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Mastering Conjectures: Unveiling Patterns and Insights

Learn how to make conjectures by analyzing patterns and images, testing assumptions, and understanding angles and shapes. Discover the power of counter-examples to refine your thinking.

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Mastering Conjectures: Unveiling Patterns and Insights

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  1. MakingConjectures

  2. How to make a conjecture • Look at the given pattern or picture. • If it’s a pattern, think of what would come next. • If it’s a picture, what other information could you assume from the picture based on knowledge you already have?

  3. Make a conjecture about the next items in each sequence. 3, 7, 3, 7, ___, ___ 3 7 42 68 2, 4, 6, 10, 16, 26, ___, ___ 2 4 1 12, 9, 10, 7, 8, 5, 6, 3, __, __, __

  4. Make a conjecture. What color blocks are missing?

  5. Answer Yellow and red

  6. Make a conjecture about the triangle. A C B

  7. Conjecture Since two sides are marked congruent, it is an isosceles triangle. Triangle ABC is an isosceles triangle.

  8. Make a conjecture about the two lines described here. Lines j and q are perpendicular.

  9. Answer • Draw the picture. • Since the lines are , they form right angles. • Conjecture: Lines j and q form 4 right angles.

  10. Angles 3 and 4 form a linear pair. Make a conjecture.

  11. Answer You should remember that linear pairs are supplementary. Conjecture: m3 + m4 = 180

  12. Make a conjecture. PQ = RS RS = TU

  13. Answer Draw a picture. P Q R S T U Conjecture: PQ = TU

  14. Conjectures can be FALSE • Sometimes you can make a conjecture that turns out to be false. • You will know it is false if you can come up with a “counter-example”. This is where you can find an example that contradicts your conjecture. • It only takes one counter-example to prove a conjecture false.

  15. Give a counter-example for this conjecture. Given: points W, X, Y and Z Conjecture: W, X, Y and Z are noncollinear. Hint: This conjecture assumes you always need to draw W, X, Y and Z so they are not on the same line. Is this true?

  16. Answer False. You can have 4 points on the same line. W Y X Z This picture is your counterexample.

  17. Find a counter-example. Given: In polygon JKLM, JK = KL = LM = MJ Conjecture: JKLM is a square. Think: Is there enough given information to ALWAYS assume 4 equal sides means a square?

  18. Answer Your conjecture is false, because a shape can have 4 equal sides, but no right angles. So showing the rhombus is your counterexample. Square Rhombus

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