Algebraic Thinking and Equations

Variables and Unknowns are important ideas in math because they help us talk about numbers we do not know yet.

Sometimes in math, a number is missing. Instead of leaving a blank space, we can use a letter or a symbol to stand for that number. That letter or symbol is called a variable.

A variable can represent:

• an unknown number we need to find, or
• a number that can change.

For example, in the equation \(x + 5 = 12\), the letter \(x\) stands for a number we do not know yet. Our job is to figure out what number makes the equation true.

In math, an unknown is the value we are trying to find. A variable is the letter or symbol that stands for that unknown value.

So in \(x + 5 = 12\):

• \(x\) is the variable.
• The unknown is the number that \(x\) represents.

Why do we use variables?

• They help us write math problems clearly.
• They help us solve missing-number problems.
• They help us describe patterns and rules.

Think of a variable like a labeled box. The box has a name, such as \(n\) or \(y\), and inside the box is a number. We do not know the number at first, but we can use clues to find it.

Main Idea 1: A letter can stand for a missing number.

You may have seen problems like:

$$\Box + 4 = 9$$

We can replace the box with a letter:

$$a + 4 = 9$$

Now \(a\) is the variable. It stands for the missing number.

To solve it, ask: What number plus 4 equals 9?

Since \(5 + 4 = 9\), we know:

$$a = 5$$

Main Idea 2: A variable can change.

Sometimes a variable does not stand for just one number. It can stand for different numbers at different times.

For example, if each notebook costs \(\$3\), then the total cost depends on how many notebooks you buy.

We can write:

$$3n$$

Here, \(n\) is the number of notebooks. If \(n = 2\), then \(3n = 6\). If \(n = 5\), then \(3n = 15\).

This means the variable \(n\) can change.

Main Idea 3: An equation is like a balance.

An equation is a math sentence with an equals sign. The two sides must have the same value.

For example:

$$m + 7 = 15$$

This says the value on the left side must equal the value on the right side.

To find \(m\), think: What number plus 7 equals 15?

The answer is \(8\), so:

$$m = 8$$

We can check by putting \(8\) back into the equation:

$$8 + 7 = 15$$

Since that is true, our answer is correct.

Main Idea 4: Solve by using the opposite operation.

When we solve for a variable, we often use the opposite operation.

• Addition and subtraction are opposites.
• Multiplication and division are opposites.

If a number is added, subtract to find the variable.

If a number is multiplied, divide to find the variable.

For example, in \(b + 6 = 10\), 6 is added to \(b\). To find \(b\), subtract 6 from 10.

$$b = 10 - 6 = 4$$

For example, in \(4c = 20\), \(c\) is multiplied by 4. To find \(c\), divide 20 by 4.

$$c = 20 \div 4 = 5$$

Worked Example 1

Solve:

$$x + 3 = 11$$

Step 1: Ask, what number plus 3 equals 11?

Step 2: Use subtraction.

$$x = 11 - 3$$

$$x = 8$$

Check:

$$8 + 3 = 11$$

The equation is true, so \(x = 8\).

Worked Example 2

Solve:

$$y - 4 = 9$$

Step 1: Ask, what number minus 4 equals 9?

Step 2: Use the opposite operation. Since 4 was subtracted, add 4.

$$y = 9 + 4$$

$$y = 13$$

Check:

$$13 - 4 = 9$$

So \(y = 13\).

Worked Example 3

Solve:

$$5n = 35$$

Step 1: Ask, 5 times what number equals 35?

Step 2: Use division.

$$n = 35 \div 5$$

$$n = 7$$

Check:

$$5 \times 7 = 35$$

So \(n = 7\).

Worked Example 4

A pack of markers costs \(\$2\) each. Write an expression for the cost of \(m\) packs. Then find the cost when \(m = 4\).

Step 1: Each pack costs 2 dollars.

If there are \(m\) packs, the total cost is:

$$2m$$

Step 2: Replace \(m\) with 4.

$$2m = 2(4)$$

$$2m = 8$$

So the total cost is 8 dollars.

This example shows that a variable can stand for different numbers. If \(m\) changes, the total cost changes too.

Tips for Working with Variables

• Read the equation slowly.
• Find the variable. Ask, “What letter stands for the number?”
• Think about what operation is happening: addition, subtraction, multiplication, or division.
• Use the opposite operation to solve.
• Always check your answer by putting it back into the equation.

Common Mistakes to Avoid

• Mixing up the variable and the answer: In \(x + 2 = 6\), \(x\) is not 2. \(x\) is the number that makes the equation true.
• Forgetting to check: Always test your answer.
• Not using the opposite operation: If the problem says \(k - 3 = 5\), do not subtract 3 again. Add 3 instead.

Let’s look at a pattern.

Suppose you make a pattern by adding 2 each time:

2, 4, 6, 8, 10

If the step number is \(n\), one rule for the pattern is:

$$2n$$

If \(n = 1\), then \(2n = 2\).

If \(n = 2\), then \(2n = 4\).

If \(n = 3\), then \(2n = 6\).

Here, the variable \(n\) helps us describe the pattern with a rule.

Summary

A variable is a letter or symbol that stands for a number. It can represent an unknown number or a number that can change.

We use variables in equations and expressions to show relationships between numbers. To solve for a variable, use the opposite operation and then check your answer.

When you see a letter in math, do not be afraid. It is just a way to name a number we need to find or a number that can change.

Writing Algebraic Expressions means turning word phrases into math expressions that use numbers, operation signs, and sometimes a variable.

