ILTS Math 110

Property of Reciprocals

any number (except for zero) multiplied by its reciprocal gives 1. (the reciprocal of a number is 1 divided by that number)

Important step to measurement

decide which area you are in, length angles, volume, mass, time, money, and temperature.

Perimeter

distance around an object

Careful use of mathematical terms and ideas is

essential to communicating mathematically.

Parallel and perpendicular

key concepts in geometry.

Problem solving strategies

Act it out.

Addends (or addenda)

"parts of addition problems" When addends are combined they produce sums. Likewise, factors can be seen as "parts of multiplication problems." When factors are multiplied they produce products. When two numbers are divided, one into another, the result is a quotient.

Counting numbers

(1,2,3,4), zero is not a counting number,

Customary to metric ratios (values are approximate)

1 inch= 2.54 centimeters, 1 yard=.91 meters, 1 mile= 1.61 kilometers, 1 ounce=28.35 grams. 1 pound=2.2kilograms, 1 quart=.94 liters.

Problem-Solving Strategies

1. Guess and Check, with this problem solving strategy , make your best guess, and then check the answer to see whether it's right. Even if the guess doesn't immediately provide the solution, it may help to get you closer to it so that you can continue to work on it. An example, 3 persons age add up to 72, and e ach person is one year older than the last person What are their ages?

Problem solving strategies

1. Make a sketch or a picture, can help you clarify a problem. Consider this problem, Mr. Rosenberg plans to put a four-foot concrete sidewalk around his backyard pool. The pool is rectangular, with dimensions of 12 by 24. The cost of the concrete is 1.28 per square foot. How much concrete is require for the job. Make a rectangle around the pool with 12 x 24 x 4. Solve

Problem solving strategies

3. Make a table or a chart

Problem solving strategies

4. Making a list, like making a table or chart, can help to organize information and perhaps provide or at least hint at a solution. Consider this problem "How many different outcomes are there if you roll two regular six-sided dice?

Problem solving strat

7. Working a simpler problem. If you need to know the product of 23 and 184, you stimate 20x200.

Problem solving strat

8. Writing an open math sentence, sometimes called translating a problem into mathematics. Consider this problem, Tianna earned 77,86,90,83 on her first four weekly science quizzes. Assuming all grades are equally weighted what score will she need on the fifth week's quiz in order to average a score of 88. You could set up the problem like this 77+86+90+83+X/5=88. Solve for X.

Problem solving strat

9. Consider working backward. If you add 12 to some number then multiply the sum by 4, you will get 60. Start with the end number.

Some pairs of operations are considered to be inverse

Addition and subtraction are inverse operations, as are multiplication and division. The operations can be thought of as "undoing" one another, multiplying 4 by 9 gives 36; diving 36 by 9 gives back 4.

Using X in a real problem

Conside the ages of two sisters, if you don't know the age of the younger sister, but know the older sister is three years older, you can show the information symbolically as follows, the age of the younger sister can be shown as x, and the age of the older sister as x + 3. If you are told that the sum of the sisters' age is say 25, you can represent the information via an equation,

Factoring can be useful especially when the equation is set to 0

Consider 2x to the 2nd power -x-1=2. It can be rewritten as 2x tot eh 2nd power-x-3=0 this allows the left to be factored into (2x-3)(x+1), giving equation solutions of 3/2 and -1.

FOIL method

FOIL stands for "first, outer, inner, last," Multiply the first terms in the parentheses, then the outermost terms, then the innermost terms, then the last terms, and then add the products together. For example, to multiply (x+3) and (2x-5), you multiply x by 2x (the first terms), x by -5 (the outer terms), 3 by 2x (inner terms) and 3 by 5 (last terms).The four products are 2x to the 2nd power, -5x,6x, and -15 adds up to 2x to the 2nd power +x-15.

Multiplying exponent terms together, the constant terms are multiplied but the exponents of terms with the same variable bases are added together, which is somewhat counterintuitive

For example 4w to the 2nd power multiplied by 8w to the 3rd power gives 32w to the 5th power (not 32 w to the 6th power as one might guess).

When like algebraic terms are divided, exponents are subtracted

For example, 2x to the 7th power divided by 5x to the 3rd power becomes 2x to the 4th power/5.

Equations are not the same as mathematical expressions

L 12+4=16 and 2x+7+12 are equations. (144-18) and 13y to the second power are expressions. Notice that expressions are "lacking a verb," so to speak (you don't say "is equal to" or "equals" when reading expressions. Inequalities are much like quations, but "greater than" r "less than" are added, such as in x<7.

