Trivia Testing with Math

How long a line can the average pencil draw?  According to my page-a-day calendar, a line that is 35 miles long.  Also, according to the same calendar entry, that same average pencil can write approximately 45,000 words.

Interesting trivia!  I decided to use this information to do a quick calculation of the average length of line needed per letter written, expecting it to be something in the 1/2 cm to 1 cm range.

I came up with an interesting result.

Here are the calculations I did:

1. I converted 35 miles to kilometres:   35 miles x 8/5 km/mile = 56 km.
2. I converted 56 km to metres:  56 km x 1000 m/km = 56,000 m
3. I converted 45,000 words to letters, based on the old typing standard of 5 letters per word:  45,000 words x 5 letters/word = 225,000 letters
4. I calculated how many metres per letter:  56,000m / 225,000 letters = 0.249 m/letter
5. I converted 0.249 m/letter to cm/letter:  0.249 m/letter x 100cm/m = 24.9 cm/letter; let's round that to 25 cm.

So that means that each letter uses a line 25 cm in length to draw it.  That, if we go back to imperial measurements (feet/yards/miles) is almost 10 inches.  10 inches or 25 centimetres - those are BIG letters!

Conclusion:  Either 35 miles is wrong, or 45,000 words is wrong, or my calculations are wrong.  I make mistakes when I do calculations.  So I redid the math - twice.  Still the same answer.  So at this point I'm saying one of those two pieces of trivia quoted is very wrong.

Another answer to "Why math?":    So we can confirm or contradict information we're given as fact!

Spirograph - Art and Math

I saw Spirograph for sale when I was Christmas shopping with my husband.  It reminded of when we had Spirograph as kids and had fun drawing the colourful, circular patterns.  I told him I needed it for Christmas as it would be a great thing to do with math students.  He obliged and I got Spirograph for Christmas.  Just like being a kid again!

A little later we had a discussion about why I considered Spirograph of value when working with kids and math.  I explained how patterns are very mathematical.  It was a hard sell.  But today my (adult) daughter and I were playing with it.  We also spent some time figuring out the math behind the patterns.

The number of teeth on each ring and each wheel are listed.  The ratio of those numbers determines the pattern.  How many points will the star have?  Will it look like a star or will it have loops at each point?

The greatest common factor (GCF) and the lowest common multiple (LCM) between the teeth on the ring and the wheel were needed to figure out how many points there would be.

The drawing to the left has 7-pointed stars. The ring we used to draw this had 105 teeth and the wheel had 45 teeth.  (Each colour star was made from a different hole in the same wheel.)   

45 = 15 x 3; 
105 = 15 x 7.  
The GCF is 15.  
The LCM is 15 x 3 x 7.  

As the 45-tooth wheel travels along the 105-tooth ring, every 45 teeth along the ring, it has completed one rotation.  The ticks on the circle to the left mark every 15 teeth.  The number 1 is 45 teeth after 0.  That marks the first rotation of the wheel.  The number 2,  45 teeth later, marks the end of the second rotation.  And so on, until we get to the end of the 7th rotation, which matches with 0 again.

The triangle shaped drawing is from a wheel that had exactly one-third the number of teeth that the ring has.

If you can get hold of Spirograph, try it out, experiment! See if you can figure out why it makes the pattern it makes!  And have fun making Art with Math!

The Cartesian Plane - where does its name come from?

I learned something from the Grade 6 text book the other day. It's so good to know that we can always learn new things. I love it best when I learn things from young people, but, as math person, it was kind of fun to learn from the Grade 6 math textbook too.

We graph things using what we call the Cartesian plane. It's an x-y grid where the x-axis is horizontal and the y-axis is vertical, and points have an x- and y-coordinate, so (3, 5) would be a point 3 units to the right of the centre "origin" and 5 units up from it.

But where did that name, the Cartesian plane, come from?

Well, there was this mathematician named Descartes. That I knew, but I never that the made the connection Cartesian Plane is named after him. It is! Rene Descartes was his name, he was a French mathematician, was born in 1596 and died in 1650. As with many learned folks back then, he did not restrict himself to one discipline. As well as being a mathematician he was a philosopher and scientist. For the math world, he was the person who developed the grid system that we now call the Cartesian plan.

Thank you M Descartes!

Bird Math

My son volunteers at the Ottawa Valley Wild Bird Care Centre.  They look after injured and sick birds that are brought in by the public.  The goal is to rehabilitate the birds and release them back into the wild. 

Here is a great example of real math in real life:

A colleague of his had to give a bird some medicine. She knew the dosage required per kg of bird mass, the mass of the bird, and the size (in mg of medicine) of the tablet she was going to use.

The dosage was 20 mg per kg body mass.
The bird's mass was 361 g.
The tablet contained 50 mg of the medicine.

What fraction of the tablet is required for this bird?

There is always more than one way to solve a math problem.  We can look at two options here.

Option 1:  Using Unit Conversion

One nice way to work with problems like this that use different units, is to use the units to help you figure out when to multiply and when to divide.  Or better yet, set it all up as multiplication.  Remember that we can "cross out" numbers that appear on the top and the bottom of our multiplication equation;  we can do the same with units.  This is also a good way to convert from one set of units to another.  For example, if we want to convert 15 km/hr to m/s (metres per second), here's what we can do:

So, coming back from that minor diversion, we can do the same for the sick bird.  We need to know what fraction of a tablet he needs to be given.  Note that is body mass is in grams, so we need to convert that to kg.

Here is the calculation:

Now, I come from a family of 5 children and 2 parents, so we grew up trying to divide pies and other goodies into 7 pieces; and I learned something from that - 1/7 is a difficult fraction to work with in practice and pies!  In the bird care case, settling for 1/8 or 1/6 of a tablet might be a reasonable solution. Not so for pieces of pie!

Option 2: Using Equivalent Ratios

Using equivalent ratios is also great way to solve many problems.  The idea is that we can pair up things that are equivalent and set them equal to each other.  Not so far off from the method above, but it breaks up the work into smaller steps, and since all our brains work differently, what makes it easy for us might be different from one person to the next.

For our quick diversion to understand equivalent ratios, let's look at a pancake batter recipe that requires 2 eggs and 3 cups of flour.  We want to make as much pancake batter for our hungry guests as we can, and we have 5 eggs.  We can set up a ratio of requirements for the big recipe to those of the regular recipe

Rearranging, and crossing out the "unit", "eggs" we get:

So for our bird awaiting his medicine, first we need to convert his mass to kg to match the dosage requirement which is in mg per kg body mass.

 Then we need to use the ratio of mg of medicine to kg of body mass to calculate how many mg of medicine he needs.

 Next we use the ratio of mg of medicine in each tablet to calculate what fraction of a tablet the bird needs.

Same answer as in the first style of calculation!  That's a good thing.  Multiple ways of solving problems should always give the same answer if we do it correctly.  And in fact solving a problem a second way is an excellent way to check that we haven't made a mistake.

I hope the birds at the Wild Bird Care Centre appreciate all the math that goes into their care!