This post will explore multiplying algebraic expressions, the area model, and factorising quadratic expressions.
This post will explore multiplying algebraic expressions, the area model, and factorising quadratic expressions.
Multiplying Expressions
Multiplying algebraic expressions by an integer
Here is an example of how to multiply an algebraic expression by an integer. This may seem like quite a trivial example, but students often struggle with the concept of multiplying each part of an expression. In this example, by laying out the expression and then repeating it the given number of times, they can clearly see that each part of the expression is multiplied. This will help set them up for the more complex multiplications to follow.
Multiplying a quadratic expression by an integer
The Area Model
Before introducing multiplying two linear expressions, it may be useful to revise the area model with students. This model is the base from which they can visualise what is happening when they do the multiplication. Relating an algorithm to a concrete and/or visual representation will deepen the students’ understanding.
Explanation of the array model for multiplication
Multiplying two linear expressions
Once students have a good grasp of the area model, you can move on to multiplying two linear expressions. The example below shows how you model this by building the area indicated by the factors (the given brackets). I show how you can visually link the concrete, representational and abstract.
Multiplying two linear expressions together
Allow the students to decide which method they want to use. Try not to give the impression that using the tiles means they are not as ‘advanced’ as others. Students should feel comfortable to pick up the tiles and use them if they meet a problem where they feel the tiles will help them. In other words, the tiles are a tool, just like a calculator, and students should feel as comfortable reaching for the tiles as they do reaching for the calculator. Similarly, drawing visual representations. Mini-whiteboards are great for these as they allow students to erase ‘mistakes’ and move on.
Factorising Quadratics
Factorising Linear Expressions
Introduce factorising with very simple examples. In the video, I explain that factoring is building a rectangle, with the given pieces, so that there is the same number of each piece in each layer. Doing a few, really simple, examples will help students to get the idea of what factorising is – dividing.
Factorising a simple linear expression
You may want to give students an example where there is more than one way to factorise and then you can discuss which way is ‘better’ and why.
Factorising a quadratic expression
Once students can multiply linear expressions to form quadratic expressions they can begin to explore factorising – the opposite operation.
Starting with simple quadratics that do factorise easily, as in the video below, will familiarise students with the process. I have tried to show how the area model relates to the generally taught algorithm.
Factorising a simple quadratic expression
In the example in the video, factorising: x2 + 5x + 6 you might explain that you are looking for:
Two numbers that multiply to +6 and that add to +5
Possible factors
multiply
add
1, 6
1 x 6 = 6
1 + 6 = 7
2, 3
2 x 3 = 6
2 + 3 = 5
Meets requirements
By relating the algorithm to the visual model, students can see why they are looking for pairs of factors that meet these requirements.
In this second example, I demonstrate zero-sum pairs’ use when building the area necessary to factorise a quadratic expression.
Factorising a quadratic expression using zero-sum pairs
Factorising a quadratic where the coefficient of x2 is not 1
When we change the coefficient of x2, the algorithm described above becomes a little more involved, and students may begin to struggle with factorising. I show how using Algebra Tiles can make this process easier to see in the two videos below.
Factorising a quadratic expression where the coefficient of x is not 1Factorising a quadratic expression where the coefficient of x is not 1, another example
In Part 5, the last of this series, I will explore perfect squares, completing the square and extending to two-variables.
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.
This blog post is going to be a little different. The Australian Council for Education Research (ACER) recently published a paper written by Geoff Masters AO called “Time for a paradigm shift in school education”. In my family, I have been teaching for over 20 years, my husband (a school Principal) for even longer and my daughter is in the middle of her Master of Teaching. Hence, this article was the focus of much discussion.
In this article Masters argues that the way the curriculum in schools is currently arranged, with students organised by age into year levels needs to be reassessed and reimagined. He focuses on the mathematics curriculum, as this is the area where there has been the most research on alternative ways of structuring the curriuclum.
In her research on proficiencies in mathematics, Di Siemon and others, found that there were eight identifiable proficiency levels that students move through as they develop their capabilities in maths. Masters suggests that these eight proficiency levels could become the basis of curriculum levels. In this way students could, regardless of their age, work through these proficiencies at their own pace. This has many far-reaching ramifications on schooling, and he addresses these in the article.
Before I add my own thoughts, here is a (very) brief summary of the article. I do suggest you read the original and have linked to the full article above.
Summary of the article:
The results from testing such as PISA show that there is a need for a change in the way we approach teaching and learning as little progress is being made and in some areas students are not reaching the expected levels of achievement.
By arranging the curriculum in year level bands those students that do reach the expected level one year will continue to fall further behind as they start the following year at a disadvantage.
The alternative paradigm that Masters puts forward, is that if the curriculum were to be based on proficiency levels then students would be able to master one level before moving to the next. They would not have to do what everyone else is doing just because they are the same age.
This would necessitate a new way of thinking about how curriculum and assessment are structured. Success would be measured by how a student is moving through the levels and not by whether they have reached an expected level for their age cohort.
Given that this model is so different to the prevailing one it is to be expected that there would be hesitancy and resistance. Masters goes on to list six concerns that he would expect teachers and others to raise on this new way of thinking:
· Concern 1: will this mean abandoning year levels?
