Kenneth ChengMathematical modeling
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Oblique drawings of solids

Read the idea, work independently, then explain what changed.

高一必修 第二册(A版).pdf · 8.2 · PDF 114 / printed page 107

Revisit first: Basic solid figures

TOPIC 01

Oblique drawings of solids

Build understanding of oblique drawings of solids through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use definitions, valid spatial reasoning and metric checks for oblique drawings of solids.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Drawing convention

The common 45° oblique convention halves receding lengths while retaining other stated scales.

Representation limits

A sketch preserves selected incidence and parallel relations, but not all angles or distances.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in oblique drawings of solids changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Schematic cuboid (not to scale): 3×4×2; volume=24, surface area=52.

Schematic cuboid (not to scale): 3×4×2; volume=24, surface area=52.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

In the 45° half-depth oblique convention, an original depth is d. Find its drawn length.

d=4d=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
The receding-axis scale is one half.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    d′=4/2=2d'=4/2=2

  3. This is a drawing convention, not a change in the solid.

The requested value is 2.

Checks and common pitfalls: This is a drawing convention, not a change in the solid.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Recover the actual depth from its half-scale drawn length.

d′=3d'=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
Undo the one-half scale factor.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    d=2d′=6d=2d'=6

  3. The same recovery is not applied to axes drawn at full scale.

The requested value is 6.

Checks and common pitfalls: The same recovery is not applied to axes drawn at full scale.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Can you read a cube’s true space-diagonal length directly with a ruler from an oblique drawing? Explain a reliable alternative.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
The drawing uses unequal directional scales and non-right projected angles.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    D=a2+a2+a2=a3D=\sqrt{a^2+a^2+a^2}=a\sqrt3

  3. A diagram represents relationships but is not generally a metric replica.

No; compute from the true three perpendicular edge lengths.

Checks and common pitfalls: A diagram represents relationships but is not generally a metric replica.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

In the 45° half-depth oblique convention, an original depth is d. Find its drawn length.

d=6d=6
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
The receding-axis scale is one half.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    d′=6/2=3d'=6/2=3

  3. This is a drawing convention, not a change in the solid.

The requested value is 3.

Checks and common pitfalls: This is a drawing convention, not a change in the solid.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Recover the actual depth from its half-scale drawn length.

d′=4d'=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
Undo the one-half scale factor.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    d=2d′=8d=2d'=8

  3. The same recovery is not applied to axes drawn at full scale.

The requested value is 8.

Checks and common pitfalls: The same recovery is not applied to axes drawn at full scale.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

A horizontal rectangle has side lengths 2 and 4. Its receding side is drawn at half length and 45°. The drawn area is c√2. Find c.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
Multiply adjacent drawn lengths by sine of the drawn angle.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    Sdraw=2⋅(4/2)⋅sin⁡45∘=22S_{\text{draw}}=2\cdot(4/2)\cdot\sin45^{\circ}=2\sqrt2

  3. The drawing area differs from the actual base area.

The requested value is 2.

Checks and common pitfalls: The drawing area differs from the actual base area.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Does a right angle in space have to appear as a right angle in an oblique drawing? Explain.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
Compare the actual coordinate axes with the drawing axes.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    90∘ in space ↦45∘ in the sketch90^{\circ}\text{ in space }\mapsto45^{\circ}\text{ in the sketch}

  3. The sketch preserves selected incidences and parallel directions, not all angles.

No. The perpendicular ground axes are deliberately drawn at 45°.

Checks and common pitfalls: The sketch preserves selected incidences and parallel directions, not all angles.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

A vertical edge has actual length 5 and vertical scale one. Find its drawn length.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
Read the scale assigned to the vertical axis.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    h′=1⋅5=5h'=1\cdot5=5

  3. The half-depth rule applies only to the specified receding direction.

The requested value is 5.

Checks and common pitfalls: The half-depth rule applies only to the specified receding direction.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

How should hidden edges be distinguished in a conventional solid sketch? Explain the purpose.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
Visibility depends on the chosen viewpoint.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Choose the viewpoint first, then decide which surfaces obstruct each edge.

  3. A hidden edge still belongs to the solid.

Use dashed segments for hidden edges and solid segments for visible edges, with a consistent viewpoint.

Checks and common pitfalls: A hidden edge still belongs to the solid.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Under a half-depth 45° oblique drawing, why is the horizontal area scale √2/4?

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Compare a rectangular area with the corresponding drawn parallelogram.
Hint 2
Area depends on both adjacent lengths and their included angle.
Worked solution

  1. Compare a rectangular area with the corresponding drawn parallelogram.

  2. Calculate or simplify this relation.

    ab(1/2)sin⁡45∘ab=2/4\frac{ab(1/2)\sin45^{\circ}}{ab}=\sqrt2/4

  3. The ratio applies to the stated horizontal drawing convention.

The depth scale contributes 1/2 and the oblique angle contributes sin45°=√2/2.

Checks and common pitfalls: The ratio applies to the stated horizontal drawing convention.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

In the 45° half-depth oblique convention, an original depth is d. Find its drawn length.

d=8d=8
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
The receding-axis scale is one half.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    d′=8/2=4d'=8/2=4

  3. This is a drawing convention, not a change in the solid.

The requested value is 4.

Checks and common pitfalls: This is a drawing convention, not a change in the solid.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Recover the actual depth from its half-scale drawn length.

d′=5d'=5
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Distinguish the actual spatial object from the stated oblique drawing convention.
Hint 2
Undo the one-half scale factor.
Worked solution

  1. Distinguish the actual spatial object from the stated oblique drawing convention.

  2. Calculate or simplify this relation.

    d=2d′=10d=2d'=10

  3. The same recovery is not applied to axes drawn at full scale.

The requested value is 10.

Checks and common pitfalls: The same recovery is not applied to axes drawn at full scale.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

A horizontal region has oblique drawn area 9√2 using the half-depth 45° convention. Find its true area.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Undo the area scale, not just the depth scale.
Hint 2
The full area factor is √2/4.
Worked solution

  1. Undo the area scale, not just the depth scale.

  2. Calculate or simplify this relation.

    S⋅2/4=92⇒S=36S\cdot\sqrt2/4=9\sqrt2\Rightarrow S=36

  3. Two-dimensional scaling includes the angle effect.

The requested value is 36.

Checks and common pitfalls: Two-dimensional scaling includes the angle effect.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for oblique drawings of solids?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use definitions, valid spatial reasoning and metric checks for oblique drawings of solids.
    • Drawing convention: The common 45° oblique convention halves receding lengths while retaining other stated scales.
    • Representation limits: A sketch preserves selected incidence and parallel relations, but not all angles or distances.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: The common 45° oblique convention halves receding lengths while retaining other stated scales.
    • Expected reasoning: A sketch preserves selected incidence and parallel relations, but not all angles or distances.
    • Expected correction: A measured sketch length need not equal a true spatial length.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