Learn with Amar
Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Use a fine enough denominator to find a rational point in an interval.
- Justify the conclusion "14145/10000=1.4145 lies strictly between the bounds" using the stated assumptions.
Before you start
Integers, inequalities, and the Archimedean property.
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
Find a rational number strictly between 1.414 and 1.415, then explain the general strategy.
Why this math matters
Use a fine enough denominator to find a rational point in an interval. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The interval endpoints are distinct.
- The displayed terminating decimals are exact.
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.
Place a rational number between two different real numbers
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find a rational number strictly between 1.414 and 1.415, then explain the general strategy.
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.
Build the model
Choose denominator 10000 so the step size is 0.0001
The grid is finer than the interval width 0.001.
Work through the mathematics
14140<14145<14150
Integer numerators identify an interior grid point.
Check the conclusion
14145/10000=1.4145 lies strictly between the bounds
More generally, sufficiently small rational grid spacing places a grid point inside every nonempty real interval.
The result
14145/10000=1.4145 lies strictly between the bounds
More generally, sufficiently small rational grid spacing places a grid point inside every nonempty real interval.
Common mistakes to catch
- Approximation and equality are different claims.
- Density is not the same as having the same set of elements.
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
Find a rational between −0.2 and −0.19.
Show a hint
Use thousandths.
Reveal answer and explanation
−0.195
The proposed decimal lies strictly inside both bounds.
Practice 2
Does density mean every real number is rational?
Show a hint
Being arbitrarily close differs from being equal.
Reveal answer and explanation
No
Irrational numbers can be approached by rationals without becoming rational themselves.
Take the idea with you
Explain why decimal grids can approximate any real measurement while using only rational values.
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: Separate an association from a controlled comparison
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