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Math equations

Math equations

Here you will learn about math equations, including what they are and how to solve them.

Students will first learn about math equations as part of expressions and equations in 6th grade. They continue to expand upon this knowledge in 7th and 8th grade.

Every week, we teach lessons on math equations to students in schools and districts across the US as part of our online one-on-one math tutoring programs. On this page we’ve broken down everything we’ve learnt about teaching this topic effectively.

What are math equations?

Math equations are statements made up of algebraic expressions and an equal sign. Math equations include defined math formulas that can be solved with substitution.

  • Math formulas are rules that connect two or more variables. They are used to work out the values of unknown variables by substituting in other known values.

For example, here is a rectangle with base katex is not defined and height katex is not defined

Math Equations image 1 US

The area of any rectangle can be found with the formula katex is not defined

Step-by-step guide: Math formulas

  • Substitution is when you give variables in an algebraic expression, equation or inequality a value. You use substitution to identify solutions.

For example,

Does katex is not defined make the equation katex is not defined true?

Substitute katex is not defined for katex is not defined and follow the order of operations:

katex is not defined

katex is not defined

katex is not defined katex is not defined

Substituting katex is not defined for katex is not defined does not make the equation true, so katex is not defined is not a solution of the equation katex is not defined

Step-by-step guide: Substitution

  • Math equations also include multi step equations and equations with fractions.

One step equations are algebraic equations with a single variable that can be solved with one step.

In order to solve, you will need to isolate the variable (get it alone) on one side of the equation. This can be done with a model or by using the inverse operation.

For example, katex is not defined

Using a model:Using inverse operations:

Math Equations table image 1

Sort the katex is not defined tiles so an equal amount
are with each katex is not defined tile.

Math Equations table image 2

There are five katex is not defined tiles for each katex is not defined tile.
katex is not defined

katex is not defined

Get the variable alone by using the
inverse operation. The inverse
operation of multiplication is division.
Divide katex is not defined on both sides of the equation.

katex is not defined

Since katex is not defined the left side can be written
as katex is not defined or katex is not defined

The right side of the equation can be
written as katex is not defined

katex is not defined

You can check your answer by
substituting the value for katex is not defined into the
original equation.

katex is not defined











Step-by-step guide: One step equations

  • Solving equations with fractions is when the unknown is part of the numerator and/or denominator of a fraction.

To solve equations with fractions, apply the inverse operation to both sides of the equation – a strategy also referred to as the “balance method.”

The inverse operation of division is multiplication.

For example, katex is not defined

katex is not defined *Multiply each side by katex is not defined

katex is not defined

katex is not defined *Subtract katex is not defined from both sides.

katex is not defined

Check your work by substituting the answer into the equation.

Math Equations image 3 US

Step-by-step guide: Solve equations with fractions

What are math equations?

What are math equations?

[FREE] Math Equations Worksheet (Grade 6 to 12)

[FREE] Math Equations Worksheet (Grade 6 to 12)

[FREE] Math Equations Worksheet (Grade 6 to 12)

Test your grades 6 to 12 students’ skills on how to solve math equations with this 15-question worksheet complete with answers for easy assessment. Save time and quickly identify learning gaps!

DOWNLOAD FREE
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[FREE] Math Equations Worksheet (Grade 6 to 12)

[FREE] Math Equations Worksheet (Grade 6 to 12)

[FREE] Math Equations Worksheet (Grade 6 to 12)

Test your grades 6 to 12 students’ skills on how to solve math equations with this 15-question worksheet complete with answers for easy assessment. Save time and quickly identify learning gaps!

DOWNLOAD FREE

Common Core State Standards

How does this relate to 6th grade math, 7th grade math, and 8th grade math?

  • Grade 6 – Expressions and Equations (6.EE.A.2c)
    Write, read, and evaluate expressions in which letters stand for numbers.

