviewing key math concepts. Categorized by skill and diffifculty level, and the mysteries can be downloaded.
Math In Your Backyard
Maths Outdoors: It's amazing what math investigations can be done with just some sticks and stones: shapes and symmetry; angles and fractals; measurement, multiplication and many more. This site is full of a lot of great ideas, not just for exploring math concepts, but for educating outside in the other subject areas.
Careers Using Math
Math Apprentice: Interactive site that shows how math relates to different interests and disciplines. A player takes the role of an intern to explore eight businesses where math is used to solve problems. Enter a building, and an employee explains the math behind a particular job, and presents you with an activity. You can also read more about a career and explore math concepts used in each profession, (eg. learning the importance of shapes and angles when designing bicycle frames).
For additional real world math activities, check out this list of sites from Education World; and this Pinterest page.
Math games using a standard deck of cards can help your kids learn and review math concepts, plus have some fun. Beyond modifications to old faves like Go Fish, War, or Concentration, the following games provide a variety of computational challenges, and can be adapted to various age groups. (Note: be sure to check out the comments at the end of this post for additional games).
Remainder Jump: This game uses playing cards plus a printable game board to review division and the concept of remainders. The object of the game is to be the first player to reach "Finish," so players must develop good strategies to move the farthest on the board. They'll be dividing, subtracting, and thinking about factors of numbers, plus honing their mental math abilities. Other free math games, puzzles, and worksheets are available at the site, called Beast Academy. You can also print out their standard deck of Beastie cards.
1000 Wins/0 Wins: For practicing addition and subtraction: player that creates combinations closest to 1000 or 0 wins.
Fast Food: For practicing multiples of numbers: make cheeseburgers, fries and sodas to score points.
Prime: Players win by making prime numbers.
OrdOp: Use a standard deck to practice order of operations and computational skills. This version includes a printable set of cards numbered 1 - 25. Here’s a video showing how to play:
Bino: Players try to make combinations of cards that will give them the most points, using both ordinary and binary numbers.
Dice and Card Games: This 11-page PDF includes a single-player solitaire-style game for studying sums or multiples of numbers; Go Fish, Concentration, and War variations; other addition and multiplication games.
Acing Math: This 69-page download features a collection of card games categorized by grade, and covering the core processes, plus other areas such as fractions, percents, decimals, positive and negative numbers, patterns, place value, exponents, and others.
Two Math Games With Cards You Can Easily Make:
Easy Piecy Decimals: You’ll need to make a deck of 20 or more playing cards with monetary decimal values between $0.00 and $1.00 to two decimal places. A 10-sided die is also needed, and you can print one here. The object is to practice adding, subtracting, and rounding simple decimals. Links to additional resources to teach decimals are listed.
Algebra Rummy: This game’s goal is to get players more familiar with algebraic terminology. You need to create a 54-card deck with algebra terms. Play is similar to standard Rummy, except you’ll be forming sets of 3-or-more like terms (3y, 5y, 6y), or like coefficients (4x, 4y, 4xy). Game can be extended by forming equations. Includes a list of algebra terms and links to additional resources.
Make Platonic Solids With Cards: Not really a game, but a twist on building with cards. Downloadable template and instructions for forming a cube, tetrahedron, octahedron, dodecahedron, and icosahedron out of playing cards. Make cuts and slide them together -- no glue needed.
Be sure to check out the comments section for more games that have been added to this article.
(Affiliate Link):
Games for learning the basics of addition.
Bridges Practice Books: Grades K - 5
Harcourt Math Practice Workbook: 2nd grade
Harcourt Math Practice Workbook: 4th grade
Math Connect Workbooks (McGraw-Hill) - workbooks and additional online resources; middle school:
Pre-algebra workbooks (McGraw-Hill)
Glencoe IMPACT Math Course 1: Workbooks and online resources; middle school
Algebra 1 and Algebra 2 workbooks (McGraw-Hill
Glencoe High School Math Workbooks: (pre-algebra; algebra 1 and 2; geometry)
MathGoodies has many free worksheets - plus a worksheet generator - to go with the site's interactive math lessons.
Homeschool Math has worksheets for grades K - 7.
MathCharts: Skip-counting charts, Roman numerals, number lines, place value, and more.
]]>Hope you are having a swell summer. After a grueling week in Orlando, I'm happy to get back to the more relaxing (and definitely less taxing) resource planning for the upcoming school year.
For those of you also in planning mode, here are some high school materials you may find useful:
Prentice Hall Algebra 2: So far, I've homeschooled two children at the high school level, and have used a variety of math resource types, including textbook-based (with a hodgepodge of videos not specifically correlated to the text); CD-ROM-based lessons with textbook from a homeschool academy; and totally online courses with no textbook at all. I have found that my high schoolers, although typical "cyber-generation," actually prefer hard-copy when it comes to certain things, math being one of them. So, I'm thinking of using the Prentice Hall Algebra 2 textbook, which appears comprehensive, and has a companion website with projects and real-world applications, and computer-scored lesson quiz.zes, and chapter tests. In addition, here are three sites with video lessons that correlate specifically to each chapter of the Prentice Hall text.
Other Math Textbook/Video Correlations: If you are looking for teaching videos related to specific textbooks for other maths, try HippoCampus. The site has also just added an Art of Problem Solving collection with videos that break down how to solve problems.
Saylor's K-12 Section: Not sure if I mentioned this before, but Saylor.org, which has hundreds of free, online, self-paced college-level courses, also offers some middle- and high-school courses. Currently, there are 9 of them in the areas of English Language Arts, high school math, and two "electives" (one on the Common Core and one on SAT prep).
