Hands-on Equations

An Introduction to Algebra

The Pieces

The following pieces we will be using for the first few lessons:

Blue Pawns –> xRed Cubes -> NumbersScale -> shows balance

Rules 1 & 2

1. You may remove the same number of blue pawns from each side of the scale.

2. You may remove the same cube amount from each side of the scale.

Lesson #1

1. X = 5 10 = 10

2. X = 7 9 = 93. X = 2 7 = 74. X = 4 12

= 125. X = 1 4 = 4

6. X = 6 14 = 14

7. X = 4 8 = 88. X = 4 14 =

149. X = 6 16 =

1610. X = 6 18 =

18

Lesson # 2

1. X = 3 6 = 6

2. X = 2 6 = 6

3. X = 1 5 = 5

4. X = 3 12 = 12

5. X = 4 12 = 126. X = 7 15 = 157. X = 8 16 = 168. X = 4 13 = 139. X = 2 5 = 510. X = 10 25 = 25

Lesson #3

1. X = 7 28 = 282. X = 4 14 = 143. X = 5 20 = 204. X = 2 13 = 13

5. X = 6 12 = 126. X = 5 20 = 207. X = 2 7 = 78. X = 3 8 = 89. X = 6 19 =

1910. X = 1 6 = 6

Rule 2a

You may remove pawns as part of the set-up. This is done for subtraction operations.

Lesson # 4

1. X = 7 26 = 26

2. X = 6 32 = 32

3. X = 8 28 = 28

4. X = 5 17 = 17

5. X = 2 9 = 96. X = 6 14 = 147. X = 5 25 = 258. X = 1 4 = 49. X = 2 12 = 1210. X = 5 20 = 20

Lesson #5

1. X = 6 16 = 16

2. X = 10 30 = 30

3. X = 10 21 = 21

4. X = 5 17 = 17

5. X = 6 15 = 156. X = 4 12 = 127. X = 2 8 = 88. X = 0 3 = 39. X = 4 24 = 2410. X = 7 29 = 29

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

Lesson #6

1. X = 2 10 = 10

2. X = 3 12 = 12

3. X = 6 26 = 26

4. X = 1 10 = 10

5. X = 5 11 = 116. X = 9 20 = 207. X = 1 5 = 58. X = 2 11 = 119. X = 7 26 = 2610. X = 3 9 = 9

Lesson #7

1. X = 2 10 = 10

2. X = 5 30 = 30

3. X = 7 26 = 26

4. X = 4 26 = 26

5. X = 6 24 = 246. X = 3 13 = 137. X = 5 27 = 278. X = 11 38 = 389. X = 7 29 = 2910. X = 2 12 = 12

Rules 3 & 3aThe following piece we will be using for the next

few lessons:

White Pawns –> -> Opposite XIf 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.

XX

Lesson #8

1. = 6 20 = 20

2. = 9 18 = 18

3. = 3 14 = 14

4. = 8 42 = 42

5. X = 7 21 = 216. X = 7 21 = 217. X = 6 27 = 278. X = 2 11 = 119. X = 1 8 = 810. X = 2 12 = 12

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Lesson #9

1. Ans. 72. Ans. 103. Ans. 84. Ans. 6

5. X = 1 6 = 66. X = 6 22 = 227. X = 8 25 = 258. X = 8 23 = 239. X = 3 21 = 2110. X = 7 24 = 24

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.

Lesson #10

1. X = 2 = -2 4 = 4

2. X = -3 = 3 9 = 9

3. X = -6 = 6 12 = 12

4. X = 8 = -8 32 = 32

5. X = 9 45 = 456. X = 4 21 = 217. X = 3 20 = 208. X = 2 15 = 159. Ans. 110. X = -5 = 5

26 = 26

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Lesson #11

1. X = -3 = 3 6 = 6

2. X = -1 = 1 2 = 2

3. X = 2 = -2 2 = 2

4. X = 2 = -2 6 = 6

5. X = 1 4 = 46. X = 4 25 = 257. X = 3 22 = 228. Ans. 89. = 8 42 = 4210. X = 5 = -5

15 = 15

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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.

