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Exploring Number of Solutions of Equations

Exploring Number of Solutions of Equations
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2022-11-30 20:42
FunctionCoordinate GeometryInequality
This activity helps students explore when linear equations have 0, 1, or infinitely many solutions. Key math concepts include properties of linear equations and determining solutions. GeoGebra commands like Shuffle, Join, Element, and If are used to create interactive elements. Suitable for grades 6-8 to understand the variety of solutions in equations.

GeoGebra Commands & Parameters

  1. Shuffle(Sequence[-5, 5]) → sols, {2, -1, -2, -4, 3, 1, 5, 0, 4, -3, -5}
  2. Shuffle(Flatten[Sequence[Sequence[3], kk, 1, 10]]) → solutions, {2, 1, 1, 3, 3, 3, 2, 2, 3, 2, 2, 1, 3, 1, 3, 2, 3, 1, 2, 1, 1, 2, 1, 3, 3, 1, 2, 1, 2, 3}
  3. Join({{1 / 2, 1 / 5, 2 / 5}, (-1 * {1 / 2, 1 / 5, 2 / 5})}) → allowedFractions
  4. Shuffle(Flatten[{Sequence[-10, 10] \ {0}, allowedFractions}]) → bNums, {1, 2, -7, 5, -9, 10, 9, -6, 4, 0.4, -1, 0.2, -5, 8, 3, -0.4, -0.5, 6, -0.2, 0.5, -4, -8, 7, -2, -3, -10}
  5. Shuffle(Flatten[{Sequence[-10, 10] \ {0, 1}, allowedFractions}]) → aNums, {-2, -7, 4, 9, 0.4, 6, -5, 2, 0.2, 8, -6, 7, -3, -0.5, -0.4, -1, 0.5, 5, -0.2, -10, -4, 3, -8, 10, -9}
  6. Element(aNums, Mod[questionCount, Length[aNums]] + 1) → a
  7. Shuffle(Flatten[{Sequence[-10, 10] \ {0, 1, a}, allowedFractions}]) → cNums, {3, -6, 6, -1, 0.2, -8, 7, -5, -9, -0.2, 5, -10, 0.4, 8, -3, -4, -0.4, 4, -0.5, 0.5, 2, 9, -7, 10}
  8. Element(sols, Mod[questionCount + 1, Length[sols]] + 1) → int
  9. Element(sols, Mod[questionCount, Length[sols]] + 1) → sol
  10. Element(solutions, Mod[questionCount, Length[solutions]] + 1) → solutionsCurrent
  11. If(solutionsCurrent ≠ 1, Element[bNums, Mod[questionCount, Length[bNums]] + 1], int - (a * sol)) → b
  12. Shuffle(Flatten[{Sequence[-10, 10] \ {0, b}, allowedFractions}]) → dNums, {7, -1, 0.2, -6, 9, 0.4, 6, 3, -2, -5, -7, -3, 2, -8, 10, -9, -10, -4, -0.2, 4, -0.5, 8, 5, -0.4, 0.5}
  13. If(Length[xNumbers] ≥ maxAllowed, 1 + maxS - slide, 1) → min
  14. Element(cNums, Mod[questionCount, Length[cNums]] + 1) → c1
  15. If(solutionsCurrent ≟ 1, c1, a) → c
  16. Element(dNums, Mod[questionCount, Length[dNums]] + 1) → d1
  17. If(solutionsCurrent ≟ 3, b, solutionsCurrent ≟ 1, int - (c * sol), d1) → d
  18. If(Length[xNumbers] ≟ 0, {}, Zip[g1(kk) - f1(kk), kk, xNumbers]) → Dif
  19. Numerator(a - c) → numer
  20. Denominator(a - c) → denom
  21. If(Length[xNumbers] ≤ maxAllowed, Length[xNumbers], min + maxAllowed - 1) → max
  22. Append("\vspace{15}\hspace{25}\bold{x}\hspace{15}\phantom{\frac{2}{5}}", If[Length[xNumbers] ≟ 0, {"\phantom{\frac{2}{5}}"}, Take[Append[Zip[(("\phantom{\frac{2}{5}}" * Text[ii]) * "\phantom{\frac{2}{5}}"), ii, xNumbers], "\phantom{\frac{2}{5}}"], min, max + 1]]) → colX
