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Hands-on Equations

Hands-on Equations. An Introduction to Algebra. The Pieces. The following pieces we will be using for the first few lessons: Blue Pawns –> x Red Cubes -> Numbers Scale -> shows balance. Rules 1 & 2. You may remove the same number of blue pawns from each side of the scale.

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Hands-on Equations

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  1. Hands-on Equations An Introduction to Algebra

  2. The Pieces The following pieces we will be using for the first few lessons: Blue Pawns –> x Red Cubes -> Numbers Scale -> shows balance

  3. Rules 1 & 2 • You may remove the same number of blue pawns from each side of the scale. • You may remove the same cube amount from each side of the scale.

  4. X = 5 10 = 10 X = 7 9 = 9 X = 2 7 = 7 X = 4 12 = 12 X = 1 4 = 4 X = 6 14 = 14 X = 4 8 = 8 X = 4 14 = 14 X = 6 16 = 16 X = 6 18 = 18 Lesson #1

  5. X = 3 6 = 6 X = 2 6 = 6 X = 1 5 = 5 X = 3 12 = 12 X = 4 12 = 12 X = 7 15 = 15 X = 8 16 = 16 X = 4 13 = 13 X = 2 5 = 5 X = 10 25 = 25 Lesson # 2

  6. X = 7 28 = 28 X = 4 14 = 14 X = 5 20 = 20 X = 2 13 = 13 X = 6 12 = 12 X = 5 20 = 20 X = 2 7 = 7 X = 3 8 = 8 X = 6 19 = 19 X = 1 6 = 6 Lesson #3

  7. Rule 2a • You may remove pawns as part of the set-up. This is done for subtraction operations.

  8. X = 7 26 = 26 X = 6 32 = 32 X = 8 28 = 28 X = 5 17 = 17 X = 2 9 = 9 X = 6 14 = 14 X = 5 25 = 25 X = 1 4 = 4 X = 2 12 = 12 X = 5 20 = 20 Lesson # 4

  9. X = 6 16 = 16 X = 10 30 = 30 X = 10 21 = 21 X = 5 17 = 17 X = 6 15 = 15 X = 4 12 = 12 X = 2 8 = 8 X = 0 3 = 3 X = 4 24 = 24 X = 7 29 = 29 Lesson #5

  10. Rule 2c • You need to rewrite problems with parentheses before solving. Ex. 1. 2(x + 3) = x + 8 2x + 6 = X + 8 Ex. 2. 2(2x + 1) = 18 4x + 2 = 18 X = 2 10 = 10 X = 4 18 = 18

  11. X = 2 10 = 10 X = 3 12 = 12 X = 6 26 = 26 X = 1 10 = 10 X = 5 11 = 11 X = 9 20 = 20 X = 1 5 = 5 X = 2 11 = 11 X = 7 26 = 26 X = 3 9 = 9 Lesson #6

  12. X = 2 10 = 10 X = 5 30 = 30 X = 7 26 = 26 X = 4 26 = 26 X = 6 24 = 24 X = 3 13 = 13 X = 5 27 = 27 X = 11 38 = 38 X = 7 29 = 29 X = 2 12 = 12 Lesson #7

  13. Rules 3 & 3a The following piece we will be using for the next few lessons: White Pawns –> -> Opposite X If X equals 2, then opposite equals -2 3. You may remove the same number of white pawns from each side of the scale. 3a. The word evaluate means: solve the problem by replacing your variable with what it equals and follow the order of operations. X X

  14. = 6 20 = 20 = 9 18 = 18 = 3 14 = 14 = 8 42 = 42 X = 7 21 = 21 X = 7 21 = 21 X = 6 27 = 27 X = 2 11 = 11 X = 1 8 = 8 X = 2 12 = 12 Lesson #8 X X X X

  15. Ans. 7 Ans. 10 Ans. 8 Ans. 6 X = 1 6 = 6 X = 6 22 = 22 X = 8 25 = 25 X = 8 23 = 23 X = 3 21 = 21 X = 7 24 = 24 Lesson #9

  16. Rules 4 & 5 • You may remove a blue and white from the same side because together they equal zero. • You may add a zero during set-up. • HINTS – add the zero to the side without cubes or the smaller cube amount • - should be added for subtraction when there is nothing to take away.

