Linear expressions and equations with rational numbers
Combining, expanding and factoring linear expressions, rewriting expressions to reveal structure, and solving multistep equations and inequalities
About four to five lessons of 45 to 60 minutes
'Price plus 8% tax' and '1.08 times the price' are the same rule, written two ways
p + 0.08p and 1.08p always give exactly the same number for any price p. They are equivalent expressions, the same rule written in two different forms, and each form reveals something different: the first shows the tax being ADDED on, the second shows the total being a single 8% INCREASE. Learning to move between forms like these, and to build, simplify and solve with expressions that have rational (fraction or decimal) coefficients, is what this unit is about.
The payoff is solving real problems: writing an equation for an unknown number of hours a garage charges for, or an inequality for how many rides a gift card can afford, then solving it exactly the way you solved simpler equations before, just with rational numbers now allowed everywhere.
- 'Price plus 8% tax' vs '1.08 times the price'two equivalent expressions for the exact same total
- A mechanic charges $45 plus $60 per houra linear expression, 45 + 60h, built directly from the words
- A gift card with $40 on it, rides cost $6 eachan inequality, 40 - 6r >= 0, for how many rides r are affordable
- Combining two garden-bed cost expressions into oneadding and factoring linear expressions with rational coefficients
What students will be able to do
Students will add, subtract, expand and factor linear expressions with rational coefficients, rewrite an expression in an equivalent form to reveal how the quantities in a problem relate, solve multistep real-world problems with positive and negative rational numbers, and write and solve one-variable equations and inequalities to model real-world situations.
- I can combine like terms in an expression with rational (fraction or decimal) coefficients.
- I can expand a(bx + c) and factor a linear expression back into that form.
- I can rewrite an expression such as p + 0.08p as a single equivalent term, 1.08p, and explain what each form reveals.
- I can solve a multistep real-world problem that combines several operations with rational numbers.
- I can write and solve a one-variable equation or inequality to model a real-world situation, including flipping the inequality symbol when needed.
Standards this unit teaches
- 7.EE.A.1Common Core (US)Work with linear expressions
Add, subtract, factor, and expand linear expressions with rational coefficients.
- 7.EE.A.2Common Core (US)Rewrite expressions in context
Rewrite an expression in different forms to show how the quantities in a problem are related.
- 7.EE.B.3Common Core (US)Multistep problems with rationals
Solve multistep real world problems with positive and negative rational numbers in any form.
- 7.EE.B.4Common Core (US)Solve equations and inequalities
Write and solve simple equations and inequalities to solve real world problems.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Year 7 linear equations & expressions teaching unitbuilding expressions from words and expanding a single bracket, the foundation this unit extends to rational coefficients
- Grade 6 GCF & LCM teaching unitfinding a greatest common factor is exactly how a linear expression is factored
- Variable in the glossarya refresher on what a letter stands for in an expression or equation
- Equation in the glossarya refresher on what it means to solve for an unknown
- Inequality in the glossarya refresher on comparing with >, <, >= and <= instead of =
Words to teach and display
- Like terms
- terms with exactly the same variable part, which can be combined by adding or subtracting their coefficients
- Coefficient
- the number multiplying a variable, which in Grade 7 can be a fraction or decimal (a rational coefficient)
- Equivalent expressions
- expressions that give the same value for every value of the variable, even though they look different
- Expand
- to multiply out a bracket, such as turning 5(2x - 3) into 10x - 15
- Factor
- to rewrite an expression as a product, the reverse of expanding
- Inequality
- a statement comparing two expressions with >, <, >= or <= instead of =
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Combining, expanding and factoring, with rational coefficients
ConcreteEvery skill from expanding and factoring whole-number expressions carries over once fractions and decimals are allowed as coefficients: combine only the terms with the same variable part, distribute a factor across every term inside a bracket, and pull out the greatest common factor to reverse it.
Combining like terms: 3/4 x + 5 - 1/4 x + 2. Combine the x-terms (3/4 x - 1/4 x = 1/2 x) and the constants (5 + 2 = 7) separately: the simplified expression is 1/2 x + 7.
Expanding: -2(3x - 5) means -2 multiplies BOTH terms inside the bracket: -2 x 3x = -6x, and -2 x (-5) = +10, giving -6x + 10. Factoring reverses this: 8x - 12 has a greatest common factor of 4, so it factors to 4(2x - 3).
Simplify 3/4 x + 5 - 1/4 x + 2, then expand -2(3x - 5), then factor 8x - 12.
- Combine like terms: (3/4 - 1/4)x + (5 + 2) = 1/2 x + 7.
- Expand -2(3x - 5): -2 x 3x = -6x, and -2 x (-5) = 10, giving -6x + 10.
- Factor 8x - 12: the greatest common factor of 8 and 12 is 4, so 8x - 12 = 4(2x - 3).
