Algebraic Thinking and Pattern Generalization

Repeating and Growing Shape Patterns

Patterns are all around us. You can see them in floor tiles, clothes, art, and even in math. A shape pattern is a group of shapes that follows a rule.

In this lesson, you will learn about two kinds of shape patterns: repeating patterns and growing patterns. You will learn how to find the rule, continue the pattern, and explain how the pattern works.

1. What is a repeating shape pattern?

A repeating pattern is a pattern that repeats the same part again and again. The part that repeats is called the pattern unit.

For example, look at this pattern:

🔺 🔵 🔺 🔵 🔺 🔵

The pattern unit is 🔺 🔵 because those two shapes repeat over and over.

To continue a repeating pattern, ask yourself:

• What part repeats?
• What shape comes next in that repeating part?

2. What is a growing shape pattern?

A growing pattern changes in a way that follows a rule. It usually gets bigger by adding shapes each time, but sometimes it can get smaller.

For example:

●    ●●    ●●●    ●●●●

This is a growing pattern because each step has 1 more shape than the step before.

To understand a growing pattern, ask yourself:

• How many shapes are in each step?
• How is each step changing?
• How many shapes are added each time?

3. How to find the rule in a pattern

A rule tells how the pattern works.

For a repeating pattern, the rule tells which shapes repeat and in what order.

For a growing pattern, the rule tells how the pattern changes each time, such as:

• add 1 shape each step
• add 2 squares each step
• add 1 triangle to the end each time

4. Steps for solving shape pattern questions

1. Look carefully at all the shapes.
2. Decide if the pattern is repeating or growing.
3. Find the rule.
4. Use the rule to figure out the missing or next shape.

Worked Example 1: Simple repeating pattern

Pattern: ⬜ ⬛ ⬜ ⬛ ⬜ ___

Step 1: Find the pattern unit.

The shapes repeat as ⬜ ⬛.

Step 2: Continue the pattern.

After ⬜ comes ⬛.

Answer: The missing shape is ⬛.

Worked Example 2: Repeating pattern with more shapes

Pattern: 🔺 🔺 ⚪ 🔺 🔺 ⚪ 🔺 ___ ___

Step 1: Find what repeats.

The pattern unit is 🔺 🔺 ⚪.

Step 2: Continue the unit.

We have 🔺, then the next shapes should be 🔺 and ⚪.

Answer: The missing shapes are 🔺, ⚪.

Worked Example 3: Simple growing pattern

Step 1: 🟦

Step 2: 🟦🟦

Step 3: 🟦🟦🟦

Step 4: ?

Step 1: Count the shapes in each step.

• Step 1 has 1 square
• Step 2 has 2 squares
• Step 3 has 3 squares

Step 2: Find the rule.

The pattern adds 1 square each time.

Step 3: Find Step 4.

Step 4 must have 4 squares.

Answer: Step 4 is 🟦🟦🟦🟦.

Worked Example 4: Growing pattern with a number rule

A shape pattern is made with triangles.

• Step 1 has 2 triangles
• Step 2 has 4 triangles
• Step 3 has 6 triangles
• Step 4 has ?

Step 1: Look at how the numbers change.

The number of triangles goes from 2 to 4 to 6.

Step 2: Find the rule.

Each step adds 2 triangles.

Step 3: Find the next step.

$$ 6 + 2 = 8 $$

Answer: Step 4 has 8 triangles.

5. Ways shapes can change in a pattern

Shape patterns do not always change in the same way. Here are some things to watch for:

• Shape: circle, square, triangle
• Color: red, blue, green
• Size: small, medium, large
• Position: up, down, left, right
• Number: more shapes added each step

For example, a pattern could repeat colors but grow in the number of shapes. Always look carefully at what is changing and what stays the same.

6. Tips for tricky pattern questions

• Do not guess too quickly. Look for the rule first.
• If it repeats, find the smallest part that repeats.
• If it grows, count how many shapes are added each time.
• Check your answer by reading the whole pattern again.

7. Let’s practice thinking

If you see this pattern:

⭐ ◼ ⭐ ◼ ⭐ ◼

It is a repeating pattern because the same two shapes repeat.

If you see this pattern:

Step 1: ⚪

Step 2: ⚪⚪⚪

Step 3: ⚪⚪⚪⚪⚪

This is a growing pattern because the number of circles increases. Here, it grows by 2 circles each step.

8. Summary

A repeating shape pattern has a part that repeats in the same order. A growing shape pattern changes each step by following a rule, often by adding shapes.

When solving pattern problems, first decide what kind of pattern it is. Then find the rule and use it to continue the pattern or fill in missing shapes.

If you remember to look, find the rule, and check your work, you can solve shape pattern questions with confidence.

Arithmetic Number Patterns are number sequences that follow a rule.

A number pattern is a list of numbers that changes in the same way each time. Your job is to find the rule and use it to continue the pattern or find missing numbers.

