086i - Consolidation Graph Relationships

Lesson 86 — Problem Solving and Consolidation: Graph Relationships

Strand: Algebra | Descriptor: AC9M7A04 | Duration: 45 minutes

Learning Intentions

• To consolidate reading, describing, interpreting and sketching graphs of real data.
• To construct a graph from data and interrogate it.

Success Criteria

I can:

1. Plot real data accurately and choose sensible scales.
2. Read, describe and interpret my own graph.
3. Compare relationships and justify conclusions with graph features.
4. Critique a graph’s construction and its use.

Warmup

(6 minutes — rapid-fire recall, mini whiteboards)

1. Which axis takes the dependent variable?
2. What does a flat section of a distance–time graph mean?
3. What is true at the point where two lines cross?
4. What is the first thing to check on an unfamiliar graph?
5. Steep versus high — which means “changing fast”?

Answers: 1. Vertical; 2. Stationary; 3. The two quantities are equal there; 4. Axes — what is plotted, units, and where the scale starts; 5. Steep.

Activities

Activity 1 — Build the Graph (16 min)

Pairs. Grid paper. The central task: construct, then interrogate.

The school fun-run trial. Two students each ran the km course; their distance was recorded every minutes:

Time (min)	0	3	6	9	12	15	18
Jess (km)	0	0.6	1.2	1.8	2.4	3.0	—
Sam (km)	0	0.9	1.5	1.8	1.8	2.4	3.0

1. Plot both runners on the same axes. Choose and justify your scales.
2. Describe each runner’s race in two sentences.
3. When were they level? Cite the graph.
4. Find each runner’s average speed for the whole run.
5. What happened to Sam between and minutes?
6. Whose pacing would you recommend for the real event, and why?

Socratic scaffolding:

Prompt	Purpose
Before plotting: ranges needed on each axis?	– min; – km. Scale, e.g., cm min and cm km.
Jess’s points — what shape do they make?	A perfect straight line: constant speed km/min.
Sam’s shape?	Fast start, slowing, a flat rest, then a fast finish.
Level moments?	At and (both km); Jess then leads until the end. Sam finishes min later.
Average speeds?	Jess: km/min ( km/h). Sam: km/min ( km/h).
The recommendation — what counts as evidence?	Jess’s even line finished sooner and needed no rest; steady pacing wins here.

Emphasise the craft points while circulating: ruled axes, even scale steps, labelled axes with units, points marked precisely, straight-edge segments between points, a key for the two runners.

Activity 2 — Mixed Consolidation Stations (14 min)

Three stations rotating the block’s skills.

Station A — Read and rate. A printed graph shows a phone downloading a file: MB vs seconds — steady MB/s for s, stalls for s, then MB/s for s.

1. File size so far at s?
2. Total downloaded by the end?
3. Average speed across the whole s.

Station B — Sketch and swap. Sketch: “a Ferris wheel car’s height above ground across two full turns.” Swap; the partner labels the peaks, troughs and one full cycle on your sketch.

Station C — Critique. A poster graph claims “membership has EXPLODED” — the -axis runs –, and the line rises from to over a year.

1. Compute the actual percentage growth.
2. Rewrite the headline honestly.
3. Redraw (rough sketch) with a -based axis: what impression now?

(Answers: A — MB; MB; MB/s. C — on ; e.g. “Membership grew about this year”; the zero-based line looks nearly flat.)

Activity 3 — Inquiry: One Graph, Three Audiences (6 min)

Whole class, quick.

The fun-run data will be shown to: (a) the runners, for training; (b) the newsletter, for celebration; (c) the first-aid team, for planning.

What would each audience want emphasised, and how might the same honest graph be presented differently for each?

Discussion targets: (a) pace per section, the rest — annotate rates; (b) finishing times and effort — simpler, headline figures; (c) where on the course runners slow or stop — time at each stage. Same data, different foregrounding; none require distortion. (Bridges to L76’s communication lesson: audience shapes presentation, honesty constrains it.)