Answer: 3/4 x + 5 - 1/4 x + 2 = 1/2 x + 7. -2(3x - 5) = -6x + 10. 8x - 12 = 4(2x - 3).
- Why can 3/4 x and -1/4 x be combined, but 5 and -1/4 x cannot?
- How can you check a factorization is correct without redoing the whole search for the GCF?
2. Rewriting an expression to reveal how quantities relate
PictorialThe same total can be written in more than one equivalent form, and each form highlights a different way of thinking about the situation. Rewriting is not 'getting a different answer', it is finding the SAME rule stated more usefully.
A shirt costs p dollars, plus 8% sales tax. Written as p + 0.08p, the expression shows the tax being added on top of the price. Since p is really 1p, combine like terms: 1p + 0.08p = 1.08p. This single-term form reveals that the total is simply 108% of the original price, a straight percent increase, without the addition needing to be done separately every time.
The same move works for a decrease: a price p reduced by 25% is p - 0.25p, which combines to 0.75p, revealing that the sale price is always 75% of the original, however large p is.
A price p is decreased by 25%. Write two equivalent expressions for the new price, and explain what the simplified one reveals.
- Written as the decrease being subtracted: p - 0.25p.
- Combine like terms (p is 1p): 1p - 0.25p = 0.75p.
Answer: p - 0.25p = 0.75p. The single-term form reveals that the sale price is always 75% of the original price p.
- Why are p + 0.08p and 1.08p equivalent, even though one has two terms and the other has one?
- What would the single-term form be for a price increased by 15%?
3. Multistep real-world problems with rational numbers
AbstractMultistep problems combine several operations in the order the story happens; translate each sentence into a calculation before combining them, rather than trying to do it all at once.
Sam has $45. He spends 2/5 of it on a book (a fraction OF the current amount, so multiply first: 45 x 2/5 = 18, then subtract: 45 - 18 = 27), then earns $12 doing chores (a one-time change, add it on: 27 + 12 = 39).
Sam has $45. He spends 2/5 of it on a book, then earns $12 doing chores. How much money does he have now?
- Amount spent on the book: 45 x 2/5 = 18.
- Remaining after the book: 45 - 18 = 27.
- After earning $12 from chores: 27 + 12 = 39.
Answer: Sam has $39 now.
- Why is '2/5 of it' calculated using the ORIGINAL $45, not some other amount?
- What operation models spending 2/5 of an amount: addition, subtraction, or multiplication?
4. Solving equations and inequalities to model real problems
AbstractWriting then solving an equation or inequality follows the same undo-in-reverse-order method from earlier grades, now with rational coefficients allowed, plus one new rule for inequalities: dividing or multiplying both sides by a NEGATIVE number flips the inequality symbol.
Equation: 'Three less than twice a number is 19' becomes 2n - 3 = 19. Undo in reverse order: add 3 to both sides (2n = 22), then divide by 2 (n = 11).
Inequality: a parking garage charges $5 plus $2 per hour; Jax has at most $17. Modeled as 5 + 2h <= 17, solving gives 2h <= 12, so h <= 6: he can park for at most 6 hours.
When the coefficient being divided out is negative, the inequality symbol flips. Solving -4x > 20 by dividing both sides by -4 gives x < -5, not x > -5.
Solve -4x > 20 for x.
- Divide both sides by -4. Because -4 is negative, the inequality symbol flips from > to <.
- 20 / -4 = -5, so x < -5.
- Check with a value in the solution set, x = -6: -4 x (-6) = 24, and 24 > 20 is true.
Answer: x < -5.
- Why does dividing both sides of an inequality by a negative number flip the symbol?
- For 5 + 2h <= 17, why is the first step to subtract 5 rather than divide by 2?
Common misconceptions and how to address them
MisconceptionAny two terms in an expression can be combined, such as simplifying 3x + 5 to 8x.
Why it happens: Students treat 'combine' as a blanket instruction to merge any nearby numbers, rather than checking the variable parts match.
How to address it: Only terms with the EXACT same variable part can be combined (3x and 5x, but not 3x and 5). Underline like terms in matching colors before combining.
MisconceptionDistributing a negative only changes the sign of the first term inside the bracket, e.g. -2(3x - 5) = -6x - 10.
Why it happens: Students apply the negative sign to the first term and then continue with the original sign for the rest.
How to address it: The number outside the bracket multiplies EVERY term inside it, sign included. Write out both multiplications separately: -2 x 3x AND -2 x (-5), before combining.
Misconceptionp + 0.08p and 1.08p must be different amounts, because one expression has two terms and looks 'more done' than the single-term one.
Why it happens: Without rewriting p as 1p first, it is not obvious that the two expressions combine to the same thing.
How to address it: Rewrite p as 1p, then combine like terms exactly as with any other expression: 1p + 0.08p = 1.08p. Test both forms with a real number for p to confirm they match.
