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Teaching video
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Grade 8 chapters and video availability01 · Read and understand
What you will learn
- Explain how to locate an irrational square root between decimals.
- Solve the two practice problems and explain how the assumptions affect the answers.
Before you start
Square numbers and decimal multiplication.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Locate √20 between consecutive tenths by comparing nearby squares.
Why this math matters
Comparing squares gives reliable bounds on a positive square root without pretending a decimal approximation is exact. Bracket a numerical answer between simple values to check whether a calculator result is plausible.
Set up the model
A useful answer starts with clear assumptions:
- Use the nonnegative square root.
- Consecutive tenths are numbers differing by 0.1.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Locate an irrational square root between decimals
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Locate √20 between consecutive tenths by comparing nearby squares.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the quantities
4² = 16 < 20 < 25 = 5²
The root lies between 4 and 5 because squaring is increasing on nonnegative numbers.
Apply the relationship
4.4² = 19.36; 4.5² = 20.25
Test nearby tenths inside the whole-number interval.
Check and interpret
4.4 < √20 < 4.5
The exact root is irrational; the interval locates it without replacing it by a terminating decimal.
The result
4.4 < √20 < 4.5
The exact root is irrational; the interval locates it without replacing it by a terminating decimal.
Common mistakes to catch
- A calculator display with finitely many digits usually gives an approximation to an irrational number.
- Use the monotonic behavior of squaring on nonnegative inputs when comparing positive roots.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Between which consecutive whole numbers does √70 lie?
Show a hint
Find perfect squares on either side of 70.
Reveal answer and explanation
8 and 9
64 < 70 < 81, so 8 < √70 < 9.
Practice 2
Is 4.47 an exact value of √20?
Show a hint
Square 4.47 and compare with 20.
Reveal answer and explanation
No
4.47² = 19.9809, so 4.47 is only an approximation.
Take the idea with you
Bracket a numerical answer between simple values to check whether a calculator result is plausible.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Recover a right triangle's missing leg
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