  23. Zip((("$\textsf{" * tt) * " }$"), tt, {"Simple: \(ax+b = cx+d\)", "More complex: \(ax+b = c(x+d)\)", "Most complex: \(a(x+b) = c(x+d)\)"}) → options
  24. SelectedIndex(options) → caseNum
  25. If(caseNum ≟ 3, left3, left12) → left
  26. If(caseNum ≟ 1, right1, right23) → right
  27. If(Length[xNumbers] ≟ 0, {}, Zip[f1(kk), kk, xNumbers]) → leftNumbers
  28. If(Length[xNumbers] ≟ 0, {}, Zip[g1(kk), kk, xNumbers]) → rightNumbers
  29. If(Length[xNumbers] ≟ 0, {"\hspace{50}"}, Zip[If[AA < BB, "<", AA > BB, ">", "="] + "\phantom{\frac{2}{5}}\hspace{-15}", AA, leftNumbers, BB, rightNumbers]) → ineq
  30. Textfield(xStudent) → InputBox1
  31. Append((("\bold{" * If[caseNum ≟ 3, left3, left12]) * "}"), If[Length[xNumbers] ≟ 0, {}, Take[Append[Zip[Text[ii], ii, leftNumbers, DD, Dif], "\hspace{50}"], min, max + 1]]) → colLeft
  32. Append((("\bold{" * If[caseNum ≟ 1, right1, right23]) * "}"), If[Length[xNumbers] ≟ 0, {}, Take[Append[Zip[Text[ii], ii, rightNumbers, DD, Dif], "\hspace{50}"], min, max + 1]]) → colRight
  33. Append("\hspace{25}", If[Length[xNumbers] ≟ 0, {}, Take[Append[Zip[ii, ii, ineq, DD, Dif], "\hspace{25}"], min, max + 1]]) → colMiddle
  34. Polygon(Zip[PixelsToPoint[Corner[1], Corner[3], Corner[5], CLIPBOARDmagicSTRINGA], CLIPBOARDmagicSTRINGA, {(pixelsLeft_{1}, pixelsTop_{1}), (pixelsLeft_{1}, y(Corner[5]) - pixelsBottom_{1}), Corner[5] - (pixelsRight_{1}, pixelsBottom_{1}), (x(Corner[5]) - pixelsRight_{1}, pixelsTop_{1})}]) → borderBox_{1}
  35. Append("\textsf\bold{Difference}", If[Length[xNumbers] ≟ 0, {}, Take[Append[Zip[Text[ii], ii, Dif], "\hspace{50}"], min, max + 1]]) → colDif
  36. Zip((("$\textsf{" * TT) * " }$"), TT, {"\phantom{One solution}", "One solution", "No solution", "Infinite solutions"}) → Answer
  37. If(caseNum ≟ 3, left3, left12) → A1
  38. If(caseNum ≟ 1, right1, right23) → B1
  39. TableText(Take[solLeft, If[caseNum ≥ 2, 1, 2], If[solutionsCurrent ≟ 1, 5, 4]], Sequence["=", kk, 1, 3 + If[caseNum ≠ 1, 1, 0] + If[solutionsCurrent ≟ 1, 1, 0]], Take[solRight, If[caseNum ≥ 2, 1, 2], If[solutionsCurrent ≟ 1, 5, 4]], "vrcl") → solSimple2
  40. Sequence(142 + (ii * (189 - 142)), ii, 0, 5) → yIB, {142, 189, 236, 283, 330, 377}
  41. Textfield(xStudent) → InputBox2
  42. Textfield(xStudent) → InputBox3
  43. Textfield(xStudent) → InputBox4
  44. Textfield(xStudent) → InputBox5
  45. If(Defined[Element[colX, C1]], Element[colX, C1], "") → D1
  46. If(Defined[Element[colLeft, C1]], Element[colLeft, C1], "") → E1
  47. If(Defined[Element[colLeft, C2]], Element[colLeft, C2], "") → E2
  48. If(Defined[Element[colLeft, C3]], Element[colLeft, C3], "") → E3
  49. If(Defined[Element[colLeft, C4]], Element[colLeft, C4], "") → E4
  50. If(Defined[Element[colLeft, C5]], Element[colLeft, C5], "") → E5
  51. If(Defined[Element[colLeft, C6]], Element[colLeft, C6], "") → E6
