Tag: square roots

Posted in Algebra

What exactly is the square root of 36?

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Mathematics is already a difficult subject so let us not make it more difficult to students to make sense of with the confusing rules we tell them. For example, we insist that the square root of 36 is 6 and -6 yet we also insist that ?36 is equal to 6 and not -6. A colleague said that by convention, when you ask ‘what is the  square root of 36?’ the answer is 6 and -6  but when you just write the symbol ‘?36 =?’ the answer must be 6. So, I said,  do you mean that if you just want the positive root, you do not read the symbol ?36 otherwise it will have 2 values? Here’s an excerpt from an article I read recently about the matter. It’s title is “What root do you want to take?” by Derek Ball.

Square root wall clock from Amazon

Teachers and books make the most extraordinary statements about quadratic equations. All quadratic equations have two solutions, they say. How about x2 – 8x + 16 = 0? Does that have two solutions? “Yes”, they say, “it has the repeated solution 4. If you put x = 4 the equation is true.” Fine. “And if you put x=4 …”  Yes, I heard you the first time.

Of course, I can mock, but it is quite useful for some purposes to think of this equation having a ‘repeated solution’, just as it is quite useful for other purposes to think of it as having just one solution. Confused? So you should be. To add to the confusion I shall ask you this question: ‘What is the square root of 9?’ Depending on context you might answer ‘3 or -3’. If I ask you: ‘What is the square root of zero’, what are you to answer: ‘Zero’ or ‘Zero and zero’ or ‘Zero and minus zero’?

Anyway, solving quadratics is where things like ?36 really come into their own. If you want to solve x2 – x – 1 = 0, you can use ‘the formula’ and obtain the solution x = (1 + ?5)/2. What I need to remember when interpreting this solution is that ?5 is positive. Or do I? Why do I? And if every collection of symbols is supposed to represent (at most) one number, what about‘+?5’?

And sooner or later you may want to solve equations like z2 – 4z + 5 = 0 and you may perhaps use the quadratic formula and obtain z = (4 +(?4))/2. Now new questions arise. Am I allowed to write ?(-4) and if I am what is its value? Is it 2i or -2i and why? Perhaps you want to say the answer is obviously 2i. In that case how about a quadratic equation whose solution involves ?(3 – 4i). Is this – 2 + i or is it 2 – i? What I am saying is that we use symbols to help us solve problems. If we use + in front of a square root sign this reminds us that in order to solve the quadratic equation completely we need to remember to take two different values for the square root. Knowing that ?36 means 6 and not -6 does not matter at all, unless we are asked silly questions in pub quizzes or GCSE exams like ‘What is ?36?’ and we are supposed (for some unknown reason) to know that we have to answer 6 and not -6.

Still not convinced? Well, am I allowed to write 3?-27 and is its value allowed to be -3? Its value could be other things too, of course. Does that complicate things? Does that make us think that we can only judge what a symbol means from the context? So, far be it from me to defend textbooks, but perhaps they have some justification for using the square root sign inconsistently. As for fractional powers, they seem to raise exactly the same issues as root signs.The moral of this tale is surely that we move away from asking questions towhich we want a correct answer, so thatwe can say ‘Right’ or ‘Wrong’, and instead solve problems that interest us, talk about mathematics and connect ideas together. And I hope we sometimes get confused, because confusion is often a spur to sorting our ideas out.

Amen to that. For more confusing rules that we give to our students read my post on Mistakes vs. Misconceptions. You may also want to know the more about Algebra Errors.

 

Posted in Algebra, Geometry, Math blogs

Math and Multimedia Carnival #7

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Welcome to the 7th edition of Math and Multimedia blog carnival.

Before we begin Carnival 7, let’s look at some of the trivias about the number seven:

Now, lets start with posts that involve mathematics sans technology.

Guillermo P. Bautista Jr., the organizer of Mathematics and Multimedia Carnival, presents Generating Pythagorean Triples posted at Mathematics and Multimedia, saying, “A simple strategy in generating Pythagorean Triples.”

