Square Xs

Session 1

The main aim of this session is to remind students that there are two solutions to a quadratic. Underlying this is the need for students to develop the habit of factorising quadratics and recognising the difference between two squares. They should have prior knowledge and experience of expanding and factorising quadratic expressions.

Teachers’ Notes

What is the square root of 25? (Is there only one answer to this question?)
What happens if you square -5? (What is -5 x -5?)
There are two square roots for any number. Only a few special numbers have two real roots. In this unit we will focus on showing this algebraically, rather than just using guess and check. We need to treat the problem as a quadratic, put everything on one side of the = sign and factorise to solve for x.
The problem is what is x when x² = 25
x² = 25
x² – 25 = 0
(x-5)(x+5) = 0
x-5 = 0 or x+5 = 0
x = 5 or x = -5
Many students will invariably suggest that it was much easier just to guess the answers. You might suggest that this way you know there are two (and only two) solutions.

Teaching Sequence

1. Start with a whole class activity.
   I'm thinking of a two digit number that is a perfect square
   It's an odd number less than 50
   The sum of the digits is 13
   What is your answer?
    
2. Repeat this type of quiz to familiarise students with perfect squares.
    
3. Now try to get the students to guess a positive root.
   I'm thinking of a number that is a square root
   The sum of the digits is 2
   The square is a three digit odd number.
   What is your answer?
    
4. Repeat this quiz but with a twist (to elicit a negative root).
   I'm thinking of a two digit number that is a square root
   It's a number less than 5
   The square is 100
   What is your answer?
    
5. Discuss what is going on here. The students will want to say that the square root of 100 is 10 and that is more than 5. They may need to look at a number line to appreciate that there are many more numbers less than 5 than just 1, 2, 3 and 4; including the -10 that you want to elicit from them.
   If I just said my two digit number is a square root less than 5, what could the answer be? If I just said my two digit number is the square root of 100, what could my answer be?
    
6. Get them to go work in pairs to make their own square root puzzles to try out on the rest of the class. To avoid using calculators, limit the possible numbers to guess being between 12 and -12.
    
7. The pairs can present a puzzle each to the class to get everyone very familiar with expecting two roots and to find these without calculators.
    
8. Show the students how the clues all fit together to give -10.
   x² = 100
   x² – 100 = 0
   (x-10)(x+10) = 0
   x-10 = 0 or x+10 = 0
   x = 10 or x = -10
    
9. Get the students to use this idea to show the two roots for a range of squares (eg 4, 25, 64, 121, etc and as an extension, a² and a²b⁴).
    
10. While the calculators are still out of reach… suggest students use this idea of the difference between two squares to calculate the square of 999. A new hint could be given each minute after the start if students get stuck.
    Hint 1: The task is to find 999²
    Hint 2: Try to find 999²-1
    Hint 3: That’s the same as finding 999²-1²
    Hint 4: This is the difference between two squares, so factorise to solve.

Session 2

In this session we look at quadratics that can be factorised in the form (x-a)(x-b)=0 and use these to solve problems. The main aim of this session is to remind students that if a product is zero, one of the factors must also be zero. Then, to use this fact, to form and solve factorised quadratic equations.

Teachers’ Notes

The aim here is to be able to tackle simple quadratics, that arise in word problem situations and that can be factorised in the form (x-a)(x-b)=0. The students may be able to form a quadratic equation, given a word problem, but should be able to put every term on one side of the equation, factorise and solve.