A variable is a letter that stands for an unknown number. For example, in the expression \(x + 4\), the letter \(x\) can be any number.

Learning to write algebraic expressions helps you describe patterns and relationships in a short, clear way. It is an important step from regular arithmetic into algebra.

Important: An expression does not have an equals sign. It shows a value. For example, \(n - 2\) is an expression.

An equation does have an equals sign. For example, \(n - 2 = 5\) is an equation.

When you write algebraic expressions, you listen for clue words that tell you which operation to use.

• Addition: sum, plus, added to, more than, increased by
• Subtraction: difference, minus, less than, decreased by, fewer than
• Multiplication: product, times, multiplied by, groups of
• Division: quotient, divided by, split into equal groups

Many algebraic expressions start with a variable because the number is unknown.

Here are some common phrase patterns:

• a number plus 5 becomes \(n + 5\)
• 7 less than a number becomes \(n - 7\)
• 3 times a number becomes \(3n\)
• a number divided by 4 becomes \(n \div 4\)

Be careful with word order. Some phrases do not go in the same order as the words.

For example, 5 more than a number means start with the number, then add 5:

But 5 less than a number means start with the number, then subtract 5:

The words more than and less than can make the order tricky, so read carefully.

Let us look at a simple step-by-step method.

1. Find the unknown number. Use a variable like \(n\), \(x\), or \(m\).
2. Find the operation the words are describing.
3. Put the parts together in the correct order.
4. Check that there is no equals sign, because you are writing an expression.

Worked Example 1

Write an algebraic expression for: a number plus 8.

Step 1: The unknown number can be \(n\).

Step 2: The words plus 8 tell us to add.

Step 3: Write the expression:

So, the algebraic expression is \(n + 8\).

Worked Example 2

Write an algebraic expression for: 6 times a number.

Step 1: Let the number be \(x\).

Step 2: The words 6 times tell us to multiply.

Step 3: Write the expression:

This means \(6 \times x\). In algebra, we usually write multiplication without the \(\times\) sign when a number and a variable are together.

So, the algebraic expression is \(6x\).

Worked Example 3

Write an algebraic expression for: 4 less than a number.

This one is a little trickier because of the words less than.

Step 1: Let the number be \(y\).

Step 2: 4 less than a number means subtract 4 from the number.

Step 3: Write the expression:

So, the algebraic expression is \(y - 4\).

Worked Example 4

Write an algebraic expression for: the quotient of a number and 3.

Step 1: Let the number be \(m\).

Step 2: The word quotient tells us to divide.

Step 3: Write the expression:

So, the algebraic expression is \(m \div 3\).

Here are some more phrase-to-expression matches to help you learn the pattern:

• 2 more than a number → \(n + 2\)
• 9 minus a number → \(9 - n\)
• a number decreased by 7 → \(n - 7\)
• the product of 5 and a number → \(5n\)
• a number divided by 2 → \(n \div 2\)

Watch out for these common mistakes:

• Mixing up the order in subtraction.
  For example, 3 less than a number is \(n - 3\), not \(3 - n\).
• Writing an equation instead of an expression.
  Do not add an equals sign unless the problem asks for an equation.
• Forgetting the variable.
  If the number is unknown, use a letter to stand for it.

One good way to check your work is to replace the variable with a number and see if the words still make sense.

For example, for 5 less than a number, we wrote \(n - 5\). If \(n = 12\), then:

That makes sense because 7 is 5 less than 12.

Now let us review the big idea.

Words like sum, difference, product, and quotient tell you which operation to use. A variable stands for the unknown number. Then you put the pieces together to write the algebraic expression.

Summary

To write an algebraic expression, first choose a variable for the unknown number. Next, look for clue words that tell whether to add, subtract, multiply, or divide. Then write the expression in the correct order, remembering that an expression does not have an equals sign.

Evaluating Expressions means finding the value of an expression when you know what number a variable stands for.

An expression is a math phrase with numbers, operation symbols, and sometimes a variable. A variable is a letter that stands for a number.

For example, in the expression \(x + 4\), the letter \(x\) is the variable. If \(x = 3\), then we can replace \(x\) with 3 and find the value.

This is called substituting. Substituting means putting the given number in place of the variable.

Here is the big idea:

• Find the variable.
• Replace it with the number you are given.
• Do the math carefully.

When evaluating expressions, it helps to follow the order the expression shows. Parentheses can help you keep track of what to replace.

For example, if \(a = 5\), then:

The value of the expression is 7.

Let’s look at the steps more closely.

1. Read the expression. Notice the variable and the operation.
2. Substitute. Replace the variable with its value.
3. Evaluate. Solve the number sentence.

Sometimes the variable appears more than once. If that happens, replace every copy of the variable with the same number.

For example, if \(n = 4\), then in the expression \(n + n\), both variables become 4:

Now let’s work through some examples.

Example 1: One-step addition

Evaluate \(m + 6\) when \(m = 2\).

Step 1: Substitute 2 for \(m\).

Step 2: Add.

Answer: The value of the expression is 8.

Example 2: Multiplication

Evaluate \(3x\) when \(x = 5\).

In math, \(3x\) means \(3 \times x\).

Step 1: Substitute 5 for \(x\).

Step 2: Multiply.

Answer: The value of the expression is 15.

Example 3: Variable used more than once

Evaluate \(y + 2y\) when \(y = 4\).

Step 1: Replace each \(y\) with 4.

Step 2: Multiply first, then add.

Answer: The value of the expression is 12.

