Mathematics Grade 5 Unitary Method

Grade 5 Mathematics: Unitary Method

1. The Driving Question

If a 6-pack of juice boxes costs $4.50, how much should a single juice box cost? And if you know the price of one, how do you figure out the price of 10, or 100, without buying them all? Why can’t you just divide the total cost by any number you want?

2. The Core Idea — Built, Not Listed

Imagine you’re at a lemonade stand with your little brother. He sells 3 cups of lemonade for $1.50. A customer wants to know how much one cup costs before deciding how many to buy. You could split the $1.50 into 3 equal parts—each part is $0.50. Now, if the customer wants 5 cups, you don’t have to guess: you know one cup is $0.50, so 5 cups must be 5 × $0.50 = $2.50. That’s the unitary method—finding the value of one unit first, then scaling it up or down to answer any question.

This works for anything that comes in groups: pencils in a pack, miles per hour, or even how many minutes it takes to read 10 pages if you know how long 1 page takes. The key is that the "unit" doesn’t have to be one thing—it could be one group (like one dozen eggs) as long as the groups are all the same size.

Key Vocabulary:
- Unit – A single fixed amount used as a standard for measurement or calculation.
Example: If a 12-ounce soda is your "unit," then a 24-ounce soda is 2 units.
- Rate – A comparison of two different units, like miles per hour or price per item.
Example: If 4 apples cost $2, the rate is $0.50 per apple.
- Scaling – Multiplying or dividing a unit to find a new total.
Example: If 1 ticket costs $8, then 7 tickets cost 7 × $8 = $56.
- Proportion – An equation that shows two rates are equal.
Example: 3 cups / $1.50 = 5 cups / $2.50 (both simplify to $0.50 per cup).

3. Assessment Translation

How This Appears in Classroom Work (Grade 5):
- Exit Tickets: "If 5 notebooks cost $12.50, how much do 8 notebooks cost?" (Show your work.) - Short Constructed Response: "A baker uses 3 cups of flour to make 24 cookies. How many cups of flour are needed for 40 cookies? Explain your steps." - Show-Your-Work Problems: Word problems with space for calculations, often involving real-world scenarios (e.g., pricing, recipes, or travel time).

Proficient vs. Developing Responses:
| Proficient | Developing | |----------------|----------------| | Finds the unit rate first (e.g., $12.50 ÷ 5 = $2.50 per notebook), then scales up (8 × $2.50 = $20). | Divides the total cost by the wrong number (e.g., $12.50 ÷ 8 = $1.56) or multiplies the original numbers directly (5 × 8 = 40, $12.50 × 40 = $500). | | Explains each step clearly: "First, I found the cost of one notebook. Then, I multiplied by 8 to get the total." | Skips steps or writes only the final answer without showing how they got there. | | Labels units (e.g., "$2.50 per notebook") to avoid confusion. | Omits units or mixes them up (e.g., writing "2.50" without specifying dollars or notebooks). |

Model Proficient Response:
Prompt: A car travels 180 miles in 3 hours. How far will it travel in 7 hours at the same speed? Response: 1. First, find the distance traveled in 1 hour: 180 miles ÷ 3 hours = 60 miles per hour.
2. Then, multiply by 7 hours: 60 miles/hour × 7 hours = 420 miles.
Answer: The car will travel 420 miles in 7 hours.

4. Mistake Taxonomy

Mistake 1: Dividing the Wrong Way
- Prompt: 4 bags of chips cost $6. How much does 1 bag cost? - Common Wrong Response: $6 ÷ 4 = $1.50 (correct), but then the student writes "1 bag costs $24" because they multiplied $6 × 4 instead of dividing.
- Why It Loses Credit: The student found the unit rate but then reversed the operation when scaling. The question asks for one bag, not four.
- Correct Approach: 1. Identify the total cost ($6) and the number of units (4 bags).
2. Divide to find the unit rate: $6 ÷ 4 = $1.50 per bag.
3. Stop here—the question only asks for one bag.

Mistake 2: Ignoring Units
- Prompt: A printer prints 120 pages in 4 minutes. How many pages does it print in 1 minute? - Common Wrong Response: 120 ÷ 4 = 30, but the student writes "30 minutes" instead of "30 pages." - Why It Loses Credit: The answer doesn’t match the question’s units. The student performed the math correctly but didn’t label the answer, making it unclear what "30" refers to.
- Correct Approach: 1. Write the rate as a fraction: 120 pages / 4 minutes.
2. Divide to find the unit rate: 30 pages per minute.
3. Label the answer: "30 pages in 1 minute."

Mistake 3: Assuming Direct Multiplication Without Finding the Unit
- Prompt: 6 markers cost $9. How much do 10 markers cost? - Common Wrong Response: $9 × 10 = $90.
- Why It Loses Credit: The student skipped finding the unit rate and assumed the cost scales directly with the number of markers. This only works if the unit rate is 1 (e.g., 1 marker = $1).
- Correct Approach: 1. Find the cost of 1 marker: $9 ÷ 6 = $1.50 per marker.
2. Multiply by 10: $1.50 × 10 = $15.

5. Connection Layer

1. Within Math: Unitary method → Ratios and Proportions
   Why it helps: The unitary method is the foundation for setting up proportions (e.g., if 3 apples cost $2, how much do 5 apples cost?). Understanding the "unit" makes it easier to see why cross-multiplication works.

2. Across Subjects: Unitary method → Science (Density and Speed)
   Why it helps: Density (mass per volume) and speed (distance per time) are rates, just like price per item. If you know the density of gold is 19.3 g/cm³, you can find the mass of any volume of gold using the unitary method.

3. Outside School: Unitary method → Shopping with Discounts
   Why it helps: Stores often advertise "3 for $5" deals. The unitary method lets you compare prices to see if it’s really a good deal (e.g., is 3 for $5 better than 1 for $1.80?). It also helps you calculate the cost of buying 2 or 4 items instead of 3.

6. The Stretch Question

If a recipe for 12 cookies calls for 2 cups of flour, how much flour would you need for 18 cookies? Now, what if the recipe is for 12 large cookies, but you want to make 18 small cookies (each small cookie is 2/3 the size of a large one)?

Pointer Toward the Answer:
Start with the first part: 12 cookies = 2 cups, so 1 cookie = 2/12 = 1/6 cup. Then 18 cookies = 18 × 1/6 = 3 cups. But for the second part, the size of the cookies changes the amount of flour per cookie. If small cookies are 2/3 the size, they might need 2/3 the flour. So 1 small cookie = (2/3) × (1/6) = 2/18 = 1/9 cup. Then 18 small cookies = 18 × 1/9 = 2 cups. The key is realizing that the "unit" isn’t just one cookie—it’s one standard-sized cookie.