Here is the problem:
Teacher notes:
Level of difficulty: The first parts of this are accessible for all. The last part would be accessible for strong HL students.
Syllabus knowledge required: None.
This is a good introduction to the need to check that all solutions are found! It is likely that students will miss lots of answers. This also demonstrates more formal mathematical techniques for proof
You can download the pdf answer here.
Week 7: A Reciprocal Sum
This is a bit of a longer problem – and could be expanded to start a nice investigation. The problem is given below:
Teacher notes:
Level of difficulty: Only HL
Syllabus knowledge required: This mixes a knowledge of graphs, ideas in formal proof and differentiation.
This is a good introduction to more formal mathematical techniques for finding upper/lower bounds for functions – and also in working with generalized functions. It would probably require longer than a usual starter. It would make a nice mini-investigation task and could be combined with using Desmos to draw some relevant graphs.
The pdf solution can be downloaded here.
Week 6: Sum of 3 Cubes
(a) How many of the following numbers in the table can you form as the sum of 3 cubes of integers? (You can use negative integers if you wish).
(b) Research what the solution is for 30 and 33!
(c) Can you find a pattern that describes the numbers that are missing from this list?
Teacher notes:
Level of difficulty: Accessible for all
Syllabus knowledge required: None
This would be a good starter which could either be done when studying proof in Analysis.
You can download a pdf of the solution here.
A shape is formed as follows:
A circle is drawn such that the points A, B, C, D are equally spaced around the circumference. The arcs AB and CD are reflected in the lines AB and CD respectively.
This makes the following shape:
(a) Find the area of the shaded shape when the radius, r, of the original circle is 1.
(b) What is the area of the shaded shape when r = a?
Teacher notes:
Level of difficulty: Accessible for SL and HL students
Syllabus knowledge required: Sectors and segments
This would be a good starter which can be done as a non-calculator exercise.
Solutions:
You can download a pdf of the solution here.
In the following grids, each blank square should be filled with the mean of the two numbers in adjacent squares.
(a) Fill in the grid below.
(b) Find x in the following grid, leaving your answer in terms of a,c:
(c) Find x in the following grid, leaving your answer in terms of a,c:
(d) For a grid of length n with first box a and last box c, make a hypothesis for the value of x, the box adjacent to c in terms of a,c,n.
(e) Check your hypothesis works when a = 10, b = 15, n = 6.
Teacher notes:
Level of difficulty: Accessible for good SL and HL students
Syllabus knowledge required: Basic averages and algebraic manipulation
This could be done when doing mean, median mode.
Solutions:
You can download the solution pdf here.
Can you prove your hypothesis?
Using the numbers 10,9,8,7,6,5,4,3,2,1 in order, the standard arithmetic operations and brackets, can you make the number 2023?
For example of the idea:
This clearly doesn’t make the correct answer – but can you find a combination that does?
Teacher notes:
Level of difficulty: Accessible for all students
Syllabus knowledge required: none.
This would be a good starter for the start of 2023!
Solutions:
One possible solution is:
Can you find any more?
You can download a pdf of this problem here.
Can you find the hidden message?
Teacher notes:
Level of difficulty: Accessible for SL Analysis and all HL students
Syllabus knowledge required: Laws of logs
This would be a good starter in December!
Solutions:
Using the laws of logs we get:
Happy Christmas!
You can download a pdf of this problem here.
Week 1: Square Dance
This puzzle is as follows:
For the unit square shown above with vertices at (0,0), (1,0), (0,1) and (1,1), how many different types of functions can you find that go through at least 2 points?
Teacher notes:
Level of difficulty: Accessible for strong SL and all HL students.
Syllabus knowledge required: Functions and transformations and/or regression techniques.
This would be a good starter which could either be done as a non GDC exercise, or using Desmos regression depending on what syllabus point you intend to support. It would be a useful exercise to support the coursework.
Solutions:
There are clearly an unlimited number of solutions. Here are a few:
You can download a pdf of the problem and the solution here.
