Math was okay until they threw the alphabet in it.
I have heard, Math was okay until they threw the alphabet in it, so many times from so many adults, who I wish were trying to be witty, but are in fact completely serious. And these are reasonable, intelligent people. Why does the abstraction of a variable or an unknown become so confusing that it creates a huge disconnect?
I’d like to share with you an assignment that I believe will help students transition from the concrete application and properties of Real Numbers to the abstractions we deal with in Algebra and mathematics beyond.
The desired outcomes of this assignment are:
- Improve mathematical literacy by encouraging students to read the mathematical meanings created by the spatial arrangement of numbers and symbols
- Improve their understanding of the Order of Operations and to help student realize that the Order of Operations is not its own topic, compartmentalized, but rather an over-arching understanding of how math is performed.
- Promote abstract thinking about numbers and their properties
- Introduce some concepts that will come into play later in Algebra, like finding the x-intercepts of a quadratic once it is factored
How to Introduce the Assignment
Students must be aware that they will be dealing with abstract ideas and that there are sometimes more than one right answer. Also, a student can be right, but not completely right, they could also be wrong, but not always. By fostering a healthy discussion about these problems you can introduce the idea that in order for something to be mathematically true, it must always be true. If a single circumstance is untrue, then the statement is untrue.
Consider the problem: Given that a and b are real numbers, and the following is true, what do you know about the numbers a and b? a×b=0
A student might say, In this case a and b are both zero.
That is correct in one case, but there are many cases where that is not true. It is true that one of them must be zero.
The First Prompt
Given that a and b are real numbers, and the following statement is true, what can you conclude at the numbers a and b?
a – b = 0
At first have them think and write on their own. Make sure they’re all working, not avoiding this uncomfortable notion.
After a given amount of time (short, maybe one minute), instruct them to talk with two different people. Be clear that the expectation is that they take turns, one person shares, the other listens and responds.
After the time is up, have a whole-class discussion, but avoid being the authority until the discussion is winding down. Only be the authority on the subject to help summarize.
The Second Prompt
With the same properties of the numbers and the statement being true, provide them with the equation:
(a + 5)(b –7) = 0
Conduct conversations as you did with the first, maybe allowing more time for them to talk together as this is a more complicated situation.
The Third Act
Switching gears from properties of variables to applying some properties of real numbers will promote their understanding of when the associative and commutative properties as well as challenge their understanding of how math is written. We are really trying to promote their ability to read and write math and their fact that the spatial arrangement has meaning in math.
Have the students try and add parenthesis, as many as they like, to the following equation, so that it will be true. Have them try it on their own first, then provide a short amount of time for peer discussion.
3•2 – 72 + 5 = 80
When you conduct a whole-class discussion, make sure it’s student lead, your role is as a mediator, not a disseminator of facts.
Instruct students to create a similar problem by making a statement they know is true and removing the parenthesis. For example, they might make up:
8(5 – 3) + 11 = 27
But would only write
8•5 – 3 + 11 = 27
When they’re completed their problem, have them show you. Once everybody has a problem, hand out 3×5 cards. On the front of the card the student will write their problem, without the parenthesis. On the back, they’ll write their name. Have them pass the cards forward and you can distribute them at to another class the following day.
Last Thing for the Lesson:
The idea here is the same as with the previous activity, but we are accessing their knowledge from a different angle. They will be given an expression with parenthesis and be asked if the parenthesis can be removed without changing the value of the expression.
For example: (5 + 4) – 2, or 11 – (4 – 9).
Introduce these problems in the same fashion, with quiet thinking first, then small group discussions, then whole-class discussion.
The homework is critical here because it will challenge students to think and examine how the way in which math is written changes the meaning. It will also force them to think about numbers in a general fashion.
The last thing I’d like to mention is that this could easily be done over two separate days depending on the aptitude of the class you’re teaching.
I hope this is helpful and food for thought.