Exercise of the Week #14

We’re back with some English after a little break!

Last Week’s Answers

This puzzle is known as a ‘cryptarithm’. Aside from being quite fun, they appear frequently in mathematical challenges, so it’s good to have a method for tackling them.

To solve cryptarithms, we must use logic and pattern recognition.

Note that F can only be 1, as even if both of the hundreds’ digits were 9, the maximum value for F would be 1.

Then, we know that the digits in the tens and ones’ positions must be the same.

We can use trial and error of values to work out the rest: given there are only a small number of possibilities, it shouldn’t take a prohibitively long time.

If we use these tips, we will find out that SUN + FUN = FANS represents 874 + 174 = 1048.

Exercise of the Week #13

A little maths exercise worthy of the heatwave.

Last Week’s Answers

In this sequence question, we mustn’t rule anything out as a red herring or distractor: part of the pattern that is only included to draw your attention from what’s actually happening.

In this puzzle, everything is important!

1. Between each pattern, we can see that the star is changing. If there is a shape which appears throughout the sequence, look to see whether it is rotating, reflecting, growing/shrinking or changing. In this sequence, the star is gaining a point each time.

2. The spiral is also consistent - as above, we need to think about how. In this sequence, the spiral is flipping each time.

3. There’s also something going on with these dots. It’s not enough to say that there’s one extra dot; that’s too surface level. Instead, notice that the new dot remains, and each time, another dot is added. Another way to think about this is that the black dot moves and is replaced by a grey dot each time.

Three things to spot! How many did you find?

Exercise of the Week #12

A little non-verbal reasoning this week.

Last Week’s Answers

This sort of question could easily be a three or four marker, but the post specified that we’re looking for five marks. That’s going to be tricky, as there isn’t much to go on.

Previously, we looked at the PETEE structure for answering extended-response comprehension questions. We could apply this here. Alternatively, we can use a similar structure: PEE.

• Point: Make your point. Ensure it is relevant to the question and that you can back it up.

• Evidence: Give evidence from the text. You’ll likely need a quote or two.

• Explanation: Explain how your evidence supports your point. Don’t just repeat the point.

Looking at the text, we know Alice is ‘very tired’ of sitting there and ‘having nothing to do’. While her sister has brought a book to read, she has not, and instead takes occasional glimpses at what her sister is reading. She dismisses the book, because it ‘had no pictures or conversations in it’, stating that a book without these does not have a use.

We need to thread these ideas together into an argument. We could say:

Alice is uninterested in literature. She paid little attention to her sister's book, occasionally glancing over but dismissing it because it 'had no pictures or conversations'. She believes that a book only has a 'use' if it contains these. This suggests that Alice believes books are only worth reading if there are pictures to look at and plenty of dialogue. Furthermore, this idea of books having a 'use' suggests she views things in a regimented and practical way: an object is only worthwhile if it has a clear 'use'. 

In this response, we make our point, give evidence, and explain the evidence. We haven’t referenced any ‘techniques’ as there aren’t any that really stand out.

Exercise of the Week #11

English this week, returning with a longer-response comprehension question. Let’s make this a five-marker. Be as detailed as possible to avoid losing marks.

Last Week’s Answers

To solve this, we first need to decrypt each dubious doubloon stratagem!

1. Thomas Tew suggests they split the pile into twenty. So, each part is worth one-twentieth. Then he tells the others to take six parts: six twentieths. That would leave eight twentieths for Tew himself! There’s a difference of two twentieths (or one tenth) between what he’d be getting and what they’d be getting.

2. John Bowen suggests they each take half of 60% of the treasure. Half of 60% is 30%, and John would get the remainder - 40%! There’s a difference of 10% between his cut and theirs.

3. Bill Teach (that’s Blackbeard, by the way) says they should each take 0.75 of one half of the treasure. We need to convert the half into four-eighths: 0.75 (or three quarters) of this is three eighths. So, Teach would be getting four eighths: a one-eighth difference.

