Investigating Earth — Chapter Practice, Grade 9

Use measurements, models, maps, and reference tools to build explanations from evidence.

Problem 1

Three readings are 8, 10, and 12 mm. Find their mean.
Answer and explanation

10 mm

(8 + 10 + 12) ÷ 3 = 10.

Problem 2

Which is an inference?
  1. The gauge reads 10 mm.
  2. The thermometer reads 18°C.
  3. The storm likely caused the river rise.
  4. The sample has mass 20 g.
Answer and explanation

The storm likely caused the river rise.

The statement interprets a cause rather than reporting a measurement.

Problem 3

Why can repeated readings all be similar yet inaccurate?
Answer and explanation

A consistent calibration error can shift every reading in the same direction.

Problem 4

Three rain measurements are 18, 20 and 22 mm. Report their mean and range, and describe a limitation. Explain the reasoning in your own words.
Answer and explanation

Mean 20 mm; range 18–22 mm; these values do not establish long-term rainfall.

Mean=(18+20+22)/3=20 mm.

Observed range is 18–22 mm; only three measurements and possible instrument bias limit the result.

Response criteria

  • Use the stated conditions.
  • Show the evidence or calculation.
  • Explain the conclusion and any stated limitation.

Problem 5

Measurements cluster at 20.0 cm but a reference is 21.0 cm. What kind of problem is suggested? Show the reasoning, not just the conclusion.
Hints

A miscalibrated instrument can give consistent wrong readings.

Answer and explanation

A systematic offset or calibration problem is plausible; small spread does not remove the 1.0 cm discrepancy.

Problem 6

A model predicts a trend that one new measurement contradicts. What should happen next?
Hints

State what the supplied observations or conditions establish, and separate that from any assumption.

Answer and explanation

Check methods and uncertainty, repeat the measurement and test the model's assumptions before revising the explanation.

Problem 7

Compare a model prediction with the measured outcomeCompare a model prediction with the measured outcomeRunoff collected (mL)04080120050100150Rain supplied (mL)Model: 80% runs offObserved comparison: 40%
Both lines are deliberately supplied model/test records rather than field observations. An 80% runoff assumption is compared with a 40% outcome over the same rainfall range. Connecting the points expresses those specified constant fractions; an actual landscape may respond nonlinearly.
Interpret the supplied visual model. At 150 mL supplied, what is the runoff mismatch and what measurement could test an explanation?
Answer and explanation

The model gives 120 mL and the comparison gives 60 mL, a 60 mL mismatch. Measure infiltration, surface storage and collection losses under matched conditions to test the assumed runoff fraction.

Read model and observation at the same horizontal rain value. The vertical gap is a difference in collected runoff, not a difference in the original rain supplied.

The model gives 120 mL and the comparison gives 60 mL, a 60 mL mismatch. Measure infiltration, surface storage and collection losses under matched conditions to test the assumed runoff fraction.

Response criteria

  • Read the labeled quantities, reference frame or objects correctly.
  • Explain the relation between the visual evidence and the mechanism.
  • Keep the drawing’s stated assumptions and limits in the conclusion.

Problem 8

A balance repeatedly reads 19.7 g for a 20.0 g reference. A student reports their average as accurate because the readings agree. Explain the error and propose a correction check.
Answer and explanation

The readings are precise but 0.3 g below the reference. Their average is still 19.7 g. Check calibration with suitable reference masses over the working range and repeat the measurement; do not assume repeatability proves accuracy.

Compare the repeated values with the independent 20.0 g reference.

Find the bias: 19.7−20.0 = −0.3 g.

Use calibration checks to distinguish a systematic offset from random spread.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 9

Elevation rises 60 m over a horizontal distance of 3 km. Find gradient.
Answer and explanation

20 m/km

60 ÷ 3 = 20.

Problem 10

On the same map, closely spaced contours indicate…
  1. gentler slope
  2. steeper slope
  3. lower rainfall
  4. higher temperature
Answer and explanation

steeper slope

The same elevation change occurs over a shorter horizontal distance.

Problem 11

Why is a map legend necessary?
Answer and explanation

It defines symbols and units; unfamiliar markings may change interpretation, such as a hill versus a depression.

Problem 12

Elevation rises from 120 m to 300 m over a horizontal distance of 3 km. Find gradient. Explain the reasoning in your own words.
Answer and explanation

60 meters of elevation change per kilometer horizontally.

Elevation change=300−120=180 m.

Gradient=180 m/3 km=60 m/km.

Response criteria

  • Use the stated conditions.
  • Show the evidence or calculation.
  • Explain the conclusion and any stated limitation.

Problem 13

Elevation rises 60 m over 300 m horizontally. Find gradient. Show the reasoning, not just the conclusion.
Hints

Use horizontal ground distance and actual elevations, not contour-line count alone.

Answer and explanation

60/300=0.20 m/m, or 20%.

Problem 14

Two maps have different contour intervals. Can line spacing alone rank their slopes?
Hints

State what the supplied observations or conditions establish, and separate that from any assumption.

Answer and explanation

No. Convert each map's elevation change and horizontal scale first; spacing is comparable only with compatible scales.

Problem 15

Closed contours can enclose either a hill or a depressionClosed contours can enclose either a hill or a depressionHillDepression150 m170 m150 m130 mHeights rise inward.Hachures and labels show inward fall.Closed contours alone are insufficient; inspect symbols and elevations.
These paired original maps isolate the difference between hills and depressions at the same 20 m interval. Hachure direction and labeled elevation together support the interpretation. The diagrams omit streams; the valley-V convention is explained in the adjoining prose rather than inferred from these circular contours.
Interpret the supplied visual model. Which central region is lower than its surrounding outer contour, and what two clues support that answer?
Answer and explanation

The right panel: its inner contour is 130 m compared with 150 m outside, and its hachures point inward toward lower ground.

The nested shape is similar in both panels. The tick marks and different height sequence change the interpretation.

The right panel: its inner contour is 130 m compared with 150 m outside, and its hachures point inward toward lower ground.

Response criteria

  • Read the labeled quantities, reference frame or objects correctly.
  • Explain the relation between the visual evidence and the mechanism.
  • Keep the drawing’s stated assumptions and limits in the conclusion.

Problem 16

A map scale is 1 cm = 0.4 km. Two points 3 cm apart along a stated map line rise from 220 to 280 m. Calculate average gradient and explain why it does not establish every local slope.
Answer and explanation

Ground distance is 1.2 km and rise is 60 m, so average gradient is 50 m/km. Local segments can be steeper or flatter; additional contour crossings or survey points are needed.

Convert 3×0.4 = 1.2 km.

Subtract 280−220 = 60 m.

Divide 60/1.2 = 50 m/km and distinguish the average from local values.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

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