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  1. 7th Grade Math

  3. Rewrite Expressions in Different Forms

7TH GRADE MATH • EXPRESSIONS AND EQUATIONS

Rewrite Expressions in Different Forms

Discover how rewriting expressions reveals hidden connections between quantities in real-world problems.

SECTION 1

Why Do We Rewrite Expressions?

People have been finding clever ways to rewrite math problems for thousands of years. Ancient merchants needed quick tricks to calculate prices, taxes, and discounts. They discovered that writing the same problem in a different way could make it much easier to solve. This idea — that the form of an expression matters just as much as its value — is at the heart of algebra.

~1800 BCE
Babylonian Merchants
Babylonian traders used clay tablets to rewrite price calculations. They found shortcuts for adding taxes and fees to the cost of goods.
~825 CE
Al-Khwarizmi's Algebra
The Persian mathematician al-Khwarizmi wrote a book on balancing and simplifying equations. His name gave us the word "algorithm."
1600s
Symbolic Algebra Emerges
Mathematicians began using letters like x and a to stand for unknown numbers. This made it much easier to rewrite and compare expressions.
Today
Everyday Applications
We rewrite expressions every day — from calculating sale prices at a store to figuring out tips at a restaurant. The same math, just written differently!

Here is the big question this lesson answers: if two expressions look different but mean the same thing, why would you pick one form over another? The answer is that different forms reveal different information about a problem. Let's find out how.

SECTION 2

Core Principles of Rewriting Expressions

Before we jump into examples, let's nail down the key ideas. An expression is a math phrase that combines numbers, variables (letters), and operations like addition or multiplication. When you rewrite an expression, you change its appearance without changing its value. Think of it like saying the same sentence in different words.

1

Equivalent Expressions

Two expressions are equivalent if they give the same result for every possible value of the variable. For example, 3x + 2x and 5x are equivalent.
2

Combining Like Terms

Like terms are parts of an expression that have the same variable raised to the same power. You can add or subtract them to simplify. For example, a + 0.05a = 1.05a.
3

Distributive Property

The distributive property lets you multiply a number across a sum: a(b + c) = ab + ac. You can also use it in reverse to factor.
4

Factoring

Factoring is the reverse of distributing. You pull out a common factor. For example, 6x + 12 = 6(x + 2). This shows that the expression is "6 times something."
✦ KEY TAKEAWAY
Rewriting an expression is like rearranging the furniture in your room. The room is the same size, and you have the same stuff — but a new layout can make things easier to find. In math, a new form of an expression can make a hidden pattern or relationship jump right out at you.
SECTION 3

Seeing Equivalent Expressions

Let's look at the classic example from the standard. Imagine you have a price a dollars, and you increase it by 5%. You can write this two ways: a + 0.05a or 1.05a. The diagram below shows why these are the same.

Two Ways to Show "Increase by 5%"FORM 1:a + 0.05aa (the original amount)5%Total = a + 0.05a= EQUALS =FORM 2:1.05a1.05 × a (multiply by 1.05)Total = 1.05aBoth bars have the SAME total length → a + 0.05a = 1.05a ✓
The top bar shows the original amount a (blue) plus 5% of a (pink). The bottom bar shows one single piece: 1.05a (cyan). Both represent the exact same total.

Notice how Form 1 shows you the two parts of the total — the original and the extra 5%. Form 2 shows you the total as a single multiplication. Both are correct. You just pick the form that is most useful for the situation.

SECTION 4

The Math Behind Rewriting

When you rewrite expressions, you rely on a few key properties. Let's see each one in action with equations.

COMBINING LIKE TERMS
a + 0.05a = 1a + 0.05a = (1 + 0.05)a = 1.05a
The variable a is really 1 × a. Add the coefficients (the numbers in front): 1 + 0.05 = 1.05.
DISTRIBUTIVE PROPERTY (EXPANDING)
3(x + 4) = 3 × x + 3 × 4 = 3x + 12
Multiply the 3 by each term inside the parentheses. The expanded form shows two separate parts.
FACTORING (REVERSE OF DISTRIBUTING)
3x + 12 = 3(x + 4)
Both 3x and 12 share a factor of 3. Pull it out. The factored form shows the expression is "3 groups of (x + 4)."
PERCENT DECREASE EXAMPLE
p − 0.20p = 1p − 0.20p = (1 − 0.20)p = 0.80p
A 20% discount on price p is the same as multiplying by 0.80. So "take off 20%" equals "pay 80%."
💡 REMEMBER THIS
Every variable without a visible number in front actually has a hidden 1. So a really means 1a. This is the secret that makes combining like terms with percents possible!
SECTION 5

Different Forms for Different Situations

Let's explore several real-world scenarios where rewriting an expression helps you understand what is really going on. The diagram below shows how the same situation can be written as an "add/subtract" form or a "multiply" form, and what each version tells you.

Three Scenarios — Two Forms Each🛒 5% SALES TAXAdd/Subtract Form:c + 0.05cMultiply Form:1.05c"increase by 5%" = "× 1.05"🏷️ 30% DISCOUNTAdd/Subtract Form:p − 0.30pMultiply Form:0.70p"decrease by 30%" = "× 0.70"💰 15% TIPAdd/Subtract Form:m + 0.15mMultiply Form:1.15m"increase by 15%" = "× 1.15"Why Choose One Form Over Another?ADD / SUBTRACT FORMShows the original and the change separatelyMULTIPLY FORMShows the result as one quick calculationTHE BIG IDEASame value, different story. Pick the form that answers YOUR question.
Three common percent scenarios. Each one has an add/subtract form (showing the parts) and a multiply form (showing the quick calculation). Both forms are equivalent expressions.
Common percent increase and decrease scenarios
SituationAdd/Subtract FormMultiply FormWhat It Tells You
8% sales tax on cost cc + 0.08c1.08cTotal is 108% of the original price
25% off price pp − 0.25p0.75pYou pay 75% of the original price
10% raise on salary ss + 0.10s1.10sNew salary is 110% of the old one
40% fewer tickets tt − 0.40t0.60tOnly 60% of the tickets remain
SECTION 6

Worked Example: The Sneaker Sale

A pair of sneakers costs d dollars. The store has a 15% off sale, and then the state adds 6% sales tax to the sale price. Write two different expressions for the final price and explain what each one shows.

