# Starter of the Week 17: WIFI password

This is the clue to solve to gain a 9-digit WIFI password. What is the password?

Teacher notes

Level of difficulty: Accessible for HL students

Syllabus knowledge required: A knowledge of odd and even functions and their link to integration. Students who are struggling can be shown the graph of the function in the integral.

A pdf of the question and solution can be downloaded from here.

# Starter of the Week 16: Stairway to Heaven

Starting with the numbers: 2,4,6,8 fill in the bottom grid in any order. Work out the box above by adding the two numbers below.

Here are some possible questions to explore:

1. What is the biggest possible total you can make in the top row?
2. Create a hypothesis for a strategy that always gives the biggest possible total.
3. Prove your result by considering integer numbers π, π, π, π
4. When the ladder is now 5 blocks high, what is the biggest possible total for the numbers 2,4,6,8,10?
5. Using letters π, π, π, π, π explain how you know you have found the largest total.
6. Explore different heights of ladders. What is the pattern that would help you predict in terms of π, π, π, π … a ladder 8 blocks high?

Teacher notes:

Level of difficulty: Accessible for all students

Syllabus knowledge required: None β but would fit well with a lesson just before or just after introducing the binomial expansion and Pascalβs triangle.

You can download the pdf of the problem and the solution here.

## Starter of the Week 15: Hollow Squares

Starter of the Week 15: Hollow Squares

A hollow square was a battle formation used in the 1800s.Β  This consisted of a hollow square inside a larger square where the generals could safely command their troops.

Here we can see that 12 men can be arranged in a hollow square formation (12 = 4 squared – 2 squared).

How many other numbers of men between 10 and 20 can be arranged in hollow squares?

Can you make a hypothesis about which numbers of men canβt be arranged in hollow squares?

Teacher notes:

Level of difficulty:Β  The first part accessible for all students, the proof part accessible to Analysis students.

Syllabus knowledge required:Β  None β but would fit well with ideas of proof or as an example of a more proof based IA (possibly combined with computing).Β  The last part requires some knowledge about ideas of proof.

You can donwnload the solution here

## Starter of the Week 14: On reflection

Starter of the Week 14: On reflection

What is the equation formed whenΒ  f(x) is subject to the following transformations:

(a)Β Β Β Β Β Β  Reflection in the line y = 2

(b)Β Β Β Β Β Β  Reflection in the line y = x+2

(c)Β Β Β Β Β Β  Reflection in the line y = x+c

Teacher notes:

Level of difficulty:Β  Accessible for strong SL and HL students

Syllabus knowledge required:Β  Quadratics and function notation

This would be a good starter when doing transformations and inverses β students could also explore how this can be plotted on Geogebra and think about possible extensions to create an exploration topic.

## Week 13 Penny for Thoughts

We make a model of a penny farthing bicycle shown above such that the small circle centred at A with radius, rΒ  and the large circle centred as B with radius, RΒ  both share a tangent with theΒ  x axis andΒ  y axis and also share a tangent at point C.

Find the value of R/r.

Teacher notes:

Level of difficulty:Β  Accessible for HL students

Syllabus knowledge required:Β  Pythagoras or similar shapes.

This would be a good starter when doing geometry β students could also explore how this can be plotted on Geogebra and think about possible extensions to create an exploration topic.

## Starter of the Week: 12 – Time Dilation

This starter is as follows:

One of the most incredible discoveries in human history has been that time is relative rather than absolute.Β  This means that all clocks β digital, mechanical and biological tick at a different speed for someone in motion relative to someone at rest.

For someone travelling at v m/s, the time they experience passing, T_0, compared to a stationary observer, T is given by:

(a)Β Β Β Β Β Β Β  A pilot brings a stop-watch onto a plane.Β  He flies at a constant speed of 300 m/s for 24 hours as measured on his watch.Β  How much more time has passed for a stationary observer on the ground?Β  [note: try this on your GDC first then try Wolfram Alpha].

(b)Β Β Β Β Β Β  The Parker Solar Probe (above) is the fastest ever man-made object.Β  It has recorded speeds of 148,000 m/s as it accelerated due to the Sunβs gravity.Β  If this speed was maintained for 10 years, how much time would have passed for a stationary observer on the Earth?

(c)Β Β Β Β Β Β Β  A rocket propelled by a solar sail is regarded as feasible technology and would be able to reach speeds of 30,000,000 m/s.Β  If this was sent on a 100 year mission how much time would have passed for a stationary observer on the Earth?

## Week 11: A \$1 million maths problem (Goldbach)

This week’s problem gives you a chance to become famous (and potentially rich)!Β  German mathematician Christian Goldbach (above) in 1742 proposed the following conjecture:

βEvery even integer greater than 2 can be written as the sum of 2 prime numbers.β

Over 250 years later no-one has been able to prove it.Β  Anyone who can prove it get \$1 million and would enter the mathematical history books!Β  Can you find make a start by finding all the solutions to following?

Teacher notes:

Level of difficulty:Β  Accessible for all students

Syllabus knowledge required:Β  None.

This would be a good starter to support a lesson on proof

## Week 10: As Smart as Pythagoras?

This week’s problem allows you to follow in the footsteps of Pythagoras!Β  Have a look at the diagram below.Β  Can you use this to discover Pythagoras’ theorem?

Teacher notes:

Level of difficulty:Β  Accessible for all students (with some potential support).

Syllabus knowledge required:Β  None.

This would be a good starter to support a lesson on proof or on Pythagoras/trig.

## Week 9: Disappearing Fractions

Here is the problem for this week:

(a)Β  What are the values of the following products?

(b) Hence find the following product in terms of n,Β  (n an integer and at least 2).

Teacher notes:

Level of difficulty:Β  First part accessible for all students.Β  Second part accessible for HL students and strong SL.

Syllabus knowledge required:Β  Manipulation of fractions (prior knowledge).

This would be a good starter to support non-calculator methods

## Week 8: Hunting Integers

Here is the problem:

Teacher notes:

Level of difficulty:Β  The first parts of this are accessible for all.Β  The last part would be accessible for strong HL students.

Syllabus knowledge required:Β  None.

This is a good introduction to the need to check that all solutions are found!Β  It is likely that students will miss lots of answers.Β  This also demonstrates more formal mathematical techniques for proof