Module 5  Equations & Formulae 
Equations & Formulae  Teacher Notes
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NRICH Tasks
The following problems are examples of open problems which give pupils an opportunity to explore and discuss their mathematics.

Forming Expressions
Explain why. Make up your own expressions and try on your friends. 
Try this excellent activity great mental starter and then try to explain why using Algebra Click here

Transforming Expressions Click here

Steps Inquiry  Level 4+ This inquiry invites students to place a number in the larger circle and calculate the results of taking the steps in the two paths. For less experienced inquirers, the teacher might label the larger circle 'input' and the two others 'output A' and 'output B'. Even if the teacher opts for the more open prompt, communication is easier when the class develops labels for the three circles.

When asked for comments or questions, students invariably reach the conclusion that, in the case of the prompt, one output is always three more than the other. This realisation acts as a sort of preliminary phase to the main part of the inquiry. Following this the common path are: § Either students select find more examples, by which they mean to experiment with different pairs of operations. (The teacher of younger classes is advised to stipulate that only multiplication, addition and subtraction are permissible in the initial phases.) Students go onto induce a relationship between two outcomes. § Or students choose prove the prompt is always true, which can lead to the development of algebra directly from one numerical case. If we start with four, for example, the top path gives 4 + 4 + 3 or 2 x 4 + 3 as an output. The bottom path gives 4 + 3 + 4 + 3 or 2 x (4 + 3). The algebraic expressions, then, are 2n + 3 and 2(n + 3), where n is the starting number. As 2(n + 3) expands to 2n + 6, it becomes clear that the difference between the outputs will always be three. Students have shown great enthusiasm for finding algebraic expressions. They have then substituted into the expressions to deduce the difference between the outcomes for a particular starting number. Changing the Prompt
§ Use three steps (to give six distinct paths). Compare the outputs, explaining (and proving) why they are the same or different. [On one notable occasion, a discussion of how many paths there would be for three steps led one student to calculate the number of permutations for all cases up to 20 steps in the search for a rule.] § Use three or four operations and the same operations in reverse  for example, x4 +2 x3 reversed to give x3 +2 x4. § Choose three or four operations and aim for equal outputs. § Use other operations such as division, squaring, cubing, and so on. § Start with two outputs and aim to devise operations that lead back to the same input. http://www.inquirymaths.com/home/algebraprompts/stepsinquiry 
Solving Linear Equations Start with an expression such as 3x + 5 and get pupils to repeatedly apply an operation such as x or +y, +3 or x2 .... Start with an expression such as 3x + 1y = 8 and get pupils to repeatedly apply an operation such as x or +2 or +y, x2, ... FlashMaths: Interactive version where you can manipulate equations Click here FlashMaths: Randomly generated equations Level 47 Click here FlashMaths: Balancing Equations +ves only on each side Visually Click here NLVM: Balancing Equations including negatives Visually Click here Bar Model Method involving Negatives Video 

Start with an equation such as 3x + 5 = 14 and try different functions such as +x which gives 4x + 5 = 14 + x then 5x + 5 = 14 + 2x etc... What happens if we 1? Is there an alternative way of simplifying this equation (5) Try with 3x + 15 = 5x + 3 ...

Simplifying
Other examples Don Steward Click here 
Mobile Puzzles This is an excellent introduction to mobiles  Click here
Leading to Algebraic problems

The following leads onto solving equations. These typically present multiple balanced collections of objects whose weights must be determined by the puzzler. The imagery helps students build the logic of balancing equations, an intuitive understanding of substitution, and commonsense strategies that become the foundation for the standard algebraic moves involved in solving equations and systems of equations. These are essentially pictorial representations of systems of several simultaneous equations in two or more variables. Such puzzles thereby help students simultaneously grasp the concept and role of a variable and develop the logic of algebraic manipulation. Gradually, shapes are replaced with variables, strings of shapes are replaced by algebraic expressions, and finally sections of mobiles are translated into equations

Pyramid equations

Linear equations
Equations with unknown on one side only 1 or 2 step 
Questions Equation Auction Idea here Resource Here Solve equations: Eight levels of difficulty. Click on the 1 to 8 buttons to generate the equations. 
Solving Equations

Solving Equations Bar or Line Method

Which is Larger?
This is a great problem, providing opportunities to use Substitution, Equations and Graphical work, could be simplified with different expressions. 
Inequalities
Solve inequalities and equations Seven levels of difficulty. Click on the 1 to 7 buttons to generate the equations.

Substitution A starter to practise algebraic substitution. Work out the value of the expressions using the given x value. Click the tiles to show or hide the answers The arrow buttons change the grid size for more or less questions. The checkboxes allows negative and decimal x values for an added challenge


Substitution Football

Write an Expression for each of the boxes (How do you know it is a square?)

Can you make a match between one of the visuals and one of the expressions? How do you know?


Algebra and the Number Line These can be solved using trial and improvement or setting up and solving equations, what if the start of the line equals something other than 0?

Pentomino Totals

Simultaneous Equations X =Length Y = Width 4X+5Y=308 and 2X=3Y So 6Y + 5Y = 308 leading to Y = 28 and X = 42 Form and Solve Simultaneous Equations Click here 
Introducing Quadratic Expansions

A complete course on Quadratics @jamestanton Click here Quadratic Solver Click here


Great Problem Using Quadratics
Understanding the Quadratic Formula

Factorising Harder Quadratics

Algebraic Fractions Great interactive resource on Algebraic Fractions 
