Hands-on Equations

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 gt x
• Red Cubes -gt Numbers
• Scale -gt 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
Lesson 1

1. X 5 10 10
2. X 7 9 9
3. X 2 7 7
4. X 4 12 12
5. X 1 4 4

1. X 6 14 14
2. X 4 8 8
3. X 4 14 14
4. X 6 16 16
5. X 6 18 18

5
Lesson 2

• X 3 6 6
• X 2 6 6
• X 1 5 5
• X 3 12 12

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

6
Lesson 3

1. X 7 28 28
2. X 4 14 14
3. X 5 20 20
4. X 2 13 13

1. X 6 12 12
2. X 5 20 20
3. X 2 7 7
4. X 3 8 8
5. X 6 19 19
6. X 1 6 6

7
Rule 2a

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

8
Lesson 4

1. X 7 26 26
2. X 6 32 32
3. X 8 28 28
4. X 5 17 17

1. X 2 9 9
2. X 6 14 14
3. X 5 25 25
4. X 1 4 4
5. X 2 12 12
6. X 5 20 20

9
Lesson 5

1. X 6 16 16
2. X 10 30 30
3. X 10 21 21
4. X 5 17 17

1. X 6 15 15
2. X 4 12 12
3. X 2 8 8
4. X 0 3 3
5. X 4 24 24
6. X 7 29 29

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
Lesson 6

1. X 2 10 10
2. X 3 12 12
3. X 6 26 26
4. X 1 10 10

1. X 5 11 11
2. X 9 20 20
3. X 1 5 5
4. X 2 11 11
5. X 7 26 26
6. X 3 9 9

12
Lesson 7

1. X 2 10 10
2. X 5 30 30
3. X 7 26 26
4. X 4 26 26

1. X 6 24 24
2. X 3 13 13
3. X 5 27 27
4. X 11 38 38
5. X 7 29 29
6. X 2 12 12

13
Rules 3 3a

• The following piece we will be using for the next
  few lessons
• White Pawns gt -gt 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
Lesson 8

1. 6 20 20
2. 9 18 18
3. 3 14 14
4. 8 42 42

1. X 7 21 21
2. X 7 21 21
3. X 6 27 27
4. X 2 11 11
5. X 1 8 8
6. X 2 12 12

X
X
X
X
15
Lesson 9

• Ans. 7
• Ans. 10
• Ans. 8
• Ans. 6

1. X 1 6 6
2. X 6 22 22
3. X 8 25 25
4. X 8 23 23
5. X 3 21 21
6. X 7 24 24

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
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

1. X 9 45 45
2. X 4 21 21
3. X 3 20 20
4. X 2 15 15
5. Ans. 1
6. X -5 5 26 26

X
X
X
X
X
18
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

1. X 1 4 4
2. X 4 25 25
3. X 3 22 22
4. Ans. 8
5. 8 42 42
6. X 5 -5 15 15

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
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

1. X 4 16 16
2. X 10 28 28
3. Ans. 1
4. 1 7 7
5. X 6 -6 12 12
6. X 8 -8 16 16

X
X
X
X
X
X
X
21
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

1. X 8 27 27
2. X 5 31 31
3. Ans. 6
4. 3 21 21
5. X 8 -8 21 21
6. X 10 -10 20 20

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
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

• 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

X
X
X
X
X
X
X
24
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

1. X 4 24 24
2. 5 27 27
3. Ans. 2
4. X 10 -10 20 20
5. X 2 -2 8 8
6. X -5 5 10 10

X
X
X
X
25
Lesson 16
X
X

1. X 5 -5 10 10
2. X -2 2 6 6
3. X -4 4 - 4 - 4
4. X 7 -7 27 27

1. 6 36 36
2. X -8 8 16 16
3. X 5 -5 15 15
4. X 7 -7 28 28
5. X -5 5 5 5
6. X 6 -6 15 15

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.
  Zeros may be added or taken away.

27
Lesson 17

• 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

1. X 6 1818
2. 1 9 9
3. X 8 -8 16 16

1. X 1 -1 6 6
2. X 2 -2 2 2
3. X 8 -8 16 16

X
X
X
X
X
28
Rule 8 9

1. You may add the same number of red dice to each
   side of the scale.
2. You may add the same number of green dice to each
   side of the scale.

29
Lesson 18

1. X -10 -6 -6
2. X -5 -13 -13
3. 6 16 16
4. -2 -11 -11

1. X 4 20 20
2. X 9 -9 18 18
3. X 2 -2 4 4
4. X -6 6 30 30
5. X 4 -X -4 8 8

X
X
X
X
X
10a. (2) (-7) -5 10b. (-3) (4) -7
30
Lesson 19

1. 13 8 8
2. X 1 -4 -4
3. -1 -8 -8
4. X 1 13 13

1. X 6 36 36
2. X -4 4 -1 -1
3. X -2 2 4 4
4. X -5 -X 5 5 5

X
X
X
X
9a. 6 2 (-4) 4 9b. (-4) (-6) 3 5 10.
X -4 -20 -20