Posted in Algebra

The multiple meanings of letter symbols in algebra- Part 2 of x

In Part 1 of this series of posts about what makes algebra difficult, I discuss the multiple meanings of equal sign learners has deal with to make sense of the subject. With the changing meaning of equal sign and equations comes the changing meaning of the letter symbols.

Teachers would oftentimes introduce algebra by telling their learners that x stands for an unknown number. It is not incorrect but that’s not all. Some teachers also introduce the word variable by saying that x can take any value that’s why x is called a variable. Again, it is not incorrect but that’s not all. I have heard teachers that say that in an equation, the x is an unknown, but in an algebraic expression, the x is a variable because it can take any value. Is it this simple? Let us consider the following example:

variable

In letter A in the figure above, x can take infinite number of values but it is not an expression. It is an equivalence. Is x a variable then? The use of x is actually as a placeholder.  In C, x can take any values so it is a variable. But f(x) is a function so x is called the argument of the function. We also have to be careful when we say that a letter symbol stands for a number (or value) because in the function in C, f does not stands for a value but simply as a name for the function that maps x to 8x +12 as I pointed out in the previous article. Because f(x)=8x+12 represents a function, we further distinguish between the values of x and f(x) as independent and dependent variable.

In letter B, x is known as unknown (pun intended) and students usually learn it so well, they apply it everywhere. I tell you a little story of a Year 7 algebra class I observed. The teacher gave the following problem:

The school library charges 3 pesos if a book is returned a day late. An additional 25 centavos is charged for each succeeding days that a book is not returned. How much will Aldo be charged if he returns a book 2 days late? 3 days late? 4 day later? 5 days late? x days late?

A student has this solution:

function table

When asked how he calculated for x days, the student explained that he only added 0.25 to 4.0. The teacher asked what about x? The students said x is an unknown but since it comes right after 5 so it must be 6.

Related to the multiple meaning of “x” are the algebraic expressions. Students learned during the introduction of algebra that 2x represents an even number and 2x+1 represents and odd number. In Equation B above, we say that 8x+12 = 2x+1. But, 8x+12=4(2x+3) so this means that 4(2x+3)=2x+1. Now, how come than an even number is now equal to an odd number? How would you now explain this to your learners? I will leave this to the readers so not to spoil the fun 🙂

Salman Usiskin has written numerous articles trying to articulate the multiple meanings of equations and letter symbols. Here are some of his ‘equations’. What is the meaning of the letter symbols in each of the following?

identity

In 1) A, L, and W stands for the quantities area, length, and width and have a feel of ‘knowns’; in 2), we say x is unknown; in 3), x is an argument; in 4), n stands for an instance of the generalized arithmetic pattern; and, in 5) x is an argument, y is the value of the function and k is a parameter. It is only in 5) that we have a feel of variability hence we say x is a variable. It has a different feel from 3) where you don’t get a sense of variability hence in this case, x is more of a placeholder.

The multiple meanings of letter symbols is a source of learners difficulty in algebra. Note, however, that this is also what makes algebra a powerful language and thinking tool.

In my next post I will discuss about the dual nature of algebraic objects as source of learners difficulty in algebra.

Posted in Algebra

Tough Algebra Questions about Equations and Expressions

Here are some questions your students have been wanting to ask you in your algebra class. Daniel Chazan and Michal Yerushalmy in their article On Appreciating the Cognitive Complexity of School Algebra posed these questions about equivalence of equations , solving equations, and equivalence of expressions for us teachers to ponder upon.

function_notationHow will you answer the following questions? What explanation will you give to the students?

Continue reading “Tough Algebra Questions about Equations and Expressions”

Posted in Humor

Christmas GeoGebra Applet

candlesHere’s wishing you the choicest blessings of the season. The applet was adapted from the work of Wengler published in GeoGebra Tube. Being an incurable math teacher and blogger I can’t help not to turn this unsuspecting christmas geogebra applet into a mathematical task.

Observe the candles.

1. When is the next candle lighted?

2. On the same coordinate axes, sketch the time vs height graph of each of the four candles?

3. What kind of function does each graph represents?  Write the equation of each function?

3. If the candles burns at the rate of 2 cm per second and all the candles are completely burned after 20 seconds, what is the height of each candle? (Note: These are fast burning candles 🙂 )

[iframe https://math4teaching.com/wp-content/uploads/2012/12/PEACE.html 650 700]

Posted in Algebra, Mathematics education

Develop your ‘Variable Sense’

Number sense refers to a person’s general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems (Burton, 1993; Reys, 1991). Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms (Howden, 1989).” – NCTM

If there’s number sense, there must also be also ‘variable sense’. Number sense is associated with arithmetic and basic to numeracy while variable sense (also called function sense) is associated with algebra. I collected the following set of tasks I believe will develop variable sense. Having a sense or feel of variables helps develop algebraic thinking and functional thinking.

Task 1

What must be true about the numbers in the blanks so that the following equation is always true?

____ + -2 = ____ + -4

Task 2

The following integers are arranged from lowest to highest:

n+1, 2n, n^2.

Do you agree? Explain why.

Task 3

What is the effect of increasing a on the value of x in each of the following equations?

1) x ? a = 0

2) ax = 1

3) ax = a

4) x/a = 1

Reason without solving.

Task 4

Drag the red point. Describe the relationship among the lengths of the line segments in each of the figure below. It would be nice if you can come up with an equation for each.

[iframe https://math4teaching.com/wp-content/uploads/2012/10/meaning_of_variable.html 600 360]

Tasks 1 and 2 are common problems. Task 3 is from a research paper I read more than five years ago. I could not anymore trace the paper and its author. Task 4 is  from Working Mathematically on Teaching Mathematics: Preparing Graduates to Teach Secondary Mathematics by Ann Watson and Liz Bills from the book Constructing Knowledge for Teaching Secondary Mathematics: Tasks to enhance prospective and practicing teacher learning (Mathematics Teacher Education). I just made it dynamic using GeoGebra.