Solving Two Step Equations using the Balance Method

An equation is when one expression, or term, is equal to another.  To solve an equation means to find the value of the variable (represented by a letter) that makes the two expressions equal.  There are two types of equations for secondary school mathematics, linear and none-linear.  In this blog I write about how I introduce solving two step equations using the balance method.

Solving Equations Involving One Step

Solving Two Step Equations using the Balance Method

The start of the lesson recaps solving one step equations.  At this stage it is important students understand y/3 means y divided by 3 and 7y means 7 multiplied by y.  Next, we discuss inverse operations, i.e., the opposite of dividing by 3 is multiplying by 3 and the opposite of multiplying by 7 is dividing by 7.   I will often use function machines to demonstrate this.

Solving Two Step Equations using the Balance Method

Solving Two Step Equations using the Balance Method

Solving Two Step Equations using the Balance MethodSolving Two Step Equations using the Balance Method

When solving two step equations using the balance method we arrange the terms, so the unknown is on one side of the scale and the numbers on the other.   Using scales as part of the working helps students to see how the equation remains balanced throughout this process.  Function machines help students identify the individual operations but, in my opinion, lack the perception of balance.

The terms move according to the inverse of the order of operations.  For instance, with 6x + 2 = 26, the addition of 2 is moved to the other side before the multiplication of 6.  The equation remains balanced by doing the same operation to both sides of the scale (or equals sign).

Developing Algebraic Reasoning

As we progress through the lesson I consolidate and extend the learning by increasing the level of challenge.  I do this by:

  • writing the equation on the right side of the equals sign rather than the left;
  • including divisions as fractions;
  • requiring students to simplify an equation before they attempt to solve it;
  • making the unknown term negative, eg., 0 =6 – 2x.

These mini-extensions all help to develop student’s algebraic fluency.

Solving Two Step Equations Plenary

Solving Two Step Equations using the Balance Method

In the plenary students match the equation to its solution.  This involves attempting a variety of questions across the ability range.   This activity takes about 10 minutes with students working independently on mini-whiteboards.   I assess student’s progress against these differentiated learning objectives:

  • All students should be able solve equations involving a multiplication and addition.
  • Most students should be able to solve equations involving divisions and multiplications.
  • Some students should be able to solve equations where the unknown term is presented as a negative.

As we progress through the scheme of work students learn how to solve equations involving brackets and begin to form their own equations from known geometrical properties.

Teach this lesson

Solving Inequalities using a Number Line

To aid the process of solving an inequality it is important to provide a clear writing frame to show how the inequality remains balanced as it is simplified. Students are familiar with the balance method from solving two-step equations.

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