Example 4: Expression with parentheses

Evaluate \(2(b + 3)\) when \(b = 1\).

Step 1: Substitute 1 for \(b\).

Step 2: Solve inside the parentheses first.

Step 3: Multiply.

Answer: The value of the expression is 8.

Here are some helpful reminders.

• A variable is just a placeholder. It stands for a number.
• Substitute carefully. Make sure you replace every variable.
• Watch the operation signs. Add, subtract, multiply, or divide correctly.
• Use parentheses if needed. They help show the substitution clearly.

Students sometimes make a few common mistakes when evaluating expressions.

• Mistake 1: Replacing only one variable when the same variable appears more than once.
• Mistake 2: Forgetting that a number next to a variable means multiplication. For example, \(4n\) means \(4 \times n\).
• Mistake 3: Doing operations in the wrong order, especially when parentheses are used.

Let’s fix a common mistake together.

Suppose you evaluate \(2k + 1\) when \(k = 3\).

A wrong way would be to think \(2k\) means 23. That is not correct.

The correct way is:

So the value is 7.

You can think of evaluating expressions like filling in a missing part of a recipe. The variable tells you where to put the number, and then you do the math.

Try these on your own:

• Evaluate \(p + 9\) when \(p = 1\).
• Evaluate \(4r\) when \(r = 2\).
• Evaluate \(c + c + 5\) when \(c = 3\).
• Evaluate \(3(d - 1)\) when \(d = 6\).

Answers:

• \(p + 9 = 1 + 9 = 10\)
• \(4r = 4 \times 2 = 8\)
• \(c + c + 5 = 3 + 3 + 5 = 11\)
• \(3(d - 1) = 3(6 - 1) = 3(5) = 15\)

Summary

To evaluate an expression, replace the variable with the given number and then solve. Always substitute for every variable, and do the operations carefully. With practice, evaluating expressions becomes quick and easy.

Balancing Equations means keeping both sides of an equation equal. Think of the equal sign, =, like the middle of a balance scale. Whatever is on the left side must have the same value as whatever is on the right side.

Many students first see the equal sign as meaning “the answer is.” But in algebraic thinking, the equal sign really means “is the same as”. That means both sides must match.

For example, in the equation \(7 + 3 = 10\), the left side has a value of 10, and the right side also has a value of 10. The equation is balanced.

In this lesson, you will learn how to check whether an equation is balanced and how to find a missing number that makes an equation true.

1. The equal sign means both sides are the same

An equation has two sides:

• the left side
• the right side

If both sides have the same value, the equation is balanced.

Here are some balanced equations:

• \(4 + 5 = 9\)
• \(12 = 7 + 5\)
• \(3 + 6 = 4 + 5\)

Notice that the answer does not always have to be on the right side. The equal sign only tells us that the two sides are equal.

2. Think of a balance scale

Imagine a balance scale at a playground or science center. If both sides have the same weight, the scale stays level. If one side is heavier, the scale tips.

Equations work the same way. If both sides have the same value, they balance. If not, they are not true equations.

For example:

\(8 + 2 = 6 + 4\)

Left side: \(8 + 2 = 10\)

Right side: \(6 + 4 = 10\)

Both sides are 10, so the equation is balanced.

Now look at this one:

\(8 + 2 = 9 + 4\)

Left side: \(8 + 2 = 10\)

Right side: \(9 + 4 = 13\)

Since 10 is not the same as 13, the equation is not balanced.

3. A missing number can make an equation balanced

Sometimes one number is missing. Your job is to find the number that makes both sides equal.

Example:

\(5 + \Box = 12\)

Ask yourself: What number added to 5 equals 12?

Since \(5 + 7 = 12\), the missing number is 7.

So the balanced equation is:

$$5 + 7 = 12$$

You can also have a missing number on the other side.

Example: \(14 = 9 + \Box\)

Ask: What number added to 9 equals 14?

The answer is 5, because \(9 + 5 = 14\).

4. You can compare both sides step by step

When an equation has numbers on both sides, solve each side first. Then compare the two values.

For example:

\(15 - 6 = 4 + 5\)

Left side: \(15 - 6 = 9\)

Right side: \(4 + 5 = 9\)

Both sides are 9, so the equation is balanced.

This is a good habit:

1. Solve the left side.
2. Solve the right side.
3. Check whether they are equal.

5. What happens when a number is missing on one side?

Sometimes you will see equations like this:

\(10 + \Box = 6 + 8\)

First solve the side you can solve easily:

\(6 + 8 = 14\)

Now the equation is:

\(10 + \Box = 14\)

What number makes 10 become 14? The answer is 4.

So:

$$10 + 4 = 6 + 8$$

6. Balanced equations can use addition or subtraction

You can balance equations with addition, subtraction, or a mix of both.

• \(7 + 2 = 9\)
• \(13 - 4 = 9\)
• \(7 + 2 = 13 - 4\)

All of these are balanced because each side equals 9.

Worked Example 1

Is \(6 + 3 = 10 - 1\) balanced?

Step 1: Solve the left side.

\(6 + 3 = 9\)

Step 2: Solve the right side.

\(10 - 1 = 9\)

Step 3: Compare.

Both sides equal 9, so yes, the equation is balanced.

Worked Example 2

Find the missing number: \(\Box + 4 = 11\)

Step 1: Ask what number plus 4 equals 11.

\(7 + 4 = 11\)

Step 2: Check.

Left side: \(7 + 4 = 11\)

Right side: 11

The missing number is 7.

