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Stick Diagrams is a new invention for visually providing a context for the solution of equations in one unknown. Worked examples follow. Examples (i) to (ix) with and Examples (a) to (d) are without stick diagrams 

(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

See if you can follow the solutions and the solution checks in them.  

Each Example comes with one or more silent movie (animated gif) solutions of further examples.

If you do not follow the use of parameters  do not worry.   Take peak at Chapters 15 in Three Skills for Algebra. It will give you a second perspective on solving linear equations starting with examples simpler than you have mastered in this site area, but that second perspective also may also help you understand the following example.  It is an example from the end of Chapter 15.

Notes for Teachers

  1. Most students understand the use of stick diagrams in solving linear equations.  One recent student could not see the equivalent between the stick diagrams and equations, but he could use the diagrams to solve.  More examples might have helped.  Not all is certain in mathematics education. 
  2. A vertical pair of equal-length sticks (line segments) with the second below the first is use to represent an equation in which the length of a line segment is the unknown.  A sequence of operations easily seen and understood may shorten or lengthen or multiply each stick by the same amount, so that a sequence of stick pairs or equation results.  The aim is obtain a pair of equal length sticks with one has the sought for length and the other have a length given by a number. That solves the equation. The solution method here employ subtraction, multiplication and division to shift from one pair of equi-length sticks to another pair in order to solve for or isolate the unknown. Some students will not catch on to the idea that operations should lead from one pair of equi-length sticks to another.
  3. A  three column table summarizing the operations appears in each example and is followed a check of the solution obtained. By checking, students know at the end of their calculation whether or not a mistake has been made in obtaining a solution.  The first column in the table presents a sequence of stick diagrams. The third column gives the corresponding equation. The  middle describes the operation in going from one pair of sticks to another (or one equation to another). Ideally the description is written in a way that it describe the operation on a pair of sticks and the corresponding equations well. The filling in of the table introduces the notion that what is done to one stick or one side of an equation has to be done to the other side as well to maintain equality of the lengths each stick or equation represents. 
  4. The equations here and in the exercises may be solved without using the stick diagrams. Some students may see the stick diagrams and decide not to use them.  But their use introduces the notion that what is done to one stick or one side of an equation has to be done to the other side as well to maintain equality of the lengths each stick or equation represents.  Ask students who see the stick diagrams and decide not to use them to have patience and to take the time to demonstrate they mastered stick diagram usage.  Implicit in their usage are all the rules of algebra for solving equations. Those  rules will be formally given later with reference and illustration by the stick diagram method of solving equations. 
  5. Students should be required to check that the solution they obtained satisfies the original equation, and be told explicitly if the right hand side does not equal the left hand side for your solution that they have to look for the error (or if time is short, acknowledge their solution is wrong). Finding that the the right hand side does not equal the left hand side and saying nothing, or worse claiming to have done the problem points to a lack of comprehension.
  6. In class or a solution given by a student, the three column table may be filled in one row at a time and one row after another with no work before it. However in these webpages, the solution is provided in paragraph form step by step before the three column table is written to summarize the proceeding.
  7. The examples here involve only  unit and  simple fractions ¼ ½ ¾ mainly because they easily inserted typed on the keyboard. Other fraction, proper and improper, and mixed numbers appear in the exercises. 
  8. The stick diagram method here employs only subtraction, division and replication of segment lengths.  Magnification and reduction of diagrams is also useful to fit them in the width of a column. Example equations are chosen so that all coefficient and terms in the stick diagram method remain non-negative. The objective of the stick diagram method is not to solve all linear equation, but to lead students to solving linear equations by operations on equations by themselves without any geometric representation by stick diagrams.  Using parallel arrows would be a method to extend the representation to include positive and negative coefficients in equations, preferably selected to have non-negative solutions.
 

Solving Linear
Equations
|(Feb 14, 2005)
with & then without stick diagrams plus testing solutions -do not hand-in untested solutions;  solving word problems; and solving systems: -  essentially one unknown, essentially triangular & general

Skill in arithmetic with fractions is a must for algebra. 

Folder Chapters -
lesson groups

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


Up
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
 



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