## A Multiplication Based Logic Puzzle

### 360 What Can You Do With Fraction Circles?

360 has more factors than any previous number. 240 and 336 held the previous record of 20 factors for each of them. How many factors do you think 360 has? Scroll down to the end of the post to find out.

360 can be evenly divided by every number from one to ten except seven, so it was a good number for the ancients to choose when they divided the circle into 360 degrees.

Magnetic Deluxe Fraction Circles

Set of 51-Magnetic shapes representing halves, thirds, fourths, fifths, sixths, eighths, tenths, twelfths and 1 whole. Grades 1-6. Circle is 3.75″ in diameter.
MSRP: 10.99
Clearance Price: \$2.50

SUPER BLOWOUT PRICE: \$0.75
10703

I received an email from Educators’ Outlet.com today informing me of their Winter Sale with Super Blowouts until February 28, 2015. One of the items they have on sale while supplies last is the magnetic deluxe fraction circles. Its incredible super blowout price is just 75 cents plus shipping. (Order more than one item for better shipping value.) The 51 piece set consists of 1 whole circle as well as circles divided into 2 halves, 3 thirds, 4 fourths, 5 fifths, 6 sixths, 8 eighths, 10 tenths, and 12 twelves.

Don’t limit yourself to the ad’s recommendation for just grades 1-6. They can be used for more than introducing students to fractions!

Areas of Parallelograms, Trapezoids, and Circles

The picture above shows what happens when the circle is divided into twelve equal wedges, and the wedges are arranged into something that resembles a parallelogram. This idea can be so easily duplicated with these fraction circles without any cutting. A somewhat similar looking figure can be just as easily made with thirds, fourths, fifths, sixth, eighths, or tenths. Here are some good questions to ask:

1. What happens to the top and bottom of the shape when the number of wedges increases?
2. Sometimes the resulting shape will look like a trapezoid, and sometimes it looks more like a parallelogram. Why does that happen?

In any event, we can calculate the area of the resulting shape. Let’s call the length of the bottom of the shape b1 and the length of the top b2. The area of the resulting shape is calculated: A = ½ · (b1 + b2) · h. Since b1 + b2 = 2πr, and the height equals the radius, we can write our formula for the area of a circle as A = ½ · 2πr · r = πr².

This exercise demonstrates that the area of rectangles, parallelograms, trapezoids, and circles are all related!

Introduction to Pie Charts

Pie charts are a great way to display data when we want to look at percentages of a whole. If you use fraction circles, you are limited to using only to certain percentages, but they can still make a good introduction to the subject. To make the pie chart work either the total of all the degrees will have to equal 360 or the total of all the per cents will have to equal 100:

After a brief introduction using the fraction circles, try Kids Zone Create a Graph.

Art and Mathematics

The fraction circle shapes can be used just as tangram shapes to create artwork. A couple designs can be found at fraction-art and fraction-circle-art. Adding rectangular fraction pieces will increase the possibilities.

Exploring Perimeter and Introducing Radians in Trigonometry

The perimeter of each fraction circle piece can be calculated. If the r = 1, the circumference of the circle is 2π, and we can see an important relationship between the degrees and the perimeter of each piece.

What experiences have YOU had with circle fractions? Did you find them frustrating or enlightening? Personally, I like them very much, but I wish they had also been cut into ninths.

The interior angles of every convex quadrilateral also total 360 degrees.

Here is all the factoring information about 360:

• 360 is a composite number.
• Prime factorization: 360 = 2 x 2 x 2 x 3 x 3 x 5, which can be written 360 = 2³·3²·5
• The exponents in the prime factorization are 3, 2 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 x 3 x 2 = 24. Therefore 360 has exactly 24 factors.
• Factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
• Factor pairs: 360 = 1 x 360, 2 x 180, 3 x 120, 4 x 90, 5 x 72, 6 x 60, 8 x 45, 9 x 40, 10 x 36, 12 x 30, 15 x 24 or 18 x 20
• Taking the factor pair with the largest square number factor, we get √360 = (√10)(√36) = 6√10 ≈ 18.974

#### Comments on: "360 What Can You Do With Fraction Circles?" (3)

1. We have some of those fraction circles (though ours aren’t magnetic, and I am a tad sad now that I can’t use my refrigerator for these things), and I love, love, love them! They come in handy for explaining tons of things (like you said, way beyond the 6th grade), and they’re just fun to play with. 😀

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2. abyssbrain said:

I prefer to use radians instead of degrees for angles especially in calculus.

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3. ivasallay said:

I love radians. There’s no reason to wait until trig or calculus to introduce them. Fraction circles can make them a natural introduction.

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