The Sneaker Sale — Step by Step

Step 1 — Apply the 15% Discount

Start with price d. A 15% discount means you subtract 15% of d. The sale price is d − 0.15d. Combine like terms: 1d − 0.15d = 0.85d.
Sale price = 0.85d

Step 2 — Apply the 6% Sales Tax

Now add 6% tax to the sale price. The sale price is 0.85d, so the tax is 0.06 × 0.85d. The total is 0.85d + 0.06(0.85d).
Total = 0.85d + 0.06(0.85d)

Step 3 — Rewrite Using the Multiply Form

Adding 6% tax is the same as multiplying by 1.06. So the total is 1.06 × 0.85d. Multiply: 1.06 × 0.85 = 0.901. The final price is 0.901d.
Total = 0.901d

Step 4 — Compare the Two Forms

Form 1: 0.85d + 0.06(0.85d) — this shows each part: the discounted price and the tax added on top. Form 2: 0.901d — this tells you that you pay about 90.1% of the original price. Both are correct!
0.85d + 0.06(0.85d) = 0.901d
✅ Check Your Answer
Try it with d = $100. The sale price is $85. The tax is 0.06 × $85 = $5.10. Total = $90.10. Does 0.901 × $100 also give $90.10? Yes! ✓
SECTION 7

When to Use Which Form

Now that you can create equivalent expressions, the next question is: which form should you choose? Here is a handy comparison.

Comparing expanded and factored expression forms
FeatureExpanded / Add-Subtract FormFactored / Multiply Form
What it looks likea + 0.05a or 3x + 121.05a or 3(x + 4)
Best forSeeing each separate part (original and change)Quick computation or seeing the overall multiplier
StrengthEasy to identify the original amount and the added/subtracted pieceEasy to calculate — just one multiplication step
LimitationTakes more steps to compute the final answerHides the individual parts so they are harder to see
Real-world useItemized receipts, budgets that show each line itemQuick mental math, repeated percent calculations
✦ KEY TAKEAWAY
Think of it like a recipe. The expanded form is like listing every ingredient and step separately. The factored form is like saying, "It's basically chocolate cake — just follow the box instructions." Both give you the same cake, but one form is more detailed and the other is faster to follow.
SECTION 8

Connection to Future Math

The skill of rewriting expressions doesn't stop in 7th grade. It is one of the most important tools in all of math. Here is a preview of where this skill goes next.

How rewriting expressions builds toward future topics
What You Learn Now (7th Grade)Where It Goes Next
Combine like terms: a + 0.05a = 1.05aIn Algebra 1, you'll combine terms with higher powers: 2x² + 3x²= 5x²
Distribute: 3(x + 4) = 3x + 12In Algebra 1, you'll distribute with two binomials: (x + 3)(x + 2) = x² + 5x + 6
Factor: 6x + 12 = 6(x + 2)In Algebra 1 and 2, you'll factor quadratics: x² + 5x + 6 = (x + 2)(x + 3)
Rewrite percent problemsIn 8th grade and beyond, you'll model exponential growth and compound interest

Every time you rewrite an expression now, you are building a muscle that you will use in every math class from here on. The better you get at seeing equivalent forms, the easier algebra, geometry, and even calculus will be in the future.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL
Maya says that "increasing a number by 20%" is the same as "multiplying that number by 1.20." Is she correct? Explain why or why not using the idea of combining like terms.
PROBLEM 2 — BASIC CALCULATION
Rewrite the expression t − 0.35t by combining like terms. Then use your simplified expression to find the value when t = 60.
PROBLEM 3 — INTERMEDIATE
A shirt costs s dollars. The store marks it up by 40% and then offers a 10% discount on the marked-up price. Write an expression for the final price in multiply form. Is the final price the same as a 30% markup? Explain.
PROBLEM 4 — APPLIED
A dog walker earns w dollars per week. She gets a 12% raise. Write two different expressions for her new weekly pay. Then calculate her new pay if she was earning $250 per week using the simpler form.
PROBLEM 5 — CRITICAL THINKING
Carlos writes the expression 5(2x + 3) − 4x. Dina writes 6x + 15. Are these equivalent? Rewrite Carlos's expression step by step to find out. Then explain what the term 6x and the number 15 each represent if x stands for the number of items someone buys.
SUMMARY

Lesson Summary

In this lesson, you learned that rewriting expressions in different forms does not change their value — it changes what you can see. You used combining like terms to turn a + 0.05a into 1.05a, showing that "increase by 5%" is the same as "multiply by 1.05." You applied the distributive property to expand expressions like 3(x + 4) = 3x + 12, and reversed the process using factoring.

The expanded (add/subtract) form helps you see each part of a problem separately, while the factored (multiply) form gives you a quick single-step calculation. Both forms are equivalent expressions — same value, different view. Choosing the right form helps you understand the problem and solve it more easily. This skill is the foundation for everything you'll do in algebra and beyond!

Varsity Tutors • 7th Grade Math • Rewrite Expressions in Different Forms