Hi Math – Learn & Discuss Together

Proof: Diameter is the Shortest Curve that bisects circular area

Post author By admin
Post date June 26, 2026
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As shown in the figure below: (please scroll down a bit to see the figure) AB’ is one diameter of the circle, and a blue curve from A to B is supposed to bisect the area of the circle. We see from the figure below:

Length of the blue curve AB is greater than: AE + EB = AE + EB’

which is certainly longer than AB’; AB’ is the diameter.

The blue curve is obvious the focus. (Note when blue curve AB is mentioned, we mean the “curved one”: it curves around, not going straight from A to B.) Besides, point E is where the blue curve intersects with CD; and CD is one diameter.

So we have proved the claim that any curve bisecting the circular area got to be longer than the diameter.

Stay with us for one more minute. Let the construction-proof process be revealed to you, as follows.

Referring back to the figure. Let us start from the circle and the blue curve only (imagine all other lines and points disappear; now we draw them step by step). Connect the two endpoints A, B of the blue curve by a line segment AB. Then draw the diameter CD // AB (i.e. line CD is parallel to line AB).

Take point O (the midpoint of CD), and then passing A and O, let another diameter AB’ be drawn.

For completing the proof , two arguments are required:

(1) The blue curve has at least one intersection point with diameter CD (thinking it: if the blue curve resides at only one side of CD, then that curve cannot divide the circular area evenly into two parts with equal area; therefore, any curves that bisects the circular area must intersects diameter CD). Now suppose the intersection point is E.

(2) B and B’ are symmetric to the diameter CD therefore EB = EB’ (trying justifying it using the property of circles and parallel lines).

The rest of the proof is straightforward (and intuitive).

Tags G 8-10, G-8

Uncategorized

Pairing up with a Perfect Match

Post author By jonah.luo
Post date November 28, 2017
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Math starts from the simple and goes a long way.

In this post, we will start from “pairing up” like 1 + 4 = 2 + 3 = 5 but there is a long way to go so that we learn the sets, working on the sets, pairing-up, observing and developing conditions for a perfect pair-up.

Pairing up for a Perfect Match

Tags english

Math Contests - Problems and Discussion Math Junior (G9 and under)

Sample Questions for Gauss Contests

Post author By jonah.luo
Post date October 14, 2014
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Questions chosen from previous Gauss contests
Gauss contests are organized by the Centre of Education for
Math and Computing, University of Waterloo

Problem 1

In the addition shown, P and Q each represent single digits, and the sum is 1PP7. What is P + Q?

(A) 9 (B) 12 (C) 14 (D) 15 (E) 13

Problem 2

In the right-angled triangle PQR, we have that PQ = QR. The three segments QS, TU and VW are perpendicular to PR, and the segments ST and UV are perpendicular to QR, as shown. What fraction of triangle PQR is shaded?

(A) 3 ⁄ 16 (B) 3 ⁄ 8 (C) 5 ⁄ 16 (D) 5 ⁄ 32 (E) 7 ⁄ 32

Problem 3

A box contains a total of 400 tickets that come in five colors: blue, green, red, yellow, and orange. The ratio of blue to green to red tickets is 1 : 2 : 4. The ratio of green to yellow to orange tickets is 1 : 3 : 6. What is the smallest number of tickets that must be drawn to ensure that at least 50 tickets of the same colour have been selected?

(A) 50 (B) 246 (C) 148 (D) 196 (E) 115

Problem 4

Greg, Charlize, and Azarah run at different but constant speeds. Each pair ran a race on a track that measured 100 m from start to finish. In the first race, when Azarah crossed the finish line, Charlize was 20 m behind. In the second race, when Charlize crossed the finish line, Greg was 10 m behind. In the third race, when Azarah crossed the finish line, how many metres was Greg behind?

(A) 20 (B) 25 (C) 28 (D) 32 (E) 40

Problem 5

In right-angled, isosceles triangle FGH, segment FH = √̅8. Arc FH is part of the circumference of a circle with centre G and radius GH. The area of the shaded region is

(A) π – 2; (B) 4 π – 2 (C) 4 π – (1 ⁄ 2) √̅8 ; (D) 4 π – 4 (E) π – √̅8

Tags G 6-7, G-8

Math for Early Ages (G7 and under)

On the Patterns -Elementary

Post author By jonah.luo
Post date March 26, 2014
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★ On the Patterns

Number Patterns * Numbers, Colors and Stars

Taking a look at the following chart.

1	2	★3	4	5	6	7	8	9	10
11	12	13	★14	15	16	17	18	19	20
21	22	23	24	★25	26	27	28	29	30
31	32	33	34	35	★36	37	38	39	40
41	42	43	44	45	46	★47	48	49	50
51	52	53	54	55	56	57	★58	59	60

The following questions are for everyone! Whether you are students in elementary schools (grade 3 and up), or in middle schools; or just parents who are tutoring your child; here are the challenges!

First, let us look up to the stars:

(1) What pattern is followed for the placement of stars ?

(2) Each number besides the star is increased by ____. (Fill in the blank)

(3) Following this pattern, the three more numbers that are besides and that comes after 58 are: __ , __ , and __.

(4) Now go back to the chart, and let us look at the colored cell (those colored by yellow)

(5) What can you do to follow the yellowed-colored cell? Please color the numbers on the chart by continue the pattern that you discovered.

