← Unit 4 Unit 4 Synthesis and Problem Solving

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom
MA5-TRG-C-01, MA5-ARE-C-01, MA5-GEO-C-01, MA5-DAT-C-01, MA5-PRO-C-01

Unit 4 Synthesis and Problem Solving

⏱ 25 min📚 Year 9📈
Think First

A survey company wants to estimate the average height of Australian Year 9 students. They measure 50 students from one school. What are the limitations? How could they improve?

💡 Revision: Ensure you are confident with all topics from Lessons 1-19.

Learning Intentions

Know

  • Integrated problem solving
  • Selecting appropriate methods
  • Checking reasonableness

Understand

  • How topics connect in real-world problems
  • When to combine multiple techniques

Can Do

  • Solve multi-step problems involving multiple topics
  • Select appropriate formulas and methods
  • Communicate reasoning clearly
SynthesisIntegrationReasonablenessCommunication
Learn Phase
1

Integrated Problem Solving

Combining skills

Real problems rarely fit into single topics. Be prepared to:

  1. Identify which skills are needed
  2. Break the problem into steps
  3. Apply appropriate formulas
  4. Check each step
  5. Combine results

Example: Find the probability that a randomly chosen student from a box plot diagram is above the median and has a height requiring a ladder at least 30° elevation to reach.

2

Checking Reasonableness

Does the answer make sense?

Always ask:

  • Are the units correct?
  • Is the magnitude reasonable?
  • Does it match a rough estimate?
  • Can I verify with a different method?

Example: If you calculate a building height as 3000 m, recheck — that is taller than any building on Earth.

3

Communicating Solutions

Show your working

Clear communication is essential:

  • State what you are finding
  • Show each step with reasons
  • Use correct mathematical notation
  • State the final answer with units
  • Include a brief conclusion

HSC markers award marks for method as well as final answers. Show all working.

✓

Check Understanding

Try it yourself

A cylindrical water tank (r=2 m, h=3 m) is 2/3 full. A model is built at 1:25 scale. Find the model volume in cm³.

Worked Example

Synthesis Problems

1

A ramp is 8 m long and rises 1.5 m. A box (0.5 m × 0.4 m × 0.3 m) needs to fit. Will it slide if the angle is too steep? Find the angle.

$sin heta = 1.5/8 = 0.1875$

$ heta = sin^{-1}(0.1875) approx 10.8°$

The box dimensions are irrelevant to the angle calculation but would matter for stability analysis.

2

In a school of 800, a stratified sample of 80 is taken: 40 from junior, 30 from middle, 10 from senior. Is this proportional? If juniors are 50%, middle 30%, senior 20%, how many should be in each group?

Expected: Junior 40, Middle 24, Senior 16

Actual middle (30) is overrepresented; senior (10) is underrepresented.

3

Two similar water tanks have volumes 8 m³ and 27 m³. The smaller has radius 1 m. Find the larger radius.

Volume ratio = $27/8 = (3/2)^3$

Scale factor $k = 3/2 = 1.5$

Larger radius = $1.5 imes 1 = 1.5$ m

Common Misconceptions

✕

Apply wrong formula in multi-step problems. Read carefully to identify what is being asked before choosing a formula.

✕

Forget to convert units in combined problems. Keep track of units throughout, especially when mixing cm, m, and litres.

✕

Skip checking steps in long problems. Errors compound. Verify each intermediate result.

Your Turn

Practice — Synthesis

Work through each question in your book or digitally. Answers are in the Questions phase.

1A building casts a 30 m shadow. The sun's angle of elevation is $35°$. Find the building height. Then, if a model is 1:100 scale, find the model height.
2A survey of 200 people shows 120 support a policy. Find P(support). If two people are chosen without replacement, find P(both support).
3A box plot shows test scores with Q1=40, med=55, Q3=70. Describe the distribution and find the IQR.
Real-World Anchor

STEM Careers

Engineers, data scientists, and researchers solve integrated problems daily. A civil engineer designing a bridge uses trigonometry (angles), measurement (material volumes), statistics (load testing data), and probability (safety factors). These interconnected skills are the foundation of quantitative careers.

📓 Copy Into Your Books

▼

Problem solving

  • Identify needed skills
  • Break into steps
  • Apply formulas
  • Check reasonableness

Units

  • Track consistently
  • Convert when needed
  • State in final answer

Communication

  • Show working
  • Use correct notation
  • State conclusion
  • Include units
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
Best first step in multi-step problem:
Cylindrical tank r=2m, h=3m. Volume in litres:
Model 1:50. Real volume 125 m³. Model volume:
Ramp 10 m long, rises 2 m. Angle:
Stratified sample of 100 from: 60% A, 30% B, 10% C. Sample sizes:
Two dice. P(sum > 9) =
Building height from shadow 20 m, elevation $40°$:
HSC markers award marks for:
1A cone has radius 3 cm, height 4 cm. Find its volume. If a similar cone has radius 6 cm, find its volume using scale factors.
2Design a survey to find the most popular sport at your school. How would you ensure it is unbiased?
3A box plot shows Q1=30, med=50, Q3=65, max=90, with an outlier at 10. Describe the distribution.

Comprehensive Answers

1Cone r=3, h=4. Volume? Similar cone r=6.
$V = rac{1}{3}pi(9)(4) = 12pi$ cm³. Scale factor 2, so $V = 8 imes 12pi = 96pi$ cm³.
2Design unbiased sport survey.
Random/stratified sample, clear unbiased questions, adequate sample size, anonymous responses.
3Box plot description.
Right-skewed (median closer to Q1), outlier at low end. IQR = 35. Most data in 30-65 range.
MC 1Best first step.
Identify given and wanted. Answer: B
MC 2Tank volume in litres.
$pi(4)(3) = 12pi$ m³ = $12,000pi$ L. Answer: B
MC 3Model volume.
$125/50^3 = 125/125000 = 0.001$ m³. Answer: B
MC 4Ramp angle.
$sin heta = 2/10 = 0.2$. Answer: A
MC 5Stratified sample.
60, 30, 10. Answer: B
MC 6P(sum > 9).
Sums 10,11,12: 3+2+1=6 outcomes. 6/36=1/6. Answer: A
MC 7Building height.
$20 imes an 40° approx 16.8$ m. Answer: A
MC 8HSC marks.
Working and method. Answer: B
SA 1Cone volumes.
$12pi$ and $96pi$ cm³.
SA 2Unbiased survey.
Random sample, unbiased questions, adequate size.
SA 3Box plot description.
Right-skewed with low outlier.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.