Tag: variables

Posted in Algebra

Fun with ‘definitions’ in algebra

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WARNING:  use the following definitions with great caution.

  • A number phrase is a mathematical phrase which does not express a complete thought.
  • An arithmetic expression is any grammatically sensible expression made up of numbers and (possibly) arithmetic operations (like addition, division, taking the absolute value, etc). Notice that it only has to be grammatically sensible; an undefined expression like 5/0 is still an arithmetic expression, but something like ‘5)+/7?’ is just nonsense. You can always work out an arithmetic expression to a specific value, unless it’s undefined (in which case you can work that out).
  • An algebraic expression is any grammatically sensible expression made up of any or all of the following:

– specific numbers (called constants);
– letters (or other symbols) standing for numbers (called variables); and
– arithmetic operations.

  • By an algebraic expression in certain variables, we mean an expression that contains only those variables, and by a constant, we mean an algebraic expression that contains no variables at all.
  • A polynomial is an algebraic sum, in which no variables appear in denominators or under radical signs, and all variables that do appear are raised only to positive-integer powers.
  • A monomial is an algebraic expression made up only from any or all of these:

– Constants;
– Variables;
– Multiplication;
– Taking opposites (optional);
– Division by nonzero constants (optional);
– Raising to constant whole exponents (optional).

  • An algebraic expression is made up of the signs and symbols of algebra. These symbols include the Arabic numerals, literal numbers, the signs of operations, and so forth.
I got these definitions from where else, www. Of course we just want to simplify things for students but … . Anyway, just make sure that you don’t start your algebra lessons with definition of terms, be they legitimate or not legitimate.
Posted in Algebra, Geogebra, High school mathematics

Embedding the idea of functions in geometry lessons

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GeoGebra is a great tool to promote a way of thinking and reasoning about shapes. It provides an environment where students can observe and describe the relationships within and among geometric shapes, analyze what changes and what stays the same when shapes are transformed, and make generalizations.

When shapes or objects are transformed or moved, their properties such as location, length, angles, perimeters, and area changes. These properties are quantifiable and may vary with each other. It is therefore possible to design a lesson with GeoGebra which can be used to teach geometry concepts and the concepts of variables and functions. Noticing varying quantities is a pre-requisite skill towards understanding function and using it to model real life situations. Noticing varying quantities is as important as pattern recognition. Below is an example of such activity. I created this worksheet to model the movement of the structure of a collapsible chair which I describe in this Collapsible  chair model.

Show angle CFB then move C. Express angle CFB in terms of ?, the measure of FCB. Show the next angle EFB then move C. Express EFB in terms of ?. Do the same for angle FBG.
[iframe https://math4teaching.com/wp-content/uploads/2011/07/locus_and_function.html 700 400]
Because CFB depends on FCB, the measure of CFB is a function of ?. That is f(?) = 180-2?. Note that the triangle formed is isosceles. Likewise, the measure of angle EFB is a function of ?. We can write this as g(?) = 2?. Let h be the function that defines the relationship between FCB and FBG. So, h(?)=180-?. Of course you would want the students to graph the function. Don’t forget to talk about domain and range. You may also ask students to find a function that relates f and g.

For the geometry use of this worksheet, read the post Problems about Perpendicular Segments. Note that you can also use this to help the students learn about exterior angle theorem.

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.

click link to view source

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, High school mathematics

Properties of equality – do you need them to solve equations?

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Should students learn the properties of equality before we ask them to solve equations in one variable? Would it be too hard for them to solve the equation 2(x + 7) = 4x without knowing the properties of equality?

In the grades, pupils learn to find equivalent ways of expressing a number. For example 8 can be written as 4 + 4, 3 + 5, 4 x 2, 10 – 2. Now, what has the pupils previous experience of expressing numbers in different ways got to do with solving equations in one variable?

Let us take this problem. What value of x will make the statement 2(x-5) = 20 true?. The strategy is to express the terms in equivalent forms.

2(x-5) = 20 can be expressed as 2(x-5) = 2(10).

2(x-5) = 2(10) implies (x – 5) = 10

x-5 = 10 can be expresses as x-5 = 15 – 5. Thus x = 15.

This way of thinking can be used to solve the equation 2(x + 7) = 4x.

2(x+7) = 2(2x)

=>    (x+7) = 2x

=>    x + 7 = x + x

=>    x = 7.

Of course not all equations can be solved by this method efficiently.   So you may asked ‘why not teach them the properties of equality first before asking them to solve equations like these?’  Here are some benefits of asking students to solve equations first before teaching the properties of equality:

1.  It makes students focus on the structure of the equation. Noticing equivalent structure is very useful in doing mathematics.

2.  It makes the equations like 3x = 18, x + 15 = 5, which are used to introduce how the properties are applied, problems for babies.

3. It is easier to do mentally. Try solving equations using the properties of equality mentally so you’ll know what I mean.

4. I hope you also notice that the technique has similarities for proving identities.

So when do we teach the properties of equality? In my opinion, after the students have been exposed to this way of solving and thinking.

Here’s on how to introduce the properties of equality via problem solving.