(6) What are the common features of these yellow cells? Color some new cells, and explain how, by coloring these cells, you have followed and extended the pattern which is already in the chart.

Take a moment to think. You can answer these questions!

Algebra

Power Mad

Post author By jonah.luo
Post date February 24, 2014
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Tags G 8-10, G-8

Geometry /Shapes

Find the Center of a Circle — Do you know how to do it?

Post author By jonah.luo
Post date July 23, 2026
No Comments on Find the Center of a Circle — Do you know how to do it?

Using a compass, you can draw a circle at any place, with any radius.

Now let’s reverse the problem. Given a circle, do you know how to find its center? (Of course, once the circle is found, there shall be no problem at all to tell its diameter, or radius.) You only see the circle itself, there is no explicit indication on where the center is.

In general, you can find the center using any convenient method, including copy-and-paste the circle onto a paper, and then fold it.

In particular (from classical Euclidean geometry), where it’s required to do so with a ruler (with which you are allowed to draw lines and line segments only) and a compass (with which you are allowed to draw circles only).

If you attempt to solve this problem with a ruler and a compass, then you are required to know how to make a perpendicular bisector. Do you know how to do it?

Tags G 6-7, G-8

Numbers

Prime numbers

Post author By jonah.luo
Post date July 23, 2026

Prime numbers are those that have 1 (one) and itself as the only two divisors. Examples of primes are 2, 3, 5, 7, 11. None of 4, 6, 9 is a prime since 4 = 2 × 2, 6 = 2 × 3, and 9 = 3 × 3.

If a number greater than one is not a prime, then it is a composite number, and can be factored into the product of primes — called prime factorization. We have given the prime factorization of 4, 6, 9 as above. For a couple of more examples:

12 = 2 × 2 × 3

36 = 2 × 3 × 3 × 3

28 = 2 × 2 × 7

So all natural numbers are divided into three classes: the number 1, the prime numbers, and the composite numbers.

A bonus point: π, besides representing in a circle, the ratio of circumference to diameter, also stands for a special function related to prime numbers. This is described as follows.

Function π(x) — for integer x, represents the number of primes less than or equal (i.e. not exceeding) x.

For example,

π(2) = 1, π(3) = 2, π(10) = 4, π(20) = 8 etc.
[To find why π(10) = 4, recall the 4 prime numbers not exceeding 10: they are 2,3,5, and 7.]

Tags G 4-6, G 6-7

Math BASICS Numbers

Number Sense – Activity: Tsunami Numbers in the News

Post author By jonah.luo
Post date July 22, 2026

About two decades ago, there was a great Tsunami happening in Asia.
What do you know about the Asian tsunami?

Read through the article first. Use the following numbers to fill in the blanks in the story. Think about which numbers make sense.

500 20 8,000; 2004 110,000 30,000 9.0

A tsunami triggered by a very large earthquake off the coast of the
Indonesian island of Sumatra on December 26, ____, has left
more than 150,000 people dead and millions homeless. Countries hit hardest by the disaster include
Sri Lanka, Indonesia, India, Thailand, and the Maldives. Almost 75% of the deaths occurred in
Indonesia, estimated at ____. Sir Lanka was second highest with about 20% of the estimated deaths, or
______ people lost that day. The rest of the deaths, approximately ____, occurred in the other nine
countries affected by the tsunami.
The ____ foot wall of water, higher than a two-story building,
swallowed entire villages. The tsunami waves were not only very high, they moved at a much faster speed
than normal. These waves were comparable in size to those you see on some of the surfing movies;
but those waves travel at 30 miles an hour, and the tsunami waves
were moving more than fifteen times as fast at ____ miles an hour.
The velocity of the force is what caused the destruction—a massive force that swept away everything in its path.

The earthquake causing this Tsunami was a destructive earthquake measuring ______ on the Richter scale,
the fourth worst earthquake in recorded history. Earthquakes are measured on a Richter scale that has
a range from 0 to 12; a 6.0 on the scale is a pretty bad earthquake.

(Story constructed from January 2005 news reports)

Tags G 4-6, G 6-7

Math for Early Ages (G7 and under) Math Modelling

cut a pizza into 11 equal slices, exactly evenly

Post author By jonah.luo
Post date July 21, 2026
No Comments on cut a pizza into 11 equal slices, exactly evenly

What is the easiest way to cut a pizza into 11 equal slices?

The step-by-step solution is:

Take a wrist watch.
Position the watch hands to noon (12:00 exact), and put in the center of pizza.
Cut in the direction of watch hands.
Advance the watch until next overlapping of its hands…

Tags G 8-10, G-8

Geometry /Shapes

3D objects with 3 views from top, front and side

Post author By jonah.luo
Post date July 18, 2026
No Comments on 3D objects with 3 views from top, front and side

For a 3D objects, given three views to you: one from top, one from front, and one from side, can you imagine what the original 3D objects looks like?

The question is not posed to a mechanic engineer, it would be trivial in that case. The question is raised to get a junior middle student to think a bit.

For a cylinder one of the three views is a circle, and the other two views are rectangles. For a cone one of the views is a circle, and the other two views are triangles. What if the three views given are a circle, a rectangle and a triangle? Can you figure out the original shape?