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Algebra

Math Review of Graphing Compound Inequalities

358 120 Deborah

Overview

Inequalities can be graphed along the number line by solving the inequality and then graphing it.  Conjunctions are true when both statements of an inequality are true.  Disjunctions are true when either one or both statements of an inequality are true.  Both conjunctions and disjunctions can form the basis for truth tables.

Review of Inequalities and the Number Line

Inequalities are expressed by relationships between numbers that are less than <, greater than>, less than or equal to≤ or greater than or equal to≥.  When just one variable is used, the sentence can be represented on the number line.  Suppose that a student wanted to show numbers greater than or equal to -40 on the number line.  That student might write a sentence such as x is ≥ -40.  Notice that the circle at -40 is filled in as the number -40 makes the inequality a true statement.

Conjunctions

A conjunction is a set of two statements joined by the word “and”, so that both statements must be true.  In other words, the points on a number line that are solutions of both inequalities are the solution set.  For example, suppose that one inequality is x ≥4 and another inequality is x> 3 +6.  The numbers that will be in common are the points larger than 9, but not including 9.  Although 9 is greater than 4, it is not included in the second inequality statement.

Disjunctions

A disjunction is a set of two statements joined by the word “or”, so that both statements could be true, or only one statement could be true.  Suppose one sentence is x> 3 and the other sentence is x ≤ 0.  The points that make that statement true are either those that are greater than 3 or those that are less than or equal to 0.

Truth Tables

Truth tables are another way of organizing statements, and are part of logic and a form of math called discrete math.  In a conjunction, a statement is true only if both statements are true.  For example, the statement “two is a prime number and three is an odd number“ is true because both parts of the statement is true.  However, the statement “two is an odd number and three is a prime number” is false because the first statement is false.  Similarly, “Four is an even number and six is an odd number” is false because the second statement is false.  “Seven is an even number and four is an odd number” is false because both statements are false.

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Math Review of Digit and Coin Problems

240 180 Deborah

Overview

Some very common word problems are problems involving digits and problems involving coins.  Like motion problems, these word problems can also be solved using equations in two variables.

Setting up a Digit Problem

Any two-digit number can be written as 10x +y where x is the number in the tens column and y is the number in the ones column.  Think of expanded notation where a number like 34 can be expressed as 3 times 10 plus 4.  The reverse, 43, can be expressed as 4 times 10 plus 3.  Common types of digit problems have the sum of the digits equal to a certain number.  For example, both 3 + 4 and 4 +3 equal 7.  However, if the number 34 is reversed to 43, it is 9 more.

Solving a Digit Problem

Suppose the sum of the digits of a two-digit number is 5.  If the digits are reversed, the new number is 27 more than the original number.  To set up the problem, find the first equation, x +y =5.  If the digits are reversed, 10y +x = 27 +10x +y.  Simplify the second equation, so that all the x’s and y’s are on one side of the equation, as 9x-9y =-27.  Then there are 2 equations, x +y = 5 and 9x-9y =-27.  9(x +y) = 5(9), or 9x +9y = 45.  They can then be solved using the addition method, so that 9x +9y =45 + 9x-9y =-27.  9x +9x = 18x, 9y-9y cancels out as zero, and 45-27 is 18.  18x = 18 means that x equals 1.  If x equals 1, then 1 +y = 5, or y equals 4.  The original number is 14, and 14 +27 is 41, so the problem checks.

Setting up a Coin Problem

Coin problems are similar to digit problems.  There are different types of coins and a total, and that relationship can be expressed with one equation.  There are also a number more for one type of coins than another type, and that relationship can be expressed with a second equation.  The system of equations can then be solved.

Solving a Coin Problem

Suppose there are 20 coins, some quarters and some dimes.  The value of all 20 coins is $3.05.  How many quarters and how many dimes are there?  Let q equal the number of quarters and d equal the number of dimes.  The first equation is q +d = 20.  The second equation is 25q +10d = 305.  Each quarter is 25 cents and each dime is 10 cents.  The number of cents total in $3.05 is 305.  The number of dimes can also be expressed in a slightly different form, as d = 20-q.  Using the substitution method, 25q +10(20-q) = 305, which can be solved as 25q +200-10q =305, rearranged as (25q-10q) or 15q = 305-200 or 105.  If 15q =105, then q equals 7.  The number of dimes is 20-7 or 13.  To check, 25(7) or 175 plus 10(13) or 130, equals 305.