Problem 1: The product of two consecutive whole numbers is 42, what are those numbers?
There are lots of ways of doing this problem.
Method 1: You might draw up a table of factors of 42 and find which have a difference of 1.
Method 2: You might multiply pairs of consecutive whole numbers until you have success.
Method 3: You might write the problem as an algebraic equation and then solve this. You will need to show the steps required to solve this problem.
x(x+1) = 42
x²+x = 42
x²+x-42 = 0
(x+7)(x-6) = 0
x+7 = 0 or x-6 = 0
x = -7 (not a whole number, so not a solution) or x = 6 (making the other number 5 or 7)
Of these possible solutions sets for the pair of whole numbers, (6,5), (6,7), which gives a product of 42?
So the possible solutions for the two numbers are 6 and 7
Discuss the pros and cons of each method. Can you be sure that your method gives all the possible solutions?
What if the numbers are bigger or fractions, would your method still be the best method to use?
Praise any correct method that works but keep leading them back to the algebraic method.
If the students have not yet mastered factorising and expanding quadratic expressions, they will need to spend time on these techniques at this stage. There are many available reinforcement exercises in textbooks and worksheets, or ten expansion and ten factorising tasks could be quickly put up on a whiteboard. Once the students are familiar with expanding and factorising, problems such as the one outlined above should be relatively straightforward to solve.

Problem 2: Two integers have a difference of 10 and a product of -16. What are those two integers?
There are lots of ways of doing this problem.
Method 1: You might draw up a table of factors of -16 and find which have a difference of 10.
Method 2: You might multiply pairs of integers, with a difference of 15, until you have success.
Method 3: You might write the problem as an algebraic equation and then solve this. You will need to show the steps required to solve this problem.
x(x+10) = -16
x²+10x = -16
x²+10x+16 = 0
(x+8)(x+2) = 0
x+8 = 0 or x+2 = 0
x = -8 (making the other number -18 or 2) or x = -2 (making the other number -12 or 8) 
Of these possible solutions sets for the pair of integers, (-8, -18), (-8, 2), (-2, -10), (-2, 8), which give a product of -16?
So the possible solutions for the two numbers are (-8, 2) and (-2, 8).
Discuss the pros and cons of each method. Because the product is negative, what does that tell you about the factors? Can you be sure that your method gives all the possible solutions? What if the numbers are bigger or fractions, would your method still be the best method to use?
Praise any correct method that works but keep leading them back to the algebraic method.
Depending on skill level and understanding, further word problems or reinforcement exercises of forming, factorising and solving quadratic equations, found in textbooks or on worksheets, should be given to the students to work on individually or in pairs.

Teaching Sequence

1. Introduce a simple problem like Problem 1. Ask:
   How would you solve this problem?
   Then discuss the methods they think up.
   Is Joe’s answer correct? Is his method correct?
   What other ways are there of doing that problem?
    
2. List the methods on the board. Be sure to cover at least the three methods above.
   Do they all give the same answer?
   Are the methods all correct?
   Which methods do you understand?
   Which is the best method for you?
   Which is the best method of all?
   How would you show your working for your favourite method? (Try to cover all the methods by asking different members of the class.)
    
3. Show the algebraic method of solution.
    
4. Now review the skills and techniques of factorising and solving quadratics.
    
5. Bring the class together to discuss what they have done.
   How did you use your basic facts?
   What patterns could you see when you were factorising?
   How did you know your answers were correct? (plugging soultions back into the original problem)
   Philly, can you show us how you did one of the problems?
   What other method can you use to solve this? (Make sure that they come up with an algebraic one.)
    
6. Introduce a simple problem like Problem 2. Ask:
   How would you solve this problem?
   Then discuss the methods they think up.
    
7. Get the students to give the steps of the algebraic solution.
    
8. At the end, ask them to summarise the steps they took with a flow chart. (This could be done in groups or pairs on poster paper)
    
9. In pairs, the students can make up their own quadratic word problem for the class to solve.

Session 3

Here we look at simple problems like the Square Xs, where there is a unique solution. In these cases, the quadratic is a perfect square.

