Lesson 8 — Which Quantity Depends on the Other

Summary

Learners decide which of two related quantities depends on the other. They name the independent quantity (the input — the one you choose) and the dependent quantity (the output — the one that follows), using pairs such as portions and cooking time, and water and plant growth. They learn that not every pair of quantities is cause and effect.

Objectives

• Tell when one of two related quantities depends on the other, naming the independent and dependent quantity. (D05.S2.06.02)

Connection

If you cook for more people, the cooking time grows — but the number of people does not grow because the cooking time grew. Cooking time depends on the number of portions; the portions are what you choose. Spotting this direction — what I choose, and what follows from it — is how a doctor reads “how much medicine depends on a child’s weight,” how a gardener knows a plant’s growth follows the water it gets, and how a shopper reasons that the total price follows the number of items. Naming the direction of a relationship is the first step toward understanding it anywhere.

Materials

• Dependency cards
• Dependency recording page
• Arrow diagram template

Preparation

• Copy dependency cards for groups, the recording page, and arrow diagram templates.
• Have examples ready: portions → cooking time (time depends on portions); water → growth (growth depends on water).

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) the independent quantity is the one you choose or control (the input); the dependent quantity is the one that follows from it (the output); (2) the test is “which do I choose, and which follows?”; and (3) correlation is not the same as causation — two quantities may rise together because a third thing drives both. Model a worked example first, then guide a second before independent work (Kirschner, Sweller & Clark, 2006). A common slip is assuming the first-mentioned quantity is always the cause; keep returning to the “which do I choose?” test. Retrieval: ask learners to recall yesterday’s table and graph (buckets → liters) — buckets are independent, liters depend on them. The critical-thinking lens is central (docs/philosophy.md §7): saying “one depends on the other” is a claim that needs a reason, and the honest habit of asking “or do they both follow something else?” is the beginning of scientific skepticism. Note the global point: farmers, fishers, and healers in every tradition have reasoned about dependence — “if the rains come, the crops follow” — long before graphs had axes.

Procedure

1. Gather (5 min). Last time you graphed two quantities growing together. Today you ask which one leads and which one follows.
2. Meet input and output (10 min). In a pair of related quantities, the independent quantity is the one you choose (the input). The dependent quantity is the one that follows from it (the output). Ask: “which do I choose, and which follows?”
3. Worked example (10 min). Portions and cooking time. Do you choose the portions? Yes. Does cooking time follow? Yes. So portions are independent and cooking time is dependent — time depends on portions, not the other way around.
4. Worked example — a caution (10 min). Rain and puddle size: puddle size depends on rain (rain is independent). But consider “umbrellas sold and rain”: umbrellas do not cause rain — both follow the weather. Not every pair is cause and effect; sometimes both follow a third thing.
5. Practice with a partner (10 min). Take your dependency cards. For each pair, draw the arrow (input → output), name the independent and dependent quantity, and say why the direction goes that way. Check each other’s reasoning.
6. Close (5 min). To read a relationship, name its direction: what I choose, and what follows from it. And remember — two things moving together are not always one causing the other.

Differentiation

• Support: Use only clear cause-and-effect pairs (water → growth, items → total price) with the arrow template; describe each direction aloud so learners with low vision can follow by listening.
• Extension: Given a pair that rises together (ice-cream sales and sunburns), explain that neither causes the other — a third thing (hot weather) drives both; write a sentence justifying each dependency.

Assessment

• Formative (observation/self): Can the learner identify the independent and dependent quantity in a real pair and explain the direction?
• Self-check: The learner asks, “Which do I choose, and which follows? Did I give a reason, and did I check whether both might follow a third thing?”

Home connection

Find a pair of related quantities at home (water and a plant, days and money saved, guests and rice cooked) and tell someone which one depends on the other and why.

Resources

• Distinguishing independent and dependent quantities is standard in middle-grades mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The caution that correlation is not causation is a core scientific idea; the habit of asking “or do both follow a third thing?” is a critical-thinking value (docs/philosophy.md §7).