Mike Dimond presents Squares ending in 5 – Two Digit Numbers posted at Education For All, saying, “Learn how to quickly calculate the square for two digit numbers ending in five. The post goes over how to quickly calculate 75 * 75.”

I also grab the post Numbers and Variables, the first in the series of post on teaching algebra to students in their first year of High School from the blog Learning and Teaching Math.

John Golden presents Math Hombre: Variable and a Problem posted at Math Hombre, saying, “This post tries to give a couple of contexts for middle school or Algebra I development of the concept of variable.”

Let me include on this list my latest post titled  Counting Smileys which shows several solutions to counting problems that are used to introduce variables and algebraic expressions.

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Now, for mathematics with technology:

David Wees presents Is Interactivity in Mathematics Important posted at Professional blog | 21st Century Educator, saying, “This blog post is a discussion of the importance of using interactive tools when teaching mathematics.” This is one way indeed to involve students in the learning.

Alexander Bogomolny presents Fascination with Tessellations posted at CTK Insights. The post presents several Java applets that illustrate various hinged tessellations and ways of inserting hinges into an existing tessellation.

Terrance Banks presents Treasure Hunt Activity posted at So I Teach Math and Coach?, saying, “Review Activity – Treasure Hunt for Algebra”

Gianluigi Filippelli presents Gravity vs height posted at Science Backstage, saying, “The dependance of gravity by height plotted with Scilab”

Tamarah Buckley presents Instant Feedback posted at Infinitely Many Solutions, saying, “My blog focuses on using iPads in a secondary math classroom.”

Pat Ballew presents Microsoft Mathematics is FREE! posted at Pat’sBlog, saying, “Software for every kid, at just the right price…”

Finally, let me share my post on Squares and Square Roots which presents a series of activities for teaching these concepts meaningfully using the free software, GeoGebra.

That concludes this edition. Submit your blog article to the next edition of mathematics and multimedia blog carnival using our carnival submission form. Past posts and future hosts can be found on our blog carnival index page

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Posted in Algebra

Teaching irrational numbers – break it to me gently

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Numbers generally emerged from the practical need to express measurement. From counting numbers to whole numbers, to the set of integers, and to the rational numbers, we have always been able to use numbers to express measures. Up to the set of rational numbers, mathematics is practical, numbers are useful and easy to make sense of. But what about the irrational numbers? You can tell by the name how it shook the rational mind of the early Greeks.


www.wombat.com

Unlike rationals that emerged out of practical need, irrational numbers emerged out of theoretical need of mathematics for logical consistency. It could therefore be a little hard for students to make sense of and hard for teachers to teach. Surds, \pi, and e are not only difficult to work with, they are also difficult to understand conceptually.

It is not surprising that some textbooks, teaching guides, and lesson plans uses the following stunts to introduce irrational numbers:

After discussing how terminating decimal numbers and repeating decimal numbers are rational, you can then announce that the NON-repeating NON-terminating decimal numbers are exactly the IRRATIONAL NUMBERS.

What’s wrong with this? Nothing, except that it doesn’t make sense to students. It assumes that students understand the real number system and that the set of real numbers can be divided into two sets – rational and irrational. But, students have yet to learn these.

Some start with definitions:

Rational numbers are all numbers of the form  \frac{p}{q} where p and q are integers and q \neq 0. Irrational numbers are all the numbers that cannot be expressed in the form of \frac{p}{q} where p and q are integers.

How would we convince a student that there is indeed a number that cannot be expressed as a quotient of two integers or that there is a number that cannot be divided by another number not equal to zero? It’s not a very good idea but even if we tell them that \sqrt{2} is an irrational number, how do we show them that it fits the definition without resorting to indirect proof or proof of impossibility? What I am saying here is it is not pedagogically sound to start with definitions because definitions are already abstraction of the concept. I would say the same for all other mathematical concepts.

Before introducing irrational numbers, students should be given tasks that raises the possibility of the existence of a number other than rational numbers. Another way is to let them realize that the set of rational numbers cannot represent the measures of all line segments. Tasks that would help them get a sense of infinitude of numbers will also help. The idea is to prepare the garden well before planting. Read my post on why I think it is bad practice to teach a mathematical concept via its definition.