Teachers' Notes

The basis for the next two sessions is to familiarise students with factorised quadratics so that they can distinguish between quadratics with two real roots, the difference between two squares and perfect squares.
Quadratics with two real roots factorised: (x-a)(x-b) = 0 and expanded x²-(a+b)x+ab = 0
The difference between two squares factorised: (x-a)(x+a) = 0 and expanded x²-a² = 0
Perfect squares factorised (x-a)² = 0 and expanded x²-2ax+a² = 0
An example of a perfect square problem is: 

Perfect Square Problem Two integers have a difference of 8 and a product of -16. What are these numbers? 

The ability to recognise such quadratics is a very useful skill for solving advanced algebraic problems that the students will encounter in later years. The steps needed to solve such problems are:

1. Write the problem as an algebraic equation
2. Expand any brackets
3. Put every term on one side of the equation (so it becomes something = 0).
4. Factorise (if possible)
5. Form the linear equation(s) that result(s) from the knowledge that if the product is 0, so too must be at least one of the factors.
6. Solve the problem. (This may include checking back with the context to eliminate solutions that lie outside of the defined range.) 

x(x-8) = -16
x²-8x = -16
x² – 8x +16 = 0
(x-4)(x-4) = 0 or (x-4)² = 0
x-4 = 0
x = 4, so the other number is 4-8 = -4

The steps above are applicable to all factorisable quadratics. It should be the students' first approach to solving a problem, as soon as they have identified that it is indeed a quadratic. Once the students have grasped the concepts of these individually, it would be beneficial to provide a mixed group of problems in reinforcement activities, such as those found in textbooks, on worksheets or that can be written on the whiteboard.
Note: Since the numbers here are relatively straightforward we do not encourage the use of calculators in this work.

Teaching Sequence

1. Introduce a simple problem like the Perfect Square Problem. Ask:
   How would you solve this problem?
   Then discuss the methods they think up. many of the students will have tried to solve the problem by 'guess and check'. Lead the class towards using the techniques of putting all the terms on one side and factorising.
   What other ways are there of doing that problem?
    
2. Show the algebraic method of solution.
    
3. Introduce another simple problem like the Perfect Square Problem. This could be as simple as 'I'm thinking of a number that when you square it and add 81, you get 18 lots of that number' . Ask:
   How would you write this problem?
   Then discuss the which suggestion matches the problem.
    
4. Get the students to solve the problem. They may need to work in pairs.
    
5. At the end, ask them to compare the steps they took with the flow chart they made in session two. Discuss how the problems today had only one solution? What might be different about today's problems that meant we only had one solution for x?
    
6. In pairs, the students can make up their own quadratic word problem for the class to solve.
    
7. Get groups to swap their problems with another group and solve the other group’s problems.
    
8. In a class discussion show the students a range of factorised quadratics and ask them to identify the perfect squares and also the difference between two squares. (These can be found in most textbooks, worksheet packs, or simply written up on the whiteboard.)
    
9. In a class discussion show the students a range of expanded quadratics and ask them to identify the perfect squares and also the difference between two squares. (These can be found in most textbooks, worksheet packs, or simply written up on the whiteboard.) Discuss the key distinguishing features.
    
10. Give the students a mixed group of factorised and expanded quadratics to solve. (These can be found in most textbooks, worksheet packs, or simply written up on the whiteboard.)

Session 4

Here we look at students using their knowledge of quadratics to solve problems independently.

Teachers’ Notes

This session is based on problems reducible to a simple quadratic equations. Students should be encouraged to use their own method to find the solution, then the range of techniques can be discussed in the context of the initial problem.
Investigations involving quadratics can be found in the Rich Learning Activities.
An activity involving the translation of a written task into an algebraic equation is Three in a row.
A problem that can be modelled practically (to increase the number of ways a student could approach the problem, is the Parabolic Investigation.
An extensive algebraic investigation is Overlapping Powers.
For any of these activities, the students should be encouraged to solve the problem using whatever approach is most comfortable for them. This may be algebraically, with arithmetic methods, or by guess and check.