In 4th grade, the most common arithmetic number patterns use:

• addition — add the same number each time
• subtraction — subtract the same number each time
• multiplication — multiply by the same number each time
• division — divide by the same number each time

When you look at a pattern, ask yourself: What is happening from one number to the next?

For example, in the pattern \(3, 6, 9, 12\), each number goes up by \(3\). The rule is add 3.

In the pattern \(20, 18, 16, 14\), each number goes down by \(2\). The rule is subtract 2.

In the pattern \(2, 4, 8, 16\), each number is multiplied by \(2\). The rule is multiply by 2.

In the pattern \(64, 32, 16, 8\), each number is divided by \(2\). The rule is divide by 2.

How to find the rule

1. Look at the first two numbers.
2. Decide what changed.
3. Check the next pair of numbers.
4. If the same change happens again, that is probably the rule.

Sometimes it helps to write the change between numbers.

Example:

\(5, 10, 15, 20\)

The changes are:

$$5 \rightarrow 10 \; (+5)$$$$10 \rightarrow 15 \; (+5)$$$$15 \rightarrow 20 \; (+5)$$

So the rule is add 5.

Worked Example 1: Adding

Find the next three numbers in the pattern:

\(7, 11, 15, 19, \dots\)

Step 1: Find the change.

\(11 - 7 = 4\)

\(15 - 11 = 4\)

\(19 - 15 = 4\)

The rule is add 4.

Step 2: Continue the pattern.

\(19 + 4 = 23\)

\(23 + 4 = 27\)

\(27 + 4 = 31\)

Answer: The next three numbers are \(23, 27, 31\).

Worked Example 2: Subtracting

Find the missing numbers:

\(30, 25, \square, 15, \square\)

Step 1: Find the rule.

\(30 \rightarrow 25\) means subtract \(5\).

Step 2: Keep subtracting \(5\).

\(25 - 5 = 20\)

\(20 - 5 = 15\)

\(15 - 5 = 10\)

Answer: The missing numbers are \(20\) and \(10\).

Worked Example 3: Multiplying

Find the next two numbers:

\(3, 6, 12, 24, \dots\)

Step 1: Look at the change.

\(3 \rightarrow 6\) means multiply by \(2\).

\(6 \rightarrow 12\) means multiply by \(2\).

\(12 \rightarrow 24\) means multiply by \(2\).

The rule is multiply by 2.

Step 2: Continue the pattern.

\(24 \times 2 = 48\)

\(48 \times 2 = 96\)

Answer: The next two numbers are \(48\) and \(96\).

Worked Example 4: Finding the Rule

Look at this pattern:

\(81, 27, 9, 3\)

Ask: Are we adding or subtracting the same number? No.

Ask: Are we multiplying or dividing by the same number? Yes.

\(81 \div 3 = 27\)

\(27 \div 3 = 9\)

\(9 \div 3 = 3\)

Rule: divide by \(3\).

Helpful Tips

• If numbers get bigger, the rule may be add or multiply.
• If numbers get smaller, the rule may be subtract or divide.
• Always check more than one pair of numbers.
• If the same change does not work each time, try a different rule.

Let’s compare two patterns

Pattern A: \(4, 8, 12, 16\)

This pattern adds 4 each time.

Pattern B: \(4, 8, 16, 32\)

This pattern multiplies by 2 each time.

Both patterns start with \(4\) and \(8\), but the rules are different. That is why it is important to check all the numbers, not just the first two.

Try this thinking:

• \(9, 18, 27, 36\) → add \(9\)
• \(50, 45, 40, 35\) → subtract \(5\)
• \(5, 10, 20, 40\) → multiply by \(2\)
• \(48, 24, 12, 6\) → divide by \(2\)

Summary

An arithmetic number pattern follows a rule. The rule tells how to move from one number to the next.

To solve number patterns, look for whether the numbers are being added, subtracted, multiplied, or divided by the same amount each time. Then use that rule to find missing numbers or continue the pattern.

Generating Patterns from Rules means making a number pattern by following a rule again and again.

A pattern is a sequence of numbers that changes in a regular way. A rule tells you how to get from one number to the next number.

For example, if the rule is add 3, and you start at 2, the pattern is:

$$2, 5, 8, 11, 14$$

Each number is 3 more than the number before it.

In 4th grade, rules are often simple, like:

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

When you generate a pattern, you must follow the rule in the same order every time.

How to generate a pattern from a rule

1. Read the rule carefully.
2. Find the starting number.
3. Apply the rule to get the next number.
4. Keep repeating the same rule.
5. Check that each step matches the rule.

Let’s look at different kinds of rules.

1. Patterns that add

If the rule says add 4, each new number is 4 more than the one before.

Worked Example 1

Start at 6. Use the rule add 4. Write the next 5 numbers.

Start with 6.