Worked Example 3

Find the missing number: \(18 - \Box = 12\)

Step 1: Ask what number must be subtracted from 18 to get 12.

Since \(18 - 6 = 12\), the missing number is 6.

Step 2: Check.

Left side: \(18 - 6 = 12\)

Right side: 12

The equation is balanced when the missing number is 6.

Worked Example 4

Find the missing number: \(9 + \Box = 5 + 8\)

Step 1: Solve the right side.

\(5 + 8 = 13\)

Now we have:

\(9 + \Box = 13\)

Step 2: Find the missing number.

\(9 + 4 = 13\)

Step 3: Check both sides.

Left side: \(9 + 4 = 13\)

Right side: \(5 + 8 = 13\)

So the missing number is 4.

Tips for balancing equations

• Remember that = means “the same as.”
• Look at both sides of the equation.
• Solve each side carefully.
• If there is a missing number, ask what makes both sides equal.
• Always check your answer at the end.

Common mistake to avoid

A common mistake is thinking the equal sign means “put the answer next.” For example, a student might think only equations like \(3 + 4 = 7\) are correct.

But equations like \(7 = 3 + 4\) and \(2 + 5 = 6 + 1\) are also correct, because both sides have the same value.

Quick check

• Is \(4 + 6 = 3 + 7\) balanced? Yes, both sides equal 10.
• What makes \(11 = 8 + \Box\) true? The missing number is 3.
• Is \(14 - 5 = 6 + 2\) balanced? No, because 9 is not the same as 8.

Summary

Balancing equations means making sure both sides of the equal sign have the same value. The equal sign means “is the same as”, not just “here comes the answer.”

To solve balancing equations, find the value on each side and compare them. If a number is missing, choose the number that makes both sides equal. When both sides match, the equation is balanced.

Solving One-Step Equations

Sometimes in math, we do not know a number yet. We can use a variable to stand for that unknown number. A variable is often a letter, like \(x\) or \(n\).

An equation is a math sentence with an equals sign. It shows that two amounts are the same. For example, \(x + 4 = 9\) is an equation.

When we solve an equation, we figure out the value of the variable that makes the equation true.

A one-step equation can be solved with just one operation. We use the opposite operation, called an inverse operation, to find the missing value.

Think of an equation like a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced.

The four main inverse operations are:

• Addition and subtraction are opposites.
• Multiplication and division are opposites.

Here is a helpful guide:

• If the variable has a number being added, subtract.
• If the variable has a number being subtracted, add.
• If the variable is being multiplied by a number, divide.
• If the variable is being divided by a number, multiply.

Goal: Get the variable by itself on one side of the equation.

Let’s look at each type.

1. Solving with addition

If a number is added to the variable, subtract that number from both sides.

Worked Example 1

Solve \(x + 5 = 12\).

The variable has 5 added to it. Use the opposite operation: subtract 5.

Check: Put 7 back into the equation.

This is true, so \(x = 7\).

2. Solving with subtraction

If a number is subtracted from the variable, add that number to both sides.

Worked Example 2

Solve \(m - 3 = 10\).

The variable has 3 subtracted from it. Use the opposite operation: add 3.

Check:

This is true, so \(m = 13\).

3. Solving with multiplication

If the variable is multiplied by a number, divide both sides by that number.

Worked Example 3

Solve \(4n = 20\).

The variable is multiplied by 4. Use the opposite operation: divide by 4.

Check:

This is true, so \(n = 5\).

4. Solving with division

If the variable is divided by a number, multiply both sides by that number.

Worked Example 4

Solve \(\frac{y}{6} = 3\).

The variable is divided by 6. Use the opposite operation: multiply by 6.

Check:

This is true, so \(y = 18\).

How to solve a one-step equation

1. Look at the operation with the variable.
2. Choose the inverse operation.
3. Do that operation to both sides of the equation.
4. Write the answer.
5. Check by plugging the answer back in.

Example of thinking:

• In \(a + 8 = 15\), 8 is added, so subtract 8.
• In \(b - 9 = 2\), 9 is subtracted, so add 9.
• In \(3c = 21\), \(c\) is multiplied by 3, so divide by 3.
• In \(\frac{d}{4} = 7\), \(d\) is divided by 4, so multiply by 4.

Common mistakes to avoid

• Only changing one side: Always do the same thing to both sides.
• Using the wrong opposite operation: Check whether you should add, subtract, multiply, or divide.
• Forgetting to check: Put your answer back into the original equation to see if it works.

Quick Practice

• \(p + 6 = 14\)
• \(q - 7 = 5\)
• \(5r = 35\)
• \(\frac{s}{8} = 4\)

Answers

• \(p = 8\)
• \(q = 12\)
• \(r = 7\)
• \(s = 32\)

Summary

To solve a one-step equation, get the variable alone by using the opposite operation. Remember to do the same thing to both sides so the equation stays balanced. Then check your answer to make sure it makes the equation true.

Order of Operations Introduction

Sometimes a math problem has more than one thing to do. You might need to add and subtract, or solve something inside parentheses first. Order of operations means doing the steps in the correct order.

This is important because if we do the steps in a different order, we can get the wrong answer.

For 3rd Grade, the most important rule to remember is this:

• Do what is inside parentheses first.
• Then solve the rest from left to right.

Parentheses look like this: \,( \)

They tell us, “Do this part first!”

Look at these two problems:

$$2 + 3 + 4$$

and

$$2 + (3 + 4)$$

In the second problem, the parentheses tell us to add 3 and 4 first.

