Lesson 7 — Tables and Graphs of Two Quantities

Summary

Learners represent the relationship between two quantities with a table and a graph. They build a table (buckets and liters: 1 → 3, 2 → 6, 3 → 9, 4 → 12), plot the ordered pairs on a first-quadrant grid, and notice the points make a straight line — one quantity grows steadily with the other.

Objectives

• Represent the relationship between two quantities with a table and a graph. (D05.S2.06.02)

Connection

A family fetching water counts how much they carry: one bucket holds 3 liters, so two buckets hold 6, three hold 9, four hold 12. Write those pairs in a table, then plot them on a grid — the points climb in a straight line, and that line shows the whole story at a glance. Tables and graphs like this are how farmers track a crop’s growth, how bus drivers read distance against time, and how nurses read a fever chart — the same idea everywhere, from a clinic in Nairobi to a field in the Punjab.

Materials

• First-quadrant grid
• Two-quantity task cards
• Recording page

Preparation

• Copy grids for pairs, task cards for groups, and the recording page for each learner.
• Have a worked example ready: buckets (1, 2, 3, 4) → liters (3, 6, 9, 12); plot (1,3), (2,6), (3,9), (4,12).

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a table lists pairs of related quantities; (2) each pair becomes an ordered pair (x, y) on the coordinate plane — x across, y up; and (3) plotted points can reveal the relationship (here, liters = 3 × buckets, a straight line). Model a worked example first, then guide a second before independent work (Kirschner, Sweller & Clark, 2006). Two common slips: reversing the ordered pair (going up first), and connecting points when the situation does not allow in-between values. Retrieval: ask learners to recall first-quadrant plotting from Grade 5 — this lesson extends it to a relationship between two quantities, with the table feeding the graph. The critical-thinking lens: a graph is evidence — reading it carefully, not guessing a trend from one point, is honest reasoning (docs/philosophy.md §7). Note the global point: coordinate grids trace to Descartes, but locating a thing by two distances is older and appears in mapmaking traditions worldwide; data tables are as old as bookkeeping and calendars.

Procedure

1. Gather (5 min). Last time you solved for a mystery number. Today you watch two quantities grow together and turn them into a picture.
2. Meet the table (10 min). A table lists pairs of related quantities. Buckets: 1, 2, 3, 4. Liters: 3, 6, 9, 12. Each bucket holds 3 liters, so liters = 3 × buckets.
3. Worked example — make the table (10 min). Fill the table together: 1 bucket → 3 liters, 2 → 6, 3 → 9, 4 → 12. Read each row as an ordered pair: (1,3), (2,6), (3,9), (4,12).
4. Worked example — graph (10 min). On the grid, put buckets along the x-axis (across) and liters up the y-axis. Plot each pair: (1,3), then (2,6), then (3,9), then (4,12). Connect them: a straight line. The line is the relationship.
5. Practice with a partner (10 min). Take your task cards. For each situation, make a table, then plot the points. Check each other: across first, then up. Describe the relationship you see.
6. Close (5 min). A table lists the pairs; a graph shows them. Together they turn a rule like “liters = 3 × buckets” into a picture anyone can read.

Differentiation

• Support: Use only whole-number pairs from 0 to 6 with a large labeled grid; describe each move aloud and use tactile grid lines so learners with low vision can follow by touch.
• Extension: Predict and plot the next pair (5, 15) and write the rule connecting the two quantities in words and symbols; make a table and graph for a situation of the learner’s own.

Assessment

• Formative (observation/self): Can the learner build a table of two related quantities and graph the ordered pairs correctly?
• Self-check: The learner asks, “Did I go across first and then up? Do my points make the shape I expected, and can I describe the relationship?”

Home connection

Track two related quantities at home over a few days (cups of water and plants’ growth, days and money saved) in a table, and plot the pairs on a grid.

Resources

• Representing two-variable relationships with tables and graphs is standard in middle-grades mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The note that locating by two distances predates Cartesian coordinates and appears in global mapmaking is a historical observation; reading graphs as evidence rather than guessing trends is a critical-thinking value (docs/philosophy.md §7).