Thinking Math@TheCulture #4

Thinking [email protected] #4

JC Mathematics, Mathematics

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be foundย here.


This is a question from 1993 Paper 1.

The positive integers, starting at 1, are grouped into sets containing 1, 2, 4, 8, \ldots integers, as indicated below, so that the number of integers in each set after the first is twice the number of integers in the previous set.

\{ 1 \}, \{ 2, 3 \}, \{ 4, 5, 6, 7 \}, \{ 8, 9, 10, 11, 12, 13, 14, 15ย \}, \ldots

(i) Write down the expressions, in terms of r for

(a) the number of integers in the r^{th} set,

(b) the first integer in the r^{th} set,

(c) the last integer in the r^{th} set.

(ii) Given that the integer 1,000,000 occurs in the r^{th} set, find the integer value of r.

(iii) The sum of all the integers in the 20^{th} set is denoted by S, and the sum of all the integers in all of the first 20 sets is denoted by T. Show that S may be expressed as 2^{18}(3 \times 2^{19} - 1).

Hence, evaluate \frac{T}{S}, correct to 4 decimal places.

 

Trick to squaring numbers

JC Mathematics, Secondary Math

Many students go wow when I evaluate workings, without a GC. I’m not showing off, but it is because I don’t really carry a calculator with me. haha. So some students do ask me how I evaluate the square of numbers so quickly. I thought, in light of A-level’s coming, I should share something interesting.

Firstly, we trace back to a formula that we saw in primary school.

(a+b)^2 = a^x + 2ab + b^2

This formula is really going to be the core of us solving any square of numbers.

Next, we just need to split the number that you want to square effectively. So consider,

63 = 60 +3
63^3 = (60+3)^2 = 60^2 + 2(60)(3) + 3^2

Some students will ask why not use (a-b)^2 instead. This is plausible, but we usually are better with addition than multiplication hah.

Arithmetic Question #4

JC Mathematics, Secondary Math

Here is an interesting question to test students if they know who to evaluate their square roots.

\sqrt{(-8)^2}

Ans: 8

First of all, students should know that this is positive square root so the resultant answer should be positive. ๐Ÿ™‚ Arithmetically speaking, we need to value the parentheses first to get \sqrt{64}.

Arithmetic Problem #7
Arithmetic Problem #6
Arithmetic Problem #5
Arithmetic Problem #4
Arithmetic Problem #3
Arithmetic Problem #2
Arithmetic Problem #1

Display of arithmetic skills

JC Mathematics

This is a video of an arithmetic challenge that took place in China in 2015. It is China against Japan, and the title is called the best brain of China. I shared a video on arithmetic previously, and a student mentioned that this only possible after years of training. So here is some kids showing us how math is done!

As you observe from the video, the participants are really young, as young as 9 years old! And the mental arithmetic skills exhibited here is truly insane. I shared this video to show students how “cool” is it to be able to do mental calculations really fast. At the same time, I hope to motivate students to brush up on their arithmetic skills.

From my experience, students with strong arithmetic backgrounds, are able to be more meticulous with their work. They can check their own work efficiently. Furthermore, it removes over-reliance on calculator. It pains me to see students do 2-6 with calculator. Just a few days back, one students got a question wrong and realised she calculated it correctly but keyed in wrongly in calculator. She trusted the calculator and went ahead. This is really a waste, and I notice this in many students.

If I do have kids, I will hope to train their arithmetic skills up. And that will begin by putting the calculator away.

Problem Solving Question #2

JC Mathematics

These series of questions target on students’ abilities to solve it. They are common questions used in job interviews and interviewers are more interested in learning how the interviewee derive the answer, instead of the answer itself. So do feel free to share your method of solving the problem. Remember, we are interested in how you solve it. ๐Ÿ™‚

A frog is in the foot of a 100m well. Each time the frog jumps, it moves up by 2m and then down by 1m. How many jumps are required for the frog to get out of the well?

Ans: 99