Let’s solve both:

• Without parentheses: \(2 + 3 = 5\), then \(5 + 4 = 9\)
• With parentheses: \(3 + 4 = 7\), then \(2 + 7 = 9\)

These happen to have the same answer, but the order is different.

Now look at a subtraction example:

$$10 - 3 - 2$$

and

$$10 - (3 - 2)$$

These will not have the same answer, so following the order is very important.

How to solve expressions with parentheses

1. Look for parentheses.
2. Solve what is inside the parentheses first.
3. Then finish the problem from left to right.

Worked Example 1

Solve:

$$5 + (2 + 1)$$

Step 1: Solve inside the parentheses.

$$2 + 1 = 3$$

Step 2: Replace the parentheses part with 3.

$$5 + 3$$

Step 3: Add.

$$5 + 3 = 8$$

Answer: \(8\)

Worked Example 2

Solve:

$$12 - (4 + 3)$$

Step 1: Solve inside the parentheses.

$$4 + 3 = 7$$

Step 2: Replace the parentheses part with 7.

$$12 - 7$$

Step 3: Subtract.

$$12 - 7 = 5$$

Answer: \(5\)

Worked Example 3

Solve:

$$15 - (6 - 2)$$

Step 1: Solve inside the parentheses.

$$6 - 2 = 4$$

Step 2: Replace the parentheses part with 4.

$$15 - 4$$

Step 3: Subtract.

$$15 - 4 = 11$$

Answer: \(11\)

Worked Example 4

Solve:

$$8 + (5 - 3) + 2$$

Step 1: Solve inside the parentheses.

$$5 - 3 = 2$$

Step 2: Replace the parentheses part with 2.

$$8 + 2 + 2$$

Step 3: Solve from left to right.

First:

$$8 + 2 = 10$$

Then:

$$10 + 2 = 12$$

Answer: \(12\)

Why parentheses matter

Parentheses can change the answer. They tell us which numbers belong together first.

Compare these:

$$14 - 6 + 1$$

From left to right:

$$14 - 6 = 8$$

$$8 + 1 = 9$$

Answer: \(9\)

Now compare with:

$$14 - (6 + 1)$$

Do parentheses first:

$$6 + 1 = 7$$

Then:

$$14 - 7 = 7$$

Answer: \(7\)

The answers are different, so the parentheses changed what we do first.

Tips to remember

• Look carefully before you start solving.
• If you see parentheses, do that part first.
• After that, move from left to right.
• Check your work to make sure you followed the order.

Brief Summary

Order of operations helps us solve math problems the right way. In 3rd Grade, remember: parentheses first, then solve from left to right. Parentheses are important because they can change the answer.

Inequalities on a Number Line

Sometimes in math, we do not want just one answer. We want to show that a number can be part of a range of answers. That is what an inequality does.

An inequality compares two amounts and shows that they are not always equal. It can tell us that a number is greater than, less than, greater than or equal to, or less than or equal to another number.

We often use a letter, like \(x\), to stand for an unknown number. Then we use an inequality symbol to show what values \(x\) can be.

Inequality symbols

• \( > \) means greater than
• \( < \) means less than
• \( \geq \) means greater than or equal to
• \( \leq \) means less than or equal to

For example:

• \(x > 3\) means \(x\) is any number greater than 3
• \(x < 8\) means \(x\) is any number less than 8
• \(x \geq 5\) means \(x\) is 5 or any number greater than 5
• \(x \leq 2\) means \(x\) is 2 or any number less than 2

How to show inequalities on a number line

A number line helps us see all the possible values of a variable. When we graph an inequality, we usually use:

• An open circle when the endpoint is not included
• A closed circle when the endpoint is included
• A shaded line or arrow to show all the values that work

Open circle

Use an open circle for \( > \) or \( < \). This means the number at that point is not part of the answer.

Example: For \(x > 4\), place an open circle at 4. Then shade to the right, because numbers greater than 4 are to the right on the number line.

Closed circle

Use a closed circle for \( \geq \) or \( \leq \). This means the number at that point is part of the answer.

Example: For \(x \leq 6\), place a closed circle at 6. Then shade to the left, because numbers less than 6 are to the left on the number line. Since it says or equal to, 6 is included.

Direction on the number line

• If the inequality says greater than, shade right
• If the inequality says less than, shade left

You can remember this because numbers get bigger as you move right and smaller as you move left.

Quick guide

• \(x > a\): open circle at \(a\), shade right
• \(x < a\): open circle at \(a\), shade left
• \(x \geq a\): closed circle at \(a\), shade right
• \(x \leq a\): closed circle at \(a\), shade left

Worked Example 1

Graph \(x > 2\) on a number line.

1. Find the endpoint: 2
2. The symbol is \( > \), so use an open circle at 2
3. \(x\) is greater than 2, so shade to the right

This means 3, 4, 5, and many more numbers would work. But 2 would not work, because the circle is open.

Worked Example 2

Graph \(x \leq 7\) on a number line.

1. Find the endpoint: 7
2. The symbol is \( \leq \), so use a closed circle at 7
3. \(x\) is less than 7, so shade to the left

This means 7, 6, 5, 4, and smaller numbers all work. Since the circle is closed, 7 is included.

Worked Example 3

Write an inequality for this situation: A student can read for at least 20 minutes.

The words at least mean the number can be 20 or more than 20.