Enjoy your summer.
]]>Signs of summer are all around us, and our family looks forward to winding down in some areas, and ramping up in others. Still, I find, with my own kids, that math is one of those things that, if not practiced all year, is easily forgotten. So, although I don’t school year-round, we do try to keep the math going, emphasizing a more game-like experience.
If you need some material for your own review endeavors, or maybe you’re planning for the fall, here are some resources that may help, including a list of links to free courses in high school math.
Math Playground: In addition to arcade style games, this site has some interesting additional features, such as:
Math Playground also has math manipulatives (pattern blocks, balance scales, geoboards), math videos and worksheets, and many games for keeping math skills sharp.
Free worksheets and worksheet generators:
www.worksheetworks.com: in addition to math, has language arts, geography, puzzles
http://www.kutasoftware.com/freemain.html: from pre-Algebra to Calculus
www.math-drills.com/: Worksheets for 1^{st} through 6^{th} grades; also has Algebra and Geometry worksheets
Math Printables: print out things like calendars, clocks, fraction circles and strips, grid paper, dot cards, pattern blocks, place value charts, symmetry (complete the figure) tangram pieces, and more
Free Textbook: Actually, it’s a CK-12 Flexbook for reviewing Middle School Math Concepts.
Free Software: Download for interactive Algebra, Geometry, Statistics, and Calculus.
Free Videos: MathTV has videos by topic or textbook. Videos of the same problem are presented by different people so if one doesn’t take, maybe another one will.
Full Math Courses:
]]>This Illuminations Lesson provides printable mat, fish and clam figures for doing number comparisons using greater than, less than and equal symbols.
ABCya Online Game for comparing numbers.
Worksheets for comparing numbers.
]]>It's a valid question educators are asking themselves more and more in the face of dismal test scores and the fact that many high school students have never gotten a good handle on basic math, the stuff we use everyday.
Yet despite the fact that most people will not go into engineering or use higher math in any way in their lives, it continues to be forced on students and hinders many from pursuing their true interests. Why should passing Algebra be such a determining factor in a person's future? (Homeschoolers, too, are affected by this either through state graduation requirements, or via college entrance exams).
This math teacher for the middle- and high school grades doesn't think that makes any sense:
"Too many of the nation’s 14-year-olds inadvertently narrow their college options before they’ve even settled into high school." The reason? They can't get past the "gateway" course, Algebra 1. Many fail a second time and never become proficient at it.
Some argue that the reason students are unsuccessful is because the course as it is generally taught is badly designed.
"Algebra, as we teach it, is a death march through endless disconnected technical tools and tips, out of context....The course has no big ideas, no direction, no purpose. And when was the last time you had to graph inequalities?"
Others argue Algebra must be taught because it is necessary for developing critical thinking. But there are alternatives, both to the way Algebra is taught, (relate it more to real-world problem solving), and for the course itself (rigorous courses in statistics and probability, or philosophy and logic to develop reasoning and analytical skills).
And wouldn't we be serving students better if we made sure they had a good, working understanding of things like decimals and percents, how to measure square footage, budgeting and personal finance, how loan amortization works -- you know, some practical stuff?
I don't think Algebra and other higher maths should be scrapped completely from high school courses of study. I just don't understand why these courses are a requirement for everyone. Hopefully, this will change.
In the meantime, the next time one of your kids asks, "Why do I have to learn this?" or "What am I ever going to use this for?", here are two sites that attempt to provide some answers:
]]>I came across some interesting math puzzles over at MathPickle. My daughter likes doing dot-to-dot puzzles, but on this site, you'll find these puzzles with a twist: you use a ruler and only connect the pairs of dots that are a specific distance apart. For example, the puzzle starts out looking like this:
And if you connect all the dots properly, (in this case, those that are 5 cm apart), you'll get a design like this:
This is one of the easy ones. There are 23 progressively harder puzzles, with solutions, that you can download from this page, along with other puzzles.
We've also been looking at symmetry:
and the way 3D shapes look when they are flattened out (noting things like the number of surface faces, sides, and vertices):
You can find both of these free downloads at Nyla's Crafty Teaching.
]]>A free Algebra 1 course just came out this week, this one from Curriki, a provider of free, K-12 open educational resources. 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. If you haven't been on Curriki yet, click on over. They've got lots of good stuff and the site's pretty easy to search.
Another free Algebra 1 course just newly released is available from SAS Curriculum Pathways. Also modular with video lessons.
Other free high school math courses are available at:
Came across this site called, Mangahigh, which offers free math challenges and games for ages 7 - 16. You can choose games by grade level and also by topic. For example, you can have a child in the elementary grades play a game that reviews certain times tables, or gives them word problems to solve. High schoolers can play games to help them brush up on Algebra, Geometry, and Statistics and Probability.
The games are designed to adapt in difficulty to the ability of each student, and they correspond to Common Core standards. You can assign exercises to a student, and then when your child logs in, he or she will be shown all the math games scheduled for them.
There's a lot to see, so it's best to just go to the site and try it out yourself. You can register for a free account by putting in a "school" name (some states require homeschoolers to register as "schools" with their own names), and upload an Excel spreadsheet of "students". It's easy to do.
]]>Here’s a sample:
A standard checker board has 64 squares alternating black and red on an 8 by 8 grid. How many of the 1296 rectangles on a checkerboard contain more than one red square?