So, Tew would get one tenth more; Bowen would get 10% more, and Teach would get one eighth more. Who’s being most devious?

We need to convert these mixed measures so that we can compare them. Let’s make them all percentages.

We can start with one eighth: Take 100 and divide this by 8: to divide by eight quickly, just halve three times.

100 goes to 50 goes to 25 goes to 12.5. So, one eighth is the same as 12.5%.

Then, we can easily find one tenth as a percentage, as 1/10 is the same as 10/100, which is 10%.

And as 12.5% is bigger than Bowen and Tew’s 10%, Teach is the most devilish, and if his method was followed, he’d be awarded a whopping £12,500,000 more than the others!

Exercise of the Week #10

Maths! Pirates! Avast!

Last Week’s Answers

To solve coded word problems such as this, you need to look for shared letters and letter groups within the words.

Two of our words start with an ‘H’, and two of our numbers start with a ‘1’, which tells us that H = 1.

One of our H words ends with S, and one of our other words begins with an S. We need to find a number that does the same thing – starts with a digit that one of the other numbers ends with.

So, take a look at the numbers which start with 1: one of them ends with a 5, but none of the other letter codes start with a 5. One of them ends with a 6, and another letter code starts with a 6.

This tells us that S = 6.

We can now see that SCOUR = 67342, as it is the only word starting with an S.

Let’s review what we now know.

1 = H

2 = R

3 = O

4 = U

5 = ?

6 = S

7 = C

8 = ?

9 = ?

We have a few letters in 13485: HOU??

The only word which matches this opening is HOUND. Which tells us that 8 = N and 5 = D.

We can work out our final letter by looking at 14896: HUNTS. This tells us that 9 = T.

We now know all the letters and the questions are easy to solve.

1 = H

2 = R

3 = O

4 = U

5 = D

6 = S

7 = C

8 = N

9 = T

1. SCOUR = 67342

2. CRUSH = 72461

3. STUNTS = 694896

4. 13961396 = HOTSHOTS

5. CONSTRUCTORS = 738692479326

Kudos if you found the last one without mistake.

Exercise of the Week #9

Verbal reasoning today - another coded problem, this time using letters and numbers.

Last Week’s Answers

We are looking for descriptive devices, so include figurative language (similes and metaphors) and literal description (five senses). Pay attention to your language and try to use alliteration or assonance to add texture to your sentences. Try to be original: avoid cliched similes or lists of adjectives. Remember that one striking word is better than a heap of mediocre vocabulary.

The marking of descriptive work is subjective. It is not as straightforward as ‘They’ve used a simile, gain a mark.’ Instead, the examiner must use their discretion to assess a response’s originality and skill. Here is a putative mark scheme which makes use of ‘bands’ to group responses. Each band would then have a range of marks in such a way that the strongest responses achieve the higher end of a higher band. Note that the only way to achieve zero marks is by doing something spectacularly silly.

Here is an example response. Try to mark down some of the devices that have been used and consider which band you would place it in.

The soothing winter sun ran its silken fingers along the water’s edge. Gently, the stream gurgled secrets to those who stopped to listen: nowadays, mostly rabbits, cautiously exposing their button-noses from the bracken. In bygone times, the waterwheel would groan and clatter; now, it rests motionless, embracing the termites and rot. The symphony of forest sounds is silent now.

Exercise of the Week #8

Back to English today; a model answer with guidance will be posted underneath next week’s exercise.

Last Week’s Answers

1)      I imagine you’ve written a cube! There are some much more complicated alternatives (such as a triangular bipyramid) but we don’t need to know about those for now…

2)      A cylinder is comprised of a rectangle and two circles. You can remember this as a cylinder is shaped like a tin: there’s a circular lid and base and a rectangular label.

3)      A pentagonal prism is effectively a 3D pentagon. The definition of a prism is a shape which has the same cross-section at all points along one of its dimensions. In this case, that shape is a pentagon.

