Dawkins Functions and Graphs Constant
Function y = = a or f ( x a ) Graph is a horizontal
line passing through the point (0, a).
Line/Linear Function y = mx + b or f ( x) = + mx
b Graph is a line with point (0,b) and slope m.
Slope Slope of the line conta
How many dots are there in the nth diagram?
SOLUTION In the 1st diagram there are 1 1 =
12 dots. In the 2nd diagram there are 2 2 =
22 dots. In the 3rd diagram there are 3 3 = 32
dots. In the nth diagram there will be n n =
n2 dots. Areas We can also repr
n if and only if kn = "m k " < m n if and only if
kn < "m We apply FFFP to the comparison of
decimals. By definition, a decimal is a fraction
whose denominator is a power of 10 written in
the special notation introduced by the German
Jesuit astronomer C.
are doing arithmetic with just one new feature
we use letters to represent numbers.
Because the letters are simply stand-ins for
numbers, arithmetic is carried out exactly as it
is with numbers. In particular the laws of
arithmetic (commutative, associat
example: x + 0 = x (Adding zero does not
change the number.) 1 x = x (Multiplying by
one does not change the number.) Algebraic
notation In algebra there are conventional
ways of writing multiplication, division and
indices. Notation for multiplication Th
and 8x are not like terms since the powers of x
are different. c 4x 2 y, 12x 2 y are like terms
Adding and subtracting like terms The
distributive law explains the addition and
subtraction of like terms. For example: 2xy +
xy = 2 xy + 1 xy =(2 + 1)xy = 3x
this number be x. Write the following using
algebra to see what you get. Multiply the
number you thought of by 2 and subtract 5.
Multiply the result by 3. Add 15. Subtract 5
times the number you first thought of.
EXERCISE 6 Show that the sum of the first
the time interval. We say she walks at a
constant speed if her average speed in any
time interval is equal to a fixed constant v
(mi/min). We then call v her speed. We go
back to our problem: Janice walked 3.6 miles
at a constant speed. It took her 35 min
fifth gets more than the fourth. The first two
workers shall get seven times fewer measures
of corn than the three others. How many
measures of corn shall each worker get? (The
answer involves fractional measures of corn.
Answer: 12 3 , 105 6 , 20, 29 1 6
treatises of the Renaissance. The book
included the solutions to the cubic and quartic
equations. The solution to one particular case
of the cubic, x 3 + ax = b (in modern notation),
was communicated to him by Niccol Fontana
Tartaglia, and the quartic was
container. EXERCISE 7 A shed contains n
tonnes of coal. An extra 1000 tonnes are then
added. a How many tonnes of coal are there in
the shed now? b It is decided to ship the coal
in 10 equal loads. How many tonnes of coal
are there in each load? EXPANDING
discovered that quadratic equations can have
two roots, including both negative and
irrational roots. Indian mathematician
Aryabhata, in his treatise Aryabhatiya, obtains
whole-number solutions to linear equations by
a method equivalent to the modern one.
are dependent on precision for its mastery.
