Week 5Rocket maths

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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. What is zero factorial?

Have you covered factorials? If not, here is brief introduction.

The factorial of a number is the result of multiplying that particular number by all the numbers smaller than it, all the way down to 1.

For example, (4 factorial) = 4 × 3 × 2 × 1 = 24.

Instead of writing “factorial”, we write “!”.

For example, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.

But, how do you begin to work out the value of 0! ?

This video explores the meaning and value of 0!.

1 mark

1.1. Choose the value of 0!

  • -1
  • 0
  • 1
  • 2

The video explains why 0! = 1.

2 marks

1.2. What the value of 5! ?

Correct Solution: 120

5! = 5 × 4 × 3 × 2 × 1 = 120

2 marks

1.3. Use a calculator to work out 10!

Correct Solution: 3,628,800

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

You can work this out by typing each number into your calculator one by one, or you can type 10 and then press the “!” button.

3 marks

1.4. How many ways can you arrange 7 different objects?

Correct Solution: 5,040

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

The video explains that the number of ways to arrange n objects is the same as n!.

2. October Sky

“October Sky” is a Hollywood film based on the true story of Homer H. Hickam, Jr., a coal miner's son who took up rocketry against his father's wishes and eventually became a NASA engineer.

In this clip, Homer’s headteacher thinks that one of his rockets caused a fire, but Homer uses maths (and physics) to explain that his rocket could not have been responsible.

Homer writes an equation on the blackboard:

s = ½ × a × t2

Crudely, this equation tells you the distance an object falls under the influence of gravity during a given time. This is how you read the equation:

  • “s” is the distance in metres, because the Latin word for space (or distance) is 'spatium'.
  • “a” is the acceleration due to gravity and has a value of 10 m/s2 (although it has a different value in the film, because Americans use different units).
  • “t” is the time in seconds.
2 marks

2.1 If an object is dropped from a cliff edge, how far will it fall in 4 seconds?

  • 4 m
  • 16 m
  • 80 m
  • 20 m
  • 160 m

s = ½ × a × t2 = ½ × 10 × 42 = 80 m

If a = 10 m/s2, then we can simplify the equation:

s = ½ × a × t2 = ½ × 10 × t2 = 5 × t2

3 marks

2.2 If an object is dropped from a cliff edge, how long will it take to fall 20 m?

  • 2 s
  • 4 s
  • 10 s
  • 100 s

There are 2 ways to solve this problem.

  1. Plug numbers into the basic question and solve it:

s=5×t2
=> 20=5×t2

  1. Re-arrange the equation, so that t is the subject of the equation:

t=s5
=> t=205
=> t=4=2 seconds.

1 mark

2.3 What caused the fire?

  • Homer’s rocket
  • The headteacher’s rocket
  • A bolt of lightning
  • A match
  • An aeronautical flare

3. Junior Maths Challenge Problem

3 marks

3.1 How many different possibilities are there for the side lengths of an isosceles triangle each of whose sides is a whole number of centimetres and whose perimeter is 24 cm?

  • 3
  • 5
  • 7
  • 9
  • 11

A bit of trial and error suggests that triangles with these sides are okay:

  • (7, 7, 10)
  • (8, 8, 8)
  • (9, 9, 6)
  • (10, 10, 4)
  • (11, 11, 2)

So the answer is 5 isosceles triangles are possible.

(6, 6, 12) is not possible, because all three points would be on a line.

(5, 5, 14) is not possible because the two short sides will not connect to the long side.

And, of course, (12, 12, 0) is not possible.

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 Parallelogams.

Cheerio, Simon.