If we use \(m\) for minutes, the inequality is:

$$m \geq 20$$

To graph it:

1. Put a closed circle at 20
2. Shade to the right, because the time can be greater than 20

Worked Example 4

Look at this inequality: \(x < 5\).

Which of these numbers make the inequality true?

\(3, 5, 8, 4\)

1. Check 3: \(3 < 5\) is true
2. Check 5: \(5 < 5\) is false because 5 is equal to 5, not less than 5
3. Check 8: \(8 < 5\) is false
4. Check 4: \(4 < 5\) is true

So the numbers that work are 3 and 4.

Reading word clues

Some words in math problems can help you choose the correct inequality symbol.

• greater than means \( > \)
• less than means \( < \)
• at least means \( \geq \)
• no more than means \( \leq \)
• or equal to means the endpoint is included, so use a closed circle

Common mistakes to avoid

• Do not use a closed circle for \( > \) or \( < \)
• Do not use an open circle for \( \geq \) or \( \leq \)
• Do not shade the wrong direction
• Remember that \(x > 4\) does not include 4, but \(x \geq 4\) does include 4

Let’s compare two inequalities

• \(x > 6\): open circle at 6, shade right
• \(x \geq 6\): closed circle at 6, shade right

These look almost the same, but there is one important difference: the second one includes 6.

How to think about it

Ask yourself two questions:

1. Is the endpoint included or not?
2. Should I shade left or right?

If the endpoint is included, use a closed circle. If it is not included, use an open circle. Then use the inequality symbol to decide the direction.

Summary

An inequality shows a range of possible values. On a number line, use an open circle for \( > \) or \( < \), and use a closed circle for \( \geq \) or \( \leq \). Shade to the right for greater than and to the left for less than. With these steps, you can read, write, and graph inequalities clearly.

Input-Output Functions help us see a rule that connects one number to another. You start with an input, follow a rule, and get an output.

You can think of this like a machine. A number goes in, the machine does something to it, and a new number comes out.

For example, if the rule is add 3, then an input of 4 gives an output of 7, because \(4 + 3 = 7\).

Learning input-output functions helps you notice patterns, describe rules, and predict missing numbers. These are important algebra skills.

Main Idea: In an input-output table, the same rule must work for every pair of numbers in the table.

• Input: the number you start with
• Rule: what you do to the input
• Output: the answer after using the rule

Here is a simple example of an input-output table:

$$\begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array}$$

If you look closely, each output is 3 more than the input. So the rule is add 3.

You can also write the rule with a letter. A letter can stand for any input number. For example, if we use \(n\) for the input, then the rule is:

$$n + 3$$

This means: start with \(n\), then add 3.

How to Find the Rule

1. Look at one input and output pair.
2. Ask: What happened to the input?
3. Check the same idea with the other pairs.
4. If the same rule works every time, you found the rule.

Many 5th grade input-output rules use simple operations like:

• add a number
• subtract a number
• multiply by a number
• sometimes two steps, like multiply and then add

Worked Example 1: Add a Number

Find the rule for this table:

$$\begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 2 & 7 \\ 5 & 10 \\ 8 & 13 \end{array}$$

Look at the first pair: \(2 \to 7\). The output is 5 more than the input.

Check the next pair: \(5 \to 10\). That is also add 5.

Check the last pair: \(8 \to 13\). That is also add 5.

So the rule is add 5.

Using a letter, the rule is:

$$n + 5$$

Worked Example 2: Multiply

Find the rule for this table:

$$\begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 3 & 12 \\ 4 & 16 \\ 6 & 24 \end{array}$$

Look at the first pair: \(3 \to 12\). Since \(3 \times 4 = 12\), the rule might be multiply by 4.

Check: \(4 \times 4 = 16\) and \(6 \times 4 = 24\).

The rule works for every pair, so the rule is multiply by 4.

Using a letter, the rule is:

$$4n$$

This means \(4 \times n\).

Worked Example 3: Two-Step Rule

Sometimes the rule has two steps. Look at this table:

$$\begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 5 \\ 2 & 7 \\ 3 & 9 \\ 4 & 11 \end{array}$$

First, test simple ideas.

Is it add 4? That works for \(1 \to 5\), but not for \(2 \to 7\), because \(2 + 4 = 6\), not 7.

Is it multiply by 2? That gives \(1 \times 2 = 2\), which is not 5.

Now try a two-step rule: multiply by 2, then add 3.

Check each pair:

• \(1 \times 2 + 3 = 5\)
• \(2 \times 2 + 3 = 7\)
• \(3 \times 2 + 3 = 9\)
• \(4 \times 2 + 3 = 11\)

The rule is multiply by 2, then add 3.

Using a letter, the rule is:

$$2n + 3$$

Worked Example 4: Find a Missing Output

Use the table to find the missing output.

$$\begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 2 & 9 \\ 3 & 11 \\ 5 & 15 \\ 7 & ? \end{array}$$

Look for the rule.

Try multiply by 2, then add 5.

• \(2 \times 2 + 5 = 9\)
• \(3 \times 2 + 5 = 11\)
• \(5 \times 2 + 5 = 15\)

The rule is \(2n + 5\).

Now use input 7:

$$7 \times 2 + 5 = 14 + 5 = 19$$

So the missing output is 19.

How to Check Your Thinking

• Make sure the same rule works for every row.
• Do not guess from only one row.
• If one-step rules do not work, try a two-step rule.
• Use the rule on a new input to see if it makes sense.

Helpful Tips

• If the output is always a little more or less than the input, the rule may be adding or subtracting.
• If the output grows much faster, the rule may be multiplying.
• If neither one-step idea works, the rule may have two steps.