6
6 + 4 = 10
10 + 4 = 14
14 + 4 = 18
18 + 4 = 22
22 + 4 = 26

The pattern is:

$$6, 10, 14, 18, 22, 26$$

You can check the pattern by seeing that each pair of numbers has a difference of 4.

2. Patterns that subtract

If the rule says subtract 5, each new number is 5 less than the one before.

Worked Example 2

Start at 30. Use the rule subtract 5. Write the next 5 numbers.

30
30 - 5 = 25
25 - 5 = 20
20 - 5 = 15
15 - 5 = 10
10 - 5 = 5

The pattern is:

$$30, 25, 20, 15, 10, 5$$

This pattern gets smaller each time because we are subtracting.

3. Patterns that multiply

If the rule says multiply by 2, each new number is twice the number before.

Worked Example 3

Start at 3. Use the rule multiply by 2. Write the next 4 numbers.

3
3 \times 2 = 6
6 \times 2 = 12
12 \times 2 = 24
24 \times 2 = 48

The pattern is:

$$3, 6, 12, 24, 48$$

Be careful: multiplying patterns can grow quickly.

4. Patterns with two steps

Sometimes a rule has more than one step. You must do the steps in the correct order each time.

For example, the rule might be add 6, then subtract 1.

This is the same as adding 5, but it is important to follow the rule exactly as written so you understand each step.

Worked Example 4

Start at 4. Use the rule add 6, then subtract 1. Write the next 4 numbers.

Start with 4.

First new number:

4 + 6 = 10
10 - 1 = 9

Second new number:

9 + 6 = 15
15 - 1 = 14

Third new number:

14 + 6 = 20
20 - 1 = 19

Fourth new number:

19 + 6 = 25
25 - 1 = 24

The pattern is:

$$4, 9, 14, 19, 24$$

Even though the rule had two steps, we repeated the same two steps each time.

How to find missing numbers in a pattern

Sometimes you are given part of a pattern and need to fill in the blanks.

Example:

$$7, 11, \_, 19, 23$$

Look at the change from 7 to 11. That is add 4.

If we keep adding 4:

11 + 4 = 15
15 + 4 = 19
19 + 4 = 23

The missing number is 15.

How to talk about a pattern

When explaining a pattern, you can say:

• what number it starts with
• what rule it follows
• whether the numbers get bigger or smaller

For example:

The pattern starts at 12 and follows the rule subtract 2. The numbers get smaller by 2 each time.

Tips for success

• Go one step at a time. Do not try to jump too far ahead.
• Use the same rule every time. Do not change the rule in the middle.
• Watch the operation. Adding, subtracting, and multiplying make different kinds of patterns.
• Check your work. Make sure every pair of numbers follows the rule.

Common mistakes to avoid

• Adding when the rule says subtract
• Forgetting the starting number
• Doing only part of a two-step rule
• Changing the amount, like adding 3 and then adding 4

Try thinking about these

• Start at 5, add 7: $$5, 12, 19, 26, 33$$
• Start at 40, subtract 8: $$40, 32, 24, 16, 8$$
• Start at 2, multiply by 3: $$2, 6, 18, 54$$

Summary

Generating patterns from rules means using a starting number and following a rule again and again. The rule can tell you to add, subtract, multiply, or do more than one step. If you follow the rule carefully and check each number, you can build the whole pattern correctly.

Constructing Mathematical Arguments means explaining why a math answer makes sense, not just giving the answer.

In 4th grade, a mathematical argument is not a fight. It is a clear math explanation. You tell what you did, why you did it, and how you know it is correct.

When you construct a mathematical argument, you use numbers, words, pictures, and math rules you already know. Your goal is to help someone else understand your thinking.

Why is this important?

• It helps you check your own work.
• It helps others follow your thinking.
• It shows that your answer is reasonable.
• It helps you solve harder problems with confidence.

The parts of a strong mathematical argument

A strong math argument usually has these parts:

1. State the answer. Tell what you think is true.
2. Show the math work. Use numbers, equations, drawings, or models.
3. Explain why. Tell how the math proves your answer.
4. Check that it makes sense. Ask if the answer is reasonable.

You can use sentence starters like these:

• I know this because...
• First, I...
• Then, I...
• This shows that...
• My answer makes sense because...

Important math habits when explaining

• Use correct numbers and symbols.
• Label what the numbers mean when needed.
• Explain each step in order.
• Make sure your words match your math work.
• Do not skip important steps.

Using equations in an argument

An equation can be part of your explanation. For example, if you say there are 4 bags with 6 marbles in each bag, you can write:

$$4 \times 6 = 24$$

Then explain: there are 24 marbles because 4 groups of 6 equals 24.

Using pictures or models in an argument

Sometimes a drawing can help prove your thinking. You might use:

• an array
• a number line
• base-ten blocks
• a bar model
• a simple labeled picture

A drawing is most helpful when you also explain what it shows.