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Math Review of Uniform Motion Problems

150 150 Deborah

Overview

Problem-solving strategies can also be used to solve uniform motion problems that use the relationship between distance, rate, and time.  Students can first understand the problem, develop and carry out a plan, find the answer, and then check the answer.

Understand the Problem

Suppose one car leaves Calgary, Alberta, Canada traveling north at 56 kilometers per hour.  Another car leaves Calgary one hour later, on the same road, but is traveling at a speed of 84 kilometers per hour.  How far will they be from Calgary when the faster car meets the first?  This is a uniform motion problem, using the equation distance (d) = rate (r) x time (t).   In Canada, the rate of speed is measured by using the metric system, or kilometers per hour.  The question is:  How far will the cars be from Calgary when they meet?  The data; the fast car leaves 1 hour after the slow car; the slow car travels at 56 km/h; the fast car travels at 84 km/h.  Both cars start at the same place, are traveling the same distance, and going in the same direction, so that information is not relevant to the problem.

Calgary

 

Develop a Plan

Draw a diagram using the data.  The distances for both the fast car and the slow car are the same, so in this case d represents the distance traveled by both cars when they meet.  If t is the time for the fast car, then t +1 is the time for the slow car.  Therefore, there are two equations, d = 56(t +1) and d = 84t.

Find the Answer

Since there are two equivalent equations, the substitution method can be used such that 56t +56 = 84t.  Simplifying, 56 = 84t-56t = 28t, 56/28 = t, t =2, or in about 2 hours.  Now that the time is known, the distance in kilometers can be calculated using either equation. Using the second equation for the faster car, the distance is 84(2) = 168 km, another 30 km or so past Red Deer.

Check the Answer

In order to check the answer, check to see if the equations make sense. 168 is equal to 56(3), or in other words, 168=56(1 +2), and 84 (2) = 168.  Incidentally, 56 km/h is almost 35 miles an hour and 84 km/h is about 52 miles an hour.

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Math Review of Problem Solving with Systems of Equations

150 150 Deborah

Overview

Systems of two equations with two variables can also be used to solve problems.  In order to solve the problem, it can be translated to a system of equations.  Once the problem is understood, students can make a plan, find the answer, and check to make sure it is correct.

Understand the Problem

In order to understand the word problem, read it very carefully and note the questions asked, the data given.  Pay attention to any word cues that indicate mathematical relationships.  Suppose a basketball team played 180 games and they won 40 more games than they lost.  How many games did the team lose?  In that case, the team played 180 games total.  40 more games were won than lost.

Make a Plan

In making a plan, write a system of equations to fit the problem.  Let x be the number of games won and y be the number of games lost.  In the first equation, x +y = 180.  In the second equation, x –y =40.  Those equations can be combined to find the answer.

Find the Answer

This system of equations can be solved by the addition method, as x +y = 180 and x-y = 40.  Therefore 2x +(y-y) = 180 +40, or 2x= 220.  Solving for x, divide both sides by 2, so x =110.  Solving for y, 180 -110 = 70.

Check the Work

To see if both sentences are true, 110 +70 equals 180, and 110-70 equals 40. The same sort of process can be used to solve equations by choosing the substitution method.  Suppose that Matilde is 13 years older than Ana.  In 9 years, Matilde will be twice as old as Ana.  Let x be Matilde’s age and y be Ana’s age.  There are two equations in the plan, x =y +13, and x +9 =2(y +9).   Using substitution, x = y + 13, and x = 2y +9, so y +13 =2y +9, so y = 13-9, or 4.  Ana is 4, and Matilde is 17.  In 9 years Matilde will be 26, and Ana will be 4+9 or 13. Matilde is twice Ana’s age in 9 years, and the answers check.

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Math Review of Solving Systems by Addition

150 150 Deborah

Overview

Besides solving systems of equations by graphing and substitution, systems of equations can also be solved by addition.  Math students can choose the best method for the problem at hand.  Sometimes this process is called “solving systems of equations by elimination”.

