Stick Diagram Solutions: First Example
(unknown on one side, whole number coefficients)

See too Animated Example(s) below

Suppose we are given 2x + 6 = 24 to solve.  

Whenever you do a multi-step problem, remember to check your answer in the manner shown in this and further lessons as an error in one step can make all the rest and your answer wrong.

Now in the equation  2x + 6 = 24 we imagine that x represents an unknown length. In the stick below, the top stick has length 2x+6 while the bottom stick has length 24. The equation say both sticks have the same length, here 24.

2x	6
24

Cutting off or subtracting 8 from both sticks gives

2x

18

a stick of length 2x on top and a stick of length 18 = 24 - 6 on bottom. This second stick diagram represents the equation 2x =18.

But 2x = x + x. So half of the length of 2x is x. To find x, we will take cut each stick into two equal pieces.

x	x
18 2	18 2

or

x	x
9	9

The stick diagram suggests that    x	=	18 2	=	9

x

9

A  three column format, summary of the operation follow. See too the solution of 2x+6 = 24

Stick Diagram	Operation	Equation
2x 6 24	Initial Situation Given	2x +6 = 24
2x 18	Subtract 6 (aka Add -6)	2x +6 = 24        6 =   6  - 2x      = 18
x x 9 9	x is half of 2x and therefore half of 15 as well	2x 2	=	18 2
x 9	To get from the  2nd row to 4th, multiply by  1 2	x	=	9

 If  I was solving 2x + 6 = 24 in class, I would just fill in the table and skip the work before it.  Each table consists of a diagram in the left column, a description of what is done or given in the middle column, and the equivalent equations in the rightmost column.  At the moment, you are required to draw the stick diagram in the solution of the equation. That is a crutch.  Later on, only the equation column is required with a few words to explain the operations.

Check: Is x = 9 a solution?

Need to verify the left hand side (LHS) and right hand side of the equation

2x + 6 = 24

have the same value when we replace x by 9.

Left Hand Side.   LHS = 2 x + 6          = 2 (9)  + 6  when x = 9          = 18 + 6          = 24 = RHS as wanted

Right Hand Side RHS  = 24 for all values of x, including x = 9

Checking your solution (or your guess) is a good way to see if your answer is right or wrong. If the check fails, the error in your calculations lies somewhere between the start of your check and the end of your solution.

Exercises: Solve the following equations with stick diagrams. Use the Three Column Format

1. What is x if  x + 8 = 28?  (Draw the diagrams & check your answer).
2. What is x if  2x + 8 = 38? (Draw the diagrams & check your answer)
3. What is x if 3x + 4 = 16? (Draw the diagrams & check your solution)

Teachers: If  students leap to the algebraic solution and have do not need to draw the diagrams,  object.  Tell the students in question that drawing the stick diagrams is a test of their skill in understanding and following instructions.  Tell them that more complicated examples will follow in which understanding the stick diagram method will improve their mastery of fractions and mixed numbers. 

Animated Example: 3x + 4 = 10 (animated gif)  

Example With  Answer Not Whole

Suppose we are given 2x + 5 = 20 to solve.  

Whenever you do a multi-step problem, remember to check your answer in the manner shown in this and further lessons as an error in one step can make all the rest and your answer wrong.

Now in the equation  2x + 5 = 20 we imagine that x represents an unknown length. In the stick below, the top stick has length 2x+5 while the bottom stick has length 20. The equation say both sticks have the same length, here 20.

2x	5
20

Cutting off or subtracting 5 from both sticks gives

2x

15

a stick of length 2x on top and a stick of length 15 = 20 - 5 on bottom. This second stick diagram represents the equation 2x =15.

But 2x = x + x. So half of the length of 2x is x. To find x, we will take cut each stick into two equal pieces.

x	x
15 2	15 2

The stick diagram suggests that    x	=		15 2

x

15 2

A static summary of the operation follow. See too the solution of 2x+5 = 20 Animated version second, or first immediately below. 

Stick Diagram	Operation	Equation
2x 5 20	Initial Situation Given	2x +5 = 20
2x 15	Subtract 5 (aka Add -5)	2x + 5 = 20         5 =   5  - 2x       = 15
x x 15 2 15 2	x is half of 2x and therefore half of 15 as well	2x 2	=	15 2
x 15 2	To get from the  2nd row to 4th, multiply by  1 2	x	=	15 2

 As in the first example above, If  I was solving 2x + 5 = 20 in class, I would just fill in the table and skip the work before it.  Each table consists of a diagram in the left column, a description of what is done or given in the middle column, and the equivalent equations in the rightmost column.  At the moment, you are required to draw the stick diagram in the solution of the equation. That is a crutch.  Later on, only the equation column is required with a few words to explain the operations.

Solution of 2x+5 = 20 Animated.

 

A  reference  for  solving linear equations and for  recognizing word problems in essentially one variable. Skill in arithmetic with fractions is a must for algebra. .

Area Intro
Proper Use of Equal Sign
A. Letters and Lengths
B.. Solving Linear Eq'ns - WS
C. Solving Linear Eq'ns - No Sticks
D. Almost One
E: 2D Systems - Sub Method.
E:  Continued
E: Still More
F. Larger Systems

Area Intro
(i) x + 20 = 29 WS
(ii) 2x + 5 = 20 WS
(iii) 3x + 10 = 32 WS
(iv) 5a + 16 = 3a+ 24 WS
(v)  (½)x + 8 = 24½ WS
(vi)  (¾)a + 16 = (¼)a+ 24 WS
(vii) (¾)q + 17 = 32 WS
(viii) 13 =[2/3]x +7 twice WS
(ix) Animated Examples WS
(a) Integral Coefficients (A)
(b) Integral Coefficients (B)
(c) Fractional Coefficients
(d) With parameters

Arithmetic Videos

Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods

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Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification

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Lesson Plans and Ideas for Teachers & Tutors:

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