Try These On Your Own

1. Find the rule:

$$\begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 6 \\ 2 & 7 \\ 3 & 8 \end{array}$$

2. Find the rule:

$$\begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 2 & 8 \\ 3 & 12 \\ 5 & 20 \end{array}$$

3. Find the missing output:

$$\begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 4 \\ 2 & 6 \\ 3 & 8 \\ 6 & ? \end{array}$$

4. Find the missing input:

$$\begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 7 \\ 2 & 10 \\ ? & 16 \end{array}$$

Answers

1. Add 5, or \(n + 5\)

2. Multiply by 4, or \(4n\)

3. The rule is \(2n + 2\), so the missing output is \(14\)

4. The rule is \(3n + 4\). Solve \(3n + 4 = 16\). Then \(3n = 12\), so \(n = 4\)

Summary

An input-output function uses a rule to turn an input into an output. The most important thing is that the same rule must work every time.

You can find the rule by looking for patterns such as adding, subtracting, multiplying, or using two steps. Once you know the rule, you can find missing inputs, missing outputs, and explain the pattern clearly.

The Coordinate Plane

The coordinate plane is a grid that helps us show the exact location of a point. It uses two number lines that cross each other.

One number line goes side to side. This is called the x-axis. The other number line goes up and down. This is called the y-axis.

The place where the two axes cross is called the origin. The origin has the coordinates \((0, 0)\).

We use the coordinate plane to name and graph points. A point is written as an ordered pair.

An ordered pair looks like this: \((x, y)\).

• The first number tells how far to move on the x-axis.
• The second number tells how far to move on the y-axis.

Order matters. The point \((3, 2)\) is not the same as \((2, 3)\).

In 5th grade, we usually work in the first quadrant. That means both numbers are positive, so points are to the right of the origin and above the origin.

How to read and graph an ordered pair

1. Start at the origin, \((0, 0)\).
2. Look at the first number, the x-coordinate. Move right that many units.
3. Look at the second number, the y-coordinate. Move up that many units.
4. Mark the point.

You can remember it like this: Across first, then up.

Main ideas to remember

• x-axis: the horizontal number line
• y-axis: the vertical number line
• origin: where the axes meet, \((0, 0)\)
• ordered pair: a pair of numbers that names a point, \((x, y)\)
• x-coordinate: the first number
• y-coordinate: the second number

Worked Example 1: Graphing a point

Graph the point \((4, 3)\).

Step 1: Start at the origin, \((0, 0)\).

Step 2: The first number is 4, so move 4 units to the right.

Step 3: The second number is 3, so move 3 units up.

Step 4: Place a dot. That dot is the point \((4, 3)\).

So, \((4, 3)\) means right 4, up 3.

Worked Example 2: Naming a point from a graph

A point is 2 units to the right of the origin and 5 units up. What is its ordered pair?

The x-coordinate is 2 because the point moved 2 units right.

The y-coordinate is 5 because the point moved 5 units up.

So the ordered pair is \((2, 5)\).

Worked Example 3: Comparing two ordered pairs

What is the difference between \((1, 6)\) and \((6, 1)\)?

For \((1, 6)\), move right 1 and up 6.

For \((6, 1)\), move right 6 and up 1.

These points are in different places because the numbers are in a different order.

This shows why the pair is called an ordered pair. The order of the numbers matters.

Worked Example 4: Finding points with the same x- or y-coordinate

Look at the points \((3, 2)\), \((3, 5)\), and \((6, 2)\).

• \((3, 2)\) and \((3, 5)\) have the same x-coordinate, which is 3.
• That means both points are lined up vertically.
• \((3, 2)\) and \((6, 2)\) have the same y-coordinate, which is 2.
• That means both points are lined up horizontally.

This is helpful when reading graphs and spotting patterns.

Tips for success

• Always start at the origin.
• Read the ordered pair from left to right: \((x, y)\).
• Move across first for x.
• Move up for y.
• Check that you did not switch the numbers.

Common mistakes

• Mixing up x and y: Remember, x comes first.
• Starting in the wrong place: Always begin at \((0, 0)\).
• Reversing the numbers: \((2, 4)\) is not the same as \((4, 2)\).
• Counting incorrectly: Move carefully, one unit at a time.

Why the coordinate plane is useful

The coordinate plane helps us describe locations clearly. It can be used to show places on a map, points in a game, or data in math problems.

It also helps students begin to see patterns in numbers and shapes. This is an important part of algebraic thinking.

Quick practice to think about

• What point is right 5 and up 2? Answer: \((5, 2)\)
• What point is right 1 and up 7? Answer: \((1, 7)\)
• Which coordinate in \((4, 6)\) is the x-coordinate? Answer: 4
• Which coordinate in \((4, 6)\) is the y-coordinate? Answer: 6

Summary

The coordinate plane is made of two number lines: the x-axis and the y-axis. We use ordered pairs, written as \((x, y)\), to name points on the plane.

To graph a point, start at the origin, move across for x, and then move up for y. If you remember across first, then up, you will be able to read and graph points correctly.

Graphing Linear Relationships

Sometimes in math, one number changes and another number changes with it. When we match an input to an output, we can put the pairs in a table and then graph them on a coordinate plane.

When the pattern changes in a steady way, it is called a linear relationship. In 5th grade, you can think of this as a pattern that makes points line up in a straight path on a graph.

