2
Rates Compare Different Units
A rate is a special ratio that compares two quantities measured in different units (e.g., miles to hours, dollars to items, words to minutes).
3
Unit Rate Denominator is Always 1
To compute a unit rate, divide the first quantity by the second quantity (\(\frac{\text{Quantity}}{\text{Units}}\)). A unit rate always expresses "how much for 1 unit."
4
Equivalent Ratios Share the Same Multiplicative Value
Multiplying or dividing both terms of a ratio by the same nonzero number produces an equivalent ratio (e.g., \(\frac{18}{6} = \frac{3}{1}\)).
5
Always State Compound Units
Never write just a bare number for a unit rate. Always attach descriptive units such as "$1.50 per bottle" or "60 miles per hour."
Model on Board:
"A printer produces 120 pages in 4 minutes. Find the unit rate."
Step 1: Set up the rate: \(\frac{120 \text{ pages}}{4 \text{ minutes}}\)
Step 2: Divide numerator by denominator: \(120 \div 4 = 30\)
Step 3: State the unit rate with units: 30 pages per minute. Check: \(30 \times 4 = 120\).
Check for Understanding: "What must be true about the denominator or reference quantity for a rate to be called a unit rate?" (It must equal 1).
Grade 6 Math • Ratios and Unit Rates Page 2
1. School Garden: 18 roses and 54 tulips. Ratio of tulips to roses.
Correct Answer: c) The ratio of tulips to roses is 3:1
Exemplar Work:
• Identify values in required order: Tulips = 54, Roses = 18.
• Write raw ratio: \(\text{tulips} : \text{roses} = 54 : 18\) or fraction \(\frac{54}{18}\).
• Simplify by dividing by common factors: Divide both by 18 \(\rightarrow \frac{54 \div 18}{18 \div 18} = \frac{3}{1}\).
• Therefore, the ratio of tulips to roses is 3:1.
Misconception Note: Choice a (1:3) is the ratio of roses to tulips. Emphasize that order cannot be swapped.
2. 6-pack of juice bottles for $9.00. Find the unit rate (cost per bottle).
Correct Answer: $1.50 per bottle
Exemplar Work:
• Set up division: \(\text{Unit price} = \frac{\text{Total Cost}}{\text{Number of bottles}} = \frac{\$9.00}{6}\).
• Long division: \(9.00 \div 6 = 1.50\).
• Check: \(6 \times \$1.50 = \$9.00\).
• Final stated answer: $1.50 per juice bottle (or \(\frac{\$1.50}{1 \text{ bottle}}\)).
3. Maya (180 words in 3 min) vs. Leo (250 words in 5 min). Faster unit rate?
Correct Answer: Maya types at a faster unit rate.
Exemplar Work:
• Maya's unit rate: \(\frac{180 \text{ words}}{3 \text{ minutes}} = 180 \div 3 = 60\) words per minute.
• Leo's unit rate: \(\frac{250 \text{ words}}{5 \text{ minutes}} = 250 \div 5 = 50\) words per minute.
• Compare: \(60 \text{ wpm} > 50 \text{ wpm}\).
• Maya is faster by 10 words per minute.
Monitor: Check that students set up unit price as \(\frac{\text{dollars}}{\text{items}}\) rather than \(\frac{\text{items}}{\text{dollars}}\). Watch for students who stop at the division step without stating the final units.
Grade 6 Math • Ratios and Unit Rates Page 3
1. Box of 4 markers for $8. Find the unit price per marker.
Answer: $2.00 per marker
Work: \(\frac{\$8.00}{4} = \$2.00\). Check: \(4 \times \$2.00 = \$8.00\).
2. Cyclist rides 36 miles in 3 hours. Find unit rate in miles per hour.
Answer: 12 miles per hour (or 12 mph)
Work: \(\frac{36 \text{ miles}}{3 \text{ hours}} = 36 \div 3 = 12\). Check: \(12 \times 3 = 36\).
3. 6 triangles and 9 squares. Ratio of triangles to squares in simplest form.
Answer: 2:3 (or 2 to 3, or \(\frac{2}{3}\))
Work: \(\frac{6 \text{ triangles}}{9 \text{ squares}} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}\). Both terms divided by GCF 3.
4. Store A (12 pencils for $3.60) vs Store B (15 pencils for $4.20). Lower price?
Answer: Store B has the lower price per pencil ($0.28 < $0.30)
Work Store A: \(\$3.60 \div 12 = \$0.30\) per pencil.
Work Store B: \(\$4.20 \div 15 = \$0.28\) per pencil.
Comparison: \(\$0.28 < \$0.30\). Store B is cheaper by $0.02 per pencil.
5. 8 cups flour to 4 cups sugar. How many cups sugar for 10 cups flour?
Answer: 5 cups of sugar
Method 1 (Unit Rate): \(\frac{4 \text{ cups sugar}}{8 \text{ cups flour}} = 0.5\) cups sugar per cup of flour. For 10 cups flour: \(10 \times 0.5 = 5\) cups sugar.
Method 2 (Equivalent Ratio): The ratio is 2 flour : 1 sugar. Half as much sugar as flour \(\rightarrow 10 \div 2 = 5\) cups sugar.
Common Errors to Watch For:
• Inverting division terms (calculating \(15 \div \$4.20 = 3.57\) pencils per dollar instead of cost per pencil).
• Writing ratios without simplifying (leaving \(\frac{6}{9}\) instead of \(\frac{2}{3}\)).
• Swapping ratio order when answering part-to-part comparisons.
Grade 6 Math • Ratios and Unit Rates Page 4
1. A runner completes 12 laps in 4 minutes. Speed in laps per minute?
Correct Answer: 3 laps per minute
Exemplar Work:
• Set up rate: \(\frac{12 \text{ laps}}{4 \text{ minutes}}\)
• Divide: \(12 \div 4 = 3\)
• Check: \(3 \text{ laps/min} \times 4 \text{ min} = 12 \text{ laps}\).
2. Lucas bought 5 notebooks for $15. Claims unit rate is "5 notebooks per dollar."
Correct Answer: Lucas is incorrect.
Explanation of Error:
• Lucas confused the quantities and divided incorrectly. Unit price is the cost for one notebook (\(\frac{\text{cost}}{\text{items}}\)).
• Correct calculation: \(\frac{\$15}{5 \text{ notebooks}} = \$3.00\) per notebook.
• If calculating notebooks per dollar, \(5 \div 15 = \frac{1}{3}\) notebook per dollar, not 5.
• The correct unit rate is $3 per notebook (or \(\frac{1}{3}\) notebook per dollar).
Assessment Look-Fors:
Reteach Prompt:
"If you want to know how much one item costs, which number must go inside the division bracket (or as the dividend)? How can thinking about 'dollars per item' help you set up the fraction?"
Grade 6 Math • Ratios and Unit Rates Page 5
