YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
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
Given2x +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 well2x
2= 18
2
x
9
To get from the
2nd row to 4th,
multiply by
1
2x = 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
- What is x if x + 8 = 28? (Draw the diagrams & check your answer).
- What is x if 2x + 8 = 38? (Draw the diagrams & check your answer)
- 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
Given2x +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 well2x
2= 15
2
x
15
2
To get from the
2nd row to 4th,
multiply by
1
2x = 15
2As 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.
www.whyslopes.com
Solving Linear Equations
|(Feb 14, 2005)a secondary I to V reference for solving linear equations and for recognizing word problems in essentially one variable whether you like it or not, skill in arithmetic with fractions is a must for algebra. .
Area Entrance Proper Use of Equal Sign A. Letters and Lengths B.. Solving Linear Eq'ns. C. Solving Linear Eq'ns D.Almost One E: 2D Systems - Sub Method. E: Continued E: Still More F. Larger Systems
Area Entrance (i) x + 20 = 29 (ii) 2x + 5 = 20 (iii) 3x + 10 = 32 (iv) 5a + 16 = 3a+ 24 (v) (½)x + 8 = 24½ (vI) (¾)a + 16 = (¼)a+ 24 (vii) (¾)q + 17 = 32 (viii) 13 =[2/3]x +7 twice (x) Animated Examples (i) Integral Coefficients (A) (ii) Integral Coefficients (B) (iii) Fractional Coefficients (iv) With parameters
Arithmetic Videos
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification
Site books and further webpages on learning and teaching mathematics and pattern based reason may develop critical thinking, improve reading and writing, and give a base for learning or teaching high school and college mathematics.
Great_Expectations: If you can learn to follow a multi-step methods in any subject precisely, you can do so in other subjects, as well.
Good news: Site pages identify what you need to study.
Bad news: Site pages do not explain everything
Worse news: Learning takes time, yours
Lesson Plans and Ideas for Teachers & Tutors:
Secondary I - fractions & allied concepts (decimals, percentages)