In this, the last post of the Algebra Tiles series, I will be looking at how you can use the tiles to demonstrate various more advanced ideas, such as difference of two squares and completing the square.
In this, the last post of the Algebra Tiles series, I will be looking at how you can use the tiles to demonstrate various more advanced ideas, such as the difference of two squares and completing the square.
Dividing a quadratic expression by a linear factor
Dividing a quadratic expression by a given linear factor is similar to factorising. Doing this is the same as for factorising, you are building a rectangle, but one of the side lengths is now fixed.
Here are three examples of such divisions in increasing complexity:
Dividing a quadratic expression by a linear factor – example 1Dividing a quadratic expression by a linear factor – example 2Dividing a quadratic expression by a linear factor – using zero-sum pairs
Perfect Squares
What is a perfect square? The result of multiplying two identical linear expressions! So we are building squares, not rectangles.
Building a perfect square
You can use the visual representation to discuss the features of a perfect square and how to recognise one when written out in full:
(x + 3)2 = (x + 3)(x + 3) = x2 + 6x + 9
Difference of Two Squares
By showing students a visual representation and using the tiles to demonstrate the addition of zero-sum pairs, students should be able to see the pattern formed when factorising the difference of two squares.
Factorising the difference of two squares
Completing the Square
What do we mean when we talk about ‘completing the square’?
By attempting to build a square, with the given expression, and showing that we may need to add or remove terms to achieve the square, we can make meaning of the term ‘completing the square’. In the second video, I show how you may need to add in zero-sum pairs to complete the square.
Completing the square – a simple exampleCompleting the square – with the use of zero-sum pairs
Extending to Two – Variables
I did mention when talking about collecting like-terms, in Part 2 of this series, that you can model expressions that contain two-variables. In this video I show how to do that:
Extending the model to two variables
Resources
There is a Padlet to accompany this blog series with links to research, virtual manipulatives and other resources.
Exploring solving linear equations using Algebra Tiles
In this post, I will look at how to use Algebra Tiles to solve Linear Equations. There are several ways you can draw diagrams to illustrate this process and also different concrete materials you can use. One of the most common will be using a balance to demonstrate that both sides of an equation must remain the same (in balance) as you manipulate the variables and constants. Using Algebra Tiles, and the zero-sum pair concept can help students understand why we ‘add/subtract the same from both sides’.
Numberless Word Problems
Students have to be comfortable with the notion that the variable an unknown amount. In order to help students I would start with Numberless Word problems (see this link and the Padlet for examples). This allows students to think about the problem before you introduce the numbers.
Here is an example:
Start with a very open scenario:
I have two baskets of apples. Each basket has the same number of apples.
Ask the students what they notice and wonder. There is no problem yet, maybe ask the students what they think the problem might be.
Then add a little more detail to the scenario:
In one of the baskets, the apples are loose. In the other basket, some of the apples are in bags.
What has this added to the problem? what questions might the students have now? What might the problem be?
I now add in some numbers to the scenario:
I have two baskets each with 11 apples. In one basket all the apples are loose. In the other basket, some apples are evenly divided into 3 bags and there are 5 loose apples.
Now you can ask the students how they could write this problem down. Maybe ask them to illustrate it using Algebra Tiles. Here are some possible abstract representations:
3 bags of apples + 5 apples = 11 apples
3b + 5 = 11
There is still no question being asked – ask the students what they think the question is. In this case, the question would probably be:
How many apples are in each of the bags?
Once the problem is set up, you can begin to discuss how to solve the problem.
Solving simple linear equations
In the video below, I show how I would solve the above problem using Algebra Tiles. I have used conventional x rather than b.
Solving a simple linear equation using Algebra Tiles
I really like to annotate what I am doing as I work. Having a surface where students can easily erase errors will encourage them to ‘give it a go’, to try ideas that they are not confident with.
This also allows them to see where the standard algorithms come from. When they put in tiles to make zero-sum pairs and then annotate that the reasoning behind the algorithm for solving equations becomes more apparent.
In this example, I have introduced negative values.
Solving a liner equation with negative values
Make sure you mix it up a little by not having the variables always on the same side of the equals sign!
Linear equations with variables on both sides of the equals sign
In the next few examples, I demonstrate how to use Algebra Tiles to solve linear equations with variables on both sides of the equals sign.
Solving a linear equation with the variable on both sides of the equals sign
Here is the same equation solved in a different way.
Same equation, different method
By encouraging students to solve in more than one way, they can begin to discover which methods are more efficient. Allow the students time to explain their methods to the class to discuss which method works ‘best’ and why. A gallery walk is a great way to do this.
One more example of solving a linear equation.
Another example with the variable on both sides
In the above example, the variable has been written after the constant term. Although there are mathematical conventions for writing expressions, students need to work with examples written in unconventional ways. Often in tests, these sorts of questions are put into ‘throw’ the students.
In Part 4 of this series, I will explore multiplying linear expressions and factorising quadratic expressions.
Resources
There is a Padlet to accompany this blog series with links to research, virtual manipulatives and other resources.
In Part 2 of this series, I will demonstrate how Algebra Tiles can be used to build expressions, model addition and subtraction of expressions, and model substitution into algebraic expressions.
In the previous blog post, I looked at how to make Algebra Tiles and use them to model integer operations.
Building an algebraic expression
If you have used Algebra Tiles to model integer operations you can now introduce the x-tile and the x2-tile.
In the video below I show how to build an expression. It is important that the students recognise the different shapes and colours and what they mean. I would take some time on this stage as you want them to be very familiar and comfortable using the tiles.