Using Addition

Although some systems of equations can be solved by substitution, other systems can be solved by adding both equations. Both equations must be written in standard form as Ax +By =C.  For example, suppose the equations in the system are x +y = 5 and 2x –y = 4.  They can be expressed as x +2x +y –y = 5 +4.  Now x +2x equals 3x, y-y = 0, and 5 +4 = 9.  The new equation is 3x +0 = 9, or x =3.  If 3 +y = 5, then y equals 2.  Checking the solution in the second equation, 2x or 6 -2 does equal 4.  The addition method can be used because the addition property for equations states that we can add the same number to both sides of the equation, and the equations are still equivalent expressions, and make a true statement.

Using Multiplication, Then Addition

The multiplication property of equations is an extension of the addition property, since multiplication is repeated addition.  Therefore, we can multiply each side of the equation by the same number or expression.  This is especially useful to eliminate a variable, before using the addition method.  For example, suppose the equations in a system were 5x +3y = 17 and 5x-2y =-3.  If they were to be added without multiplying, the new equation would be 5x +5x +3y – 2y = 17-3, or 10x-y =14.  That is not closer to a solution, as there are still 2 variables in the system.  Suppose both sides of the second equation are multiplied by -1, so that the new equation is -1(5x -2y) = -1(-3), or -5x +2y = 3.  Using the addition method, 5x – 5x +3y +2y = 17 +3, or 5y =20, or y =4.  If 5x +12 = 17, then 5x = 5, or x =1.  Using the second equation to check, 5-8 = -3.

 

Using Multiplication More than Once

Sometimes the multiplication property needs to be used more than once in order to use the addition method.  Suppose the system of equations is 5x +3y = 2 and 3x +5y =-2.  Using the multiplication property once, 5(5x +3y) = 5(2) = 25x +15y = 10.  The second equation can be multiplied by -3, so that    -3(3x +5y) = -3(-2), or -9x -15y =6.  Then the addition method can be used, so that 25x -9x +15y -15y = 10+6 or 16x = 16, or x =1.  Solving for y, 3y = 2-5, or -3, so y= -1.  Checking the second equation, 3 (1) + 5 (-1) = -2.  The solution is (1, -1).

 

 

Problem-Solving Using the Addition Method

Suppose that the problem were to translate a word problem to a system of equations, then solve.  The sum of two numbers is 115, and their difference is 21.  Understanding the problem, the first equation is x +y = 115, and the second equation is x-y = 21.  Using the addition method, x + x +y-y = 115 +21, or 2x =136, or x = 68.  Substituting for x, 68 +y = 115, or y = 47.  Checking with the second equation, 68-47 = 21.

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Math Review of Solving Systems by Substitution

150 150 Deborah

Overview

One of the ways to solve systems of equations is by graphing the equations.  However, graphing the equations is not always the most accurate method to solve them.  If one variable in a system is represented in terms of the other variable in the system, the systems can be solved by substitution.

Using Substitution

Suppose one of the equations in the system is x + y = 5 and the other equation is x = y +1.  The expression y +1 can be substituted for x, so that y +1 +y =5.  Then, there is just one variable so that 2y +1 =5, 2y +1 -1 = 5-1, or 2y = 4, or y =2.  In order to check, substitute the value of y to solve for x, such that x +2 = 5, or x +2-2 = 5-2, or x = 3.  Check the second equation also, so that 3 =2 +1. That is the way to use substitution to solve a system of equations.

Isolating the Variables

Sometimes, the variables cannot be isolated as easily in a system of equations, but the system of substitution can still be used.  Suppose the equations were x-2y = 8 and 2x +y = 8.  The first equation can be rearranged such that x = 8 +2y.  Using substitution, the second equation then becomes 2(8 +2y) +y =8, or 16 +4y +y =8.  As before, there is only one variable, such that 5y = 8-16 or 5y=-8, or y = -8/5.  Again, check the value of x, so that x – (2)(-8/5) =8, or      x +16/5 =8.  (Notice how the sign changes when two negative values are multiplied.) Then multiply both sides by 5, so that 5x +16 = 40, or 5x =24 or x = 24/5.  To check the first equation, 24/5 – 2[-8/5] equals 24/5 +16/5 = 40/5, or 8.  To check the second equation 2 (24/5) – (8/5) = 48/5 – 8/5) = 40/5 = 8.