Worked Example 1: Explaining addition

Problem: Mia says that \(38 + 25 = 63\). How can we explain why she is correct?

Step 1: State the answer.

Mia is correct. The sum is 63.

Step 2: Show the math work.

Break apart the numbers into tens and ones:

$$38 = 30 + 8$$

$$25 = 20 + 5$$

Add tens and ones:

$$30 + 20 = 50$$

$$8 + 5 = 13$$

Then add those parts:

$$50 + 13 = 63$$

Step 3: Explain why.

We added the tens and the ones. The tens made 50, and the ones made 13. Together, that is 63.

Step 4: Check that it makes sense.

Since \(38\) is close to \(40\) and \(25\) is close to \(20\), the answer should be close to \(60\). The answer \(63\) is reasonable.

Worked Example 2: Explaining multiplication

Problem: There are 5 boxes with 7 crayons in each box. Prove that there are 35 crayons in all.

Step 1: State the answer.

There are 35 crayons.

Step 2: Show the math work.

This is 5 equal groups of 7:

$$5 \times 7 = 35$$

We can also use repeated addition:

$$7 + 7 + 7 + 7 + 7 = 35$$

Step 3: Explain why.

Multiplication means equal groups. Since there are 5 groups and each group has 7 crayons, multiplying 5 by 7 gives the total number of crayons.

Step 4: Check that it makes sense.

If 5 boxes had 5 crayons each, there would be 25 crayons. Since each box has 7 crayons, the total should be more than 25. So 35 makes sense.

Worked Example 3: Explaining a comparison

Problem: A student says \(402 > 399\). How can we prove this?

Step 1: State the answer.

The student is correct. \(402\) is greater than \(399\).

Step 2: Show the math work.

Compare place values:

• 402 has 4 hundreds, 0 tens, and 2 ones.
• 399 has 3 hundreds, 9 tens, and 9 ones.

Step 3: Explain why.

The hundreds place is most important here. Since 4 hundreds is greater than 3 hundreds, \(402\) is greater than \(399\), even though 399 has more tens and ones.

Step 4: Check that it makes sense.

On a number line, 402 comes after 399. Numbers farther right are greater.

Worked Example 4: Explaining a shape idea

Problem: Sam says a rectangle with 4 equal sides is also a square. Is Sam correct? Explain.

Step 1: State the answer.

Yes, Sam is correct.

Step 2: Show the facts we know.

• A rectangle has 4 sides and 4 corners.
• A rectangle has 4 right angles.
• A square has 4 equal sides and 4 right angles.

Step 3: Explain why.

If a rectangle has 4 equal sides, and rectangles already have 4 right angles, then it has the same features as a square. That means it is a square.

Step 4: Check that it makes sense.

A square is a special kind of rectangle. So a rectangle with 4 equal sides fits the rules for a square.

How to answer when you disagree

Sometimes you may need to explain why an answer is not correct. Be respectful and use math.

You can say:

• I disagree because...
• The mistake is...
• Here is the correct way...

Example:

A student says \(46 + 18 = 54\).

You could explain: I disagree because adding 10 to 46 gives 56, and adding 8 more gives 64. So the correct sum is:

$$46 + 18 = 64$$

Finding and explaining mistakes

Looking for mistakes is part of constructing mathematical arguments. It helps you understand math better.

Common mistakes include:

• adding or subtracting the wrong place values
• forgeting to count all groups
• mixing up the meaning of numbers in a word problem
• giving an answer that is too big or too small

When you find a mistake, explain:

1. what the mistake is
2. where it happened
3. how to fix it

Example of fixing a mistake

A student writes:

$$3 \times 8 = 24$$

Then says, "So 4 groups of 8 is 24."

This explanation has a mistake. The equation \(3 \times 8 = 24\) shows 3 groups of 8, not 4 groups of 8.

To fix it, write:

$$4 \times 8 = 32$$

Now the equation matches the words "4 groups of 8."

Tips for writing your own mathematical argument

• Read the problem carefully.
• Decide what you need to prove.
• Solve the problem step by step.
• Use equations, pictures, or models.
• Explain how your work proves the answer.
• Check if your answer is reasonable.

A simple math argument frame

You can use this pattern:

1. My answer is...
2. First, I...
3. Next, I...
4. This proves...
5. My answer makes sense because...

Example using the frame

Problem: Is \(9 \times 4 = 36\)?

My answer is yes.

First, I know multiplication means equal groups.

Next, 9 groups of 4 can be shown by repeated addition:

$$4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 36$$

This proves that \(9 \times 4 = 36\).

My answer makes sense because \(10 \times 4 = 40\), so \(9 \times 4\) should be 4 less, which is 36.

Summary

Constructing mathematical arguments means explaining your math thinking clearly.

A strong argument tells the answer, shows the work, explains why it is correct, and checks that it makes sense.

You can use words, equations, pictures, and known math facts to support your thinking.

