Module 5 Equations & Formulae
Module Content Lesson Resources Lesson Plans Exam Questions Probing Questions

Equations & Formulae - Teacher Notes

Solvemymaths has produced pages like this for lots of topics

NRICH Tasks

NRICH: Multiplication Square

NRICH: Cubes Within Cubes Revisited 

NRICH: Plus Minus
NRICH: 2-digit square
NRICH: Creating Pictures
NRICH: Letterland
NRICH: CD Heaven
NRICH: What's it Worth?
NRICH: Matchless
NRICH: Intersections 
NRICH: Tablecloth
NRICH: Square Pizza 
NRICH: For What?
NRICH: Why 24?
NRICH: Perfectly Square

Extras

Function Prompt
Interactive Function Maker
What is the function?
Forming
Equations Activity
Random Questions - No Brackets
Random Questions - Brackets
Random Questions - Expanding Brackets
Equation Balance
Balancing positives only
Balancing positive and negatives
Randomly generated questions
Random Substitution Grid
Loop Cards - Randomly Generated
 

Online Card Matching
Random Questions - Factorising
Random Questions - Factorising 1
Random Questions - Factorising 2
Video Worked Examples:
Change subject 1  Easy
Change subject 2  Medium
Quads 2  Factorising when a is not 1
Quads 3   Solving from factorising 1
Quads 4   Solving from factorising 2
Quads 5   Quadratic formula 1
Quads 6   Quadratic formula 2
Quads 7   Complete the square 1
Quads 8   Complete the square 2
 
Random Completing Square 1
Random Completing Square 2
Random Completing Square 3
Quadratics 9   Algebraic fractions (1)
Quadratics 10   Algebraic fractions (2)
Change subject 3  Complicated

 

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

   
Students have changed the prompt in the following ways:

§  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/algebra-prompts/steps-inquiry

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 4-7 – 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 common-sense 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


In each pyramid each number is the sum of the two numbers immediately below. Find the value of n – we have these resources in the scheme of work: To preview -
Pyramid 1: positive whole number solutions
Pyramid 2: positive and negative whole number solutions
Pyramid 3: positive whole number solutions.
Pyramid 4: positive and negative whole number solutions
Pyramid 5: positive whole number solutions
Pyramid 6: positive and negative whole number solution

Linear equations

Solving Equations using the Manipulatives by @MissNorledge – Click here

Equations with unknown on one side only 1 or 2 step - Questions
Equations with one unknown involving a bracket -
Questions
Equations with unknowns on both sides -
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 check-boxes 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?

https://harmath.wordpress.com

Pentomino Totals

mrorr-isageek.com/pentomino-puzzles/

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