Unit rates, proportional relationships and percent
Computing unit rates (including fractional ones), spotting proportional relationships, and solving multistep percent problems
About four lessons of 45 to 60 minutes
Which job pays more: $18/hour, or $650 for a 36-hour week?
You cannot compare those two job offers until you turn them into the same kind of number: dollars for ONE hour. $650 for 36 hours is $18.06 per hour, just barely more than $18/hour flat. That single-unit comparison, a unit rate, is how you compare almost anything: prices, speeds, pay, fuel use.
Ratios show up everywhere business and everyday life meet: a $1.05 markup on every dollar (tax), a recipe that scales up or down, a graph of cost against quantity that either passes through the origin (proportional) or does not. This unit builds all three: finding a unit rate, telling a proportional relationship from an imposter, and solving the percent problems (tax, tips, discounts) that use the same proportional reasoning underneath.
- Miles per gallon, or dollars per hourany 'per' quantity is a unit rate: one number, for exactly one unit of the other
- A candle business charging $2.50 per candlerevenue = 2.5 x candles, a proportional relationship through (0, 0)
- A restaurant bill with tax and tip added ona multistep percent problem, computed on the same original price
- A jacket marked '20% off, then an extra 10% off'successive percent changes do not just add together, as this unit shows
What students will be able to do
Students will compute a unit rate from a ratio, including one written as a fraction, decide whether a table or relationship is proportional and find its constant of proportionality, and use proportional reasoning to solve multistep percent problems involving tax, tips, discounts and percent increase or decrease.
- I can compute a unit rate, including one where the given quantities are fractions.
- I can decide whether a table of values represents a proportional relationship by checking whether y/x is constant for every pair.
- I can find the constant of proportionality k from a table or a real-world description, and use it to write y = kx.
- I can solve a multistep percent problem such as finding a sale price, adding tax and tip, or finding percent increase or decrease.
- I can explain why two successive percent changes do not simply add together.
Standards this unit teaches
- 7.RP.A.1Common Core (US)Compute unit rates
Compute unit rates, including rates given as fractions, in real world problems.
- 7.RP.A.2Common Core (US)Recognise proportional relationships
Decide whether two quantities are proportional and identify the constant of proportionality.
- 7.RP.A.3Common Core (US)Percent problems
Use proportional reasoning to solve multistep percent problems such as tax, tips, and discounts.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 6 ratios & rates teaching unitunit rates are built directly from the ratio and rate ideas introduced here
- Grade 6 percentages teaching unitfinding a percent of a number is the core skill multistep percent problems build on
- Ratio in the glossarya quick refresher on what a ratio compares
- Percentage in the glossarya percent is a rate out of 100, the idea behind every tax/tip/discount problem
Words to teach and display
- Unit rate
- a rate written for a single unit of the other quantity, such as dollars per 1 hour
- Proportional relationship
- a relationship where y/x is the same constant value for every pair, so the graph is a straight line through (0, 0)
- Constant of proportionality (k)
- the constant value of y/x in a proportional relationship, written as y = kx
- Percent
- a rate out of 100
- Discount
- an amount subtracted from a price, usually given as a percent of the original price
- Markup / percent increase
- an amount added to a starting value, usually given as a percent of that starting value
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. Unit rates, including ones written as fractions
ConcreteA unit rate answers 'how much for exactly ONE'. Divide the first quantity by the second, keeping careful track of which unit each number belongs to: the word right after 'per' is always the divisor.
Jenna earns $36 for 4 hours of babysitting. Her unit rate is 36 / 4 = $9 per hour. Once you have that single number, you can answer any related question by multiplying: 7 hours would earn 7 x $9 = $63.
Grade 7 unit rates can involve fractions on either side. A runner who covers 2 3/4 miles in 1/2 hour is still just dividing: 2.75 miles / 0.5 hours. Convert to decimals or keep everything as fractions, but divide the first quantity by the second exactly the same way.
A runner covers 2 3/4 miles in 1/2 hour, running at a steady pace. Find her unit rate in miles per hour.
- Write 2 3/4 as a decimal or improper fraction: 2 3/4 = 11/4 = 2.75 miles.
- The unit rate is distance divided by time: 2.75 / 0.5.
- 2.75 / 0.5 = 5.5.
Answer: The unit rate is 5.5 miles per hour.
- In '$36 for 4 hours', which number divides which to get the unit rate?
- Why does dividing by 1/2 (or 0.5) give a bigger number than the original quantity?
