Category: Algebra

Posted in Algebra, Mathematics education

When is a math problem a problem?

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One of the main objectives of mathematics education is for students to acquire mathematical habits of mind. One of the ways of achieving this objective is to engage students in problem solving tasks. What is a problem solving task? And when is a math problem a problem and not an exercise?

What  is a problem solving task?
A problem solving task refers to a task requiring a solution or answer, the strategy for finding such is still unknown to the solver. The solver still has to think of a strategy. For example, if the task,

If x^2 - 7 = 18, what is x^2 - 9 equal to?

is given before the lesson on solving equation, then clearly it is a problem to the students. However, if this is given after the lesson on solving equation and students have been exposed to a problem similar in structure, then it cease to be a problem for the students because they have been taught a procedure for solving it. All the students need to do is to practice the algorithm to get the answer.

What is a good math problem?

The ideal math problem for teaching mathematics through problem solving is one that can be solved using the students’ previously learned concepts/skills but can still be solved more efficiently using a new algorithm or new concept that they will be learning later in the lesson. If the example above is given before the lesson about the properties of equality, the students can still solve this by their knowledge of the concept of subtraction and the meaning of the equal sign even if they have not been taught the properties of equality or solving quadratic equation (Most teachers I give this question to will plunge right away to solving for x. They always have a good laugh when they realize as they solve the problem that they don’t even have to do it. They say, “ah, … habit”.)

Given enough time, a Year 7 student can solve this problem with this reasoning: If I take away 7 from x^2 and gives me 18 then if I take away a bigger number from x^2 it should give me something less than 18. Because 9 is 2 more than 7 then x^2 - 9 should be 2 less than 18. This is 16.

Why use problem solving as context to teach mathematics?

You may ask why let the students go through all these when we there is a shorter way. Why not teach them first the properties of equality so it would be easier for them to solve this problem? All they need to do is to subtract 2 from both sides of the equal sign and this will yield x^2 - 9 = 16. True. But teaching mathematics is not only about teaching students how to get an answer or find the shortest way of getting an answer. Teaching mathematics is about building a powerful form of mathematical knowledge. Mathematical knowledge is powerful when it is deeply understood, when concepts are connected with other concepts. In the example above, the problem has given the students the opportunity to use their understanding of the concept of subtraction and equality in a problem that one will later solve without even being conscious of the operation that is involved. Yet, it is precisely equations like these that they need to learn to construct in order to represent problems usually presented in words. These expressions should therefore be meaningful. Translating phrases to sentences will not be enough develop this skill. Every opportunity need to be taken to make algebraic expressions meaningful to students especially in beginning algebra course. More importantly, teaching mathematics is not also only about acquiring mathematical knowledge but more about acquiring the thinking skills and disposition for solving problems and problem posing. This can only happen when they are engage in these kind of activities. For sample lesson, read how to teach the properties of equality through problem solving.

Finally, and I know teachers already know this but I’m going to say it just the same. Not all ‘word problems’ are problems. If a teacher solves a problem in the class and then gives a similar ‘problem’ changing only the situation or the given ‘numbers’ but not the structure of the problem or some of the condition then the latter is no longer a problem but an exercise for practicing a particular solution to a ‘problem’. It may still be a problem of course to those students who did not understand the teacher’s solution. I’m not saying that this is not a good practice, I am just saying that this is not problem solving but an exercise.

You may also want to read How to Solve It: Modern Heuristicsto further develop your problem solving skills.

Posted in Algebra

Summation problems with solutions

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Series  is the sum of the terms of a sequence. The operation of getting this sum is called summation. A series can be represented in a compact form, called summation notation or sigma notation. The Greek capital letter sigma, , is used to indicate a sum. In this post, Mr. Frederick Koh shares an example of a task involving summation which he believes can help advanced level students appreciate summation notation. Mr. Koh says:

Many students get thoroughly irked by the sigma notation and reconciling it properly with the notion of mathematical sequences, so I shall put forth a detailed worked example to demonstrate how one must be flexible in obtaining the required solutions.

summation of series

Frederick Koh is a teacher residing in Singapore who specialises in teaching the A level maths curriculum. He has accumulated more than a decade of tutoring experience and loves to share his passion for mathematics on his personal site www.whitegroupmaths.com .

Mr. Koh is also the author of the following posts:

A challenging complex number problem with solution

Differentiation in parametric context with solution

Posted in Algebra, Geogebra

Making connections: Square of a sum

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One of the ways to connect concepts in mathematics is to make use of the same representation to teach mathematics. Let me take for an example the visual representation of the square of a sum, (a+b)^2 = a^2+2ab+b^2 . This concept is usually ‘concretized’ using algebra tiles. However, if you have facility for computer technology I would recommend using the GeoGebra applet below: [iframe https://math4teaching.com/wp-content/uploads/2011/09/square_of_a_sum.html 650 550]

Suggested tasks:

1. You can show the grid (click view to do that) to teach Grade 3 pupils about area. You can change the dimensions of the rectangles and squares by dragging D.

2. For older students you can give this task (don’t show grid but you can show lengths then drag D): The square AGHC is dissected into rectangles and squares.If the sides of square AEFD is 2 units and that of square DIJC is 3 units,

a) calculate the area of the other rectangles and square?

b) write two numerical expressions representing two ways of getting the area of the big square.

4. Having worked with numerical expressions, students will be ready to work with variables: If AD is x and DC is y, find two expressions for the area of the square AGHC. This of course leads to the identity (x+y)^2 = x^2 + 2xy + y^2 . The popular FOIL method should only come after this activity. For the record, I’m not a fan of this method.

5. This representation can also be used to teach how to calculate expressions such as 0.75 x 0.75 +0.25 x 0.75 x 2 + 0.25 x 0.25. This is one way to help students appreciate an application of the square of a binomial.

6. You can also use this figure to teach the idea of function: If you drag D along AC, how will a change in the distance of AD affect the area of the rectangles and squares?

a) Create a table comparing the area of the quadrilaterals as the side AD increases from 0 to 5 units.

b) Do as in a) but this time compare the perimeters.

c) Express the area/perimeter of each quadrilateral as a function of the length of AD.

I will write about square of a difference in the next post.

Posted in Algebra, Geogebra, GeoGebra worksheets

Teaching mathematics with GeoGebra

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GeoGebra constructions are great and it’s fun ‘watching’ them especially if you know the mathematics they are demonstrating. If you don’t and most students don’t then we have a bit of a problem. Even if the applet demonstrates the mathematics to students I don’t think there’ll be much learning there. No one learns mathematics by watching. We know that ‘mathematics is not a spectator sport’. You have to play the game. In Geogebra  and Mathematics I proposed that if GeoGebra is to help students in learning mathematics with meaning and understanding, then students should know how to use it. But these GeoGebra tools will be most useful only to students if they know the mathematics behind the tools and why they work and behave like that.  And so we teach students the mathematics first? Where’s the fun in that?

I believe (Translation: I’ve yet to do a study if my theory works) that it is possible to learn mathematics and the tools of GeoGebra at the same time .  I will be sharing in this blog GeoGebra activities where students learn to use GeoGebra as they learn mathematics. The main objective is of course to learn mathematics. The learning of GeoGebra is secondary. I will start with the most basic of mathematics and the most basic of the tools in GeoGebra: points, lines, and the coordinates system.

The lesson includes four GeoGebra activities:

Activity 1 – What are coordinates of points? Read the introduction about coordinates system here.

Activity 2 – What are the coordinates of points under reflection in x and y axes?

Activity 3 – How to describe sets of points algebraically Part 1?

Activity 4 – How to describe sets of points algebraically Part 2? (under construction)