Eighth Grade · Math · week 25

Estimation with roots

Estimate the value of irrational square roots to the nearest tenth using the perfect-square sandwich method.

Building on last week's sandwich method: once you know the two whole numbers a root falls between, test tenths in that range to narrow the estimate further.

A helpful shortcut before testing decimals: compare how close the number under the root is to each nearby perfect square. If it is much closer to one, the root will lean toward that side.

📋 For the teacher
Teach it (10–15 min):
  • This is the same sandwich method as last week, pushed one decimal place further -- after finding the two whole numbers, test tenths between them.
  • The fast shortcut: compare how close the target number is to each perfect square (55 is 6 away from 49 but 9 away from 64, so it leans toward 7) -- this predicts the right direction before testing decimals.
  • Use estimation for a real reason this week: finding a side length from an area (a square garden, a square rug) is this exact skill in a practical context, and previews the Pythagorean theorem work coming soon.
  • A calculator can always CHECK an estimate, but the by-hand sandwich instinct is the real point -- it is what makes an unexpected calculator answer feel "reasonable" or "suspicious."
Talk about it:
  • How do you decide which tenth to try first when narrowing down an estimate?
  • Why is comparing the target number's distance to each perfect square a useful shortcut?
  • Why does finding a square rug's side length from its area require a square root?

Mastery looks like: They estimate a non-perfect-square root to the nearest tenth using the sandwich method and apply it to a real area-to-side-length context.

Common stumbles: Stopping at the whole-number sandwich without narrowing to a tenth; testing decimals in a direction that does not follow from which perfect square the number is closer to.

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