MisconceptionIn a word problem like 'spends 2/5 of it', the fraction is subtracted directly from the total instead of being multiplied by it first, e.g. 45 - 2/5 = 44.6.
Why it happens: Students see 'spends... of it' and jump straight to subtraction, skipping the multiplication that finds the actual amount spent.
How to address it: Translate word by word: find the PART first by multiplying the fraction by the whole amount (45 x 2/5 = 18), and only then subtract that part from the total.
MisconceptionYou never need to flip the inequality symbol when solving.
Why it happens: Most early equation-solving practice never involves dividing by a negative number, so the flip rule is never triggered and gets forgotten.
How to address it: Whenever a step divides or multiplies BOTH sides by a negative number, the inequality symbol flips direction. Check the sign of the coefficient before that step every time.
Guided practice (with answers)
1. Simplify: 2/3 x + 4 - 1/3 x - 1
Answer: 1/3 x + 3, because 2/3 x - 1/3 x = 1/3 x, and 4 - 1 = 3.
2. Expand: 5(2x - 3)
Answer: 10x - 15.
3. Factor: 12x + 18
Answer: 6(2x + 3), because the greatest common factor of 12 and 18 is 6.
4. Write two equivalent expressions for a price p decreased by 15%, and simplify one into a single term.
Answer: p - 0.15p, which simplifies to 0.85p (the sale price is always 85% of the original).
5. Maria has $60. She spends 1/3 of it on a game, then her aunt gives her $15. How much does she have now?
Answer: $55, because she spends 60 x 1/3 = $20 (leaving $40), then 40 + 15 = $55.
6. Solve: 4n + 7 = -9
Answer: n = -4, because 4n = -16, so n = -16 / 4 = -4.
7. Solve the inequality: -4x > 20
Answer: x < -5, because dividing both sides by -4 (negative) flips the symbol.
Independent practice worksheets
Practise combining, expanding, factoring, and solving equations and inequalities with computed, never-wrong answer keys.
Differentiation
- Color-code like terms before combining, so which terms belong together is visual rather than something to track mentally.
- Practice distributing a negative as its own drill: write out -2 x 3x and -2 x (-5) as two separate multiplications every time, before combining into one line.
- For equivalent-expression work, always rewrite p as 1p first, so combining p and 0.08p looks identical to any other like-terms problem.
- Give the inequality flip rule its own short checklist step: 'Am I dividing or multiplying by a negative? If yes, flip the symbol.'
- Introduce expressions requiring both expanding AND combining like terms in the same problem, e.g. -3(x - 4) + 2x.
- Ask students to find and simplify the single-term equivalent expression for two successive percent changes (e.g. 20% off, then a further 5% loyalty discount) and explain what the combined multiplier represents.
- Pose a multistep word problem that requires writing an inequality with more than one operation, such as a budget with both a flat fee and a per-item cost.
- Explore why -4x > 20 and 4x < -20 have the exact same solution set, without solving either one first.
Assessment: exit ticket
A three-question exit ticket sampling expanding and combining, an equivalent-expression, and solving an equation.
1. Expand and simplify: -3(x - 4) + 2x
Answer: -x + 12, because -3(x - 4) = -3x + 12, and -3x + 12 + 2x = -x + 12.
2. A price p increases by 6% tax. Write the total as a single-term expression.
Answer: 1.06p, because p + 0.06p = 1p + 0.06p = 1.06p.
3. Solve: -2/3 x - 5 = 1
Answer: x = -9, because -2/3 x = 6, and dividing by -2/3 (multiplying by its reciprocal, -3/2) gives x = 6 x (-3/2) = -9.
Teacher notes and timings
- Rough timing across four to five lessons: Lesson 1 combining/expanding/factoring (section 1), Lesson 2 equivalent expressions (section 2), Lesson 3 multistep word problems (section 3), Lesson 4-5 equations and inequalities plus the exit ticket (section 4 and assessment).
- This unit assumes comfort with expanding a single bracket and building expressions from words (the Year 7 linear equations unit) and finding a GCF (the Grade 6 GCF/LCM unit). Revisit either if shaky before adding rational coefficients.
- Language to keep repeating: only combine terms with the SAME variable part; a factor outside a bracket multiplies EVERY term inside; two expressions are equivalent if they give the same value for every value of the variable; flip the inequality symbol only when dividing or multiplying by a negative.
- Section 2's tax/discount examples deliberately reuse the same percent-of-a-price idea from the ratios and percent unit, so students see 7.EE.A.2 as the algebraic form of reasoning they already trust numerically.
- Curriculum note: 7.EE.A.1-2 (expression skills) and 7.EE.B.3-4 (equation/inequality skills applied to real problems) are grouped as one unit here because a Grade 7 word problem typically needs both: building the right expression, then solving the equation or inequality it becomes.
- Present mode and print both work: use Present to build the equation from the word problem live with the class in section 4, then print the worksheets for independent practice.