Problem solving strats

Look for patterns. This technique encourages you to ask, "what's happening here?" Spotting a pattern would be helpful in solving a problem such as, Nevin's weekly savings account balance for 15 weeks are as follows, $125, $135, $ 148, $72, $85, $96, $105, $50, $64, $74, $87, $42, $51, $60, $70, If the pattern holds, what might Nevin's balance be? Do an average.

Properties of Triangles

Sum of the measures of the three angles of any triangle is 180 degrees. If, therefore, the measures of two angles are known, the third can be deduced using addition, then subtraction.

Some numbers are real numbers and cannot be accurately represented by fractions

The ratio of length of the diameter of any circle to it circumference or pi for instance is irrational. There are useful approximations of pi such as 3.14159 but pi cannot be pinned down in either fraction or decimal notation.

Hundreds v. Hundreths

Three hundred means 300 wheras three hundredths means 0.03.

Translating a problem into an equation

Three teachers who are retiring are said to have 78 years of experience among them. You don't know how many years of experience Teacher A has, but you know that Teacher B has twice as many as A, and Teacher C has three more years of experience than B. How many years of experience does each have?

A trend is

a pattern over time

A point (fundamental concept in geometry)

a point is a specific location, taking up no space, having no area, and frequently represented by a dot. A point is considered one dimensional.

Mathematics is

a science of precision.

Decimals

aka decimal fractions, which come to an end when represented exactly, are terminating decimals. Repeating decimals are those in which the digits repeat a pattern endlessly (3.33333 for example) To us the shorthand notation to shoe repeating decimals, you can write the repeating block just once putting a bar over it.

Integers are

all of the qhole numbers and their negative counterparts (...-2,-1,0,1,2,....)Note that negative and positive fractions are not considered integers (unless they are equivalent to whole numbers or their negative counterparts).

Multiplicative Identity Property of One states

any number multiply Property of One states, any number multiplied b 1 remains the same. (34 x 1 =34, for instance). The number 1 is called the Multiplicative Identity.

Factors

are any of the numbers or symbols in mathematics that when ultiplied together form a product. (the whole number factors of 12 are 1,2,3,4,6,12). A number with exactly two whole number factors (1 and the number itself) is a prime number) the first few primes are 2,3,5,7,11,13, and 17/ Most other whole numbers are composite numbers, because they are composed of several whole number factors (1, is neither a prime or composite, it has only one whole number factor)

Customary units

are generally the same as US units. Customary units of length include inches, feet, yards, and miles. Customary units of weight include ounces, pounds, and tons. Customary units of capacity (or volume) include teaspoons, tablespoons, cups, pints, quarts, and gallons.

Two important coordinate systems

are the number line and the coordinate plane, and both systems can be used to solve certain problems. A particularly useful tool related to the coordinate plane is the distance formula, which allows you to compute the distance between any two points on the plane.

Binomials

are two algebraic expressions of two terms. The FOIL method is one way to multiply binomials.

Absolute value of a number

can be thought of as its distance from zero on a number line.

Operations with algebraic expression are governed by rules and conventions such as

for instance only like algebraic terms can be added or subtracted to produce similar expressions. For example, 2x to the 3rd power plus 3x to the third power can be added together to give you 5x to the third power, because the terms are like terms. They both have a base of x to the third power. You cannot say add 7m to the third power and 6m to the 2 power because m to the third power and m to the second power are unlike bases. (note, to evaluate an algebraic expression means to simplify it using conventional rules)

There is an agreed upon order of operations

for simplifying complex expressions (PEMDAS).

Metric units of weight include

grams and kilograms. The gram is the basic metric unit of mass. A large paper clip weighs about 1 gram. It takes about 28 grams to make 1 ounce. Metric units of capacity include milliliters and liters. The liter is the basic metric unit of volume (or capacity). A liter is slightly smaller than a quart, so it takes more than four liters to make a gallon.

Metric units

include millimeters, centimeters, and kilometers. The centimeter is the basic metric unit of length, at least for short distances. There are about 2.5 centimeters to 1 inch. The kilometer is a metric unit of length used for longer distances. It takes more than 1.5 km to make a mile. A very fast adult can run a km in about 3 minutes.

A polygon

is a closed plane figure bounded by straight lines or a closed figure on a sphere bounded by arcs of great circles. In a plane, three-sided polygons are triangles, four-sided polygons are quadrilaterals, five sides make pentagons, six sides are hexagons and eight-sided polygons are octagons (note that not all quadrilaterals are squares). If two polygons (or any figures) have exactly the same size and shape, they are congruent. If they are the same shape, but different sizes, they are similar.

