Objectives to be covered: 1.01a-c Develop number sense, define and use irrational numbers, compare and order real numbers, and estimate with rational/irrational numbers (includes square and cube roots). 3.01 Represent problem situations with geometric models. 3.02 Using geometric properties to include Pythagorean Theorem. 3.03 Identify, describe, and predict dilations in a coordinate plane. 4.01 Collect, organize, and analyze data to include scatter plots. **Students must also be able to differentiate between stratified, random, and systematic surveys** 5.01a-c Develop an understanding of a function. Translate among verbal and algebraic representations of functions Identify relations and functions as linear or non-linear Find, identify, and interpret intercepts or linear equations. Interpret and compare properties of linear functions using tables, graphs, and equations. 5.02 Write an equation of a linear equation given two points, one point and slope, or the slope and y-intercept.
Tuesday’s review – 1.01a-c Warm-up Maria has a square garden that she wants to divide in half diagonally using bricks that are 1 foot long. The area of the garden is 64 square feet. How many whole bricks would she need to make the division? (Obj. 3.02)
Changing fractions to decimals and decimals to fractions • When changing a fraction to a decimal, divide the top number by the bottom number. (i.e. ¾ is 3 divided by 4 in the calculator) • - When changing a decimal to a fraction, type in decimal as written and press “math, enter, enter” in the calculator. (*remember, if decimal is a repeating number, type number until cursor goes to next line on calculator*).
Ordering fractions and decimals • When ordering fractions, 1st change the fraction to a decimal • Once in decimal form, stack the decimals • Add ending zeros so that columns line up • Look at numbers from left to right and put in order (make sure you pay attention to greatest to least and least to greatest). • Example: ¼, ½, ¾, 2/3, 3/5 and 1/12 • ¼ = 0.25 • ½ = 0.50 • ¾ = 0.75 Least to Greatest: • 2/3 = 0.666666… 0.08333…, 0.25, 0.50, 0.60, 0.6666..., 0.75 • 3/5 = 0.6 • 1/12 = 0.083333…
You try one… 1/3, 5/3, √72, -4/7, and 2 1/3 = .33333… 5/3 = 1.66666… √72 = 8.485 -4/7 = -.571 2 = 2 -4/7, 1/3, 5/3, 2, √72
Greater than and less than with integers • When determining which number is greater, change all values to a decimal (i.e. ¾ would become 0.75, √63 would become 7.94, etc.) • Remember the “mouth” points towards the bigger value. Ex. 4 > 3.98 • Be careful of what is being asked – which is true vs. which is NOT true.
Rational Numbers • (including Integers, Whole Numbers) • Rational Numbers – any number that can be expressed as a ratio of two integers. • Ex. 6 = 6/1 • 0.5 = 5/10 = 1/2 • *This includes decimals that can be changed back to a fraction.* • Ex. 0.625 = 5/8 • 0.2222… = 2/9 • - Integers – all whole numbers and their opposites. • Ex. 1 and -1 • 576 and -576 • Whole Numbers – “counting numbers” – no fractions or decimals, and no negatives. • Ex. 0, 1, 2, 3, …
Irrational Numbers • A number that CANNOT be expressed as a ratio of two integers or as a repeating or terminating decimal. An irrational number, as a decimal, DOES NOT REPEAT AND DOES NOT END • Ex. .856498768454397860… • • √2
Determine if the following are rational or irrational: • 3/8 • 15 • 0.4555… • 1.838383… • 7.6532154… • 5/12 • 7. 4√9 • 8. 12 √55 • 8 √44 • 10. 5 √1 Rational Rational Rational Rational Irrational Rational Rational Irrational Irrational Rational
Squares and Square Roots Complete the following: Solve. Solve. Simplify. 1. 42 6. √4 11. √20 2. 92 7. √81 12. √400 3. 252 8. √225 13. √720 4. 102 9. √625 14. √32 5. 162 10. √10000 15. √289
