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Master the multiplication combinations once and for all

Learn ’em Forwards and Backwards

Current Version 1.5 — May 19, 2016


(Scroll down to read instructions)

Sample Problem

Click image (or title) to start the program.

Learning the multiplication and division combinations has the same relation to mathematics as doing push-ups has to playing a sport.  This isn’t “real mathematics” but it is a necessary skill, and if you get good at it, the rest of mathematics becomes easier.  This program can help you achieve mastery painlessly, let you know how you’re doing along the way, and let you know when you get there.

“Learn ’em Forwards and Backwards”  is based on a variation of the Leitner System that teaches all of the multiplication and division combinations efficiently and allows you to see your progress graphically as you go along.

I wrote the original version of this program while working with a remedial class of middle school and high school students who had experienced repeated failures in trying to learn their basic multiplication combinations over several years.  They entered the class very discouraged, thinking they were simply not good at math.  The program was very popular with them (almost addictive) and they made rapid progress, most of them achieving mastery after several weeks.

This program measures response time on each individual problem and adjusts how many days before asking the same problem again, based on how consistently and how quickly the user is able to give the correct answer.  Commit to doing a short (10 or 15 minute) session every day until you achieve mastery and you will know the basic multiplication combinations, forwards and backwards, for life!


The program runs right in the web page.  There is nothing to download or install.  Just bookmark the site.

  1. Create a user name.  Always log in with the same user name because all of your progress records are saved under this name in your own computer’s local storage.  (I can’t see your results.)  You will have to use the same computer every time to be able to access your records.  Several people can use the program on the same computer, as long as each one has a different user name.  (I have not attempted to implement high security with passwords, etc., so you are on the honor system to respect each other’s data.)
  2. Select the problems you want to practice.  If you are learning the multiplication combinations for the first time you may want to just practice up to the 3’s, or 5’s, etc.  If you have been taught all of the combinations in the past and are using this program for review, select “All.”
  3. Set a time limit for the session that does not seem burdensome: something between 10 and 15 minutes is probably enough for one sitting.  (There is also an option for an open-ended session.)  See how you feel and how many problems you can finish in the time you selected.  The program will keep track and take up in the next session where you left off.  You don’t need to do all the problems at once.
  4. Problems are presented in two forms like this:   3 x 4 = ?,   and   3 x ? = 12.  The second form is equivalent to division.  Presenting division as “finding the missing factor” emphasises the connection between multiplication and division.  These two forms are presented mixed together. That’s why the program is called “Learn ’em Forwards and Backwards.”
  5. Make sure the keyboard cursor (vertical line) is in the input box.  It should be there automatically, but if not, click inside the box to put it there.
  6. When you see the red ?, type your answer in the input box, then type the <Enter> key (…labled the <Return> key on some keyboards.)
  7. If you get the problem right, the computer will display the problem with the answer in blue in place of the question mark.  Type <Enter> to continue.
  8. If you miss a problem, the computer will restate it with the answer in red in place of the question mark.  You will have to retype the correct answer before you can proceed.  This forces you to stop and focus on the right answer.  Once you have given the right answer and you see it in blue, type <Enter> to continue to the next problem.
  9. When you are ready to stop for the day, click the “Report” button at the top of the screen.  This will show you your mastery level and the response time for each problem.  If you need to report your progress to someone (such as a teacher) by email, you can type <Ctrl-a> (to Select All), then <Ctrl-c> to copy what has been selected to your computer’s “Clipboard.”  Now start an email and in the message area type <Ctrl-v) to paste the contents of the clipboard.  (On a Mac, use the Cmd key rather than the Ctrl key.)
  10. The mastery level ranges from 0 (untried), to 5 (mastery).  Advancement depends on getting the correct answer quickly and consistently.  This activity is designed to be done every day.  As your proficiency grows, the interval between repetitions of the same problem becomes greater.
    –Mastery level 0:  The problem has not yet been presented.
    –Mastery level 1:  You either missed the problem or took too long to answer it.  The problem may be asked again on the same day.
    –Mastery level 2:  You answered the problem correctly, but you had to think about it awhile.  The problem won’t be presented again until the next day.
    –Mastery level 3:  You answered quickly, without having to think too long.  (It is possible to go directly from level 0 to 3 if you already know the answer well.)  You won’t see the same problem again for at least 2 days.
    –Mastery level 4:  You have previously achieved level 3 and have again answered correctly and quickly.  You won’t see this problem again for 4 days.
    –Mastery Level 5:  You have previously achieved mastery level 4 and continue to answer correctly and quickly.  The goal is to reach level 5 for all problems.  When you reach level 5 you won’t see the same problem again for 8 days.

Increasing the “asking interval” for problems you have answered correctly does two things:

  1. It keeps you focused on the problems that need the most work, and
  2. It moves the learning to long-term memory, making the memory permanent.

As your mastery level increases for more and more problems, there may be fewer problems available in a session.  This is normal.  Take it as a reward!  Continue every day until you achieve complete mastery … 5‘s across the board.

The sequence in which the problems are presented may seem a little strange.  Rather than go from smaller to larger numbers, we go from easier to harder combinations.  (The 10’s are easier to learn than the 3’s, for instance, so they come earlier.)  The presentation sequence is modelled on the Math Mammoth curriculum by Maria Miller, which is a good choice for homeschooling in the elementary grades.  (Math Mammoth has also added a Prealgebra course, which I recommend as good preparation for our HSC sequence of courses.)

The sequence taught here has a couple of added enhancements.  In addition to the multiplication tables up to 12 x 12, it covers four additional combinations: the perfect squares up to 16 x 16.  Also, division by zero is covered in “missing factor” form:  0 x ? = 1″ is “undefined.”  There is no solution, so the expected answer is “N”.  0 x ? = 0, is also undefined, but in this case any answer will work, so the expected answer is “A”.  These exceptions are not usually covered at this level, but this seems like as good a time as any to introduce these concepts.

“Learn ’em Forwards and Backwards”

(To delete all users and free up the local storage when you no longer want to use this program, click here, but be sure not to do that unless you are really done.  Deletion cannot be undone.)