In this lesson, you will learn how to:

• read input-output tables,
• write ordered pairs,
• plot points on a coordinate plane,
• notice when points make a straight-line pattern.

1. Input and output

An input is the number you start with. An output is the number you get after following a rule.

For example, if the rule is add 2, then:

• input 1 gives output 3,
• input 2 gives output 4,
• input 3 gives output 5.

We can show this in a table.

Example table for the rule add 2:

$$ \begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 3 \\ 2 & 4 \\ 3 & 5 \\ 4 & 6 \end{array} $$

2. Ordered pairs

Each input-output pair can be written as an ordered pair. An ordered pair looks like \((x, y)\).

In 5th grade, you can think of:

• the first number as the input,
• the second number as the output.

From the table above, the ordered pairs are:

• \((1, 3)\)
• \((2, 4)\)
• \((3, 5)\)
• \((4, 6)\)

Important: Order matters. The pair \((2, 5)\) is not the same as \((5, 2)\).

3. The coordinate plane

A coordinate plane is a graph with two number lines:

• the horizontal line,
• the vertical line.

The point where they cross is called the origin. The origin is \((0, 0)\).

To graph an ordered pair \((x, y)\):

1. Start at the origin.
2. Move along the horizontal line to the first number.
3. Then move up to the second number.
4. Place a point.

So to graph \((3, 5)\):

• start at \((0,0)\),
• move right 3,
• move up 5,
• mark the point.

4. What makes a relationship linear?

A relationship is linear when the output changes by the same amount each time the input changes by 1.

For example, look at this table:

$$ \begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 2 \\ 2 & 4 \\ 3 & 6 \\ 4 & 8 \end{array} $$

The outputs go up by 2 each time:

• from 2 to 4 is +2,
• from 4 to 6 is +2,
• from 6 to 8 is +2.

Because the change is steady, the points make a straight-line pattern when graphed.

5. How to graph a linear relationship from a table

Use these steps:

1. Read the input and output values.
2. Write each row as an ordered pair.
3. Plot each ordered pair on the coordinate plane.
4. Look at the pattern of points.
5. If the points line up in a straight path, the relationship is linear.

Worked Example 1: A simple add pattern

The rule is: output = input + 1.

Table:

$$ \begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 0 & 1 \\ 1 & 2 \\ 2 & 3 \\ 3 & 4 \end{array} $$

Step 1: Write the ordered pairs.

• \((0,1)\)
• \((1,2)\)
• \((2,3)\)
• \((3,4)\)

Step 2: Plot the points.

Graph each pair by moving to the input first and the output second.

Step 3: Look for the pattern.

The points follow a straight path upward. This is a linear relationship.

Worked Example 2: Multiplying by 2

The rule is: output = 2 \times input.

Table:

$$ \begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 2 \\ 2 & 4 \\ 3 & 6 \\ 4 & 8 \end{array} $$

Step 1: Write the ordered pairs.

• \((1,2)\)
• \((2,4)\)
• \((3,6)\)
• \((4,8)\)

Step 2: Plot the points.

Place all four points on the graph.

Step 3: Check the outputs.

The outputs increase by 2 each time the inputs increase by 1.

Step 4: Decide if it is linear.

Yes. The change is steady, so the points make a straight-line pattern.

Worked Example 3: Starting at a number other than 0

The rule is: output = input + 4.

Table:

$$ \begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 5 \\ 2 & 6 \\ 3 & 7 \\ 4 & 8 \end{array} $$

Step 1: Write the ordered pairs.

• \((1,5)\)
• \((2,6)\)
• \((3,7)\)
• \((4,8)\)

Step 2: Plot the points.

Graph each point carefully.

Step 3: Notice the pattern.

The outputs go up by 1 each time:

• 5 to 6 is +1,
• 6 to 7 is +1,
• 7 to 8 is +1.

Because the change is the same each time, the relationship is linear.

Worked Example 4: Is this linear?

Look at this table:

$$ \begin{array}{c|c} \text{Input} & \text{Output} \\ \hline 1 & 3 \\ 2 & 5 \\ 3 & 7 \\ 4 & 9 \end{array} $$

Step 1: Check how the outputs change.

• 3 to 5 is +2,
• 5 to 7 is +2,
• 7 to 9 is +2.

Step 2: Write the ordered pairs.

• \((1,3)\)
• \((2,5)\)
• \((3,7)\)
• \((4,9)\)

Step 3: Graph them.

The points line up in a straight path, so this is also a linear relationship.

6. Tips for avoiding mistakes

• Do not switch the numbers in an ordered pair. \((2,5)\) and \((5,2)\) are different points.
• Start at the origin each time you plot a new point.
• Use the input first and the output second.
• Check the pattern in the outputs. If the change is the same each time, it is a good sign the graph is linear.
• Plot carefully so your points land on the correct grid intersections.

7. What to look for on the graph

When you graph a linear relationship, the points show a steady pattern. They may go upward as you move right. They may also form a straight path in another direction. The important idea is that the pattern is straight and steady.

You do not need to know every rule right away. If you can read the table, write ordered pairs, and graph the points correctly, you can understand the relationship.

Quick practice ideas

• Make a table where the rule is add 3.
• Write the ordered pairs from the table.
• Plot the points.
• Ask: Do the points make a straight-line pattern?

Summary

Graphing linear relationships means taking input-output pairs from a table and plotting them on a coordinate plane. First, write each row as an ordered pair \((input, output)\). Then graph the points. If the outputs change by the same amount each time and the points line up in a straight path, the relationship is linear.