2. Spotting a proportional relationship and its constant
PictorialA relationship between x and y is proportional only if y/x gives the exact same number for every single pair in the table, and that shared value is the constant of proportionality, k. Graphed, a proportional relationship is always a straight line that passes through (0, 0).
A small candle business sells candles for $2.50 each: 2 candles earn $5, 4 candles earn $10, 6 candles earn $15, 8 candles earn $20. Check y/x for every pair: 5/2 = 2.5, 10/4 = 2.5, 15/6 = 2.5, 20/8 = 2.5. Every ratio matches, so this IS proportional, with k = 2.5, giving the rule revenue = 2.5 x candles.
Compare that with a delivery service that charges a flat $1 booking fee plus $2 per package: 1 package costs $3, 2 packages cost $5, 3 packages cost $7, 4 packages cost $9. Check y/x: 3/1 = 3, 5/2 = 2.5, 7/3 is not a whole number, 9/4 = 2.25. The ratio keeps changing, so this is NOT proportional, even though it also increases steadily. The flat $1 fee is the giveaway: a proportional relationship never has a fee added on top, because it must pass through (0, 0).
A table shows x: 3, 6, 9 and y: 12, 24, 36. Is this proportional? If so, find k.
- Check y/x for every pair: 12/3 = 4, 24/6 = 4, 36/9 = 4.
- Every ratio gives the same value, 4, so the relationship is proportional.
Answer: Yes, it is proportional, with constant of proportionality k = 4 (so y = 4x).
- Why does a flat booking fee stop a relationship from being proportional?
- If y/x is 4 for one pair but 5 for another pair in the same table, is the relationship proportional?
3. Multistep percent problems: tax, tips and discounts
AbstractEvery percent problem is proportional reasoning: 'percent of' means multiply by the percent written as a decimal. A multistep problem just repeats that move more than once, on the correct base amount each time.
A meal costs $42.00 before tax. Sales tax is 8%, and you leave a 20% tip on the pre-tax amount. Both the tax and the tip are separate percentages of the SAME original $42, so compute each one and add them on: tax = 42 x 0.08 = $3.36, tip = 42 x 0.20 = $8.40, total = 42 + 3.36 + 8.40 = $53.76.
Watch out when percent changes happen one after another instead of on the same base: a second discount is a percent OF the already-discounted price, not of the original. That single idea is exactly what the misconception below is built around.
A meal costs $42.00. Tax is 8%, and you leave a 20% tip on the pre-tax amount. Find the total.
- Tax: 42 x 0.08 = 3.36.
- Tip: 42 x 0.20 = 8.40.
- Total: 42 + 3.36 + 8.40 = 53.76.
Answer: The total is $53.76.
- Why are tax and tip both calculated on the ORIGINAL $42 in this problem, rather than one on top of the other?
- How would you find the sale price of an item after a 25% discount, in one multiplication?
Common misconceptions and how to address them
MisconceptionAny table where the numbers keep increasing is a proportional relationship.
Why it happens: Students focus on 'both go up together' and miss that a proportional relationship needs the SAME ratio y/x every time, with no extra fee or starting amount.
How to address it: Compute y/x for every single pair, not just one. If a flat fee or starting amount is added on top of a rate, the relationship is not proportional, because it does not pass through (0, 0).
MisconceptionA unit rate is found by dividing whichever way feels natural, sometimes dividing hours by dollars instead of dollars by hours.
Why it happens: Without anchoring to the word 'per', it is easy to divide in the wrong order and get an answer with the wrong units.
How to address it: The word right after 'per' (or 'for each') tells you the divisor. '$9 per hour' means dollars divided by hours; always write the units next to the numbers to check the division is the right way round.
MisconceptionSuccessive percent changes just add together, so a 20% discount followed by a 10% discount is a 30% discount.
Why it happens: Adding the percentages feels natural, but the second discount applies to the already-reduced price, a smaller amount, so it removes less than 10% of the ORIGINAL price.
How to address it: Work through it with real numbers: a $50 item after 20% off is $40 (saved $10). A further 10% off $40 removes $4, giving $36. Total saved is $14, which is 28% of $50, not 30%. Apply each percent change to the current price, one step at a time, never add the percentages first.
Misconception'Percent off' or 'percent more' is subtracted or added directly as a plain number, e.g. a $60 item with 25% off becomes $60 - 25 = $35.
Why it happens: Students treat the percent like a dollar amount instead of converting it to a fraction or decimal OF the price first.