 

Understanding the Problem and Developing a Plan

Math problems that are written in words can often be translated into systems of equations, then solved by using substitution.  Suppose the statement were “The sum of two numbers is 82.  One number is 12 more than the other.  What is the larger number?”  The first sentence can be represented by the equation x +y = 82.  The second sentence can be represented by the equation x=12 +y.

Problem-Solving:  Solving the Problem and Checking the Answer

To solve the problem, take the system of equations and use substitution, so that 12 +y +y = 82, then 2y = 82-12, or 2y = 70, then y = 70/2, or 35.  Using the second equation to solve for x, 12 +35 = 47, and using the first equation, 47 +35 = 82.

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Math and Physics Review of Curling and Bobsled

150 150 Deborah
Overview

Curling and bobsled are winter sports that rely on friction against an icy track or surface. In curling, a special stone is moved by a combination of momentum and friction against a sheet of ice. Team players use a variety of strategies to determine which of their stones will score the highest. In bobsledding and related sports, two and 4 person teams gain maximum velocity at the start, steering their sleds to minimize the effects of gravity, wind resistance, and drag.

Curling

The sport of curling probably originated in Scotland in medieval times. Weavers used their large granite stones from warp beams to skim across the ice. They built special shallow ponds and used frozen rivers. The object of the game is to move the stone down the pond to a target, called “home.” Once the team member pushes off the block to give momentum to the stone, it is not kicked or thrown. The team captain, or “skip”, guides the stone with one broom, while two other players sweep the ice back and forth in front of the stone with special brooms, generating enough heat from friction to melt a thin film of water for the stone to glide upon. If stones collide, they exchange momentum.

curling

Equipment for Curling

The stones weigh around 45 pounds, and are made from a type of granite that resists water, so any melting ice becomes the glide path. Absorbed water would slow the stone’s movement. At one time, curling brooms were made of corn husks, but curling brooms used today are made from synthetic materials to stand up to the rapid sweeping action across the ice.


curling-stone

Bobsledding

The winter sport of bobsledding calls for two or four person teams. They push an aerodynamically-designed sled down a 50 meter start course, jump into the sled, keeping it steady and straight, and careen down the course against the force of gravity. The force of gravity can reach as much as 5G, similar to the forces on fighter pilots.

bobsled

Minimizing Drag and Air Resistance

After the bobsled team pushes off the sled, no further acceleration is possible, as the vehicles are not motorized. The smooth design of the bobsled, as well as the rubber surface of the suits competitors wear, are designed to minimize the amount of drag and air resistance that would slow them down.

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Math Review of Representing Solid Figures

150 150 Deborah
Overview

Three-dimensional solid figures can be represented by the two-dimensional pattern of polygons that create them. The pattern, called a net, is a visual representation that illustrates the formula for the surface area of the three-dimensional figure. If the net were folded, it would produce that figure.

Representation of Cylinders

The net for a cylinder consists of two circles adjoining a rectangle. This relates to the formula for the surface area of a cylinder; 2 πr2+ 2πrh. The 2 πr2 is the formula for the area of the two circles. In order for the rectangle to fit around the circumference of the circle, the width of the rectangle is 2πr. The height of the rectangle is the height of the cylinder. Since the formula for the area of a rectangle is length multiplied by width, the area of the rectangle will have the measurement 2πrh.

 

Representation of Rectangular Prisms

The net for a cube consists of the six square faces that make up the cube. A square has the same length, width, and height. It is a special type of a rectangular prism, also known as a rectangular cuboid. A rectangular prism also has 6 faces, in three parallel pairs that meet at right angles. The top and bottom faces are congruent, as are the two other pairs of opposite sides. The formula for the surface area of a rectangular prism is 2(lw +wh +lh), which the net illustrates perfectly.

Representations of Pyramids

Pyramids are solid figures with triangular faces that meet at a single point called an apex, and a polygon base. A tetrahedron is a special type of pyramid with 4 triangular faces, and a regular tetrahedron, with all triangles equilateral and congruent, is a Platonic solid. Another type of pyramid has a square or rectangular base and three triangular sides. The net that illustrates the pyramids has the base bounded by triangles on each side. A regular tetrahedron has a net with all four triangles inside a larger triangle.

Representations of Other Figures

Many other solid figures can be represented by their nets. For example, a cone with a circular base is represented by the circular base adjoined by a quarter circle. Solid figures have been extrapolated into more than three dimensions. A tesseract is a four-dimensional figure with three-dimensional faces. It has been used in surreal art, science fiction, music, and popular culture.

