Proportional Relationships & Slope
Aligned to 8.EE.B.5 — Common Core State Standards for Mathematics.
What this lesson teaches
A proportional relationship graphs as a straight line through the origin, and its unit rate is the slope. Comparing the slopes of two relationships tells you which one grows faster.
Worked examples
Example 1
Car A travels 120 miles in 2 hours, so its unit rate is 120 ÷ 2 = 60 mph (slope 60). Car B's graph rises 50 miles every 1 hour (slope 50). Since 60 > 50, Car A is faster.
Example 2
A faucet fills 3 gallons in 4 minutes (slope 0.75 gal/min); a hose fills 5 gallons in 8 minutes (slope 0.625). The faucet is faster.
Example 3
A proportional graph must pass through the origin. A phone plan with a $20 monthly fee starts at 20, so it has slope but is not proportional.
Practice questions
- Find the unit rate (slope) for 45 words typed in 3 minutes.
- A line passes through (0, 0) and (4, 10); what is its slope?
- Store A charges $9 for 3 lb; Store B's graph shows $8 for 2 lb. Which has the better unit price?
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
Relaciones Proporcionales y Pendiente
Una relación proporcional se grafica como una recta que pasa por el origen, y su tasa unitaria es la pendiente. Comparar las pendientes de dos relaciones indica cuál crece más rápido.
Ejemplo 1: El auto A recorre 120 millas en 2 horas, así que su tasa unitaria es 120 ÷ 2 = 60 mph (pendiente 60). La gráfica del auto B sube 50 millas por cada hora (pendiente 50). Como 60 > 50, el auto A es más rápido.
Ejemplo 2: Una llave llena 3 galones en 4 minutos (pendiente 0.75 gal/min); una manguera llena 5 galones en 8 minutos (pendiente 0.625). La llave es más rápida.
Ejemplo 3: Una gráfica proporcional debe pasar por el origen. Un plan de celular con $20 de cuota mensual empieza en 20, así que tiene pendiente pero no es proporcional.
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