When you practice explaining and proving your ideas, you become a stronger math thinker.

Identifying the Rule in an Input-Output Table

An input-output table shows how one number changes into another number.

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

Your job is to figure out the rule. The rule tells what happens to every input to make the output.

For example, if the input is \(3\) and the output is \(7\), the rule might be add 4, because \(3 + 4 = 7\).

Sometimes the rule is adding. Sometimes it is subtracting, multiplying, or dividing. We can find the rule by looking carefully at how the numbers change.

How to find the rule

1. Look at one input and its matching output.

2. Ask: What happened to the input?

3. Try a math action like add, subtract, multiply, or divide.

4. Check the same rule with the other pairs in the table.

If the same rule works for every pair, then you found the rule.

Helpful clues

• If the output is a little bigger than the input, the rule may be add.

• If the output is a little smaller than the input, the rule may be subtract.

• If the output is much bigger, the rule may be multiply.

• If the output is smaller but still related in equal groups, the rule may be divide.

Worked Example 1: Add

Look at this table:

Start with the first pair: \(2\) becomes \(5\).

Ask: What happened to \(2\) to make \(5\)?

We can see that:

Now check the next pair:

Check the last pair:

The same rule works every time. The rule is add 3.

Worked Example 2: Subtract

Look at this table:

Start with \(9\) and \(5\).

Ask: What happened to \(9\) to make \(5\)?

Check the next pair:

Check the last pair:

The rule is subtract 4.

Worked Example 3: Multiply

Look at this table:

Start with \(3\) and \(12\).

Ask: What can we do to \(3\) to get \(12\)?

Check the next pair:

Check the last pair:

The rule is multiply by 4.

Worked Example 4: Divide

Look at this table:

Start with \(8\) and \(2\).

Ask: What can we do to \(8\) to get \(2\)?

Check the next pair:

Check the last pair:

The rule is divide by 4.

What if the rule does not work?

Sometimes your first guess is wrong. That is okay.

For example, if \(4\) becomes \(12\), you might first guess add 8.

But then if \(5\) becomes \(15\), adding \(8\) does not work, because:

So try another rule. In this case:

And:

Now the rule works. The rule is multiply by 3.

Tips for success

• Use one input-output pair to make a guess.

• Always check your guess with the other pairs.

• If your rule does not work for every pair, try a different rule.

• Look for simple rules first: add, subtract, multiply, or divide.

Let’s practice thinking

If the table shows \(1 \to 6\), \(2 \to 7\), and \(3 \to 8\), what is happening?

Each output is \(5\) more than the input, so the rule is add 5.

If the table shows \(2 \to 10\), \(3 \to 15\), and \(4 \to 20\), what is happening?

Each output is \(5\) times the input, so the rule is multiply by 5.

Summary

An input-output table shows how numbers change by a rule.

To find the rule, look at an input and output, decide what happened, and then check that rule on the whole table.

The rule might be add, subtract, multiply, or divide.

If the rule works for every pair, you found the correct rule.

Comparing Two Related Number Patterns

Number patterns are lists of numbers that follow a rule. When we compare two related number patterns, we look at two lists of numbers side by side and study how they are the same and how they are different.

For example, one pattern might add 2 each time, and another pattern might add 3 each time. We can compare them to see which numbers are bigger, how fast they grow, and whether they ever match.

This helps us notice rules and make smart guesses about what comes next.

What is a number pattern?

A number pattern is a sequence of numbers that follows a rule. The rule tells how to get from one number to the next.

• Pattern A: \(2, 4, 6, 8, 10\) → add 2 each time
• Pattern B: \(5, 8, 11, 14, 17\) → add 3 each time

These are called related patterns because both are growing patterns, and we can compare them step by step.

How to compare two number patterns

When comparing two patterns, it helps to follow these steps:

1. Find the rule for each pattern.
2. Write the patterns in order.
3. Match the terms by place: first with first, second with second, and so on.
4. Look for what changes and what stays the same.

You can ask questions like these:

• Which pattern starts bigger?
• Which pattern grows faster?
• How much bigger is one pattern than the other?
• Do the patterns ever have the same number?

Useful words

• Term: one number in the pattern
• Rule: what you do each time
• Increase: how much the pattern goes up
• Compare: tell how two things are alike or different

Looking at patterns side by side

It is often easiest to compare patterns in a chart.

Example chart:

Now we can compare each row. Pattern B is always 1 more than Pattern A at the start? Let’s check carefully. In the first row, \(4")==

Writing Mathematical Expressions from Text means turning number words and clue words into a math expression.

An expression is a math phrase made of numbers, symbols, and sometimes a letter for an unknown number. An expression does not have an equals sign.

For example, the words "5 more than 8" can be written as \(8 + 5\).

This skill helps us read math in words and write it in symbols. Once we can do that, solving problems becomes much easier.

Step 1: Find the numbers and the action words.

