Original Site Title: Appetizers and Lessons for Mathematics and
Reason, June 1995 to April 2012. New site title:
Logic and Mathematics Skill & Concept
Development with How-TOs Français: 26
pages
for college students,
gifted teens, home-tutoring and K1-12 schooling. Avid readers in school and out may like Site Volumes.
|
Logic
5 Chapters Arithmetic 10 Steps
Algebra 12
Starter Steps & 5
Advanced Steps |
Welcome: Site content may develop
critical thinking, improve reading and writing, and build
mathematics skills. See online chapters on
on
logic and pattern
based reason.
|
Home < Algebra Starter Lessons < 7 Axioms Logic and Equivalent Equations << 1 Equivalent Computation Rules
[1][2] [3] [4] [5] [6]
Equivalent Computation RulesEvaluation of the computation rules \begin{eqnarray*} f(a,b,c) &=& a+b +c \\ h(a,b,c) &=& (b+c)a \\ g(a,b,c) &=& ba +ca \\ \end{eqnarray*} describe calculations with three numbers a, b and c. But the arithmetic in each evaluation will be different. Unless you are familar with the distributive property, there is no reason to expect all three or any two of these calcuation to give the same result when evaluated. As a rule, different formulas give different result. But there are exceptions. Recognizing when two different computations give the same result allows us to do one calculation in place of the other, which ever is easiest, or whichever is the most convenient. Motivational ExercisesCalculate the following $A = 4 \times 5 + 8\times 5 - (4+8)5+ 4\times2$ $B = 8+ 16 \cdot 10 + 9 \cdot 10 - (16+9)10$ $C = 2^3+ \frac 45 \times \frac 8{11} +\frac 45 \times \frac 34 - \frac 45 \left(\frac 8{11} + \frac 34\right)$ $D = 4+ 8 + 12+ 16 + 20 + 24 -\frac12 \cdot\frac{4+24}6 +\frac 6\pi\frac {4\pi}3 $ $E = 10^0+ 10^1+10^2+10^34 -\frac{10^4-1}{10-1}+2 +\frac{16}2$ Did you see any patterns? The calculations will be long without a knowledge of the distributive property of addition for whole numbers and fractions, and without a knowledge of summation methods for arithmetic and geometric sums. Here as we learn more, some work involve, less work will be required in doing arithmetic. The Basic Distributive LawThe division of a single rectangle into non-overlapping sub-rectangles gives two different way to compute the single rectangle's areas: directly from width times length; and indirectly as the sum of areas of the sub-rectangles. The equality of the two different ways implies the distributive law for products of sums, where the sums involve positive terms. Distributive Law, Geometrically implied.
\[(b+c)a = ba +ca \] as the area of the largest rectangle can be computed in two different ways, directly or as the sum of the areas ab and bc of the sub-rectangles. Now recall the computation rules: Evaluation of the computation rules \begin{eqnarray*} h(a,b,c) &=& (b+c)a \\ g(a,b,c) &=& = ba + ca \\ \end{eqnarray*} From the equality of two different ways to calculate the area of a rectangle, we expect these two very different calculations to give the same result. T The equality \[ (b+c)a = ba +ca \] holds whenever a, b and c give lengths or the number of units in each length In function function, we may expect \[ h(a,b,c) = g(a,b,c)\] whenever a, b and c can serve as the number of units of length in the rectangles drawn above. The equality\[ (b+c)a = ba +ca \] is an identity - an computational identity. The calculation on the left gives the same result as the calculation on the right. Substitution: The replacement of one calculation by another which gives the same result is the key "substitution" step in solving equations and in deriving or justifying formulas. Previously, you have have thought of formulas and computational rules as algebraic descriptions of calculations that might be done, when or if convenient. One step beyond that is to learn that different computation rules - algebraically described or not - may themselves have the same result. Further step is to see the equality \[ (b+c)a = ba +ca \] as computational identity. In it, the left hand side may replace the right hand side whenever a, b and c are given as numbers, are given by expression that yield numbers. The Counting View Point - optionalIn the case where the numbers a, b and c are whole numbers, the computational identify\[ (b+c)a = ba +ca \] comes from the equality of different ways to count.Suppose we have b groups of a objects and c groups of a objects. Then in total, we have ba + ca objects. That is one way to count the objects. Another way to count them is say we have (b+c) groups of a objects. That implies have (b+c)a objects. But in counting objects, experience implies different ways to count do not or should not change the total count. So the two different computation rules ba + ca and (b+c)a for the count should be equal. Area measurement where the side lengths are fractional multiples of a unit a length extends the notion of counting. The assumption that two different ways to compute areas is a consequence of counting practices and assumptions directly for lengths that are fractional multiples of a unit length, and in the limit for lengths given by irrational multiples. ExercisesWhat value does the distributive law give for each of the following $P = 23.5 \times 12.1 + 34. 5\times 12.1 - ( 23.5 +34.5) 812.1 $ $Q = 160 \cdot 892 + 9 \cdot 892 - (160+9)892$ $R = 8 \cdot \pi + 92 \cdot \pi - (8+92)\pi$ $S = (8+92)\pi - [8 \cdot \pi + 92 \cdot \pi] $ $T = (10+8) \cdot 201 + (10-8) \cdot 201 - (20)201$ $U= (10+8) \cdot (4 +56) + (10-8) \cdot (4 +56) $ $- (10+8 +10-8)(4 +56) $ $Z = (uv) \cdot mn + (s+t) \cdot mn - (uv+ s+t)mn$ Key Point. The distributive property \[ (b+c)a = ba +ca \] can be applied to numbers and to letters or algebraic expressions whose computation would give numbers. Each time you apply the distributive law above, identify the values assigned to a, b and c in the application. |
|
Home < Algebra Starter Lessons < 7 Axioms Logic and Equivalent Equations << 1 Equivalent Computation Rules
[1][2] [3] [4] [5] [6]
|
Logic-Reason for all Careful Thinking Chains of Reason Mathematical Induction Responsibility Bodies-of-Knowledge |
Arithmetic - Ages 10+ |
Geometry 1 Maps + Plans Use 2 Euclidean Geometry 3 Rct +Polr Coordinates 4 Lines-Slopes [I] 5. What is Similarity |
Algebra Starters - the base 1. Better Work Format 2. Solve Linear Eqns 3. Computation Rules 4. Axioms, Item 3 Viewpnt 5. Formulas Backwards |
More Algebra Logarithms-ax & m/nth roots Five Polynomial Operations Quadratics Geometrically Functions || Vectors too Arith. Skill Check+Answers |
Calculus Prep/Preview What is a Variable Why study slopes Why factor polynomials Complex Numbers Limits + Continuity |
All trademarks and copyrights in this are owned by their
respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest
© 1995-2011, by site author, Alan Selby, Ph. D., Montreal,
All Rights Reserved ---
Skype
or Email to contact.