## Core Concept 1.4

## Simplifying and manipulating algebra

1.4.1.1 Understand that a letter can be used to represent a generalised number

1.4.1.2 Understand that algebraic notation follows particular conventions and that following these aids clear communication

1.4.1.3 Know the meaning of and identify: term, coefficient, factor, product, expression, formula and equation

1.4.1.4* Understand and recognise that a letter can be used to represent a specific unknown value or a variable

1.4.1.5 Understand that relationships can be generalised using algebraic statements

1.4.1.6 Understand that substituting particular values into a generalised algebraic statement gives a sense of how the value of the expression changes

### 1.4.1.4 Understand and recognise that a letter can be used to represent a specific unknown value or a variable

●Understand that unknown quantities can be named and operated on.

●Understand that a letter stands for a variable and can take a range of values.

The dynamic representations below use dynamic algebra tiles.

Moving the slider changes the value of x.

The example above is from the NCETM Secondary PD material

What would the value of each expression be is x = 5?

What do you notice when x = 4

When is 3x+2 = 2(x+3)?

When is 3x+2 < 2(x+3)?

When is 3x+2 > 2(x+3)?

When is 2x+5 = 3x+2?

When is 2x+5 < 3x+2?

When is 2x+5 > 3x+2?

For each of the following statements, use a letter to represent the number Isla is thinking of, write the statement using letters and numbers, and find the number she is thinking of. Draw a diagram to represent each statement.

a) ‘I am thinking of a number; I add four and the answer is 12. What number am I thinking of?’

b) ’‘I am thinking of a number; I add four, multiply by three and the answer is 12. What number am I thinking of?’

Move the slider to change the value of x.

Open in Desmos by clicking edit graph on Desmos on the bottom right corner of the image.

For each of the following statements, use a letter to represent the number Isla is thinking of, write the statement using letters and numbers, and find the number she is thinking of. Draw a diagram to represent each statement.

c) ‘I am thinking of a number; I add four, multiply by three, subtract six and the answer is 12. What number am I thinking of?’

d) ‘I am thinking of a number; I add four, multiply by three, divide by two and the answer is 12. What number am I thinking of?’

The red tiles represent -1. In this representation -6 is placed underneath the positive tiles to represent zero pairs.

Which is bigger? 3x or x+3

Combing the dynamic representation with the graphical representation allows students to see connections.

When you move the slider you can see when

3x > x+3

3x = x+3

3x < x+3

Arrange these cards in order.

Use the base blocks below to help. See more on Base Blocks and Algebra Tiles here.

Here students can see base x when x varies.

These can be used to support students understanding of the card sort activity above.

### 1.4.1.5 Understand that relationships can be generalised using algebraic statements

●Use letter symbols to represent mathematical relationships.

●Use letter symbols to model situations.

●Interpret the impact of changing one variable on another within a generalised relationship.

Describe how each of the following is represented in this bar model.

a) x + 2 = 10

b) 10 – x = 2

c) 10 – 2 = x

What statements would represent this bar model?

1.4.2.1 Identify like terms in an expression, generalising an understanding of unitising

1.4.2.2 Simplify expressions by collecting like terms

1.4.3.1* Understand how to use the distributive law to multiply an expression by a term such as 3(a + 4b) and 3p2(2p + 3b)

1.4.3.2 Understand how to use the distributive law to factorise expressions where there is a common factor, such as 3a + 12b and 6p3 + 9p2b

1.4.3.3 Apply understanding of the distributive law to a range of problem-solving situations and contexts (including collecting like terms, multiplying an expression by a single term and factorising), e.g. 10 – 2(3a + 5), 3(a ± 2b) ± 4(2ab ± 6b), etc.

### 1.4.3.1 Understand how to use the distributive law to multiply an expression by a term such as 3(a + 4b) and 3p2(2p + 3b)

●Understand the structure of the distributive law.

●Understand the impact of the multiplier.

●Understand that the multiplier can be a variable.

●Understand the importance of the sign (positive or negative) of each term in an expression and how it affects the final result.

●Understand the impact of a negative multiplier on the result.

●Apply knowledge of fractions and decimals when expanding brackets.

●Apply the use of algebra to a different context.

For each of these expressions, write another expression without brackets that will always have the same value.

a) 1(3x + 5)

b) 2(3x + 5)

c) 3(3x + 5)

d) 10(3x + 5)

a(bx+c)

In this dynamic representation you can change the values of a,b, c and x

1.4.4.1 Use the distributive law to find the product of two binomials

1.4.4.2 Understand and use the special case when the product of two binomials is the difference of two squares

1.4.4.3 Find more complex binomial products

### 1.4.4.1 Use the distributive law to find the product of two binomials

●Recognise that the product of two binomials is an expression with four terms.

●Appreciate when the product of two binomials can be simplified.

●Understand that the product of (x + a)(x – b) is an expression of the form x2 + cx – d or x2 – cx – d.

●Solve problems involving the product of pairs of binomials.

For each of these expressions, write another expression without brackets that will always have the same value.

What area model would represent these expressions?

a) 3(x + 5)

b) 3(x+y)

c) x(x+5)

d) y(x+5)

e) x(x+y)

f) y(x+y)

For each of these expressions, write another expression without brackets that will always have the same value.

What area model would represent these expressions?

a) 3(x - 5)

b) -3(x-5)

c) x(x-5)

d) x(5-x)

e) -x(5-x)

1.4.5.1* Understand that an additive relationship between variables can be written in a number of different ways

1.4.5.2 Understand that a multiplicative relationship between variables can be written in a number of different ways

1.4.5.3 Apply an understanding of inverse operations to a formula in order to make a specific variable the subject (in a wide variety of increasingly complex mix of operations)

### 1.4.5.1 Understand that an additive relationship between variables can be written in a number of different ways

●Every addition can be rewritten as a subtraction and every subtraction as an addition.

Identify two addends and their sum in the following equations and show them on a bar model (as below).

a) 126 + 437 = 563

b) 2x + 17 = y

c) r = p + q

d) x2 + 6x = 4p2 + 9

e) 3m − 2n + r = V

Algebra discs on MathsBot (below)

3m - 2n + r = V

3m - 2n + r + 2n = V + 2n

3m + r = V + 2n

3m - 2n + r + 2n -3m = V + 2n -3m

r = V + 2n - 3m

3m - 2n + r + 2n - r = V + 2n - r

3m = V + 2n -r

m = V + 2n -r

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