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Hands-on Equations

This hands-on introduction to algebra uses visual aids like blue pawns and red cubes to teach the foundational concepts of algebra through simple equations. Learn how to balance equations, apply operations, and understand variables with practical examples. Follow along with interactive lessons that provide step-by-step guidance and practice problems to reinforce learning. Ideal for beginners or anyone seeking to strengthen their algebra skills, this program makes math engaging and accessible.

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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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