  17. X = 2 = -2 4 = 4 X = -3 = 3 9 = 9 X = -6 = 6 12 = 12 X = 8 = -8 32 = 32 X = 9 45 = 45 X = 4 21 = 21 X = 3 20 = 20 X = 2 15 = 15 Ans. 1 X = -5 = 5 26 = 26 Lesson #10 X X X X X

  18. X = -3 = 3 6 = 6 X = -1 = 1 2 = 2 X = 2 = -2 2 = 2 X = 2 = -2 6 = 6 X = 1 4 = 4 X = 4 25 = 25 X = 3 22 = 22 Ans. 8 = 8 42 = 42 X = 5 = -5 15 = 15 Lesson #11 X X X X X X

  19. Rule 5a & 5b 5a. You may add a blue pawn to each side of the scale. By adding the pawn to each side, the scale stays balanced 5b. You may add a white pawn to each side of the scale. By adding the pawn to each side, the scale stays balanced.

  20. X = -4 = 4 12 = 12 X = -3 = 3 6 = 6 X = 12 = -12 12 = 12 X = 2 = -2 2 = 2 X = 4 16 = 16 X = 10 28 = 28 Ans. 1 = 1 7 = 7 X = 6 = -6 12 = 12 X = 8 = -8 16 = 16 Lesson 12 X X X X X X X

  21. X = -3 = 3 11 = 11 X = -6 = 6 8 = 8 X = -2 = 2 8 = 8 X = -1 = 1 2 = 2 X = 8 27 = 27 X = 5 31 = 31 Ans. 6 = 3 21 = 21 X = 8 = -8 21 = 21 X = 10 = -10 20 = 20 Lesson 13 X X X X X X X

  22. Review and New • A zero should be added for subtraction when there is nothing to take away. • (-x) is another way to write . X

  23. X = -3 = 3 12 = 12 X = 10 = -10 30 = 30 X = 2 = -2 3 = 3 X = 6 = -6 -2 = -2 X = 7 38 = 38 X = 4 24 = 24 Ans. 3 X = 5 = -5 19 = 19 X = 1 = -1 2 = 2 X = 3 = -3 10 = 10 Lesson 14 X X X X X X X

  24. -x = 3 12 = 12 -x = 7 16 = 16 x = 2 -x = -2 3 = 3 x = 1 -x = -1 8 = 8 X = 4 24 = 24 = 5 27 = 27 Ans. 2 X = 10 = -10 20 = 20 X = 2 = -2 8 = 8 X = -5 = 5 10 = 10 Lesson 15 X X X X

  25. X = 5 = -5 10 = 10 X = -2 = 2 6 = 6 X = -4 = 4 - 4 = - 4 X = 7 = -7 27 = 27 = 6 36 = 36 X = -8 = 8 16 = 16 X = 5 = -5 15 = 15 X = 7 = -7 28 = 28 X = -5 = 5 5 = 5 X = 6 = -6 15 = 15 Lesson 16 X X X X X X X X X

  26. Rules 6 & 7 • The Green Dice represent negative numbers • You may remove the same amount from the green dice on both sides. • The same amount of red and green dice equal zero. Zero’s may be added or taken away.

  27. 1a. (-3) + (-4) = -7 2a. (+5) + (-3) = 2 3a. (-3) - (-1) = -2 4a. (-3) - (+2) = -5 1b. (-2) + (-5) = -7 2b. (-5) + (+2) = -3 3b. (-4) - (-2) = -2 4b. (+3) - (-2) = 5 Lesson 17 • X = 6 18=18 • = 1 9 = 9 • X = 8 = -8 16 = 16 • X = 1 = -1 6 = 6 • X = 2 = -2 2 = 2 • X = 8 = -8 16 = 16 X X X X X

  28. Rule 8 & 9 • You may add the same number of red dice to each side of the scale. • You may add the same number of green dice to each side of the scale.

  29. X = -10 -6 = -6 X = -5 -13 = -13 = 6 16 = 16 = -2 -11 = -11 X = 4 20 = 20 X = 9 = -9 18 = 18 X = 2 = -2 4 = 4 X = -6 = 6 30 = 30 X = 4 -X = -4 8 = 8 Lesson 18 X X X X X 10a. (+2) + (-7) = -5 10b. (-3) – (+4) = -7

  30. = 13 8 = 8 X = 1 -4 = -4 = -1 -8 = -8 X = 1 13 = 13 X = 6 36 = 36 X = -4 = 4 -1 = -1 X = -2 = 2 4 = 4 X = -5 -X = 5 5 = 5 Lesson 19 X X X X 9a. 6 + 2 + (-4) = 4 9b. (-4) – (-6) + 3 = 5 10. X = -4 -20 = -20

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