When you read a math sentence, look for the numbers first. Then look for words that tell what operation to use.

• add, plus, sum, more than often mean addition
• subtract, minus, less than, fewer than often mean subtraction
• times, groups of, product, multiplied by often mean multiplication
• shared equally, divided by, quotient of often mean division

Step 2: Look for the unknown number.

Sometimes the words talk about a number we do not know yet. We can use a letter, like \(n\), \(x\), or \(m\), to stand for that unknown number.

For example, "a number plus 6" can be written as \(n + 6\).

Step 3: Be careful with word order.

Some phrases tell the numbers in a different order than we write them in math.

For example, "3 more than 7" means start with 7, then add 3. So it is written as:

$$7 + 3$$

Another example is "4 less than 12". This means start with 12, then subtract 4. So it is written as:

$$12 - 4$$

The words more than and less than can be tricky. Read them slowly.

Helpful clue words

• more than means add to the number that comes after it
• less than means subtract from the number that comes after it
• a number means use a letter for the unknown
• the sum of means an addition expression
• the difference of means a subtraction expression
• the product of means a multiplication expression
• the quotient of means a division expression

Worked Example 1

Write an expression for: "9 plus 4"

The word plus tells us to add.

So the expression is:

$$9 + 4$$

Worked Example 2

Write an expression for: "6 less than 15"

The phrase less than can be tricky. It means subtract 6 from 15.

So the expression is:

$$15 - 6$$

Worked Example 3

Write an expression for: "a number multiplied by 8"

We do not know the number, so we use a letter such as \(n\).

The words multiplied by tell us to multiply.

So the expression is:

$$n \times 8$$

Worked Example 4

Write an expression for: "the sum of 12 and a number"

The phrase the sum of means addition.

Use \(n\) for the unknown number.

So the expression is:

$$12 + n$$

More examples to learn from

• "7 more than a number" becomes \(n + 7\)
• "the difference of 20 and 5" becomes \(20 - 5\)
• "3 times a number" becomes \(3 \times n\)
• "the quotient of 18 and 3" becomes \(18 \div 3\)

Watch out for these common mistakes

• Do not use an equals sign unless the problem says two sides are equal. Expressions usually do not have \(=\).
• Be careful with less than. "2 less than 10" is \(10 - 2\), not \(2 - 10\).
• Be careful with more than. "5 more than 9" is \(9 + 5\).
• If you see a number or an unknown number, use a letter.

How to check your work

1. Did I find the numbers or the unknown number?
2. Did I choose the correct operation?
3. Did I put the numbers in the correct order?
4. Did I write an expression without an equals sign?

Let’s practice thinking

If the words say "4 more than a number", start with the unknown number and add 4: \(n + 4\).

If the words say "9 less than a number", start with the unknown number and subtract 9: \(n - 9\).

If the words say "the product of 5 and a number", multiply: \(5 \times n\).

If the words say "the quotient of a number and 2", divide: \(n \div 2\).

Summary

To write a mathematical expression from text, look for the numbers, the operation words, and any unknown number. Use a letter for the unknown. Then write the expression carefully, especially when you see phrases like more than and less than.

With practice, you will get better at turning word phrases into math expressions quickly and correctly.

Solving Equations with Unknown Variables

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

Sometimes an equation has a letter or a box standing for a number we do not know yet. This is called an unknown variable. Our job is to figure out what number makes the equation true.

For example, in \(x + 4 = 9\), the letter \(x\) is the unknown. We need to find the number that can be added to 4 to make 9.

Think of it like a mystery number. We look at the equation and ask, “What number fits?”

Why does this work?

An equation is like a balanced scale. If both sides are equal, the equation is true. We want to find the missing number that keeps the equation balanced.

To solve an equation, we use the opposite of the operation we see.

• If we see addition, we use subtraction.
• If we see subtraction, we use addition.
• If we see multiplication, we use division.
• If we see division, we use multiplication.

These are called opposite operations. They help us undo what is happening to the unknown number.

Steps for solving an equation

1. Look at the equation carefully.
2. Find the operation next to the unknown.
3. Use the opposite operation to undo it.
4. Check your answer by putting it back into the equation.

Worked Example 1: Addition

Solve \(x + 5 = 12\).

We want to know what number plus 5 equals 12.

The operation is addition, so we use subtraction.

Check the answer:

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

Worked Example 2: Subtraction

Solve \(m - 3 = 8\).

We want to know what number minus 3 equals 8.

The operation is subtraction, so we use addition.

Check the answer:

This is true, so the answer is \(m = 11\).

Worked Example 3: Multiplication

Solve \(4n = 20\).

This means \(4 \times n = 20\). We need to find the number that makes this true.

The operation is multiplication, so we use division.

Check the answer:

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

Worked Example 4: Division

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

This means the unknown number is being divided by 6.

The operation is division, so we use multiplication.

Check the answer:

This is true, so the answer is \(p = 18\).

Important idea: Always check

After you solve an equation, put your answer back into the original equation. If both sides are equal, your answer is correct.

Checking helps you catch mistakes and feel sure about your work.

How to think about equations

• Ask: “What is happening to the unknown?”
• Then ask: “What is the opposite operation?”
• Solve for the unknown.
• Check the answer.

Try these in your head

• \(a + 2 = 10\) so \(a = 8\)
• \(b - 7 = 4\) so \(b = 11\)
• \(3c = 15\) so \(c = 5\)
• \(\frac{d}{5} = 2\) so \(d = 10\)

Common mistakes to avoid

• Do not forget to use the opposite operation.
• Do not guess without checking.
• Be careful with subtraction and division. Think about what operation is happening to the unknown.

Summary

Solving equations means finding the unknown number that makes a math sentence true.

Use opposite operations to undo what is happening to the unknown:

• Addition goes with subtraction.
• Subtraction goes with addition.
• Multiplication goes with division.
• Division goes with multiplication.

Then always check your answer in the original equation. If both sides are equal, you solved it correctly.

Balancing True and False Equations

In math, an equation is a number sentence with an equal sign, like \(8 + 4 = 12\).

The equal sign \(=\) means “has the same value as”. It does not just mean “the answer is next.” It means the amount on the left side and the amount on the right side must be the same.

When both sides have the same value, the equation is true. When the two sides have different values, the equation is false.

Learning to check whether an equation is true or false helps you understand how math stays balanced, like a seesaw with the same weight on both sides.

Main Idea: Check Both Sides

To decide if an equation is true or false, follow these steps:

1. Find the value of the left side.
2. Find the value of the right side.
3. Compare the two values.
4. If they are the same, the equation is true. If they are different, the equation is false.

For example, look at \(6 + 3 = 10 - 1\).

• Left side: \(6 + 3 = 9\)
• Right side: \(10 - 1 = 9\)

Both sides equal \(9\), so the equation is true.

Why “Balancing” Matters

An equation is like a balance scale. If one side has the same amount as the other side, it is balanced.

Here is a balanced equation:

Check each side:

• Left side: \(7 + 5 = 12\)
• Right side: \(9 + 3 = 12\)

Since both sides are \(12\), the equation is balanced and true.

Here is an unbalanced equation:

• Left side: \(14 - 6 = 8\)
• Right side: \(5 + 2 = 7\)

Since \(8 \ne 7\), the equation is not balanced. It is false.

You Can Have Numbers on Both Sides

Sometimes students think the answer must always come after the equal sign. But in equations, both sides can have operations.

For example:

This is true because \(7 + 8 = 15\). The equal sign shows both sides are the same.

Also:

This is true because:

• Left side: \(20 - 5 = 15\)
• Right side: \(9 + 6 = 15\)

Worked Examples

Example 1: A simple true equation

Is \(4 + 5 = 9\) true or false?

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

Both sides are \(9\), so the equation is true.

Example 2: A simple false equation

Is \(13 - 4 = 10\) true or false?

• Left side: \(13 - 4 = 9\)
• Right side: \(10\)

Since \(9 \ne 10\), the equation is false.

Example 3: Operations on both sides

Is \(6 + 7 = 20 - 7\) true or false?

• Left side: \(6 + 7 = 13\)
• Right side: \(20 - 7 = 13\)

Since both sides are \(13\), the equation is true.

Example 4: Find the missing number to balance the equation

What number makes this equation true?

We need the left side to equal \(14\).

Since \(8 + 6 = 14\), the missing number is \(6\).

Now the equation is:

This equation is true because both sides equal \(14\).

Another Missing Number Example

What number makes this equation true?

We want the left side to equal \(11\). Ask: what number must be subtracted from \(18\) to get \(11\)?

Since \(18 - 7 = 11\), the missing number is \(7\).

A Helpful Strategy

• Solve the left side first.
• Solve the right side next.
• Do not guess.
• Compare carefully.
• If there is a missing number, think: “What number makes both sides match?”

Watch Out for These Mistakes

• Mistake 1: Thinking the equal sign means “put the answer here.” It really means both sides are the same.
• Mistake 2: Solving only one side. You must check both sides.
• Mistake 3: Rushing and making an adding or subtracting mistake. Work carefully.

Practice Thinking

Try asking yourself these questions when you see an equation:

• What is the value on the left side?
• What is the value on the right side?
• Are they equal?
• If not, what number would make them equal?

Summary

An equation is true when both sides have the same value. An equation is false when the two sides have different values.

To check an equation, solve each side and compare. To balance an equation with a missing number, find the number that makes both sides equal.

When you remember that the equal sign means “the same as”, equations make much more sense.

Multi-Step Problem Solving Strategies means solving a word problem by doing more than one step. Sometimes a problem does not tell you the final math right away. You have to find one answer first, then use it to find the next answer.