  • Grade 6 – Expressions and Equations (6.EE.B.5)
    Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

  • Grade 7 – Expressions and Equations (7.EE.A.1)
    Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

  • Grade 8 – Expressions and Equations (8.EE.C.7)
    Solve linear equations in one variable.

  • Grade 8 – Expressions and Equations (8.EE.C.7b)
    Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

How to solve with math equations

There are a lot of ways to use math equations. For more specific step-by-step guides, check out the pages linked in the “What are math equations?” section above or read through the examples below.

Math equations examples

Example 1: volume of a cube

Find the volume of the cube using the formula, katex is not defined or katex is not defined

Math Equations image 5 US

  1. Use the formula given in the question.

Use the formula katex is not defined to find the volume of the cube.

The sides of the cube are all 8. Substitute the measurements into the given formula.

katex is not defined

2Work carefully to answer the question, one step at a time.

katex is not defined

3Write the final answer clearly.

The final answer is katex is not defined unitskatex is not defined

Example 2: substitution basic inequality

Does katex is not defined make the inequality katex is not defined true?

Substitute each variable with its given value.

Show step

Solve using the order of operations.

Show step

Example 3: substitution equation

Find the value of katex is not defined when katex is not defined and katex is not defined

Substitute each variable with its given value.

Show step

Solve using the order of operations.

Show step

Example 4: solve for the variable using addition/subtraction

Solve the equation for katex is not defined

katex is not defined

Identify the inverse operation to use.

Show step

Do the inverse operation to both sides of the equation.

Show step

Solve for the unknown variable.

Show step

Check the answer.

Show step

Example 5: solve the equation using multiplication/division

Solve the equation, katex is not defined for katex is not defined

Identify the inverse operation to use.

Show step

Do the inverse operation on both sides of the equation.

Show step

Solve for the unknown variable.

Show step

Check the answer.

Show step

Example 6: solving equations with fractions – two steps

Solve for katex is not defined

Identify the operations that are being applied to the unknown variable.

Show step

Apply the inverse operations, one at a time, to both sides of the equation.

Show step

Write the final answer, checking that it is correct.

Show step

[FREE] Math Equations Worksheet (Grade 6 to 8)

[FREE] Math Equations Worksheet (Grade 6 to 8)

[FREE] Math Equations Worksheet (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of math equations. 10+ questions with answers covering grades 6, 7 and 8 math equations topics to diagnose learning gaps!

DOWNLOAD FREE
x
[FREE] Math Equations Worksheet (Grade 6 to 8)

[FREE] Math Equations Worksheet (Grade 6 to 8)

[FREE] Math Equations Worksheet (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of math equations. 10+ questions with answers covering grades 6, 7 and 8 math equations topics to diagnose learning gaps!

DOWNLOAD FREE

Teaching tips for math equations

  • Instead of introducing this topic through worksheets and rote memorization of the rules, use manipulatives, such as algebra tiles. This can help students build conceptual understanding of these abstract concepts.

    However, it is important not to rush the work with these tools – while it may take more time in the short term to develop understanding with manipulatives versus memorizing a procedure, in the long term students will be more flexible and more easily able to adapt their strategies to more complex topics.

    **Note: It is also useful to discuss the limitations of algebra tiles with students, such as they can only be used to represent smaller integers.

  • Whenever possible, consider using project based learning or real-life assessments, such as having students measure items within the classroom and then using the formulas to calculate the area of the items.

  • When teaching students how to solve math problems such as these, the traditional focus is often solely on the solution of an equation. While this is a crucial piece, also give students time to explore what numbers are not a part of the solution set as well.

    In middle school, students are still building foundational ideas around algebraic expressions, equations and inequalities, so it is important not to always rush them to the answer.

  • Always have students check to make sure their answer is correct by substituting the value back into the original equation. This helps them catch simple mistakes and gives them practice substituting values into the equation.

Easy mistakes to make

  • Thinking the variable can only be on the left-hand side
    While the variable is commonly seen on the left side of the equation, it can also be on the right-hand side.