  52. If(Defined[Element[colMiddle, C1]], Element[colMiddle, C1], "") → F1
  53. If(solutionsCurrent ≟ 3 ∧ Defined[Element[xNumbers, I2]], Element[colors, 1], If[solutionsCurrent ≟ 2 ∧ Defined[Element[xNumbers, I2]] ∨ J2 ≥ 10, Element[colors, 4], If[J2 ≥ 5, Element[colors, 3], If[J2 > 0, Element[colors, 2], If[J2 ≟ 0, Element[colors, 1], "white"]]]]) → K2
  54. If(solutionsCurrent ≟ 3 ∧ Defined[Element[xNumbers, I3]], Element[colors, 1], If[solutionsCurrent ≟ 2 ∧ Defined[Element[xNumbers, I3]] ∨ J3 ≥ 10, Element[colors, 4], If[J3 ≥ 5, Element[colors, 3], If[J3 > 0, Element[colors, 2], If[J3 ≟ 0, Element[colors, 1], "white"]]]]) → K3
  55. If(solutionsCurrent ≟ 3 ∧ Defined[Element[xNumbers, I4]], Element[colors, 1], If[solutionsCurrent ≟ 2 ∧ Defined[Element[xNumbers, I4]] ∨ J4 ≥ 10, Element[colors, 4], If[J4 ≥ 5, Element[colors, 3], If[J4 > 0, Element[colors, 2], If[J4 ≟ 0, Element[colors, 1], "white"]]]]) → K4
  56. If(solutionsCurrent ≟ 3 ∧ Defined[Element[xNumbers, I5]], Element[colors, 1], If[solutionsCurrent ≟ 2 ∧ Defined[Element[xNumbers, I5]] ∨ J5 ≥ 10, Element[colors, 4], If[J5 ≥ 5, Element[colors, 3], If[J5 > 0, Element[colors, 2], If[J5 ≟ 0, Element[colors, 1], "white"]]]]) → K5
  57. If(solutionsCurrent ≟ 3 ∧ Defined[Element[xNumbers, I6]], Element[colors, 1], If[solutionsCurrent ≟ 2 ∧ Defined[Element[xNumbers, I6]] ∨ J6 ≥ 10, Element[colors, 4], If[J6 ≥ 5, Element[colors, 3], If[J6 > 0, Element[colors, 2], If[J6 ≟ 0, Element[colors, 1], "white"]]]]) → K6
  58. If(Defined[Element[colRight, C1]], Element[colRight, C1], "") → G1
  59. If(Defined[Element[colDif, C1]], Element[colDif, C1], "") → H1
  60. If(Defined[Element[colRight, C2]], Element[colRight, C2], "") → G2
  61. If(Defined[Element[colRight, C3]], Element[colRight, C3], "") → G3
  62. If(Defined[Element[colRight, C4]], Element[colRight, C4], "") → G4
  63. If(Defined[Element[colRight, C5]], Element[colRight, C5], "") → G5
  64. If(Defined[Element[colRight, C6]], Element[colRight, C6], "") → G6
  65. TableText(Take[{{D1, E1, F1, G1, H1}, {D2, E2, F2, G2, H2}, {D3, E3, F3, G3, H3}, {D4, E4, F4, G4, H4}, {D5, E5, F5, G5, H5}, {D6, E6, F6, G6, H6}}, 1, Min[max + 2, 6]], "c|_lllll") → text2

AI Drawing Prompt Reference

The graphic explores the solutions of a linear equation. It displays an equation expression, possibly in the form ax+b=cx+d. A list of possible solutions, including integer and fractional solutions, is shown beside it. Classification labels indicate three cases: one solution, no solution, and infinite solutions. The construction order is: first draw the equation, then generate the solution list, then add classification labels, and finally display the solution type based on the number of solutions. The graphic aims to help understand the diversity of equation solutions.
The prompt above was compiled from the original drawing commands and is for reference only.