Year 10 Maths • AC9M10N01 • Authored homework
Exact values, surds and approximation error
Compare exact and approximate representations of real numbers, including surds and pi, and explain how rounding error can accumulate in repeated calculations. Show working, use correct notation and check reasonableness.
Part A
Short-answer questions
State whether sqrt(50) is rational or irrational, and explain your answer.
Simplify sqrt(72) into exact surd form.
Write sqrt(18) as a simplified surd and as a decimal rounded to 2 decimal places.
Explain why pi should often be kept exact until the final step of a calculation.
Round 7.3864 to 3 significant figures.
A length is rounded from 12.48 cm to 12.5 cm. Find the absolute rounding error.
Explain the difference between an exact value and an approximation.
Estimate sqrt(63) without a calculator by comparing it to nearby square numbers.
Correct this error: sqrt(20) = 10 because half of 20 is 10.
Give one situation where repeated rounding could noticeably affect a final answer.
Part B
Long-answer questions
Simplify sqrt(48) + sqrt(75) exactly, then give a decimal approximation to 3 significant figures.
A circle has radius 7 cm. Calculate its area exactly in terms of pi, then approximate it to 2 decimal places. Explain why both forms are useful.
A student rounds sqrt(2) to 1.41 before multiplying by 100. Compare this with using sqrt(2) exactly and rounding only at the end.
A calculation uses 3.14 instead of pi in five repeated steps. Explain why the final error may be larger than a single rounding error.
Simplify 3sqrt(12) - 2sqrt(27) + sqrt(75), showing all surd steps.
A diagonal of a square has exact length 8sqrt(2) cm. Approximate it to 1, 2 and 3 decimal places and compare the accuracy.
Create a measurement problem where exact surd form is more precise than a rounded decimal, then solve it.
Explain how significant figures, decimal places and exact values are different using examples.
Evaluate a solution where a student rounded every intermediate answer to one decimal place. Describe the risk and a better process.
Write a full worked response showing exact calculation first, then final rounding with a reason for the chosen accuracy.
Part C
Research and understanding task
Find a real context involving repeated calculations, such as construction, GPS distance, circle measurement, engineering or finance. Explain where approximations occur and how keeping exact values longer can improve accuracy.