4)      A square-based pyramid has five vertices and eight edges. Multiply these to get 40. A trickier one to spot.

5)      A sphere, when cut, will always give a circle. This circle is largest when you cut directly through the centre.

6)      Which 2D shape has angles which add to 720? You may know that this is a hexagon. Each of the internal angles of a hexagon is 120, and there are 6. As it looks like a rectangle from above, we must be thinking of a hexagonal prism.

Exercise of the Week #7

We’re back after a holiday hiatus, returning with some shapes.

Last Week’s Answers

Use the key to work out which lines must be added to the empty cells. If you got it correct, they should spell out ‘HAPPY EASTER’!

Exercise of the Week #6

And we’re back with a springtime reasoning puzzle! Good luck.

Last Week’s Answers

These sorts of questions can be nefarious. Let’s look at a means of tackling them.

PETEE

You might’ve been taught this at school. If not, PETEE stands for: Point, Evidence, Technique, Effect, Emphasis.

Your point is the statement you intend to answer the question with. Everything that follows exists to support the point you have to make.

Evidence is a quote from the text. Pick a good one - ideally something rich that supports your point. When I say ‘rich’, I mean that there should be enough in your quote to explain further. If it has a technique, that’s perfect, as we should then discuss what technique is being used. Don’t worry if it isn’t obvious, like clear use of alliteration or a striking simile: keep your eye out for things like repetition, or juxtaposition.

We next explain what effect this technique has on the reader. What does it make you think of? How does it make you feel?

Finally, we explain why the writer uses the technique - what are they trying to emphasise?

If you use this structure, you’re giving yourself the best chance of securing all marks on these (typically thee to four mark) questions.

Model Answer

The writer creates the impression of a superfically commonplace man with a sinister depth to him. They use a striking simile to describe his glance, which falls ‘as trenchant and heavy as an axe.’ This suggests that he is capable of doing harm with merely a stern look. This emphasises that although he is ‘commonplace’, ‘ordinary’ and ‘usual’, something darker lurks within.

There are other techniques you might prefer to reference.

• There is juxtaposition between his ‘commonplace’ features and ‘remarkably cold’ eyes.

• There is much use of parenthesis in the final lines, drawing attention to the narrator’s unease: he can’t make sense of this man.

• There is sibilance in the last line: ‘expression of his lips, something stealthy - a smile - not a smile’. Perhaps the writer is drawing attention to the character’s snakelike qualities.

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’Heart of Darkness’ is for older readers: something to pick up when you are in years 9 or 10. It’s a critique of European colonialism in Africa in the 19th and 20th centuries. The character described in the extract is the unnamed ‘manager’ running an ivory trade on the Congo River.

Exercise of the Week #5

Today we’re looking at a comprehension question - the variety that’ll make a student shudder!

As ever, answers will be posted next week. Good luck.

Last Week’s Answers

You’ve probably seen this (or a similar problem) on your travels. It’s one of those where you might get a different answer each time you look at it, so is really an exercise is being systematic. This means employing a strategy to ensure you don’t miss anything.

So, let’s start with the smallest triangles.

As we can see, there are sixteen of those.

Don’t forget to include the triangles which are vertical - they still count.

Next up, we need to look at the medium-sized triangles: each one is comprised of four smaller triangles…

So now we’re up to 20. But wait, there are more of those medium-sized triangles! Don’t forget that they can overlap.

And now we’ve got 23. There aren’t any other medium-sized triangles, so now we’re onto the large triangles. These are made up of nine smaller triangles, and there are three of them.

So that’s it; surely we’ve got them all?

Not quite! There’s one more we mustn’t forget:

That’s right, the entire diagram is itself a triangle, which brings our final total to 27 triangles.

If you counted the little fellow on the top right as a triangle too, that’s fine! Technically he’s a square-based pyramid, or perhaps a triangle-based pyramid, so either way, that’s another four triangles there… eep.