Teaching Sequence

1. Give the students some warm up exercises of forming and solving a quadratic equation. This could be: A rectangle has long sides that are two units longer than the shorter sides, and a total area of 48. What is the length of a shorter side? Encourage the students to solve this problem in whatever manner they see best.
    
2. Discuss the different methods that the students used. How did you tackle this task? Why did you choose to do that? You thought that it would be the quickest way, but was it? How important was the setting out of your work? How useful was it to write the problem as an algebraic equation?
    
3. Introduce the students to a Rich Learning Activity, such as Three in a row. Encourage the students to solve this problem in whatever manner they see best.
4. Review: Discuss the different approaches the students took, appealing to comfort of using familiar techniques but the need to ensure that each step is valid and correct mathematically.

Session 5 (Extension)

Here we look at 'completing the square' to solve quadratic. This is aimed at extending students' appreciation of the elegance of algebra and is the basis for deriving the quadratic formula that students will be introduced to in Level 7. At this level (5), the examples need to be carefully scaffolded to be as simple as possible. Completing the square allows students to find the solutions for a quadratic that is not easily factorised – but it for such cases, the students need to have an appreciation of square roots and expressing such irrational numbers in surd form.

Teachers’ Notes

This is aimed at extending students' appreciation of the elegance of algebra and is the basis for deriving the quadratic formula that students will be introduced to in Level 7. At this level (5), the examples need to be carefully scaffolded to be as simple as possible, eliminating any need for calculators.

Sample Problem 1 A teacher has given a student a problem that says x² - 8x = -7

1. Find the perfect square that matched the terms with x in them. ie Find an expression for x² - 8x
   x² - 8x + 16 = (x-4)² 
   x² - 8x = (x-4)² - 16
    
2. Substitute this expression into the original problem. ie 'complete the square'
   (x-4)² -16 = -7
    
3. With the 'square' on one side of the equation, solve the problem.
   (x-4)² = 9
   x-4 = ±√9
   so x-4 = 3 or x-4 = -3
   x = 7 or 1

Sample Problem 2 – not easily factorised A teacher has given a student a problem that says x² + 6x = 5

1. Does this factorise? x² + 6x – 5 = 0 No, there aren't any pairs of factors of 5 with a difference of 6.
    
2. Find the perfect square that matched the terms with x in them. ie Find an expression for x² + 6x
   x² + 6x + 9 = (x+3)²
   x² + 6x = (x+3)² - 9
    
3. Substitute this expression into the original problem. ie 'complete the square'
   (x+3)² - 9 = 5
    
4. With the 'square' on one side of the equation, solve the problem.
   (x+3)² = 14
   x+3 = ±√14
   x = 3±√14

Teaching Sequence

1. Give your class a range of quadratic problems (factorising and solving) that have many perfect squares. and the difference between two squares amongst them. Discuss the patterns that they see. When we have the difference between two squares, x²-36 = 0, isn't it easier to just write x² = 36 and know that there are two answers… x = ±6? What makes it allowable to just take a square root? Is this only when there is just one term on each side of the equation? This discussion should lead the class into seeing the action of taking the square root of each side as a valid algebraic step as long as you are taking the square root of the whole side and know that there are two roots, the positive and the negative. How can we recognise the perfect squares? Can we rewrite any such term, like x²+10x in terms of a perfect square? But x²+10x is smaller than (x+5)² by how much?
    
2. Give Sample Problem 1 to your class and discuss and show how it may be solved.
    
3. Now try several more similar exercises. The same exercises used in session 2 could be solved in this way.
    
4. In pairs or groups, the students should be able to make several problems up for the class. They could start with a factorised quadratic, expand and rearrange it, to have the constant term isolated on one side of the equation.
    
5. The students who have a solid understanding of the process, could be shown Sample Problem 2, and given several other quadratic problems with irrational roots.
    
6. Review: The class can be shown a range of different quadratic problems and discuss, building a mind map of, the different techniques they could use to solve the problems and how recognising the difference between two squares and perfect squares.