PG 28 9 May 2019Blackboard Equation

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Hidden Figures – the blackboard equation

Last week, we featured the film “Hidden Figures”, about the black women who played a crucial role in the space race, and you watched a trailer for the film.

A couple of you mentioned the equation that the young Katherine Johnson was trying to solve on the school blackboard, which you can see in this still from the film.

In case it is not clear, the question is:

Solve this equation for x: x2+6x72x25x3=0.

Some of you might have come across this sort of equation before, or perhaps a simpler version of it. Either way, here is a step-by-step method for solving it, or rather two methods.

Method 1 (Brute force)

Brute force means doing it the hard way. If you want to open a locked door, then the smart way is to find the key or pick the lock, but the brute force way is to just smash through the door. In this case, brute force means testing every possible value of x until you find some solutions.

For x=0,

For x=1,

For x=2,

So, after checking three values of x, we found one solution, i.e., the equation equals zero when x=1. In this case, the brute force approach has delivered a solution, but there are 4 solutions in total and it might take ages to find the others if we followed the brute force approach, particularly if the other values of x are very large or fractions or decimals.

Method 2 (Smart Maths)

If we have an expression involving something like x2+6x7, then we can often factor it into a pair of simpler bracketed terms, which in this case would be x+7x1.

If you multiply out x+7x1, you get x2+7xx7=x2+6x+7, which is what we had to start with.

Similarly, we can rewrite 2x25x3 as2x+1x3.

Therefore, x2+6x72x25x3=0 can be re-written as x+7x12x+1x3=0.

Of course, if the four brackets multiply to make zero, then at least one of the brackets must equal zero, which means x=7, 1, –½ or 3.

What are values for x in this equation? Try the Smart Maths method first, but if you are stuck then try brute force.

x23x+2x2+3x+2=0

There are 4 possible answers, two positive and two negative.

1 mark

1.1 Larger positive solution

Correct Solution: 2

1 mark

1.2 Smaller positive solution

Correct Solution: 1

1 mark

1.3 Less negative solution

Correct Solution: -1

1 mark

1.4 More negative solution

Correct Solution: -2

x23x+2x2+3x+2=x1x2x+1x+2=0

So, x = 1, 2, -1, -2.

You might not have come across this sort of maths in your classroom – it is known as factoring quadratics. This video gives you an idea of how you factor a quadratic.

2. Some Random Philosophy

Each Parallelogram contains challenges about maths, because I know you are keen mathematicians, but it is also important to sometimes stray beyond numbers and geometry and explore other aspects of the world. You should be curious about lots of things. Ultimately, and for unknown reasons, these apparently odd challenges will help you to be a better mathematician.

This part of the Parallelogram is all about a philosophical puzzle called the Trolley Problem. Watch the following short video. There is no question to answer at the end, but the Challenge is to understand the Trolley Problem and realise that sometimes it is hard to decide what is right and what is wrong.

3. Add it up

3 marks

3.1 In Parallelogram #22 (section 3), we described a quick way to add up all the numbers from, say, 1 to 100. Can you remember it?

Show Hint (–1 mark)
–1 mark

Sometimes it’s easier to pair numbers - try starting with the smallest and the biggest.

The sum of all the numbers from 1 to 40 is:

  • 800
  • 810
  • 820
  • 822

If we pair up all the numbers we get 20 pairs and each one totals 41, namely (40 + 1), (39 + 2), (38, 3) and so on. And 20 x 41 = 820.

4. Double trouble

3 marks

4.1 In Parallelogram #27, we covered Moore’s Law and the power of doubling, so here is a doubling question. You have a large piece of paper, which is 0.1 mm thick, and you can fold it over and over again as many times as you like. Each time you fold the paper, it doubles in thickness. How many times do you have to fold the paper in order for it to become so thick that it will reach the Moon? (The distance from the Earth to the Moon is 384,000 Km.)

Show Hint (–1 mark)
–1 mark

If you double a number, say 1, ten times, then how much bigger does it get? Use this value as a stepping stone to solving the whole problem.

  • 42
  • 42,000
  • 42,000,000
  • 42,000,000,000

This video clip from Chris Seber (at MathMeeting.com) explains how to work out the answer:

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge... if not, then maybe next week.
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • The next Parallelogram is next week, at 3pm on Thursday.
  • Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon.