Modeling with Expressions — I can model a real situation with an expression or equation and interpret units and constraints. - Pre Algebra

SECTION 1

Why Do We Write Math for Real Life?

People have been turning real-life problems into math for thousands of years. Ancient farmers needed to figure out how much grain to store. Builders needed to calculate how many bricks to buy. Merchants needed to track profits. In every case, they were doing the same thing you are about to learn: modeling (using math to represent a real situation).

~1800 BCE • Babylonian Word Problems

Ancient Babylonians carved word problems onto clay tablets. They wrote about fields, wages, and trade — the first known examples of turning everyday situations into math.

~300 BCE • Greek Geometry & Formulas

Greek mathematicians like Euclid created formulas for areas and volumes. These formulas are expressions that model real shapes in the physical world.

~825 CE • Al-Khwarizmi & Algebra

The Persian mathematician al-Khwarizmi wrote a book about solving equations from word problems. The word 'algebra' comes from the title of his book!

1600s • Variables Get Letters

René Descartes popularized using letters like x and y to stand for unknown quantities. This made it much easier to write expressions and equations.

Today • Math Models Everywhere

Scientists, engineers, game designers, and business owners all use expressions and equations to model the real world every single day.

The big question this lesson answers is: How do I take a situation described in words and write it as a math expression or equation — and how do I know what the numbers and units mean?

SECTION 2

Core Ideas You Need to Know

Before we start building expressions, let's nail down a few key ideas. These are the building blocks for everything else in this lesson.

Expression

A math phrase that uses numbers, variables, and operations (like +, −, ×, ÷) but does not have an equals sign. Example: 3x + 5.

Equation

A math sentence that says two things are equal, using an = sign. Example: 3x + 5 = 20. An equation is like a balanced scale.

Variable

A letter (like x, n, or t) that stands for a number you don't know yet. It's the mystery value you're trying to find or describe.

Units

The labels that tell you what a number measures — dollars, miles, hours, pounds, etc. Without units, a number like '12' is meaningless in a real problem.

Constraints

Rules or limits that a variable must follow. For example, 'you can't buy a negative number of tickets' means the variable must be 0 or greater.

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KEY TAKEAWAY

Think of writing an expression like writing a recipe. The variable is the ingredient you can change (like the number of cookies). The operations (+, −, ×) are the steps in the recipe. The units are like whether you're measuring cups or tablespoons. And the constraints are the rules — like 'no more than 2 cups of sugar.'

SECTION 3

From Words to Math — A Visual Guide

The diagram below shows how to translate a word problem into an expression, step by step. Follow the arrows from the English sentence on the left to the finished expression on the right.

Follow the five steps: read, identify the variable, spot key words, build the expression, and check units and constraints. Notice how "per" signals multiplication and "plus" signals addition.

The five steps in the diagram work for almost any word problem. The trickiest part is usually Step 3 — finding the key words that tell you which operation to use. Don't worry, we'll practice a lot more of that!

SECTION 4

Key Words → Math Operations

Certain English words almost always point to a specific math operation. Learning this 'translation dictionary' is the most important skill for modeling.

ADDITION PATTERN

a + b

Key words: plus, more than, increased by, total, sum, combined, together. Example: '5 more than a number n' → n + 5.

SUBTRACTION PATTERN

a − b

Key words: minus, less than, decreased by, fewer, difference, left over. Watch out: '7 less than x' is x − 7, not 7 − x!

MULTIPLICATION PATTERN

a × b or ab

Key words: times, per, each, of, product, double, triple. Example: '$5 per ticket for t tickets' → 5t.

DIVISION PATTERN

a ÷ b or a / b

Key words: divided by, split equally, per (sometimes), ratio, quotient. Example: 'Share 24 cards equally among p players' → 24 ÷ p.

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Watch Out!

The phrase "less than" flips the order. '9 less than n' is n − 9, not 9 − n. Think of it as 'start with n, then take away 9.'

SECTION 5

Understanding Units and Constraints

Writing an expression is only half the job. You also need to know what the answer means. That's where units and constraints come in.

This diagram shows how to check that your expression makes sense. The units check confirms the answer is in dollars. The constraints limit the variable to values that make sense in real life.

A quick units check is like a spell-check for math. If your answer's units don't match what the question asks for, something went wrong. And constraints keep your answer realistic. You can't have −3 tickets or 2.7 people!

SECTION 6

Worked Example — Pizza Party Budget

Let's walk through a full problem together. Read slowly and follow each step.

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Problem

Your class is having a pizza party. Each pizza costs $12, and you also need to buy a $15 pack of plates and cups. Write an expression for the total cost if you buy p pizzas. Then write an equation if the class collected $75 total. What are the units and constraints?

Pizza Party — Step by Step

1
Step 1 — Identify What Changes and What Stays Fixed

The number of pizzas can change — that's our variable. Let p = number of pizzas. The cost per pizza ($12) and the plate/cup pack ($15) are fixed numbers.

Variable: p (pizzas)
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Step 2 — Translate Words to Math

"Each pizza costs $12" means we multiply: 12 × p, or 12p. "Also need" means we add the $15. So the total cost expression is 12p + 15.

Expression: 12p + 15
3
Step 3 — Write the Equation

The class collected $75. Set the expression equal to 75 to get an equation.

Equation: 12p + 15 = 75
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Step 4 — Check the Units

12 is in dollars per pizza. p is in pizzas. 12p gives us dollars. 15 is in dollars. Adding dollars + dollars = dollars. The result is in dollars. ✓

Units: dollars ✓
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Step 5 — State the Constraints

p must be a whole number (you can't order half a pizza from most places). p must be at least 1 (you need at least one pizza for a party!). Also, 12p + 15 ≤ 75, so 12p ≤ 60, meaning p ≤ 5. You can buy at most 5 pizzas.

Constraints: p is a whole number, 1 ≤ p ≤ 5

SECTION 7

Common Mistakes and How to Avoid Them

Even strong math students make mistakes when modeling. Here are the most common pitfalls and how to avoid them.

Common Mistake	Why It Happens	How to Fix It
Writing '5 less than x' as 5 − x	You translate words left to right, but 'less than' flips the order.	Ask: 'Am I starting with x and removing 5?' If yes, it's x − 5.
Forgetting units entirely	Numbers feel complete on their own, so units seem optional.	Always label your variable's unit when you define it (e.g., t = hours).
Ignoring constraints	The math 'works' even with negative or decimal answers.	Ask: 'Does this answer make sense in real life?' Check for negative, fractional, or impossibly large values.
Mixing up expressions and equations	Both use variables and operations, so they look similar.	Check: is there an = sign? Equals sign → equation. No equals sign → expression.

Four common modeling mistakes and how to fix them

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KEY TAKEAWAY

Think of constraints like the rules of a video game. In a racing game, your car can't drive through walls or go below 0 mph. In a math model, your variable can't be negative people or fractional tickets. The real world has rules — your math should follow them too!

SECTION 8

From Expressions to Equations and Beyond

The skills you're learning now are the foundation for everything that comes next in algebra. Here's a quick look at how this topic grows.

What You Learn Now	What Comes Next
Write expressions from word problems	Solve equations to find unknown values
Identify constraints like x ≥ 0	Write and solve inequalities (e.g., 3x + 2 < 20)
Use one variable (like t or p)	Use two variables and graph them on a coordinate plane
Check units by hand	Use dimensional analysis in science (chemistry, physics)
Model simple situations (cost, distance)	Model complex real-world systems (population growth, budgets)

How today's skills connect to future math topics

Every time you write an expression from a word problem, you are practicing the exact same thinking that engineers, scientists, and data analysts use every day. The problems get harder, but the steps stay the same: read, define variables, translate, check units, check constraints.

SECTION 9

Practice Problems

Try these five problems on your own. They go from easy to challenging. After each one, check the answer to see how you did.

Question 1: CONCEPTUAL

What is the difference between an expression and an equation? Give one example of each using the variable n.

Show answer

An expression is a math phrase without an equals sign, like 4n + 3. An equation is a math sentence with an equals sign, like 4n + 3 = 19. The expression describes a quantity, while the equation says two quantities are the same. Think of an expression as a recipe and an equation as a statement that the recipe equals a specific amount.

Question 2: BASIC CALCULATION

A gym membership costs $25 per month plus a one-time sign-up fee of $40. Write an expression for the total cost after m months. What are the units of each part?

Show answer

The expression is 25m + 40. The 25 has units of dollars per month (the rate). The variable m is in months. So 25m gives you dollars. The 40 is in dollars (a fixed fee). Adding them gives the total cost in dollars. The result makes sense because dollars + dollars = dollars.

Question 3: INTERMEDIATE

Kai has 120 stickers and gives away s stickers to each of his 4 friends. Write an expression for how many stickers Kai has left. Then list the constraints on s.

Show answer

Kai gives away s stickers to each of 4 friends, so he gives away 4s stickers total. The expression for stickers remaining is 120 − 4s. The constraints are: s must be a whole number (you can't give half a sticker), s ≥ 0 (you can't give a negative number), and 4s ≤ 120, which means s ≤ 30. So s can be 0, 1, 2, … up to 30.

Question 4: APPLIED

A phone plan charges $0.10 per text message and $35 per month for calls. Last month, Priya's total bill was $52. Write an equation and find how many texts Priya sent. State the units and constraints.

Show answer

Let t = number of text messages. The cost for texts is 0.10t (dollars per text × texts = dollars). Adding the monthly fee gives the equation 0.10t + 35 = 52. Solving: 0.10t = 17, so t = 170. Priya sent 170 texts. The units check: 0.10 is dollars/text, t is texts, 35 is dollars, and 52 is dollars — everything is in dollars. Constraints: t must be a whole number and t ≥ 0. Since the bill is exactly $52, t = 170 is the only answer.

Question 5: CRITICAL THINKING

Two car rental companies offer these deals. Company A: $30 per day plus $0.15 per mile. Company B: $50 per day with unlimited miles. Write an expression for the daily cost of Company A if you drive d miles. Then write an equation to find how many miles make the two companies cost the same. What constraint does the real world place on d?

Show answer

Company A's daily cost is 30 + 0.15d, where d is miles driven. Company B's daily cost is just 50. Setting them equal: 30 + 0.15d = 50. Solving: 0.15d = 20, so d ≈ 133.3 miles. If you drive fewer than about 133 miles, Company A is cheaper. If you drive more, Company B is cheaper. The constraint is d ≥ 0 because you can't drive negative miles. Note that d does not have to be a whole number — you can drive 133.5 miles! This shows that some constraints depend on what the variable represents.

SUMMARY

Lesson Summary

Summary

In this lesson, you learned how to turn real-world situations into math. An expression uses numbers, variables, and operations to describe a quantity. An equation sets an expression equal to a value. You practiced translating key words like 'per,' 'more than,' and 'less than' into the correct operations (+, −, ×, ÷).

You also learned to check your work by verifying units (the labels on your numbers, like dollars or hours) and identifying constraints (limits on your variable, like 'must be a whole number' or 'cannot be negative'). These five steps — read, define, translate, check units, check constraints — will carry you through algebra and beyond!