Between which two integers do the following lay? • 1. √59 • 2. √780 • 3. √360 • - √540 • - √3450 • 6. √62 • 7. √90
Simplifying a radical √84 The 2’s will pair up and go outside the radical and the 7 and 3 will stay inside the radical. 42 2 • Factor the number down until you have all prime numbers. • If there is a pair put them outside the radical as written. • If there isn’t a pair, leave it inside the radical. • If there is more that one number, inside or outside the radical, multiply. 7 6 2√21 2 3 **When in doubt, solve the radical using the calculator and solve all the answer choices – see which decimals match.**
Cubes and Cube Roots • Complete the following: • Solve. • 53 • 2. 73 • 3. 43 • 4. 93 • 5. 123 • Find the cube roots of the following: • 6. 64 9. 125 • 7. 8 10. 1 • 8. 27 11. 729
Absolute Value: • ** The distance a given number is from zero on the number line** • Ex. I-4I = 4 • I7I = 7 • 1. What is the absolute value of -9? • What is the absolute value of 15? • 3. I -4 + 2 I • 4. I – 25 +17 I • 5. I 26 + 5 I – I -30 + 5 I
Central Tendency • Mean – the average of the set of data • Median – the middle number in the set of data (if two numbers – find average) • Mode – the number that shows up the most • Range – the difference between the greatest and smallest values • Make sure to put the numbers in order from least to greatest first.
Reading a box-and-whisker plot 30 31 32 33 34 35 36 37 38 39 40 What is the median of this box and whisker? What is the range of this box and whisker?
Wednesday’s review – 3.01-3.03 Warm-up Mike is 6 feet tall and casts a 9 foot shadow. At the same time of day, what is the length of the shadow cast by a 2 foot fire hydrant? (Obj. 3.01)
Formulas you need to know: Area and Perimeter: Surface Area: Volume: CircleRectangular PrismPrisms & Cylinders 2r = Circumference SA = 2lw+2lh+2wh V = Bh r2 = Area Cube (B= area of base) Rectangle/square SA = 6s2Pyramids & Cones l * w = Area Cylinder V = 1/3Bh Triangle 2 r2 + 2 rh = SA (B=area of base) ½bh = Area PyramidCube: Trapezoid B + ½ pl V = s3 ½ h(b1 + b2) = Area (B= l*w; p = perimeter; l = slant) *** The angles of a triangle add up to a total of 180 degrees ***
Now you try some: 1. Find the area: Triangle h = 14m, b = 6m 2. Find the area: Triangle h = 4m, b = 2m 3. Find the area: Rectangle l = 8cm, w = 4cm 4. Find the area: Rectangle l = 7ft, w = 45 ft 5. Find the area: Circle diameter = 10cm 6. Find the area: Circle radius = 18ft 7. Find the circumference: r = 5m 8. Find the circumference: d = 28cm 9. Find the volume: Cube s = 8in 10. Find the volume: Rectangular prism l = 2ft, w = 6ft, h = 8ft 11. Find the volume: Cylinder r = 6 in, h = 10 in 12. Find the volume: Triangular pyramid b = 8m, h = 12m, height of pyramid = 50m 13. Find the volume: Cone d = 24ft, height of cone = 6ft
Changing dimensions • When changing dimensions remember the following: * Perimeter – what ever is done to the dimension is what happens to the perimeter. * Area – multiply what is done to one dimension by what is being done to the other dimension. * Volume – multiply what is being done to all three dimensions.
Examples… • If the radius of a circle is doubled, what will happen to its area? • If the length of a rectangle is tripled and the width is doubled, what affect will this have on the area? • If the radius and height of a cone are tripled, what will happen to the volume? • If the radius is multiplied by 3 and the height is doubled, how will this affect the volume of a cylinder?
Finding area using other information… My mom’s picture frame has sides that measure 49 inches. The wooden frame is 3 inches wide. The photograph is behind a copper matting that is 6 inches wide. What is the maximum area of the photograph?
