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!

Strategies for basic addition and subtraction facts

In this video I show several strategies for learning the basic addition and subtraction facts:

• finding all the possible sums for a certain number; e.g. for 6, these would be 0 + 6, 1 + 5, 2 + 4, and 3 + 3;
• writing out all the fact families where the sum is a certain number;
• the 9 "trick"; 9 + 7 is the same as 10 + 6 (9 wants so badly to be 10 so that it "steals" one from 7);
• doubles and the accompanying addition facts;
• number rainbows and how to use them to practice subtraction;
• a structured drill for addition facts.
  

Enjoy!



Strategies for Addition and Subtraction Facts - Video

Unit studies in math - States by the Numbers

There is a book for
each state of the US.

Each book has 38 pages
(80 activities)

Price per book:
$2.99 PDF download

Free Sample (PDF):
North Dakota

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Bundle of 50 workbooks $19.99
($0.40 per state)

Your order is secure
Shopper help at Kagi

Recently I've had the pleasure of adding to MathMammoth.com a series of workbooks titled States by the Numbers, by Make It Real Learning company.

States by the Numbers is a series of workbooks where the problems in each workbook are based on data from the Census Bureau's 2008 Statistical Abstract of the United States.

These workbooks offer a real-data math adventure across the United States! You can use the workbook for your state along with other resources as a unit study about your own state, or your other favorite states. Meanwhile the students will also learn and practice place values, rounding, estimation, fractions and percentages.

There are 50 workbooks in the series — one for each state. Each workbook includes basic instruction and 80 practice problems. The problems can be used on grades 3-7; probably the best fit is for grades 4-6.

Since the activities teach both mathematics and social studies, many teachers and families enjoy using the workbooks to reinforce mathematics across the curriculum. Although the activities may be effectively used in a formal teaching setting, they are designed specifically for the independent learner.

Please see this page for a list of mathematical objectives in each of the books.

Math Teachers At Play - Sep 4, 2009 edition

Welcome to another edition of "Math Teachers At Play"! It is again a very engaging and interesting assortment of posts, so feel free to stay a while and relax. Thank you for everyone who submitted!

We'll start out in the early years of kindergarten. What happens when a research mathematician goes into a kindergarten class? Something interesting, creative ... and best of all, kids love it. Go check Sequences and Creative Math for Kindergartners.

Next in line, kids in the elementary school (grades 1-4) spend a lot of time with math facts. Consider the fun Joey-Joey math game to help them practice!

We advance to 5th grade. With "Math Buddy" you can explore Fractions as part of life.

"Discrete Ideas" presents The Shortest Path - or mental math multiplication shortcuts for all of us. As a tie-in, I provide a PROOF for one of those shortcuts, which could be used as an exercise in algebra class.

Next, we advance to middle school. Quadrilateral Property Combo Chart shows how the Number Warrior finally untangled the dilemma of explaining the relationships between the quadrilaterals.

Tom from "I Want to Teach Forever" offers us an Erasing Debt Activity with a printable activity sheet. This is based on real-life data (actually offers he's received in the mail), and can get students really engaged in comparing the various options for paying back a debt. For middle school on up.

And lastly, time for miscellaneous math teaching articles.

What is the shape of the Earth? Do you think it's a sphere (a ball)? John Cook from "The Endeavour" answers the question. He also submitted Three algorithms for converting color to grayscale. It actually sounds pretty simple!

Well, now we get off math for a bit. An urban, high school mathematics teacher rambles on about something annoying... but gives us a glimpse to the issues teachers face.
From "Math and Logic Play" we get The Woman, Dog, and Flea riddle.
A complex-sounding question has a surprisingly simple answer.

We end the carnival with something fun. Here's something that most of us have probably not tried... using SKITS to teach math! From "I Want to Teach Forever"...

Thanks for reading - I hope you enjoyed something! Here's where you can submit to the next carnival.