Using Algebra Tiles to build an expression
Collecting Like-Terms
Once students can build a simple expression you can move on to more complex expressions and looking how they can be simplified by collecting together like-terms. Students often find it difficult to recognise what is meant by a like-term. By using the tiles it is easy to see which terms are similar by their size and shape. I would start with positive values only and then when students are confident with those introduce negative values, which are the same size and shape but a different colour.
Collecting like-terms
Multi-variable expressions
You can use Algebra Tiles to model expressions that have more than one variable. Below are videos of two ways this can be done. The first is using different coloured tile-sets for each variable. This can get a little messy and I would suggest the students write a key for each variable showing which colour is positive and which negative. The beauty of this method is that the students can really identify what a ‘like-term’ is. They can collect the tiles of the same shape and related colours.
The second method uses only one set of tiles as follows:
How to use Algebra Tiles to illustrate multiple variables
Using the square tiles to represent different variables can help the students to appreciate that each variable represents a different value.
Using Algebra Tiles to represent a multi-variable expression
Addition and Subtraction of Expressions
Once students can collect like-terms together you can move on to addition and subtraction of expressions. Below I use the same expressions with each operation and show how the result is different.
Adding expressions
Addition of expressions
Subtracting expressions
Subtracting multivariable expressions
Using the second method of modelling a multivariable expressions I showed above, I will demonstrate how you can use Algebra Tiles to subtract two expressions.
(Spot the mistake!)
Subtraction of multivariable expressions
Substitution
I have found that students often struggle with substitution. They do not understand what it means. Using Algebra Tiles the students can physically exchange the variable tiles for the unit tiles. The x2-tile is replaced by a square with side length of the given value. This allows the students to build a clear visual representation of what they are doing when they substitute the variable with a given value.
Substitution
Other ideas and suggestions
You may find that students will move away from using the tiles quite quickly and start drawing diagrams or using algebraic notation. I would suggest that you encourage them to use the tiles for the first few examples of each new type of question/activity. Students should feel comfortable to use the tiles whenever they feel the need.
I would also allow students access to the tiles during tests as well. Tests and quizzes can cause students increased stress level. Just knowing that they have the manipulative available can reduce their anxiety.
In Part 3 of this series, I will explore solving simple linear equations.
There is a Padlet to accompany this blog series with links to research, virtual manipulatives and other resources.
I am putting together a series of blog posts demonstrating how to use Algebra Tiles for the new year. As algebra tiles are on my mind, when I saw this Open Middle problem posed in a tweet from Robert Kaplinsky I decided to give it a go using Algebra Tiles.
Algebra Tiles are a great manipulative to use when you introduce algebra to students. They allow the students to explore concepts such as adding and subtracting expressions, factorisation and substitution. There are three different tiles representing the constant, the variable and the variable squared. Each tile is in two colours representing positive and negative values.
The first step was to model what I knew from the given problem. I used a laminated sheet to place the tiles on and wrote what I knew on it.
An Algebra Tile model of the given problem
I arranged the twelve x2-tiles in a rectangle and this gave me my starting point in finding a solution. The arrangement of these tiles was just a random choice.
I noticed that two of the given numbers in the problem limited my choice of the constant in the second factor pair. The -3 in the top factor pair and the -15 in the final expression, restricted my choice of value in the second-factor pair. It had to be +5.
I had 15 negative block that had to be -3 x something – so the something had to +5
I then filled in all the other tiles:
All the tiles laid out in a rectangle
A basic feature of Algebra Tiles is Zero-Sum Pairs. When you have two tiles of the same size and shape, but different colour, they form a zero-sum pair and can be removed. (They cancel each other out)
I formed all the zero-sum pairs of the x-tiles:
The zero-sum pairs
There were three more positive x-tiles than negative x-tiles remaining. as the coefficient of the x-term has to be negative this was not an allowable solution.
My next strategy was to reverse the 3 and the 4 to see if that would leaver me with more negative x-tiles.
The tiles arranged 3 x 4
Again, I could quickly see that there were too many positive x-tiles.
I then started to explore other factors of 12: 6 x 2, 2 x 6
First I arranged the tiles 2 x 6:
Tiles arranged 2 x 6
But there were too many positive x-tiles.
So I switched to 6 x 2:
Tiles arranged 6 x 2
At last! In this arrangement there were more negative x-tiles! I arranged the x-tiles in zero-sum pairs to see how many positive x-tiles were remaining.
Zero-sum pairs from the 6 x 2 arrangement
There were 8 negative x-tiles remaining so my solution was:
(6x + 5) (2x – 3) = 12x2 – 8x – 15
I thought there could be more than one solution to this problem, there normally is in Open Middle problems. So I tried the final factors: 1 x 12 and 12 x 1.
First I tried a 1 x 12 arrangement:
Tiles arranged 1 x 12
This time I ran out of tiles and room! There were too many positive x-tiles so I move on to my last arrangement, 12 x 1:
Tiles arranged 12 x 1
It was easy to see that there were a lot more negative x-tiles in this arrangement. This gave me the solution:
(12x + 5) (x – 3) = 12x2 – 31x – 15
I had found two solutions to the problem.
For many students, the original problem would look inaccessible. Algebra and the notation used is scary and off-putting. By using the Algebra Tiles and building a visual representation of the problem students have a starting point, a route into the problem. Some students may not need to explore all the possibilities with the tiles. Once they have modelled a couple of possibilities many students will start to make predictions and be able to relate what they are seeing to the notation that is used.
Show the students how the arrangement of the tiles relates to the terms they get when they multiply out the brackets – connecting FOIL and the Area Model.
A nice extension of this problem would be to ask students to explore the possibilities if they were not restricted to the -3.
This and many other insights and suggestions for using Algebra Tiles to deepen students’ understanding of algebra concepts will be explored in my series next year.
Below is a link to the Algebra Tiles Learning Sequence I wrote for AMSI. The blog series will be based around this sequence.