How to address it: Always compute the percent of the price first (25% of $60 = $15), and only then subtract or add that dollar amount. The percent is never subtracted as a bare number.
MisconceptionIn a 'find the whole' problem like '18 is 30% of what number', students multiply 18 by 0.30 instead of dividing.
Why it happens: Multiplying is the more practiced move in easier percent-of problems, so it gets reused here even though the unknown has moved.
How to address it: Identify what the unknown actually is: if the PART (18) and the PERCENT (30%) are known but the WHOLE is missing, divide the part by the percent as a decimal: 18 / 0.30 = 60.
Guided practice (with answers)
1. Find the unit rate: $4.50 for 3/4 lb of grapes.
Answer: $6 per lb, because 4.50 / 0.75 = 6.
2. Is this table proportional? x: 3, 6, 9 y: 12, 24, 36. If yes, give k.
Answer: Yes, proportional, k = 4, because 12/3 = 24/6 = 36/9 = 4.
3. Is this table proportional? x: 3, 6, 9 y: 10, 22, 36.
Answer: No, because 10/3 is about 3.33, 22/6 is about 3.67, and 36/9 = 4 — the ratio is not constant.
4. An $80 jacket is discounted 15%. Find the sale price.
Answer: $68, because the discount is 80 x 0.15 = $12, and 80 - 12 = $68.
5. 60 is increased by 25%. What is the new value?
Answer: 75, because 60 x 1.25 = 75 (or 60 + 60 x 0.25 = 60 + 15 = 75).
6. 18 is 30% of what number?
Answer: 60, because 18 / 0.30 = 60.
Independent practice worksheets
Practise unit rates, proportional relationships and percent problems with computed, never-wrong answer keys.
Differentiation
- Start unit rates with whole-number quantities only, and introduce the fractional case (2 3/4 miles in 1/2 hour) once the division routine is secure.
- Give a ready-made table with a y/x column already drawn, so checking for proportionality is a single division per row rather than a two-step process.
- For percent problems, always convert the percent to a decimal on paper before multiplying (write '20% = 0.20' every time) until the conversion is automatic.
- Work the successive-discount misconception with concrete dollar amounts (a $50 item) before any algebraic or general version.
- Introduce unit rates with two fractional quantities (e.g. 3/4 mile in 2/5 hour) requiring fraction division.
- Ask students to write the equation y = kx for a real proportional scenario they design themselves, and use it to predict an unlisted value.
- Pose a three-step percent problem (discount, then tax, then a delivery fee that is itself a percent of the new subtotal) and have students justify the order of operations.
- Investigate: what percent discount, applied twice, is equivalent to a single 36% discount? (Answer: 20%, since 0.8 x 0.8 = 0.64, an overall 36% reduction.)
Assessment: exit ticket
A three-question exit ticket sampling unit rates, proportionality, and a multistep percent problem.
1. 160 words are typed in 4 minutes. Find the unit rate in words per minute.
Answer: 40 words per minute, because 160 / 4 = 40.
2. Is this table proportional? x: 5, 10, 15 y: 8, 16, 24. If yes, give k.
Answer: Yes, proportional, k = 1.6, because 8/5 = 16/10 = 24/15 = 1.6.
3. A $25 shirt has a 20% discount, then 6% sales tax is added to the discounted price. Find the final price.
Answer: $21.20, because the discount is 25 x 0.20 = $5, giving a discounted price of $20; tax is 20 x 0.06 = $1.20, so the final price is 20 + 1.20 = $21.20.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 unit rates including fractions (section 1), Lesson 2 proportional relationships (section 2), Lesson 3 multistep percent problems (section 3), Lesson 4 mixed practice plus the exit ticket.
- This unit assumes comfort with basic ratios and finding a simple percent of a number (Grade 6 ratios/rates and percentages units). Revisit those first if either foundation is shaky.
- Language to keep repeating: 'per' tells you the divisor for a unit rate; a proportional relationship has a CONSTANT y/x and passes through (0, 0); percent changes apply to the CURRENT amount, one step at a time.
- The successive-discount misconception is one of the most common Grade 7 percent errors in the research literature; work it with concrete numbers before any general rule, and revisit it explicitly in the assessment.
- Curriculum note: 7.RP.A.1-3 (Common Core) form a natural single unit, unit rates and proportionality being the mathematical foundation that multistep percent reasoning is built on.
- Present mode and print both work: use Present to build the y/x table live with the class for section 2, then print the worksheets for independent practice.