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PROBLEM:

A circle is inscribed in a quadrant of a circle whose radius is 10cm. Find the radius of the inscribed circle.

Share your solution in the comment section below.

Proving the central angle – inscribed angle relationship

When two secants intersect on the circle, it forms an inscribed angle. There are three cases in which the secants may be positioned as shown in the figure below. These positions form different cases for the central angle-inscribed angle relationship. So when you prove the theorem that the measure of the inscribed angle is half the measure of the central angle intercepting the same arc as the…

Six ways to give feedback to students to keep them in the task of learning

Assessment is embedded in instruction in two ways. One is what I call assessment in the service of teaching. The second is what I call assessment in the service of learning. Assessment in the service of teaching refers to the use of assessment information to improve teaching while assessment in the service of learning refers to the use of assessment information in the form of feedback to keep the…

Four equal triangles problem

Challenge yourself with this triangle problem. Squares were constructed around each side of triangle ABC. The free points of the square were connected forming three more triangles. It is claimed that all the four triangle have equal areas. Do they? Show that they do or don’t. Hint: Make a similar problem. Use an isosceles triangle or right triangle for the triangle ABC.

How to orchestrate mathematical and productive class discussion

Show and tell activity (aka lecture method) may work for some but never in a mathematics class. Getting students to explain and ask questions are nice but only when the explanation and the questions are mathematical. Reasoning and justifying are good habits of mind but they are only productive if they are based on mathematical principles.  Explaining, asking questions, and substantiating one’s…

Reasoning abilities and concepts for learning calculus

This post describes foundational reasoning abilities and mathematical knowledge the students need to develop before beginning a course in calculus. Covariational reasoning – this involves recognition of two quantities that are changing together. A student who considers how two quantities in a dynamic situation change together is said to be engaging in covariational reasoning. This is a…

Can students learn math when we teach them algorithms?

It is a bunch of procedures. That’s how people perceive algorithms are. And they are right. Algorithm has been defined as 1) “step-by-step procedures that are carried out routinely”; 2) “a precisely-defined sequence of rules telling how to produce specified output information from given input information in a finite number of steps”. Its no wonder then that teaching its algorithms is perceived as…

Van-Aubel’s Theorem

Problem

In the figure, H is the midpoint of side BC of ΔABC. The points I and J are the intersection of the the diagonals of square ABDE and square ACGF respectively, that is they are the centers of the square. Prove that IH and HJ are congruent and that angle IHJ is a right angle.

Proof

You supply the text :-)

Challenge Problem:

Squares are drawn on the the side of the quadrilateral. K, O, P,…

Teaching the Derivative Function to Grade 10

Most Grade 10 syllabus do not yet include the concept and calculation of derivative. At this level, the study of function which started in Year 7 and Year 8 on linear function is simple extended to investigating other function classes such as polynomial function to which linear and quadratics belong, exponential function and its inverse, rational function etc. There is no mention of derivative.…

Point of tangency – no calculus, please

Problem

Two lines intersect the graph . The secant line intersects the graph at point A and B and the tangent line at point E. You can read the coordinates of A and B from the graph but not E. What are the coordinates of E?

You can use calculus, that is, the derivative but this is a Year 9-10 problem. So, please solve it using the mathematics at these levels.

Happy Solving!!!

Hint: Application…

Puzzle: Minimum time to cross the river

Puzzle

Jose, Carlos, Linda, and Anna need to cross a bridge. It was dark and they only have one torch which they need to use to cross the bridge. Each of them have crossed the bridge before alone.

Jose can run and cross it for one minute.

Carlos can also do it in two minutes.

Linda needs five minutes to cross the bridge while Anna needs ten minutes.

The bridge is narrow and not too strong so…

Application of the Discriminant

The discriminant of a quadratic equation, ax2 + bx + c = 0 is D = b2 – 4ac. If D>0, the quadratic equation has two distinct roots; if D<0, then the equation has no real roots; and, if D=0, the we have two equal roots. Let’s apply it in the following problem.

What is the equation of the line tangent to f(x)=x2 at the point (2,4)?

Solution using the idea of discriminant

The function f(x) and the…

4 Awesome Kids’ Math Games for the Classroom

Photo Courtesy – Official U. S. Navy Page

If you’re a math teacher and are looking for fun ways to motivate your class or lighten the atmosphere, here are some great kids’ math games to help you along. Needless to say, the kids are definitely going to love them!

Fun Kids’ Math Games – Fun Classroom Ideas for Teachers

1. Simple Mental Math Game

Skills – Number sense, computational fluency

Grade…

Algebraic Expressions Divisible by 3

Problem

If a+2 is divisible by 3, which of the following expressions is/are also divisible by 3?

A. 3+2a

B. 5a-2

C. 8+7a

D. 1+5a

Solution

When a number is divisible by 3 then it is a multiple of 3. This means that a+2 is a multiple of 3. Hence, we can write a+2 = 3N, where N is an integer. This further implies that a=3N-2. So to test if an expression in ais divisible by 3, we just substitute 3N-2…

Types of Problem Solving Tasks

The phrase ‘problem solving’ has different meanings in mathematics education. Even its role in mathematics teaching and learning is not clear cut. Some view problem solving as an end in itself. Others see it as starting point for learning. Nevertheless, here are some of the types of problem solving tasks we would see in textbooks and teaching. They are arranged according to cognitive demand.…

When is it Algebra and When is it Arithmetic

In the post Algebra vs Arithmetic, I distinguished between arithmetic and algebra by arguing that it has nothing to do with the use of letters. That algebra is about letters and arithmetic is about numbers is an oversimplified view of algebra and can create misconceptions. Here are additional way you would know if you are teaching algebra or not.

The following excerpts is from Paper 6 of Nuffield…

How to find an equation given the roots

This is what you do when you solve an equation of a quadratic equation like :

Solution:

=> (x-5)(x+2)=0

=> If x-5 = 0, then x = 5; if x+2 = 0, then x =-2.

The solution is x=5 or x=-2. If you solved the equation, these roots are of course the x-intercepts.

If the question is “Find an equation given the roots 5 and -2.”, then all you need to do is to get the product of (x-5)(x+2) and equate to 0.…