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If you’d like a more in-depth explanation of this problem, as well as a complicated formula for working out the number of triangles in a similar diagram of any size, YouTuber ‘MindYourDecisions’ has an excellent video solution.

Exercise of the Week #4

Back to mathematics this week, as we look at a viral problem you might’ve encountered before.

Last Week’s Answers

Agnetha says she is a liar. It’s a bit of a paradox, but if you’re a liar, you can’t admit to it, because then you’d be telling the truth. Likewise, you can’t be a truth-teller and call yourself a liar.

However, Agnetha actually says: ‘Brillon and I are also liars’. This statement can be a lie if Brillon is a truth-teller.

As we know Brillon is a truth-teller, what Charlton says is true, making him a truth-teller too!

So:

Agnetha is a liar, and the others are both truth-tellers.

Exercise of the Week #3

This week we’ll be looking at a logic puzzle. These can appear in verbal reasoning or critical thinking examinations, requiring students to analyse each statement and consider whether an assumption can hold true for the rest of it. Let’s take a peek:

Last Week’s Answers

Here are a few quick techniques you can employ to write persuasively:

1. Lists of three: Humans are quite attached to the number three. Using lists of three allows you to pack in more information, to be more memorable, and to be more persuasive. (See?)

2. Emotive language: Pull on the heart strings! If you’re writing to persuade someone to donate to a dog’s shelter, describe those poor, lonely puppies, who cry for a good home!

3. Rhetorical questions: Haven’t you heard of them? These are questions which don’t require an answer. Would you like to live on the streets?

Here’s an example paragraph arguing against the points given:

It’s not unusual to dislike homework. Many students consider it to be a burden, further constraining their already busy schedules so that they are unable to spend time doing the things they enjoy. However, homework that is well-thought-out by teachers needn’t take students hours to complete: in fact, studies have shown that most students spend no longer than an hour each evening on homework. That’s not too honorous, is it? 

It has been said that scrapping homework would provide teachers more time to prepare their lessons and mark other work. While this is true, it is the job of teachers to both set and mark homework (as well as planning lessons.) If marking homework is getting in the way of other responsibilities, teachers have not been considering the homework they are setting thoroughly. A well-considered task shouldn’t take forever to mark!

Finally, there is an argument that homework does more harm than good. But, as the school day is limited, these extra weekly hours are vital to ensure topics covered in class are consolidated. Not everything can happen in the classroom. 

Well-considered homework prepares students to learn independently, engages them, and provides critical consolidation of work already covered. If it were banned, I am certain we would see grades fall and achievements dwindle. Would you want that for your child?

Exercise of the Week #2

Our exercise this week looks at persuasive writing. A model answer will be provided next week. Good luck!

Last Week’s Answers

1) She starts with £3000, but the King takes half, leaving her with £1500. However, he then gives her £500, which means she now has £2000. Then, the Chancellor takes one quarter, which is £500, leaving her with £1500.

2) Work backwards: if the Chancellor takes one quarter, Fred must have had £4000 after receiving money from the King. Now, take away the £500 he received. He’s got £3500. This is after the King has taken half - so we need to double it, giving us our final answer of £7000.

3) If I don’t have any money, the King takes half of 0, which is 0. But then, he gives us £500 regardless! But don’t get too excited: the Chancellor takes one quarter (£125) leaving us with £375. Nice!

4) If I’m left with £4125, this is after one quarter has been taken. So, £4125 is three-quarters of what we had before the Chancellor took his tithe. We need to know the full amount, four-quarters, so we can divide by 3 and multiply by 4 to give £5500. This is after the King gave us £500, so take that off: £5000. Finally, this is after the King halved our money, so double it: £10,000.

Exercise of the Week #1

I’ve decided to launch exercise of the week!

Be sure to check back each week for a new mathematics, English or reasoning exercise.

The answers will be posted next time. Good luck!