Sheet № 129 · Foundation + Higher · AQA · Edexcel · OCR
Estimating Square Roots –
Estimating square roots is a non-calculator skill that appears on both Foundation and Higher GCSE Maths papers. You need to find which two consecutive whole numbers a square root lies between, and sometimes give a closer estimate.
§Key definitions
Question:
Estimate √40 to one decimal place.
Answer:
√40 ≈ 6.3
Q1 (Foundation):
Between which two consecutive whole numbers does √60 lie?
Q2 (Foundation):
Estimate √20 to one decimal place.
Q3 (Higher):
Show that √90 is between 9.4 and 9.5.
§Formulas to memorise
If a² < n < b² where b = a + 1, then a < √n < b
For a closer estimate, find how far n is between a² and b²: estimate ≈ a + (n − a²) / (b² − a²)
Worked example
Estimate √40 to one decimal place.
Working:
⚠ Common mistakes
- ✗Using the wrong pair of perfect squares. Make sure you identify the correct consecutive squares. Memorise all squares up to 15² = 225.
- ✗Forgetting that estimation is approximate. The linear interpolation method gives a reasonable estimate but not the exact value.
- ✗Mixing up square roots and halving. √36 = 6, not 18. The square root asks "what number times itself gives 36?"
✦ Exam tips
- →Learn your square numbers up to at least 15² = 225 so you can identify the surrounding squares quickly.
- →For "show that" questions, you must square both boundary values and demonstrate the number lies between them.
- →A common question format asks you to place √n on a number line — use estimation to position it accurately.