Diameter of a circle

is a straight line segment that goes from one edge of a circle to the other side, passing through the center. The radius of circle is half of its diameter (from the center to an edge). A chord is any segment that goes from one spot on a circle to any other spot (all diameters are chords, but not all chords are diameters).

Exponential notation

is a way to show repeated multiplication more simply. 2 x 2 x2 for instance can be shown as 2 to the 3rd power and is equal to 8.

A segment

is any portion of a line between two point on the line. It has a definite start and a definite end. The notation for a segment extending from point A to point B is AB (line should be over AB). A ray is like a straight segment, except it extends on forever, so one end of the line has an arrow.

Property of real numbers

is called the density property. It states that given any two real numbers there is always another real number between them. (think of the number line where no matter how close two points are, there is always a point in between them)

Distributive property of multiplication over addition

is shown hereafter in simple notation form, a(b+c)=(axb)+(aXc)

Ratio notation

is simple an alternative method for showing fractions. For example, 2/5 can be rewritten as "2/5."

Factoring polynomials

it is sometimes necessary to factor out any factor that might be common to all terms first. The two terms in 5x to the 2nd power-10 for example both contain the factor 5. This means that the expression can be rewritten as 5(xto the 2nd power-2)

All illustration of the distributive property is this

multiplying6 by 47 will give the same result as multiply 6 by 40 will give the same resulting as multiplying 6 by 40,, multiplying 6 times 7 then adding the products. That is 6 x (47) = (6x 40)+(6x7)

Rational numbers

numbers that can be written as fractions (this include integers 12 for instance can be written as 12/1.

Scientific Notation

provides a method for showing numbers using exponents. Thus the number of 75000 in scientific notation is 7.5 x 10 to the 4 power.

Volume

refers to how much space is inside three dimensional, closed containers. It is useful to think of volume as how many cubic units could fits into a solid. If the container is a rectangular solid, multiplying width, length, and height together computes the volume. If all six faces (sides) of a rectangular solid are squares, then the object is a cube.

Factoring a polynomial means

rewriting it as he product of factors (often two binomials). The trinomial x to the 2nd power-11x+28 for instance can be factored into (x-4)(x-7). You can check your work by foiling the binomials.

Variable is

simply a symbol that represents an unknown value. Most typically x is the letter used, although any letter can be used. By "translating" real problems to algebraic form containing one or more variables (often as equations or inequalities), solutions to many problems can be found mathematically.

Decimal numbers

simply certain fractions written in special notation. All decimal numbers are actually fractions whose demonimators are powers of 10 (10, 100, 1000) 0.033 for instance can be though of as he fraction 33/1000.

Pythagorean Theorem

states that in any right triangle with legs (shorter sides) a and b, and hypotenuse (longest side) c, the sum of the squares of the sides will be equal to the square of the hypotenuse. In algebraic notation the Pythagorean theorem is given as a to the 2nd power + b to the second power=c to the second power.

Measurement division

the number of groups is not known. Using the example above, if you knew that there were 14 bolts per container, and that there were 98 bolts altogether, finding the number of containers would require measurement division.

The commutative property of addition and multiplication states

the order in which addends are added or factors are multiplied does not determine the sum or product. (6x9 gives the same product as 9x6) Division and subtraction are not commutative.

translation tips

the word "is" often suggests an equal sign; "of" may suggest multiplication, as does product. "sum" refers to addition; "difference" suggests subtraction; and a quotient is obtained after dividing. The key when translating is to make sure that the equation matches the information given in the word problem.

Negative numbers

those less than zero. Fractions less than zero are negative too.

Angle

two rays share an endpoint. A degree is a unit of measure of the angle created. If a circle has 360 even slices, each slice has an angle measure of 1 degree. If an angle has 90 degrees it is called a right angle. Angles of less than 90 degrees are call acute angles. Angles greater than 90 degrees are obtuse angles. If two angles have the same size, they are congruent. Congruence is shown this way <m=<n (read angle m is congruent to angle n)

Division is partitive

when you know the total and the number of parts or groups but you don't know ho many are in each part. Consider,"You have 7 containers of bolts and a total of 98 bolts. How many bolts are in each container (assuming the same number in each) Arriving at the answer is an example of a partitive division.

X (teacher a), 2x (teacher b), 2x+3 (teacher c) translates as

x+2x+(2x+3)=78