In many word problems, the important information is hidden inside the story. Good problem solvers slow down, read carefully, and ask, “What do I need to find first?”

This lesson will help you learn how to break a problem into parts, choose the right operations, and solve in order.

Why multi-step problems can feel tricky

A multi-step problem may include:

• more than one operation, like addition and subtraction
• information in a certain order
• extra words that can distract you
• a hidden question you must answer before the final question

That is why it helps to use a strategy every time.

A simple plan for solving multi-step problems

1. Read the whole problem carefully.
2. Circle or notice the question. What is the problem asking for?
3. Underline important facts and numbers.
4. Decide what must happen first, next, and last.
5. Write an equation or number sentence for each step.
6. Solve carefully.
7. Check: Does your answer make sense?

Look for clue words, but think carefully

Clue words can help, but they do not always tell the whole story. You must understand what is happening in the problem.

• altogether, in all, total often mean add
• left, remain, how many more, fewer often mean subtract
• groups of, each, every often mean multiply
• shared equally often means divide

Important: In a multi-step problem, you may need to use different operations in different steps.

Step-by-step thinking

When you read a word problem, try asking yourself these questions:

• What is happening first?
• Do I need to combine, compare, multiply, or divide?
• Is there a hidden amount I need to find?
• What do I do with that answer next?

Example 1: Two-step problem with addition and subtraction

Lena picked 18 apples in the morning and 15 apples in the afternoon. She gave 9 apples to her neighbor. How many apples does she have left?

Step 1: Find how many apples Lena picked in all.

She picked 18 in the morning and 15 in the afternoon.

$$18 + 15 = 33$$

So Lena picked 33 apples in all.

Step 2: Subtract the apples she gave away.

$$33 - 9 = 24$$

Answer: Lena has 24 apples left.

Why this is multi-step: You could not subtract 9 first. You had to find the total number of apples before finding how many were left.

Example 2: Multiply, then subtract

There are 6 tables in a classroom. Each table has 4 pencils. The teacher gives away 5 pencils. How many pencils are left?

Step 1: Find the total number of pencils.

There are 6 tables with 4 pencils each.

$$6 \times 4 = 24$$

So there are 24 pencils at first.

Step 2: Subtract the pencils given away.

$$24 - 5 = 19$$

Answer: There are 19 pencils left.

What hidden job did we do? We first found the total number of pencils. That total was not stated directly in the problem.

Example 3: Add first, then divide

Three friends collected stickers. Ava collected 12 stickers, Ben collected 16 stickers, and Cara collected 20 stickers. They put all the stickers into 4 equal bags. How many stickers are in each bag?

Step 1: Add all the stickers.

$$12 + 16 + 20 = 48$$

There are 48 stickers total.

Step 2: Divide the stickers into 4 equal bags.

$$48 \div 4 = 12$$

Answer: There are 12 stickers in each bag.

Why order matters: You must know the total number of stickers before you can share them equally.

Example 4: Finding a hidden amount first

A toy store had 50 kites. On Monday, it sold 18 kites. On Tuesday, it sold 7 more kites. Then the store got 12 new kites. How many kites does the store have now?

This problem has more than one change, so we go in order.

Step 1: Subtract the kites sold on Monday.

$$50 - 18 = 32$$

Step 2: Subtract the kites sold on Tuesday.

$$32 - 7 = 25$$

Step 3: Add the new kites.

$$25 + 12 = 37$$

Answer: The store has 37 kites now.

How to organize your work

It helps to write your thinking in a neat way. You can use words, equations, or both.

For example, you might write:

• First: Find the total.
• Next: Subtract what was used.
• Last: Write the answer with a label.

A label tells what your answer means, like 24 apples or 12 stickers in each bag.

Check your answer

After solving, ask yourself:

• Did I answer the question that was asked?
• Did I do the steps in the right order?
• Does my answer make sense?
• Did I remember the label?

For example, if a problem asks how many are left, your answer should not be more than the starting amount unless more items were added later.

Common mistakes to avoid

• Using only one step when the problem needs two or more steps
• Choosing the wrong operation
• Doing steps in the wrong order
• Forgetting hidden work, like finding a total first
• Forgetting the label in the final answer

Helpful strategy: Retell the story

If a problem feels confusing, say it in your own words.

For example:

“First there were 50 kites. Some were sold on Monday. More were sold on Tuesday. Then new kites arrived.”

Retelling helps you see the order of events.

Helpful strategy: Draw or model

You can also draw quick pictures, boxes, or number lines to show what happens first and next. You do not need fancy drawings. Simple sketches can help your brain organize the problem.

When solving multi-step problems, remember:

• Read carefully.
• Find the question.
• Look for the hidden step.
• Solve in order.
• Check your answer.

Summary

Multi-step problem solving means using two or more steps to solve a word problem. First, understand the story and figure out what you need to find. Then solve each part in the correct order, and check that your final answer makes sense.