Parallelogram 26 Level 5 1 Mar 2024Evolution Rap

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteau 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.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Semi-circle and squares

2 marks

1.1 What's the total area of these two squares?

Correct Solution: 256

Show Hint (–1 mark)
–1 mark

We are not given the dimensions of the squares, so the answer must be independent of the size of the two squares. So, imagine that the squares are of equal size.

We are not given the dimensions of the squares, so the answer must be independent of the size of the two squares.

So, imagine that the squares are of equal size. Then we have two squares with diagonals equal to the radius of 16. The side of each square (by Pythagoras) must be 12×162=128.

Therefore, the area of each square must be 128, and the total area for both squares must be 256.

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 You are given that 2786 × 231 = 643566. What is the value of 643566 / 27.86

  • 23100
  • 2310
  • 23
  • 23.1
  • 2.31
  • (Not answered)

643556 ÷ 2786 = 231, so 643566 ÷ 27.86 = 231 × 100 = 23100.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 What is the maximum number of pieces of shape T which can be placed within the 5 × 5 grid shown, without overlapping, and with their edges along the lines of the grid?

  • 3
  • 4
  • 5
  • 6
  • 7
  • (Not answered)
Show Hint (–1 mark)
–1 mark

There is a mathsy way to approach this problem, but you can also draw a quick sketch and move forward with trial and error.

Show Hint (–1 mark)
–1 mark

What options can you exclude? 7 tiles is impossible, because 7 × 4 > 25.

And you can ignore 3 tiles, because it is fairly easy to place 4 tiles. But can you fit 5 or even 6 tiles?

The diagram shows that it is possible to fit five T shapes in the square. In order to fit six T shapes into the square, exactly one of the 25 squares would be left uncovered, hence at least three corner squares must be covered.

We now label a corner square H or V if it is covered by a T shape which has the top part of the T horizontal or vertical respectively. If all four corner squares are covered then there must be at least two cases of an H corner with an adjacent V corner.

Each such combination produces a non-corner square which cannot be covered, eg the second square from the right on the diagram. If only three corner squares are covered, there must again be at least one H corner with an adjacent V corner and therefore a non-corner square uncovered, which means that it is impossible to fit six T shapes into the square.

4. Intermediate Maths Challenge Problem (UKMT)

5 marks

4.1 Given that it takes a men b hours to paint c square metres of the Forth Bridge, how long would it take d men to paint e square metres of the bridge?

  • abecd
  • abdce
  • abcde
  • acdbe
  • acebd
  • (Not answered)
Show Hint (–2 mark)
–2 mark

I took a different approach to the one in the official solution. The rate (area per painter per hour) will be the same for both scenarios.

Show Hint (–1 mark)
–1 mark

Another approach – again not the official one – is to see whether the equations make any sense. For example, the more painters (d), the shorter the time, but answers the second and fourth options have d on the top of the expression, which implies the that if you increase the number of painters then it will take proportionally longer. So, you can exclude the second and fourth options as sensible answers. Perhaps you can exclude one or two other options.

It takes ab man-hours to paint c square metres of the bridge and hence abc man-hours to paint 1 square metre. So d men will take abcd hours to paint 1 square metre and abecd hours to paint e square metres.

5. Who is the mother?

3 marks

5.1 The four statements below refer to a mother and her four daughters.

One statement is true, three statements are false.

  1. Alice is the mother.
  2. Carol and Ella are both daughters.
  3. Beth is the mother.
  4. One of Alice, Diana or Ella is the mother.

Who is the mother?

Correct Solution: Ella

Show Hint (–1 mark)
–1 mark

If statement (1) is true, then so is (2), so (1) = false.

  • If statement (1) is true, then so is (2), so (1) = false.
  • If statement (2) is true, then (1), (3) or (4) is true, so (2) = false.
  • If statement (3) is true, then so is (2), so (3) = false.
  • If statement (4) is true, then it can be true in 3 ways:
    1. Alice is the mother, but then (1) is also true.
    2. Diana is the mother, bur then (2) is also true.
    3. Ella is the mother, in which case all the others are false.

6. Evolution Rap

No questions, just something interesting to finish with. A rap by Baba Brinkman, which explains the process of evolution by comparing it to the writing of a rap. Performance. Feedback. Revision.

(If you have problems watching the video, right click to open it in a new window)

There will be more next week, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.