Mix It, Match It: Ratios & Unit Rates
You just got hired at the Berry Blast Smoothie Cart. Every recipe, price tag, and refill on the menu is really a ratio — and your job is to keep the flavor the same no matter how big the batch gets. Cada receta y cada precio es una razón (ratio).
1 Engage
Hook & essential question
The cart's most popular drink, the Berry Blast, uses 2 scoops of berries for every 3 cups of juice. A big group orders a giant batch and you pour in 6 cups of juice. How many scoops of berries do you need so the drink still tastes exactly the same?
The juice doubled from 3 cups to 6 cups. What has to happen to the berries?
Quick think (no calculator): Make a first guess. Is the answer 3 scoops, 4 scoops, or 5 scoops? Write your guess and one reason in your notebook. You will check it in the Apply section.
Essential question: When I change the amount of one thing in a recipe or price, how do I change the other thing so the relationship stays equal?
2 Explore
Investigate before the lesson
Explore each tool below for a few minutes. As you go, watch for one big idea: a ratio keeps tasting (or costing) the same as long as you multiply both numbers by the same amount.
Try it: the Berry Blast mixer
Slide to change how many batches of the 2 : 3 recipe you make. Notice that the ratio of berries to juice never changes — only the totals grow. That is what equivalent ratios look like.
Berries: 2 scoops · Juice: 3 cups
Ratio stays: 2:3 = 2:3
Red = berries · Yellow = juice
3 Explain
The math, in plain language
A ratio compares two amounts. We can write the Berry Blast ratio three ways, and they all mean the same thing: 2:3, or "2 to 3", or the fraction 2/3. To make an equivalent ratio, multiply (or divide) both numbers by the same value.
- ratio
- a comparison of two amounts · la razón
- equivalent ratios
- ratios that show the same relationship · razones equivalentes
- rate
- a ratio of two different units (miles, dollars) · la tasa
- unit rate
- the rate for exactly 1 · la tasa por unidad
Worked example — keeping the flavor equal
Recipe ratio: 2:3 berries to juice. You poured 6 cups of juice. The juice went from 3 to 6, which is ×2. So multiply the berries by 2 too:
| Berries (scoops) | 2 | 4 |
|---|---|---|
| Juice (cups) | 3 | 6 |
2 × 2 = 4 scoops of berries. The ratio 4:6 is equivalent to 2:3, so it tastes the same. (How close was your Engage guess?)
Worked example — finding a unit rate (best deal)
A 4-cup bottle of juice costs $6. What is the cost for 1 cup? A unit rate means "per 1", so divide both numbers until the second number is 1:
$6 ÷ 4 = $1.50 per 1 cup
The unit rate is $1.50 per cup. Unit rates make it easy to compare deals: the lower the price per cup, the better the buy.
Level 1 · support Step-by-step helper / Ayuda paso a paso
- Write the ratio as a table with two rows (label each row with its unit).
- Look at the row that changed. Ask: "Times how much?" (e.g., 3 → 6 is ×2).
- Do the same multiplication to the other row. Multiplica las dos filas por el mismo número.
- For a unit rate, divide both numbers so the second number becomes 1.
Level 2 · enrichment Push your thinking
Two carts sell lemonade. Cart A: 3 cups for $4. Cart B: 5 cups for $6. Without a calculator, decide which is the better deal by comparing unit rates ($ per cup). Then explain why comparing "per 1" is fairer than just comparing total prices.
4 Apply
Show what you can do — auto-checked
Level 1 · support Sentence starters for explaining
"The ratio is ____ to ____. The ____ changed by × ____, so I multiplied the other amount by ____ too. My answer is ____."
"La razón es ____ a ____. Multipliqué las dos cantidades por ____. Mi respuesta es ____."
Level 2 · enrichment Reverse it
A smoothie batch uses berries and juice in a 2 : 3 ratio and contains 20 cups of juice. Work backwards to find the scoops of berries. Then write a one-sentence rule for scaling a ratio when you know the larger amount.
Teacher Notes & Answer Key (not printed)
Stage-by-Stage Notes (Engage → Explore → Explain → Apply → Reflect)
- Engage: students guess whether a new mix tastes the same; surface intuition about ratios before formal work.
- Explore: the live mixer lets students build equivalent ratios; have them notice both numbers scale together.
- Explain: define ratio, equivalent ratio, and unit rate; connect to the simplified form.
- Apply: see answer key below.
- Reflect: students explain why both numbers must scale and where unit rates help.
Apply — Answer Key
- Q1 — Equivalent ratio: 2 : 3 with 6 cups juice → 6 ÷ 3 = 2, so 2 × 2 = 4 scoops berries.
- Q2 — Equivalent ratio (MC): answer c.
- Q3 — Unit rate: 1.5 (e.g. 3 ÷ 2).
- Q4 — Scale up: 15.
- Q5 — Better-deal compare (MC): answer b.
Standard
CCSS 6.RP.A.1–3.
5 Reflect
Think about your thinking
A. To keep a recipe tasting the same, what do you have to do to both numbers in the ratio? Explain in your own words.
B. Why is a unit rate ("per 1") useful when you are deciding which deal to buy? Give a real example.
C. One thing that is now clear to me / Una cosa que ahora entiendo bien:
Your reflections save with this HyperDoc when you use Save as PDF or Save as DOC above.
6 Extend
Optional challenge project
Design a smoothie menu for the Berry Blast Cart
Invent three new smoothies. For each one:
- Write the ingredient ratio (for example, 3 : 2 berries to juice).
- Scale it up to a "family batch" that serves 6 — show your equivalent-ratio table.
- Set a price, then calculate the unit rate (price per cup) so a customer can compare deals.
Inventa tres batidos. Escribe la razón, haz una tabla de razones equivalentes y calcula el precio por taza (tasa por unidad).