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Visualizing Square Roots Between Natural Numbers

Visualizing Square Roots Between Natural Numbers
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2026-05-18 19:28
In collections Quadratzahlen und Quadratwurzeln
This example demonstrates how to locate a square root between two consecutive natural numbers by comparing numeric bounds visually and interactively. It focuses on the mathematical idea that if n < x < n+1 then √x falls between √n and √(n+1), supporting reasoning about real-number order and square-root properties. The key concepts include the definition of square roots, monotonicity of the square-

GeoGebra Commands & Parameters

  1. Remove(2…143, Sequence[k^(2), k, 1, 12]) → NumberListLevel2
  2. Shuffle(NumberListLevel2) → NumberLevel2, {93, 8, 54, 85, 63, 50, 11, 89, 76, 15, 29, 56, 13, 69, 108, 31, 94, 99, 40, 24, 122, 18, 117, 114, 75, 138, 39, 12, 128, 124, 80, 118, 136, 37, 106, 14, 77, 142, 103, 95, 140, 72, 91, 52, 141, 46, 74, 38, 90, 119, 102, 45, 113, 17, 101, 132, 34, 127, 27, 30, 73, 55, 61, 58, 10, 67, 26, 96, 131, 137, 43, 78, 68, 7, 62, 123, 104, 3, 71, 98, 84, 129, 28, 42, 48, 107, 133, 57, 134, 53, 120, 86, 139, 22, 143, 105, 32, 97, 47, 116, 5, 20, 2, 115, 79, 6, 21, 130, 19, 126, 135, 65, 60, 35, 88, 59, 125, 33, 82, 109, 66, 51, 23, 70, 83, 110, 87, 41, 44, 92, 112, 111}
  3. Shuffle(NumberListLevel3) → NumberLevel3, {2, 3, 6, 11, 5, 10, 9, 4, 1, 8, 7}
  4. If(IsInteger[sqrt(number2)], number2 - 1, floor(sqrt(number2))) → sol21
  5. If(IsInteger[sqrt(number2)], number2 + 1, ceil(sqrt(number2))) → sol22
  6. KeepIf(number31 < sqrt(a) < number32, a, Sequence[k, k, 2, 143]) → SolutionList3
  7. If(level ≟ 2, correct21 ∧ correct22, level ≟ 3, correct3) → correct
  8. Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {4, 9, 10, 7, 2, 1, 10, 9, 7, 9}
  9. Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
  10. Textfield(input) → inputBox
  11. Join({Sequence[Segment[Corner[4] + (RasterPixMat * ((k * RasterStepXY), 0)), Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY))], k, 0, RasterMaxX / RasterStepXY], Sequence[Segment[Corner[4] + (RasterPixMat * (0, (k * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], k, 0, RasterMaxY / RasterStepXY]}) → RasterStrecken
  12. Join({Sequence[Text[UnicodeToLetter[65 + k], Midpoint[Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY)), Corner[4] + (RasterPixMat * (((k + 1) * RasterStepXY), RasterMaxY))], false, true, 0, -1], k, 0, RasterMaxX / RasterStepXY - 1], Sequence[Text[("" * k), Midpoint[Corner[4] + (RasterPixMat * (RasterMaxX, ((k - 1) * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], false, true, 1, 0], k, 1, RasterMaxY / RasterStepXY]}) → RasterBeschriftung
  13. Sequence(Sequence[Corner[4] + (RasterPixMat * ((k * RasterStepXY), (m * RasterStepXY))), k, 0, RasterMaxX / RasterStepXY], m, 0, RasterMaxY / RasterStepXY) → RasterPunkte
  14. Translate(imgChilliGrau1, Vector[(1 * Vector[((0.8 * dimChilli), 0)])]) → imgChilliGrau2
  15. Translate(imgChilliRot1, Vector[(1 * Vector[((0.8 * dimChilli), 0)])]) → imgChilliRot2
  16. Translate(imgChilliRot1, Vector[(2 * Vector[((0.8 * dimChilli), 0)])]) → imgChilliRot3
  17. Text(txtCapLevel2, PosCalc, false, true, 0, 0) → txtInputLevel2
  18. Text(txtCapLevel3, PosCalc, false, true, 0, 0) → txtInputLevel3
  19. Textfield(input21) → InputBox21
  20. Textfield(input22) → InputBox22
  21. Textfield(input3) → InputBox3
  22. Text(InputBox21) → txtIB21
  23. Text(InputBox22) → txtIB22
  24. If(level ≟ 2, Defined[input21] ∧ IsInteger[input21] ∧ input21 ≥ 0 ∧ (¬isSqrt21) ∧ Defined[input22] ∧ IsInteger[input22] ∧ input22 ≥ 0 ∧ (¬isSqrt22), level ≟ 3, Defined[input3] ∧ IsInteger[input3] ∧ input3 ≥ 0) → clickableBtnValidate
  25. Textfield(inputH21) → InputBoxH21
  26. Textfield(inputH22) → InputBoxH22
  27. Textfield(inputH31) → InputBoxH31
  28. Textfield(inputH32) → InputBoxH32
  29. If(number2 < 4^(2), 1, number2 < 9^(2), 6, number2 < 10^(2), 7, 8, number2 < 5^(2), 2, number2 < 6^(2), 3, number2 < 7^(2), 4, number2 < 8^(2), 5) → firstNumberHint1
  30. Textfield(inputH11) → InputBoxH11
  31. Textfield(inputH12) → InputBoxH12
  32. Textfield(inputH13) → InputBoxH13
  33. Textfield(inputH14) → InputBoxH14
  34. Textfield(inputH15) → InputBoxH15