Using Units to Guide Solutions
Aligned to N-Q.A.1 — Common Core State Standards for Mathematics.
What this lesson teaches
Units carry meaning: tracking them through a calculation tells you whether an answer is set up correctly. When units cancel and leave the ones you want, the operation is right. Choosing sensible scales for a graph or formula keeps the model readable.
Worked examples
Example 1
A car travels 150 miles on 6 gallons. Find miles per gallon. Set up: 150 miles / 6 gallons. Divide: 150/6 = 25. The units are miles/gallon, exactly what 'per gallon' asks for. Answer: 25 miles per gallon.
Example 2
A recipe needs 3 cups of flour for 24 cookies, and you want 60 cookies. Track the units: (3 cups / 24 cookies) × 60 cookies = 7.5 cups. The word cookies cancels top and bottom, leaving cups — which is what the question asked for, so the setup was right before any arithmetic happened.
Example 3
Common mistake: flipping the conversion factor. Converting 90 minutes to hours, writing 90 min × (60 min / 1 hr) gives 5,400 with units of min²/hr — nonsense. The units are the check: you want min to cancel, so it must appear on the BOTTOM of the factor: 90 min × (1 hr / 60 min) = 1.5 hr. If the leftover units are not the ones you were asked for, the factor is upside down.
Practice questions
- A runner covers 400 meters in 50 seconds. Find the speed in meters per second.
- Convert 90 kilometers per hour into meters per second using 1 km = 1000 m and 1 h = 3600 s.
- A recipe needs 3 cups of flour per 12 cookies. How many cups are needed for 40 cookies, and what units confirm your setup?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Usar unidades para guiar soluciones
Las unidades tienen significado: seguirlas a lo largo de un cálculo indica si el planteamiento es correcto. Cuando las unidades se cancelan y quedan las que buscas, la operación está bien. Elegir escalas sensatas para una gráfica o fórmula mantiene el modelo legible.
Ejemplo 1: Un auto recorre 150 millas con 6 galones. Halla las millas por galón. Planteamiento: 150 millas / 6 galones. Divide: 150/6 = 25. Las unidades son millas/galón, justo lo que pide 'por galón'. Respuesta: 25 millas por galón.
Ejemplo 2: Una receta necesita 3 tazas de harina para 24 galletas y quieres 60 galletas. Sigue las unidades: (3 tazas / 24 galletas) × 60 galletas = 7.5 tazas. La palabra galletas se cancela arriba y abajo y queda tazas — que es lo que pedía la pregunta, así que el planteamiento era correcto antes de hacer cuentas.
Ejemplo 3: Error común: invertir el factor de conversión. Al pasar 90 minutos a horas, escribir 90 min × (60 min / 1 h) da 5,400 con unidades de min²/h — un sinsentido. Las unidades son la comprobación: quieres que min se cancele, así que debe ir ABAJO en el factor: 90 min × (1 h / 60 min) = 1.5 h. Si las unidades que sobran no son las pedidas, el factor está al revés.
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