While all of mathematics demands precision,
the need for precision is far greater in algebra
than in arithmetic. Here is an example of the
kind of precision necessary in algebra. We are
told that to solve a syst
precise definition of the graph of an equation,
we cannot explain this fact. Consider another
example of the need for precision in algebra:
the laws of exponents. These are: For all
positive numbers x, y, and for all rational
numbers r and s, xrxs = xr+s
sides of a balance, with x2 x on one side and
1 on the other. Finally, even ignoring all the
questionable steps, how do we know 1 2(1
5) are solutions of x2 x 1 ? In other words,
have we proved that the following is true? ( 1
2 (1 5)2 1 2 (1 5) 1=0 Let u
use of symbols to state and explain basic facts
about the four operations on fractions, we
reduce most explanations only to those
amenable to picturedrawings or hands-on
activities. Thus only single-digit numbers are
used most of the time for numerators a
asymptotes that pass through center with
slope b a . Hyperbola ( ) ( ) 2 2 2 2 1 y k x h b
a - - - = Graph is a hyperbola that opens up and
down, has a center at (h k, ) , vertices b units
up/down from the center and asymptotes that
pass through center wi
these definitions in place, we are now at least
in a position to ask whether the following is
true: For all positive numbers x, y, and for all
rational numbers r and s, xrxs = xr+s (xr)s = xrs
(xy)r = xryr The proof that these laws of
exponents for ration
4) + 6(x 1) = 6x + 12 + 6x 6 = 12x + 6
EXERCISE 8 Expand the brackets and collect
like terms: a 5(x + 2) + 2(x 3) b 2(7 + 5x) + 4(x
+ 6) c 3(2x + 7) + 2(x 5) LINKS FORWARD A
sound understanding of algebra is essential for
virtually all areas of mathematic
fixed collection of numbers by the use of the
four operations +, , , , together with n (for
any positive integer n) and the usual rules of
arithmetic, is called an expression in x and y.
E.g., 85xy2 7 + xy 3 ! x5 y. An expression
in other symbols a, b, .
explain setting up a proportion? Janices
walking routine was actually the following:
Walk briskly for 2.1 miles in 35 minutes, rest
40 minutes, and walk another 1.5 miles in 45
minutes. In this case, it actually took her 35 +
40 + 45 = 120 minutes to walk
bc b d bd b d bd a b b a a b a b c d d c c c c a ab
ac b ad b c a a c bc d + = + =
=+-+=-=-+=+-
+ = + = Exponent Properties ( )
( ) ( ) ( ) 1 1 0 1 1, 0 1 1 n m m m n n m n m n m
m m n m n nm n n n n n n n n n n n n n n n n a
a a a a a a a a a a a a ab a
from fractions to algebra? There is more.
Suppose we let x = , then x x + 1 + 1 x 1 =
x(x 1) + (x + 1) (x + 1)(x 1) = x2 + 1 x2 1
implies + 1 + 1 1 = ( 1) + ( + 1) ( +
1)( 1) = 2 + 1 x2 1 Now we are talking
about the addition of fractions whose
numerators
notation? From al-Khwarizmi (circa 780-850):
What must be the square which, when
increased by 10 of its own roots, amounts to
thirty-nine? The solution is this: You halve the
number of roots, which in the present instance
yields five. This you multiply by
appealing to another metaphor. It must be
recognized that the difficulty lies in having to
compare the 19th point (to the right of 0) in
the sequence of 54ths with the 6th point (to
the right of 0) in the sequence of 17ths. How
to compare a 54th with a 17
do the following computation as is, 1.5 0.028 +
42 1.03 = (1.5 1.03) + (42 0.028) 0.028
1.03 , or would you prefer to change all the
complex fractions to ordinary fractions before
adding? The validity of the formula for adding
fractions, k " + m n = kn "
an unknown number of pencils. He has three
other pencils. Let x be the number of pencils in
the pencil case. Then Joe has x + 3 pencils
altogether. Theresa has a box with least 5
pencils in it, and 5 are removed. We do not
know how many pencils there are
mathematics must respect this WYSIWYG
characteristic. 2g. Coherence Mathematics is
more learnable if there is continuity from topic
to topic, and from grade to grade. Consider the
addition of rational expressions, e.g., x x + 1 +
1 x 1 = x(x 1) + (x + 1)
power of x is x) x o = 1 1x = x 0x = 0 The
Improving Mathematics Education in Schools
(TIMES) Project cfw_9 SUBSTITUTION Assigning
values to a pronumeral is called substitution.
EXAMPLE If x = 4, what is the value of: a 5x b x
+ 3 c x 1 d x 2 SOLUTION a 5
can be an infinite number of x, yet to be
determined, that satisfy this equality. For this
reason, this x is traditionally called a variable,
or an unknown. It is important to realize that
the precise quantification of x in the meaning
of a quadratic equa