Parallelogram 31 Year 7 11 Jun 2020Top of the League

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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. Top of the league

Four of the greatest English football teams compete against each other to see who is the best – Manchester City, Liverpool, Derby County and Chelsea. After every team has played every other team in a mini league competition, they finish as follows.

  • Chelsea finished above Manchester City.
  • Liverpool were not third.
  • There were two teams between Manchester City and Derby County.
1 mark

1.1 Who finished first?

  • Manchester City
  • Liverpool
  • Derby County
  • Chelsea
1 mark

1.2 Who finished second?

  • Manchester City
  • Liverpool
  • Derby County
  • Chelsea
1 mark

1.3 Who finished third?

  • Manchester City
  • Liverpool
  • Derby County
  • Chelsea

If “there were two teams between Manchester City and Derby County”, then they must have finished 1st and 4th. And if “Chelsea finished above Manchester City”, then Manchester must have come 4th and Derby 1st. Liverpool and Chelsea must, therefore, have finished 2nd and 3rd, but we know Liverpool did not finish 3rd, so they must have finished 2nd.

2. Junior Maths Challenge Problem (UKMT)

3 marks

2.1 What is the value of (222 + 22) ÷ 2?

  • 111
  • 112
  • 122
  • 133
  • 233

(222 + 22) ÷ 2 = 222 ÷ 2 + 22 ÷ 2 = 111 + 11 = 122.

3. Seeing dots

This is a really interesting video by Jo Boaler. I think it illustrates how we might all see a problem differently, and how there are many ways to solve a problem, but they all lead to the same answer.

2 marks

3.1 Megan visualises the 7 dots as two dots on the sides, and a third dot inside a...

  • Circle
  • Square
  • Triangle
  • Hexagon
  • Bracket

4. Junior Maths Challenge Problem (UKMT)

4 marks

4.1 Last year, at the school where Gill teaches Mathematics, 315 out of the 600 pupils were girls. This year, the number of pupils in the school has increased to 640. The proportion of girls is the same as it was last year.

How many girls are there at the school this year?

  • 339
  • 338
  • 337
  • 336
  • 335

The fraction of girls in the school last year was 315600. By dividing both the numerator and denominator by 3 and then by 5 we see that

315600=105200=2140.

The proportion of girls this year is the same. Therefore, of the 640 pupils in the school this year, the number that are girls is

2140×640=21×16=336.

5. Junior Maths Challenge Problem (UKMT)

4 marks

5.1 Consider the following three statements.

  1. Doubling a positive number always makes it larger.
  2. Squaring a positive number always makes it larger.
  3. Taking the positive square root of a number always makes it smaller.

Which statements are true?

  • All three
  • None
  • Only (1)
  • (1) and (2)
  • (2) and (3)
Show Hint (–1 mark)
–1 mark

Remember some numbers are less than 1. Think about what happens to 12.

Statement (1) is true, because, if x is a positive number, then

2x=x+x>x.

Statement (2) is false. If a positive number which is less than 1 is squared, then the answer is smaller than the original number. For example, with x=12, we have

x2=14<12.

Similarly, if we take the square root of a positive number that is less than 1, then the answer is larger than the original number. For example

14=12>14.

Hence statement (3) is false.

Therefore statement (1) is the only one that is true.

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.