· Concern 2: teachers would be unable to manage classrooms in which students are not all working on the same content at the same time.
· Concern 3: some students will be disadvantaged if students are not all taught the same content at the same time.
· Concern 4: a restructured curriculum will result in ‘streaming’ and/or require the development of individual learning plans.
· Concern 5: a restructured curriculum will lower the educational standards
· Concern 6: it will not be possible to do this in some subjects.
In response to these concerns Masters argues that there are valid social reasons for students to be grouped in year levels and that this would not need to change. Teachers are already faced with a wide spectrum of ability and achievement in their classrooms. What would change is the way in which teachers would be differentiating – they would have a different curriculum/set of expectations for students at different levels.
Teaching all students, the same concept, when not all are ready for it is not equitable. Teaching each student at their level would enable the individual student to progress and hence the overall level of achievement for a cohort would also improve.
To introduce this change, Masters suggests that one subject be tackled first – Mathematics. Mathematics is considered to be a sequential subject and there has already been much research on learning trajectories/progressions in this subject. Other subjects could follow as their curricula are researched and designed in a similar manner.
My response
I do not disagree with the premise – that a reimagining of how we design, deliver and assess the curriculum is needed. The system of schooling we have was devised over 100 years ago to suit the needs of the universities and employers at the time. The needs of the modern university and employer, and employee, have changed. There are new subjects that do not fit easily in the subject ‘silos’ most education systems use, such as coding and STEM. The present curriculum is very crowded; the amount of knowledge a student is expected to know at the end of their schooling is far wider than in previous years. Educational systems struggle to accommodate all interests and needs of their stakeholders.
What does worry me though is how these changes will be enacted in the classroom. Today teachers are not just educators; they are also nurses, social workers, negotiators, councillors . . . We need to think about the teachers when big changes are talked about in curriculum design.
Having worked in both the UK and Australia I have experienced several major changes in curriculum design. For such a radical change to be successful all stakeholders, and when talking about education that is everyone, need to be well informed of what the changes are, why they are happening and what students will be doing.
Teachers: Teacher professional development is key. Not just a few PowerPoint sessions and reams of documents to read. To be successful there would need to be generous investment in teacher training and development and teachers will need to be given the time, away from the classroom, they need to plan such a radically different program of study.
Many teachers teach the way they were taught. To shift the way a teacher teaches is not simple. It takes time, in-class mentoring and coaching, and any change needs to demonstrate that it will lead to improved student outcomes very quickly if teachers are to persevere with a new pedagogy.
Teachers work hard and will not be attracted by an idea that seems to involve them working harder. They are already differentiating and trying to deliver the curriculum to students working at many levels in their classrooms. This is difficult, time consuming to plan, and exhausting to deliver. In the system as outlined by Masters, to plan and assess multiple programs, based on different curricula, in the one classroom, looks like it will be harder and more work.
Students: Students are not stupid! However you try to arrange students in differentiated groups, whatever you call them (the red table, blue table . . . the lions, the penguins etc.) they know who is working at a higher level and who at a lower. Having students in the same room working on different programs will make these distinctions more obvious. Students will have to be convinced that doing something different does not mean doing something better/worse. Peer pressure and the need to ‘fit in’ is stronger now than ever before, due to the pervasiveness of social media. Although students may experience more success, as they will be working on material that is accessible to them, that feeling could be cancelled by the feelings of being behind as they see their peers working on different content.
Parents: Parents will need to be convinced that this will be better for their children. And be willing to have the school decide at what level their child is currently at. So often I have heard parents request that their child is put in a different group/stream as they know that the child is capable of more than the school thinks they are. Parents will need to be comfortable that their child is not being offered the same program as others in the classroom. They will need to be educated in a different reporting system and measures of progress/success.
Employers: Depending on how students are assessed at the end of their schooling employers would also need to have these levels of proficiency explained clearly. Their future employees will be coming to them having completed schooling at different levels of proficiency.
I am not against change. The present system is not working for all our students, some change is necessary. In many schools there is already a move towards a more personalised and student led curriculum. Having a curriculum that is less rigid in what is expected to be delivered each year would support this personalisation.
School systems, teachers and students would have to be more flexible than they are now. As students master a proficiency level they would be moving to work with a different group. This may happen quicker for some than for others. Students would have to be willing to move, to work with others.
When I try to imagine what this would like I think that the architecture of the school may need to change too. Rather than groups of 20-30 students in a small room maybe this style of teaching would work better in a larger area with 50+ students and two or three teachers moving around working with small groups. As a student masters a concept they would then move to another group, to work on a different problem or concept. This would require the students to be motivated and self-disciplined. There would have to be a good system in place to track students’ progress and to make sure that no one fell through the cracks. This would relieve some of the burden of planning and assessing as teachers would be working in teams.
All of this sounds, to me, exciting and full of possibility. It needs people of vision who can communicate this vision and to show how it can be successfully executed on mass.
I do not believe that schools in 50 years’ time will look like the schools of the past 100 years. Change is coming. And as with all change it is both exciting and daunting. But, without the investment in stakeholder engagement and teacher professional development such a move could end up being another failed educational experiment that could set back the cause of quality teaching and learning.