Checks for Understanding

(6 minutes — exit ticket, collected)

1. From your Activity 1 graph: when was Sam ahead of Jess?
2. State Jess’s speed in km/h and how the graph shows it is constant.
3. Two cost lines cross at . Interpret this sentence-length.
4. A graph’s -axis starts at rather than . State one legitimate reason and one misleading use.
5. Reasoning. Why does an average speed for a whole journey generally differ from every speed shown on the graph?

Answers: 1. From just after the start until min; 2. km/h; the points lie on one straight line; 3. At items the two options cost the same — below it one is cheaper, above it the other; 4. Legitimate: revealing small real variation; misleading: inflating a trivial change into a dramatic one; 5. It blends fast, slow and stationary phases into one overall rate — total distance over total time — which need not equal any single phase’s rate.

Common Misconceptions

Misconception	How to pre-empt it
Uneven or unlabelled scales when constructing.	Scale justification is question 1 of the build task, and marked.
Joining points with a curve when data justifies segments (or vice versa).	Discuss: between measurements we interpolate cautiously — straight segments state minimal assumptions.
”They were level” read only at obvious crossings.	Sam and Jess are level at too — starts count.
Averaging the phase speeds to get overall speed.	Station A’s stall makes the error visible; total-over-total is the rule.
Believing honest data cannot mislead.	Station C: accurate points, distorting axis.
One presentation fits all audiences.	The three-audiences inquiry.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A snail climbs m up a wall each day and slips m each night. The wall is m. Sketch height vs time. When does it reach the top?

Answer

A sawtooth rising net m/day — but on day it climbs from m to m and stays (no slip at the top). Day , not day : the graph makes the classic trap visible.

E2 (AMC Junior style). From a graph, a train covers km in the first hour, waits min, then km in the final hour. Find its average speed.

Answer

km in h km/h.

E3 (Challenge). Sketch a single journey whose distance–time graph contains: a steep straight section, a gentler straight section, a flat section, and no downward section — then explain why distance-travelled graphs never slope down.

Answer

E.g. sprint, jog, rest. Total distance accumulates; it cannot un-happen, so the graph never falls. (Distance-from-home graphs can fall — the distinction matters and is worth drawing.)

E4 (Challenge). Jess runs the real km fun-run at her trial pace. Sam improves: no rest, and every -minute split at km. Who wins now, and by how long?

Answer

Sam: km at km per min min… check: splits km, so min. Jess: min. Jess still wins by minutes — Sam’s improvement removed the rest but not the gap.

Homework

1. Plot this bath’s depth data and answer: | min | 0 | 2 | 4 | 6 | 8 | 10 | 12 | — | depth (cm) | 0 | 8 | 16 | 24 | 24 | 24 | 4 | (a) Describe the three phases. (b) Fill rate? (c) What happened at ? (d) Drain rate, assuming it continued steadily to empty — when is the bath empty?
2. Two printers: QuickJet prints pages/min but takes min to warm up; SteadyPrint starts instantly at pages/min. (a) Sketch pages vs time for both. (b) When does QuickJet catch up? (c) Which finishes a -page job first, and by how much?
3. From last week’s canteen sales graph (sketch provided): sales rise Monday–Wednesday, dip Thursday, peak Friday. Write a two-sentence summary for the newsletter and one caution about concluding too much.
4. Reasoning. Explain, with a sketch, the difference between “distance travelled” and “distance from home” graphs for a walk to the shop and back.
5. Challenge. Design a misleading-but-accurate graph of your own invented data, then write the two-line exposé that reveals the trick.

Answers: Q1 — (a) steady fill, plateau (taps off, bathing), rapid drain (b) cm/min (c) the plug was pulled (d) from cm falling cm/min ( in min): empty at min. Q2 — (b) QuickJet: ; SteadyPrint: ; equal at min ( pages) (c) pages: QuickJet min; SteadyPrint min — QuickJet by s. Q4 — travelled: rises, flat (in shop), rises again; from home: rises, flat, falls to zero.