    For example,
    katex is not defined

  • Confusing the difference between a mathematical expression and a mathematical equation
    A mathematical expression is a statement that does not include an equal sign, like katex is not defined A mathematical equation is a complete mathematical sentence that includes an equals sign, like katex is not defined

  • Forgetting to follow the order of operations (PEMDAS)
    When substituting values into an equation to solve, always follow the order of operations: parentheses first, then exponents, then from left to right multiplication and division, and then from left to right addition and subtraction.

    For example,
    Solve katex is not defined when katex is not defined

    Math Equations image 9 US

  • Forgetting that the solution to an equation can be any type of number
    The unknowns do not have to be whole numbers or integers. In fact, the solutions often involve fractions or decimals.

Practice math equations questions

1. Find the area of the triangle using the appropriate formula.

 

Math Equations image 10 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

Use the formula katex is not defined to find the area of the triangle.

 

Math Equations image 11 US

 

Substitute the measurements into the formula and solve.

 

katex is not defined

 

The area of the triangle is katex is not defined

2. Find the value of katex is not defined when katex is not defined

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

Substitute each variable with its given value.

 

katex is not defined

 

Solve using the order of operations.

 

katex is not defined

3. Which number is a solution for katex is not defined

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

Substitute each variable with its given value and then read each inequality:

 

katex is not defined is greater than katex is not defined

 

katex is not defined is greater than katex is not defined

 

katex is not defined is greater than katex is not defined

 

katex is not defined is greater than katex is not defined

 

Substituting katex is not defined for katex is not defined makes the inequality true, so katex is not defined is a solution for katex is not defined

4. Solve the equation for katex is not defined

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined The inverse operation is division, so divide both sides of the equation by katex is not defined

 

katex is not defined

 

On the left side of the equation: katex is not defined and katex is not defined

 

On the right side of the equation: katex is not defined

 

katex is not defined

 

Check:

 

Math Equations image 12 US

5. Solve the equation for katex is not defined

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined The inverse operation is addition, so add katex is not defined to both sides of the equation.

 

katex is not defined

 

On the left side of the equation: katex is not defined so katex is not defined is left.

 

On the right side of the equation: katex is not defined

 

katex is not defined

 

Check:

 

Math Equations image 13 US

6. Solve: katex is not defined

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

First, multiply both sides of the equation by katex is not defined

 

Next, add katex is not defined to both sides.

 

Finally, divide both sides by katex is not defined

 

Math Equations image 14 US

 

The final answer is katex is not defined

 

You can check the answer by substituting the answer back into the original equation.

 

katex is not defined

Math equations FAQs

Why are math formulas important?

Math formulas make work in mathematics more efficient.

While students always spend ample time working to build ideas around a mathematical concept, formalizing it with the use of a formula (like the area of a triangle as katex is not defined allows them to move forward in their understanding and answer more complex questions.

How can shape patterns be helpful when learning about math formulas?

Growing shape patterns start with a shape and grow in a constant way from term to term. Students learn how to write general rules for these patterns. For example, Start with 2 circles and add 3 circles each time.” This can then be translated into a math equation for the pattern: katex is not defined, where n = the pattern term number and t = the total number of circles.

See also: Shape patterns

When do students learn the Pythagorean theorem?

Students will learn about the Pythagorean theorem in the 8th grade.

What types of equations do students learn about in mathematics?

Students start with simple equations and math formulas in elementary school and then progress to solving linear equations and systems of equations in middle school.

In high school, students continue to expand their knowledge by learning more advanced equations such as quadratic equations (including the quadratic formula), exponential equations, and trigonometric equations.

As they move into college level courses such as precalculus and calculus, they will solve rational equations, logarithm equations, parametric equations and differential equations

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

One on one math tuition

Find out how we can help your students achieve success with our math tutoring programs.

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