Points to bring out when we teach algebra with cups and counters

This the fourth post in a series about using cups and counters to introduce concepts in algebra. In my last post I started to describe a teaching sequence. We got to the point where the students were writing expressions for the number of counters in a diagram.

It’s a good idea to mention to the class that if two cups are marked with different variables they may have a different number of counters but they also could have the same number of counters. One misconception among students who are being introduced to algebra is that different variables always stand for different numbers.

As the lesson progresses students may offer different correct answers to a question. They may change the order of addition (3+x or x+3). If this happens I often refer them back to the first example and point out ’3+5′ and ’5+3′  are both correct sums. You might also have students who suggest multiplication as an alternative to repeated addition. For example:

 Repeated addition or multiplication?

Here some students may answer ’a+b+b’ and other might answer ’a+2 x b’ . It’s a good opportunity to remind students of the relationship between multiplication and repeated addition. I usually provide some more examples and ask for both answers. I always leave the multiplication symbol in the answers at in my early algebra lessons as it can mimise confusion. At this stage we’re not concered about simplification so there’s no need to omit the multiplication symbol.

With these points in mind it’s time to start handing out the cups and counters to the class. I’ll describe how I like to do this in my next post in this series.

Introducing variables using cups and counters

This is part 3 of a series of posts  to complement the cups and counters animations.

How can we make a clear distinction between a variable and the quantity that it represents?

One method is to the let the variable represent the number of counters in a cup.

I start by asking students the total number of counters in this diagram:
How many counters are in this diagram?

 

 

 

 

When I ask them how they came up with their answer I end up with this on the board:

The number of counters is 3+5
i.e. the number of counters is 8.

Then I change the graphic and ask the same question again. How many counters are in this new diagram?

 How many counters are in this diagram?

 

 

 

Not so easy this time! I explain that in algebra there are often quantities that we do not know. We often use letters to represent these quantities.

I add an ‘x’ below the cup and explain that when we write ‘x’ below a cup it means that there are ‘x’ counters in the cup.

 How many counters are in this diagram?

 

 

 

I ask again, how many counters are in the whole diagram. We come up with the expression ‘x+7′. I do a lot of reinforcing and explaining at this point, and provide some other examples. (I’ll add a link here for you in a few weeks once I create a relevant animation).

There are some points that are quite important to bring out at this point. I will discuss them in my next post in this series.

Does ‘a’ equal 3 or do we have 3 apples?

I’m writing some posts to explain the rationale behind my cups and counters animations. In my last post I discussed a problem with the ‘apples and bananas’ approach to algebra.

Another common problem that students have is the inability to distinguish between a variable and the quantity it represents. For a student to understand the expression ’3a’ they need to understand:

  1. ‘a’ is a variable which stands for some number.
  2. Whatever number ‘a’ is we are multiplying it by 3.
  3. ’3a’ represents the result when we do this multiplication.

If we tell students that ‘a’ is an apple and ’3a’ represents 3 apples, none of these points are made clear (remembering that we can’t multiply by an apple).  How will they make sense of a sentence like ‘Fred has 3a apples.”? Does Fred have 3 apples? Does Fred have ’3 apples’ apples?

The ‘apples and bananas’ approach is simple and concrete but leads to misconceptions. In my next post I’ll introduce an alternative approach that is simple and concrete and avoids these misconceptions. I didn’t invent the approach but I’ve worked out a teaching sequence and  I will share it with you.

Cups and counters not apples and bananas

I’ve added some animations that use cups and counters to model variables.

I’ll be adding more in coming weeks but I thought I would share with you the rationale behind this approach.

The “apples and bananas” approach equates variables with objects. For example ’3a+4a’  is explained as ”3 apples plus 4 apples” or ’3a+2b’ as “3 apples + 2 bananas”. Although it might lead to the right answer, this approach quickly fails. Consider the expression ‘ab’ . How do you multiply an apple with a banana?

In fact, a variable  must represent a number, not an object. We too often see statements like “Let f be Fred.” rather than “Let f be Fred’s age.” when we mark papers.

In my next post I’ll explain another problem students often have with variables.