Pythagorean Theorem a2 + b2 = c2 ** “a” and “b” are the “legs” of the right triangle ** “c” is the “hypotenuse” (the longest side in a right triangle) Using the calculator: *** When given sides “a” and “b” type the following in your calculator to find the value of “c”: √(a2 + b2) *** When given sides “a” and “c” or “b” and “c” type the following in your calculator to find the missing value: √(c2 – a2) or √(c2 – b2)
Let’s try a few: 1. 2. x 10 Find x: x 3 4 8 • A rectangle is 6 m wide and 11 m long. How long is the diagonal of the rectangle? • A rope is stretched from the top of a 7 foot tent pole to a point on the ground 12 ft from the base of the pole. How long is the rope? • The bases on a baseball diamond are 90 feet apart. How far is it from home plate to second base? • 6. The leg of a triangle measures 7 in. The hypotenuse measures 12 in. What is the length of the other leg? What is the area of the triangle (round answer to the nearest tenth)?
More examples of Pythagorean Theorem If George builds a ramp and places the support beams vertically every 2 ft, how long is the tallest board? 10ft 8ft
Dilations • When dilating an object, use the scale factor given and multiply all coordinates/side measurements of the object by given scale factor. • Ex. Given coordinates (2,3) (4,9) (5,7) and (2,8) • You have a scale factor of 3. The new coordinates would be: • (6,9) (12,27) (15, 21) and (6,24). • Given measurements of : • If there is a scale factor of ½ , what would the new measurements be? • 1.5, 2, and 2.5 • ***A scale factor greater than 1 will increase the size of the object. A scale factor equal to 1 will result in the same size object. A scale factor less than 1 will decrease the size of the object.*** 5 3 4
Other important formulas to know: • To find the sum of the angles of a polygon: • (n-2)180 where “n” is the number of angles within the polygon • - To find a each individual angle measurement of a regular polygon: • (n-2)180 where “n” is the number of angles within the polygon • n • - Simple interest: • I = Prt • (I = simple interest; P = principle/amount started with; r = interest rate; t = time in years) • Distance: • D = r * t
Given a formula, solve… • To find the minimum speed and maximum speed of a vehicle using skid marks, police use the following formulas: Minimum S=5.5√0.7L S = 5.5 √0.7(23) = 22mph Maximum S=5.5√0.8L S = 5.5√0.8(23) = 24mph Find the minimum and maximum speed if the skid marks are 23 feet long.
Working with Proportions: 1. John is 6 ft tall and casts a 10 ft shadow. His mailbox is 4 ft tall. At the same time of day, how long would the shadow be that the mailbox casts? 2. A map has a scale of 1 cm equals 50 kilometers. How many kilometers are represented by 8 cm on the map? 3. The school building is 20 feet high and casts a 40 ft shadow. A tree that is next to the school casts a 50 ft shadow. How tall is the tree? 4. One package of bubble gum costs $1.50. How many packages can you buy if you have $7?
Knowing when to use proportions instead of Pythagorean Theorem • If the question contains the word “similar”, regardless of the kind of triangle, use proportions. • If the question contains the word “shadow”, use proportions. • If the question contains the word “diagonal” use Pythagorean Theorem.
Examples 4 4 8 x 8 4 3 ? When a triangle is inside another triangle, even if they are both right triangles, use proportions because the triangles are “similar” to each other.
More examples… 5 3 8 4 X If two triangles are “connected” like above, they are similar and you use proportions to solve.
Continued… 8 X 6 3 If there are two right triangles, like above, use proportions because they are similar.
Finding the missing angle of a shape using properties of parallel lines… 7 6 5 60˚ 1 2 3 4 Find the measure of angle 3.
