Understanding basic division

Denise has made a good post on the concept of division, which I heartily recommend. She deals with a study where Finnish researchers gave this problem about division and remainders to high school students and pre-service teachers:

• We know that:
  
  498 ÷ 6 = 83.
  
  How could you use this relationship (without using long-division) to discover the answer to:
  
  491 ÷ 6 = ?
  
  [No calculators allowed!]
  

I really like the question. To solve it, you need to TRULY understand what DIVISION and remainders are all about!

Now, let's think about it. Have you ever seen a pattern in division and remainders, like the one below?

20 ÷ 4 = 5
21 ÷ 4 = 5 R1, or 5 1/4
22 ÷ 4 = 5 R2, or 5 2/4
23 ÷ 4 = 5 R3, or 5 3/4
24 ÷ 4 = 6
25 ÷ 4 = 6 R1, or 6 1/4
26 ÷ 4 = 6 R2, or 6 2/4
27 ÷ 4 = 6 R3, or 6 3/4
28 ÷ 4 = 7
29 ÷ 4 = 7 R1, or 7 1/4
30 ÷ 4 = 7 R2, or 7 2/4
31 ÷ 4 = 7 R3, or 7 3/4

Students need to see and do such patterns when they are first learning basic division.

The pattern shows that every fourth number is evenly divisible by 4, and the ones in between have remainders 1, 2, or 3 in order. If the answer is given as a mixed number, the remainder is the numerator.

Back to 498 ÷ 6 = 83. Since 498 is divisible by 6, so is the number just 6 less than 498, or 492. In fact, 492 ÷ 6 = 82, or in other words, the quotient is one less than 83.

This makes sense when thinking of division as, "How many times does it fit?" If 6 fits into 498 exactly 83 times, then it fits into 492 one less time, or 82 times.

Continuing, also 492 − 6 = 486 is divisible by 6, and this time 486 ÷ 6 = 81.

We can now build the pattern from 486 onward until we have 491 on our list:

486 ÷ 6 = 81
487 ÷ 6 = 81 R1 or 81 1/6
488 ÷ 6 = 81 R2 or 81 2/6
489 ÷ 6 = 81 R3 or 81 3/6
490 ÷ 6 = 81 R4 or 81 4/6
491 ÷ 6 = 81 R5 or 81 5/6
492 ÷ 6 = 82

So, 491 ÷ 6 = 81 R5 or 81 5/6. Problem solved.

Review of Mangahigh games

I got an advance notice of a new games website called Mangahigh.com. It's more than just simple math games, though. These games are designed extremely well, both from mathematical and "enjoyment" perspective. Mangahigh is led by a team of mathematicians, educationalists, and games designers so the games feature commercial-quality gameplay.

I wanted to highlight two of the free games here:

Flower Power
This is the one that I liked most... it's addictive! Grow flowers and harvest them to make money. Practice ordering decimals, fractions, and percentages. The game starts with ordering decimals (daisies), and proceeds into fractions (tulips or roses) and percents. Each time you get a full stem, you need to decide whether to pick the flowers to sell (earn money) or to let them be pollinated and thus get more flowers to grow. Grades 3-8.


Save Our Dumb Planet
Defend Earth from deadly meteorites using missiles. A team of dumb scientists are on hand to suggest possible trajectories. Practice drawing lines, quadratic curves, and some harder curves using their equations. The game has many levels, and you can stay at the easier levels if you so wish. Don't listen to the dumb scientists' talk - they mislead you! Grades 8-11 (algebra).

If I understood right, the games that are there now is just the beginning, and there are more to come.

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Lastly, Mangahigh also has a powerful game called Prodigi. This is a math learning engine (or a math practice environment) that has thousands of maths problems with worked solutions and hints that adapt to each student's ability and learning speed. You can try Prodigi for free; however, only on a certain "easy" and set level. To enjoy the full possibilities of Prodigi, you need to subscribe.

Carnival time

"Math Teachers at Play" carnival #15 is posted at mathfuture.wikispaces.com/Math+teachers+at+play. Lots of interesting posts, once again. Go check it out!