Defining Quantities for Modeling
Aligned to N-Q.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
When you model a real situation, you first choose which quantities to measure and how to combine them. A good quantity is measurable, relevant, and makes the comparison you care about fair. Rates and ratios (per person, per square foot) let you compare unequal things.
Worked examples
Example 1
Two towns want to compare crowding. Town A has 9,000 people; Town B has 40,000. Raw counts mislead, so define density = people / area. Town A covers 3 sq mi: 9000/3 = 3000 per sq mi. Town B covers 20 sq mi: 40000/20 = 2000 per sq mi. Town A is more crowded.
Example 2
Two streaming services want to compare growth. Service A added 2 million subscribers this year; Service B added 400,000. Raw counts favor A, but define growth rate = added / starting count. A started with 100 million: 2/100 = 2%. B started with 5 million: 0.4/5 = 8%. B is growing four times faster.
Example 3
Common mistake: choosing a quantity that cannot actually be measured. Defining school quality as student happiness sounds reasonable but has no agreed scale and no data source. A usable quantity needs three things: a clear unit, a way to collect it, and a level of precision. Graduation rate (percent of entering ninth graders who finish in four years) has all three; happiness has none of them.
Practice questions
- To rank two pizzas by value, what single quantity (a ratio) should you compute?
- A team scored 240 points in 12 games. Define and compute a quantity that lets you compare it to a team with 300 points in 20 games.
- Explain why 'total library books' is a poor quantity for comparing how well two schools serve their students, and give a better one.
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
Definir cantidades para modelar
Al modelar una situación real, primero eliges qué cantidades medir y cómo combinarlas. Una buena cantidad es medible, relevante y hace justa la comparación que te importa. Las tasas y razones (por persona, por pie cuadrado) permiten comparar cosas desiguales.
Ejemplo 1: Dos pueblos quieren comparar su congestión. El pueblo A tiene 9,000 personas; el B, 40,000. Los conteos crudos engañan, así que define densidad = personas / área. A cubre 3 mi²: 9000/3 = 3000 por mi². B cubre 20 mi²: 40000/20 = 2000 por mi². El pueblo A está más congestionado.
Ejemplo 2: Dos servicios de streaming quieren comparar su crecimiento. El servicio A sumó 2 millones de suscriptores este año; el servicio B sumó 400,000. Los conteos brutos favorecen a A, pero define tasa de crecimiento = sumados / cantidad inicial. A empezó con 100 millones: 2/100 = 2%. B empezó con 5 millones: 0.4/5 = 8%. B crece cuatro veces más rápido.
Ejemplo 3: Error común: elegir una cantidad que en realidad no se puede medir. Definir la calidad escolar como la felicidad del estudiante suena razonable pero no tiene escala acordada ni fuente de datos. Una cantidad utilizable necesita tres cosas: una unidad clara, una forma de recolectarla y un nivel de precisión. La tasa de graduación (porcentaje de estudiantes de noveno que terminan en cuatro años) tiene las tres; la felicidad no tiene ninguna.
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