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Math Review of Graphing Systems of Equations

150 150 Deborah

Overview

One of the ways to solve systems of equations is by graphing the equations on the same coordinate plane.  By graphing the equations, it is possible to tell whether they have no solutions in common, one solution in common, or an infinite number of solutions in common.

No Solutions in Common

These linear equations are also known as parallel lines, those with the same slope and different y-intercepts.  Another way to describe them is that the solutions of that particular system are inconsistent.  For example, suppose that the equations are y = 3x – 1 and 2y = 6x +4.  Solving the second equation, y = 3x +2.    Both equations have the same slope, but their y-intercepts are different, so they are parallel.

One Solution in Common

Some of these linear equations are perpendicular lines, where the product of their slopes is equal to -1, but lines can also meet at other angles and still have one solution in common.  A system of equations that has at least one solution in common is consistent.  Both equations have one point in common, although it is the only solution of the system.

Identifying Solutions

One way to identify if a particular point is a solution of both equations in a system is to see if its coordinates solve both equations.  For example, check to see if a point with the coordinates (1, 2) is a solution of the system y= x +1 and 2x +y = 4.  The point is a solution of the equation y = x +1, because 2 = 1 +1, and it is also a solution of the equation 2x +y = 4, because 2 +2 = 4.  It is a solution of that system of equations.  A point with the coordinates (5, 6) is a solution of the equation y=x +1, but is it a solution of the equation 2x +y = 4?  10 +6 is equal to 16, which is not equal to 4.  The point (5, 6) is not a solution of that system of equations.

All Solutions in Common

Some systems of equations have all solutions in common, so that any solution of one equation is also a solution of the other equation.  The lines coincide along the same graph.  They are both consistent and dependent.  Suppose the system of equations is x +y =9 and 3x +3y =27.  The simplest form of 3x +3y = 27 is x +y =9, just by dividing every member of the equation 3x +3y =27 by 3.

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Math Review of Parallel and Perpendicular Lines

150 150 Deborah

Overview

Parallel lines never intersect when they are graphed on the same plane, while perpendicular lines are lines that intersect at one point at right angles to each other.  Their linear equations have special relationships.

Parallel Lines

Parallel lines are lines in the same plane that have no points in common.  Suppose that one line has the equation y = 2x.  In slope-intercept form, its slope would be 2 and the y-intercept would be 0.  Suppose that another line in the same plane has the equation y = 2x +4.  In that case, its slope is still 2 but the y-intercept is 4.  Those lines would have no points in common, because there isn’t any point that would be a solution of both equations.  Therefore the lines would not intersect, and they are parallel.

Solving Equations for Parallel Lines

In the example of y=2x and y=2x +4, both lines have the same slope, 2, and the y-intercepts are different.  Both equations are already solved for y.  Given pairs of equations, they can both be put in slope-intercept form and solved for y to determine the slope and the y-intercept.  If the slopes of the lines are equal and the y-intercepts are not the same, the lines are parallel.  Suppose the equations for the lines are y = -3x +4 and 6x +2y = -10.  Are those lines parallel?  The slope of the line y = -3x +4 is already -3 and the y intercept is +4.  Solving the second equation for y takes place in 2 steps, because 2y = -6x -10, moving the 6x, so y equals (-6/2) x – (10/2), or -3x -5.  The slope of both lines is -3 but the y-intercepts are different, so they are parallel.

Perpendicular Lines

Perpendicular lines are lines that are in the same plane that intersect at one point, forming a 90° angle (a right angle).  Slopes that have a product of -1 are perpendicular. Suppose a line has the equation y = 2x -3 and another line has the equation y = ( -1/2) x -4.  The product of the slopes, 2(-1/2) is -1, so they are perpendicular.

Solving Equations for Perpendicular Lines

In order to determine of two equations are for perpendicular lines, solve for y and determine the product of the slopes.  Suppose the equations are 3y = 9x +3 and 6y +2x =6 are perpendicular.  Solving for y, 3y=9x +3 can be simplified to y = 3x +1 by dividing both sides by 3.  Solving for y, 6y = -2x +6, or y = (-1/3) x +1.  The product of the slopes, 3 (-1/3) = -1.

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