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YOU are better than YOU think. Show yourself how:
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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 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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-/[]\- 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. |
Finished 1. Solving Linear EquationsThe site section Solving Linear Equations with and then without Stick Diagrams covers solution of (i) linear equations ax+b = cx + d in one unknown, (ii) triangular systems of equations and (iii) systems of equations in essentially one unknown before covering the solution of systems of equations in two unknowns by the (a) substitution, (b) comparison and (c) row-addition-multiplication method. The site section emphasize exact arithmetic with whole numbers and fractions, and the checking or verification of solutions. With the latter students can check and/or correct their work before submission for grading. Warn the student that if a check fails, the error is somewhere between the start of their solution and the end of their check.
NotesThe remarks or their suggestions, hindsights, will be incorporated into site lessons on Solving Linear Equations with and then without Stick Diagrams.
1 Fraction Sense versus Fraction Skills
Students at this level should not need the stick diagrams. However, if you see that students have a weak command of fractions, examples if not exercises with the stick diagrams may develop the missing fraction sense. Students may need to review the fraction summary and more in the site section Fractions, Ratios, Rates, Proportions & Units. Explain to students that efficient arithmetic skills with fractions is a must. Test and test students on their comprehension of fractions and their mastery of efficient ways to add, multiply and divide fractions in proper, improper and mixed form. Teachers: Coefficients in solving one equation in one or more unknowns should be chosen to imply integer coefficients in the first instance and fraction coefficients and even fraction coefficients in the second instance. Algebra requires an efficient command of exact arithmetic with whole numbers and fractions. Some students may need to review the fraction summary and more in the site section Fractions, Ratios, Rates, Proportions & Units 2 Notation For solving ax + b = c, in place of or besides stick diagrams, I would use the column format
as a hint of and preparation the format seen in the row multiplication and addition method for solving linear equations in two unknowns. The site section Solving Linear Equations with and then without Stick Diagrams does not yet use this format in its algebraic solution of equations. A correction or improvement may follow later.
I would also use a similar format for reducing ax+ b = cx + d to the form ax + b = cx+d 3. Triangular Systems. In Solving Linear Equations with and then without Stick Diagrams, introduction of triangular systems (optional in 436) provides a quick and easy way to meet the concept of simultaneous equations. Students in meeting triangular systems of equations may be surprised that one unknown, say x or y, has the same value in two or more simultaneous equations. The objective here is to minimize that surprise. Solving triangle and scramble triangle systems of equations is not yet part of the course, so it inclusion is optional. But inclusion may develop students algebraic and arithmetic skills. 4. Systems of Equations in Essentially One Unknown
where there no need for the expansion use of the distributive law a(b+c) = ab + ac. (The contraction use ax+bx = (a+b)x is present, and might be used without mention.) Students will have to be reminded to calculate A, B and C after obtaining the value of the unknown x, otherwise their solution is incomplete. Substitution is more complicated in systems of equation like the following
where there is a need for the expansion use of the distributive law a(b+c) = ab + ac and the collection of terms involving the essentially unknown x. C choosing coefficients so solutions are whole numbers or integers will ease or avoid difficulties for students in the first instance. Students may need some practice here to ensure or check that their use of the distributive law is correct. The ability to check their solutions will lead students to question their own calculations and so seek help in repairing faulty arithmetic skills.
5 Solving Systems of Equations in two Unknowns.
(5a) The substitution method introduced for systems of equations in essentially one unknown applies to equations of systems of the form
I might give the above system as one question and thene the equivalent system
as another system in the same lesson or same exercise set. So the substitution method for reducing the number of unknowns (here eliminating the y) to obtain an equation in one unknown appears as special case of the method for solving systems of equations in one unknown. After find the "essential" unknown, students need to calculate the other from one equation, and then check the find values for x and y satisfy both equations. In the case of equations of the form y = 2 x + 5 the first equation is satisfied automatically by the computation of y. So only the second equation 3x + 2y = 27 needs to be checked. That is a shortcut. If students do not understand it, have them check for both equations. Use the following format: Check Format:
(5b) The comparison method gives another method for eliminating one unknown to obtain a single equation in one unknown. Here two simultaneous equations
imply the two expressions for y should have the same value. So
Here the ability of students to solve one equation in one unknow (step 1) becomes useful. After finding x, student need to check that the two right-hand side expression ax + b and cx + d give the same value for y. Check Format:
(5c) The equation-multiplication-addition method is yet another method for eliminating one unknown and obtaining one equation in one unknown. It can be applied to systems of equations of the form
to obtain 0 = (a-c)x + (b-d). It can be applied to systems of equations of form
In the case where a, b, c and d are integers, the choice of multipliers can be based on comparison of the least common multiples of the x coefficients a and c, and the least common multiples of the y coefficient b and d. If the former is smaller than the latter lcm, eliminate x and obtain an single equation in y, while if the former is greater than the latter, eliminate y to to obtain an equation in x. The foregoing elimination decisions result in smaller coefficients. Have the students to use the follow check format: Check Format:
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