Lesson 12

1. X = -4 = 4 12 = 12

2. X = -3 = 3 6 = 6

3. X = 12 = -12 12 = 12

4. X = 2 = -2 2 = 2

5. X = 4 16 = 166. X = 10 28 = 287. Ans. 18. = 1 7 = 79. X = 6 = -6

12 = 1210. X = 8 = -8

16 = 16

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Lesson 13

1. X = -3 = 3 11 = 11

2. X = -6 = 6 8 = 8

3. X = -2 = 2 8 = 8

4. X = -1 = 1 2 = 2

5. X = 8 27 = 276. X = 5 31 = 317. Ans. 68. = 3 21 = 219. X = 8 = -8

21 = 2110. X = 10 = -10

20 = 20

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Review and New

A zero should be added for subtraction when there is nothing to take away.

(-x) is another way to write .X

Lesson 14

1. X = -3 = 3 12 = 12

2. X = 10 = -10 30 = 30

3. X = 2 = -2 3 = 3

4. X = 6 = -6 -2 = -2

5. X = 7 38 = 386. X = 4 24 = 247. Ans. 3 8. X = 5 = -5

19 = 199. X = 1 = -1

2 = 210. X = 3 = -3

10 = 10

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Lesson 15

1. -x = 3 12 = 12

2. -x = 7 16 = 16

3. x = 2 -x = -2 3 = 3

4. x = 1 -x = -1 8 = 8

5. X = 4 24 = 246. = 5 27 = 277. Ans. 28. X = 10 = -

10 20 = 20

9. X = 2 = -2 8 = 8

10. X = -5 = 5 10 = 10

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Lesson 161. X = 5 = -5

10 = 102. X = -2 = 2

6 = 63. X = -4 = 4

- 4 = - 44. X = 7 = -7

27 = 27

5. = 6 36 = 366. X = -8 = 8

16 = 167. X = 5 = -5

15 = 158. X = 7 = -7

28 = 289. X = -5 = 5

5 = 510. X = 6 = -6

15 = 15

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Rules 6 & 7

The Green Dice represent negative numbers

6. You may remove the same amount from the green dice on both sides.

7. The same amount of red and green dice equal zero. Zero’s may be added or taken away.

Lesson 17

1a. (-3) + (-4) = -72a. (+5) + (-3) = 2 3a. (-3) - (-1) = -24a. (-3) - (+2) = -5

1b. (-2) + (-5) = -72b. (-5) + (+2) = -33b. (-4) - (-2) = -24b. (+3) - (-2) = 5

5.X = 6 18=18

6. = 1 9 = 9

7.X = 8 = -8 16 = 16

8. X = 1 = -1 6 = 6

9. X = 2 = -2 2 = 2

10.X = 8 = -8 16 = 16

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Rule 8 & 9

8. You may add the same number of red dice to each side of the scale.

9. You may add the same number of green dice to each side of the scale.

Lesson 18

1. X = -10 -6 = -62. X = -5 -13 = -

133. = 6 16 = 164. = -2 -11 = -

11

5. X = 4 20 = 20

6. X = 9 = -9 18 = 18

7. X = 2 = -2 4 = 4

8. X = -6 = 6 30 = 30

9. X = 4 -X = -4 8 = 8

10a. (+2) + (-7) = -5

10b. (-3) – (+4) = -7

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Lesson 19

1. = 13 8 = 82. X = 1 -4 = -43. = -1 -8 = -84. X = 1 13 =

13

5. X = 6 36 = 366. X = -4 = 4

-1 = -17. X = -2 = 2

4 = 48. X = -5 -X = 5

5 = 59a. 6 + 2 + (-4) = 4

9b. (-4) – (-6) + 3 = 5

10. X = -4 -20 = -20

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