What does it mean to balance an equation?

Understanding the concept of “equal” is a critical part of developing number sense and algebraic thinking. An equation is like a weighing scale – both sides balance because they represent the same amount. In other words, the answer to the expression on the left side of the equals sign (=) should be equal to the value on the right side of the equals sign in a balanced equation. 

In an unbalanced equation, either the left hand side of the equation has a greater value than the right hand side

or the right hand side of the equation has a greater value than the left side

Example 1: Addition and Subtraction

Suppose you are trying to find out how many sweets are in the bag shown here. Each of the sweets weighs the same amount. The bag of sweets is represented by x. The weight of the bag is zero. To solve the equation you need to find the value of the missing number by performing the same operation on each side.

By subtracting one sweet from each side, the scales remain balanced.

You can now see that one bag is equivalent to two sweets. 

Written algebraically, this is: x+1=3

Subtract 1  from both sides, to get: x=2

Example 2: Multiplication and Division

In this case, two bags of sweets are equivalent to eight sweets. Remember – the weight of the bag is zero, so you’ll only need to consider the weight of the sweets inside the bag.

To find the equivalent of one bag, divide both sides by two.

Written algebraically, this is: 2x=8

Divide both sides by 2 to get: x=4

Example 3: Solving More Complex Equations

Sometimes an equation will have multiples of an unknown and other numbers, as in: 4x+2=10. In equations of this type, your aim is to get all the xs (or unknowns) on one side and all the numbers on the other.

Let’s solve the equation 4x+2=10. We can represent this in the following diagram, where each bag represents the unknown value x.

We want to get the x on its own. Start by subtracting 2  from both sides.

4x +2 – 2 = 10 – 2

4x = 8

Then divide both sides by 4.

 x=2

Example 3: Solving Equations with Unknowns on Both Sides of Equation

Sometimes there are unknown numbers on both the left and the right hand side of an equal sign. This requires you to do some extra steps. This example will work to solve the equation: 2x+2=x+4 

The equation  2x+2=x+4  is represented by the following diagram.

Just like before, the bags represent the unknown value (x) and the sweets represent the numbers in the equation. Our goal is to get the unknown value on only one side of the equation, so we can begin by subtracting x (taking one bag away) from each side.

Now that there is the unknown value remaining on only one side, all you need to do is subtract 2 from both sides.

Written algebraically, this becomes: 2x+2=x+4

Subtract x from both sides to get: x+2=4

Subtract 2 from both sides to get: x=2

1st and 2nd Grade Balancing Equations

In first and second grade, students are learning on mastering their addition and subtraction facts. Through the process of learning these facts, they need to also be familiar with how to go about finding an unknown number that can solve the addition and subtraction equations. 

CCSS.MATH.CONTENT.1.OA.A.1

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.1

CCSS.MATH.CONTENT.2.OA.A.1

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

Students in first grade should be able to show mastery with finding the value of an unknown digit when adding and subtracting within 20 and those that are in second grade should be able to master this task within 100.

3rd Grade Balancing Equations

In third grade, students begin to learn their multiplication and division facts. With this mastery, it is expected that students are able to determine unknown digits when dealing with those multiplication and division facts. 

In addition to multiplication and division, students should also be able to find the missing digits within addition and subtraction problems with digits greater than 100.

CCSS.MATH.CONTENT.3.OA.A.4

Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

Students in third grade should also be able to recall how to add and subtract to determine the unknown number within an equation as well. The following worksheets work to guide students to add, subtract, multiply, and divide to find the value of an unknown number. At this stage in the curriculum, the unknown number is often depicted with a blank space (__) or  a question mark (?). Occasionally you will see the unknown number being displayed as an (x) but it is less typical at this stage in their development.

4th Grade Balancing Equations

In fourth grade, students continue to work with all four properties, but the problems that they are working with become more complex. Students will have to solve both sides of the equation and use multiple properties to determine the final answer of the unknown number. The missing number in fourth grade can have a letter standing for the unknown quantity. Fractions are also introduced in the third grade. As students advance to fourth grade, they build a better fractional understanding. Students can also be expected to determine the value of a missing number when fractional concepts are introduced into being a part of the equation.

CCSS.MATH.CONTENT.4.OA.A.3

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

5th Grade Balancing Equations

In fifth grade, students are introduced to parentheses and brackets and are taught to interpret these expressions and determine answers to equations that include them. Students also move beyond the fraction to develop a deeper understanding of decimals. You should expect students with a mastery in these skills to be able to apply their thinking to finding the value of unknown numbers in expressions that include parentheses, brackets, and decimals.

CCSS.MATH.CONTENT.5.OA.A.1

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

CCSS.MATH.CONTENT.5.NBT.A.3

Read, write, and compare decimals to thousandths.

6th Grade Evaluating Expressions and Equations

In sixth grade, the terminology turns from balancing equations to evaluating expressions and equations. Students in sixth grade are gaining the skills that will be necessary to complete pre-algebraic thinking. An expression is a mathematical phrase that contains numbers, variables, or both. Expressions never have an equal sign whereas an equation is a mathematical sentence that says two expressions are equal.

CCSS.MATH.CONTENT.6.EE.A.2

Write, read, and evaluate expressions in which letters stand for numbers.

CCSS.MATH.CONTENT.6.EE.A.2.A

Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.

CCSS.MATH.CONTENT.6.EE.A.2.B

Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

CCSS.MATH.CONTENT.6.EE.A.2.C

Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.

7th Grade Evaluating Expressions and Equations

In seventh grade, students are to create their own expressions and equations that can be used to solve real life problems that they may come across. By having a strong understanding of how to make an equation balanced, students are able to use that thinking to create problems and solve problems that require such thinking. By having the students practice writing the equations and expressions on their own, they are proving that they have that higher level of understanding that will be essential to algebra in high school. Many of the problems included in the worksheets below outline this process of writing the equations.

CCSS.MATH.CONTENT.7.EE.B.4

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.