Thursday’s review – 4.01 Warm-up You have a principle of $500 on a loan. The interest you must pay on the loan is 4.5%. You want to pay off the loan within 3 years, how much money will you pay back total? A = P(1 + r)t (Obj. 1.02)
Scatter plots ~ When drawing a scatter plot, place the independent variable on the X axis and the dependent variable on the Y axis. ~ There are three types of correlation: positive, negative and no correlation. ~ Positive correlation – as one data set increases, the other data set increases. When the data is graphed, there is an upward trend from left to right. ~ Negative correlation – as one data set increases, the other data set decreases, or vice versa. When the data is graphed, there is a downward trend to the right. ~ No correlation – one data set has no effect on the other. When graphed, the data is scattered all over the graph – there is no pattern.
Making a prediction given a graph or scatterplot • When given a graph or scatterplot and asked to make a prediction, draw a line of best fit (a line that comes closest to the majority of the points. • Find the amount/ value given on the axis, and draw a straight line up to the line of best fit. • Then draw another straight line directly over to the other axis – this will make a rectangle.
Making a prediction given a table… Find how many segments you would have if there were 9 points.
Determining correlation using word problems: 1. The amount of time you spend studying for the 4.5 week assessments and the score you receive. 2. The distance from a destination and the time it takes to get there. 3. The outdoor temperature and hot chocolate sales. 4. Your height and your IQ. 5. Your age and the color of your hair.
Circle Graphs 2 1 What percent of the circle is section 1? Section 2? 12.5% and 25%
Surveys There are three types of surveys: random, systematic, and stratified. ~ Random samples – A sample in which each individual or object in the entire population has an equal chance of being selected. ~ Systematic samples – A sample of a population that has been selected using a pattern or rule. ~ Stratified samples – A sample of a population that has been divided into subgroups (will usually contain the word “randomly” in the question/example).
Misleading information or graphs • If a graph does not begin at zero, it is misleading. • Make sure to read all answer choices – only one will make sense. • When surveys are taken the following have to be in place: • Need large enough sample. • Information needs to be taken at the same time of year and for the same amount of time. • The objects being compared need to be the same size.
What does biased mean? UNFAIR – why the sample taken would not be a good representation of the population. **Remember – population is the BIG picture and sample is WHO you are asking.**
Samples and Surveys: Identify the population and the sample. Tell why the sample may not be a good representation of the population. 1. The first ten people leaving the theater are asked to give their feedback about the movie. 2. The professional baseball league All-Star ballots are handed out at the stadium on game days. 3. At a convention of science teachers, various attendees are asked to name their favorite subject in high school. 4. Interviewers at the mall are surveying girls with red hair to find out if a correlation exists between personality and red hair. 5. A teacher polls all of the students who are in detention on Friday about their opinions on the amount of homework students should have each night.
Samples and Surveys • Identify the sampling method used. • 1. At a frozen pizza manufacturing plant, a coupon for a free pizza is put inside the package of every 100th pizza. • The teacher asks that students with birthdays from July to December go to the chalkboard to work the next problem. • 3. A surveyor opens the yellow pages and calls the last person on each page. • At a 1 mile marker of a marathon, a timekeeper shouts out the time elapsed to every 10th runner that passes by. A statistician records the times shouted. • Every twentieth student on a list is chosen to participate in a poll. • Seat numbers are drawn from a hat to identify passengers on an airplane that will be surveyed. • 7. A geologist visits 10 randomly selected lakes in the region and collects soil samples in randomly selected areas along each shoreline.
Friday’s Review – Obj. 5.01a-c Warm-up You are an employee at Kmart. You get 25% off all merchandise. There is a sale where everything is marked 75% off. How much will you pay for a sweater that has an original cost of $100? (Obj. 1.02)
Graphing Linear Equations * A linear equation is an equation whose solutions fall on a line on the coordinate plane. * If the equation is linear, a constant change in the “x” value corresponds to a constant change in the “y” value. This creates a straight line. * In order to determine the coordinates to plot, use a function table. (see next slide for example)
Now you try: y = x2 * Now use the (x, y) coordinates to plot your equation*