A Multiplication Based Logic Puzzle

Posts tagged ‘trapezoids’

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

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




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:

Pie Chart Pieces

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.

Perimeter of Fraction Circle Pieces

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

37 and How Many Trapezoids I Can Draw

37 is a prime number. 37 = 1 x 37. Its only factors are 1 and 37. Prime factorization: none.

How do we know that 37 is a prime number? The square root of 37 is an irrational number approximately equal to 6.08. If 37 were not a prime number, then it would be divisible by at least one prime number less than or equal to 6.08. Since 37 is not divisible by 2, 3, or 5, it is a prime number.

37 is never a clue in the FIND THE FACTORS puzzles.

A trapezoid is often defined for young students as a four-sided shape with EXACTLY two parallel sides. Once a person studies higher level math, the definition changes: A trapezoid is a four-sided shape with AT LEAST two parallel sides. How many different kinds of trapezoid can a person draw? It depends on which definition you use. If you use the second definition, you can also include parallelograms, rectangles, rhombuses, and squares. Either definition will allow the standard isosceles trapezoid and several others. But how many? Whichever definition you use, figuring out how many different ones can be drawn is a nice puzzle to solve. This blog post does a nice job explaining the different ones, and it even came up with ones I hadn’t considered!


I guess this is a good time to give my answer for the challenge of how many different trapezoids there are to draw. At the least it’ll provide an answer to people who seek on Google the answer to how many trapezoids there are to draw. In principle there’s an infinite number that can be drawn, of course, but I wanted to cut down the ways that seem to multiply cases without really being different shapes. For example, rotating a trapezoid doesn’t make it new, and just stretching it out longer in one direction or another shouldn’t. And just enlarging or shrinking the whole thing doesn’t change it. So given that, how many kinds of trapezoids do I see?

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