Ratio Reasoning: Tables, Graphs & the Better Deal
Welcome to Maple Street Smoothie Co. — a small shop that mixes fruit and yogurt by ratio. Today you are the shift manager: you will scale recipes with ratio tables, graph them, and decide which supplier offers the better deal. Hoy eres el encargado del turno en una tienda de batidos.
1 Engage
Hook & essential question
Two stands are selling the same lemonade at the school fair. Look at their signs. Without a calculator, which stand is the better deal — or are they the same?
Sunny Stand · 3 cups for $6
Cool Cart · 5 cups for $8
Quick think (no calculator): Make a first guess. Which is the better deal for a buyer — Sunny Stand, Cool Cart, or the same? Write your guess and one reason in your notebook. You will test it in the Apply section using a unit rate (price for 1 cup).
Essential question: How can ratio tables, graphs, and unit rates help me scale a recipe and decide which deal is best?
2 Explore
Investigate before the lesson
Explore each tool for a few minutes. Watch for one big idea: equivalent ratios make a straight line through the origin when you graph them, and the steepness of that line is the unit rate.
Live explorer — scale the smoothie recipe
The house recipe is 2 cups of fruit for every 3 cups of yogurt (a 2 : 3 ratio). Drag the slider to make more batches. Notice the two numbers grow together, but the fruit-per-yogurt rate never changes.
2 cups fruit : 3 cups yogurt
Fruit per 1 cup yogurt: 0.67 cups · la tasa unitaria
Each batch plots a new point — all on one straight line through (0, 0).
3 Explain
The math, in plain language
A ratio compares two amounts. Two ratios are equivalent when you can multiply (or divide) both numbers by the same factor. A unit rate is the amount for exactly one of something — it is the fastest way to compare two ratios.
- ratio
- compares two amounts, like 2 : 3 · la razón
- equivalent ratios
- same comparison, scaled up or down · razones equivalentes
- ratio table
- a table of equivalent ratios · tabla de razones
- unit rate
- amount for exactly 1 · la tasa unitaria
Build a ratio table
Start with the recipe 2 cups fruit : 3 cups yogurt. Multiply both rows by the same number to scale.
| Fruit (cups) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Yogurt (cups) | 3 | 6 | 9 | 12 |
Every column is the same ratio: 2 : 3 = 4 : 6 = 6 : 9 = 8 : 12.
Graph the ratios
Equivalent ratios always form a straight line through (0, 0).
Worked example — the better deal (unit rate)
Back to the Engage signs. Find the price for 1 cup at each stand.
Sunny Stand: $6 ÷ 3 cups = $2.00 per cup
Cool Cart: $8 ÷ 5 cups = $1.60 per cup
$1.60 < $2.00, so Cool Cart is the better deal — you pay less for each cup. (How close was your Engage guess?)
Level 1 · support Step-by-step helper / Ayuda paso a paso
- To make an equivalent ratio, multiply both numbers by the same factor (×2, ×3, …).
- To find a unit rate, divide to get the amount for 1 (price ÷ cups).
- To compare deals, the lower price per cup is the better buy.
- Keep the labels with your numbers (cups, dollars). Mantén las etiquetas con tus números.
Level 2 · enrichment Push your thinking
You can compare ratios two ways: price per cup (lower is better) or cups per dollar (higher is better). Compute Cool Cart both ways and explain why both methods agree on the same winner. Which way is more useful when you have a fixed budget?
4 Apply
Show what you can do — auto-checked
Level 1 · support Sentence starters for explaining
"I made an equivalent ratio by multiplying both numbers by ____. To find the unit rate I divided ____ ÷ ____ = ____ per one. The better deal is ____ because ____."
"Hice una razón equivalente multiplicando ambos números por ____. Para hallar la tasa unitaria dividí ____ ÷ ____. La mejor oferta es ____ porque ____."
Level 2 · enrichment Reverse it
A juice has a unit rate of $1.25 per cup. The store sells it only in packs that cost a whole number of dollars. List two possible cups-and-price combinations that match this rate, then write the rule you used.
Teacher Notes & Answer Key (not printed)
Stage-by-Stage Notes (Engage → Explore → Explain → Apply → Reflect)
- Engage: compare two smoothie mixes and predict which is fruitier.
- Explore: the slider builds a ratio table and plots points on a line.
- Explain: equivalent ratios lie on one line through the origin; the unit rate is the slope.
- Apply: answer key below.
- Reflect: students connect table, graph, and unit rate.
Apply — Answer Key
- Q1 — Ratio table (2:3), 15 yogurt: 15 ÷ 3 = 5, fruit = 2 × 5 = 10.
- Q2 — Unit rate: 1.6.
- Q3 (MC): answer b.
- Q4 (MC): answer c.
- Q5 — Scale: 35.
- Q6 (MC): answer b.
Standard
CCSS 6.RP.A.3a.
5 Reflect
Think about your thinking
A. A ratio table, a graph, and a unit rate can all show the same relationship. Which one would you reach for first to compare two deals, and why?
B. Why do equivalent ratios always make a straight line through the origin (0, 0) when you graph them?
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
Run a price-comparison for Maple Street Smoothie Co.
Find three real prices for the same item (online or in a store) sold in different sizes — for example juice, snacks, or notebooks. Then:
- Make a ratio table for each item (amount and price).
- Find the unit rate (price for 1) of each.
- Rank them from best deal to worst, and write one sentence defending your #1 pick.
- Challenge: Graph one item's ratio table and label the point that shows the unit rate.
Compara tres precios reales del mismo producto, halla la tasa unitaria de cada uno y ordénalos de la mejor a la peor oferta.