Simplify linear expressions without brackets

What you will learn

• Simplify linear expressions without brackets

Models used: number sentences, stories, tables.

Before you start

• Form simple expressions involving an unknown

Count the bags, not what is inside

Two bags and three more identical bags contain five bags’ worth of counters.

1. Matching bags hold the same amount. A level balance means both sides have equal value.

   Two bags and three more identical bags contain five bags’ worth of counters.

   2x + 3x

   A letter can take a numerical value. Like terms count the same unknown-sized unit; substitution replaces it consistently.

   xx

   xx

   2x = 2x · balanced

2. Matching bags hold the same amount. A level balance means both sides have equal value.

   Join the bag counts. We have not opened a bag or changed what x means.

   2x + 3x = 5x

   Equality is a relationship between both sides. Adding or removing the same amount on both sides keeps that relationship true.

   xxxxx

   xxxxx

   5x = 5x · balanced

3. Matching bags hold the same amount. A level balance means both sides have equal value.

   Five bags still means five copies of x. It does not mean x × x.

   5x = x + x + x + x + x

   Equality is a relationship between both sides. Adding or removing the same amount on both sides keeps that relationship true.

   xxxxx

   xxxxx

   5x = 5x · balanced

Worked example: A new example, together

Simplify 2x + 6x.

1. Count equal unknown units. Read each value with the heading above its column.

   Say what one letter means before grouping terms or substituting a value.

   Like terms can be combined by adding their coefficients.

   Count equal unknown units

   Quantity	Expression
   2 copies of x	2x
   6 copies of x	6x

2. Count equal unknown units. Read each value with the heading above its column.

   Both terms count the same unknown unit x.

   Count equal unknown units

   Quantity	Expression
   2 copies of x	2x
   6 copies of x	6x

   Simplify 2x + 6x.

   8x

3. Count equal unknown units. Read each value with the heading above its column.

   Both terms count the same unknown unit x.

   Count equal unknown units

   Quantity	Expression
   2 copies of x	2x
   6 copies of x	6x

4. Count equal unknown units. Read each value with the heading above its column.

   Join the counts: 2 + 6 = 8.

   Count equal unknown units

   Quantity	Expression
   2 copies of x	2x
   6 copies of x	6x

5. Count equal unknown units. Read each value with the heading above its column.

   The expression is 8x.

   Count equal unknown units

   Quantity	Expression
   2 copies of x	2x
   6 copies of x	6x

6. Count equal unknown units. Read each value with the heading above its column.

   Check the result against the question we started with.

   For x = 7, both expressions equal 56.

   We found 8x. Like terms can be combined by adding their coefficients.

   Count equal unknown units

   Quantity	Expression
   2 copies of x	2x
   6 copies of x	6x

Common mistakes to discuss

• Collecting unlike terms. Only like terms combine. 2x + 6x is 8x, but 2x + 6 stays as it is.

Check your understanding

• Simplify linear expressions without brackets
• interpret and use an expression through the meaning of its unknown
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.