Some Manipulatives
Calculator
Calculators will be required for use on Math A and B assessments. Scientific calculators are required for the Math A Regents examinations. Graphing calculators that do not allow for symbolic manipulation will be required for the Math B Regents examination and will be permitted (not required) for the Math A Regents examination starting in June 2000.
Note
The Math A exam may include any given topic listed in the Core Curriculum with any performance indicator. The content includes most of the topics in the present Course I and a selection of topics from Course II. Programs other than Course I and II could be used as long as all the performance indicators and topics in the curriculum are part of the program. Examples of assessment items for Math A have been provided for most performance indicators. The items were taken from the 1997 pilot test and 1998 Test Sampler. Suggestions for classroom activities are substituted for any performance indicator that was not included in the sample test.
The Math B exam may include any given topic listed in the Core Curriculum with any performance indicator. Programs other than Course II and III could be used as long as the performance indicators and topics mentioned are part of the program. Since there is no Math B exam at this time, no assessment items have been included for Math B. Suggestions for possible classroom activities or problems are given instead to provide clarification of most performance indicators.
Math A (to the top)
Key Idea 1
Mathematical Reasoning
Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 1A. Construct valid arguments. |
|
See Assessment Example 1A. |
| 1B. Follow and judge the validity of arguments. |
|
See Assessment Example 1B. |
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 2A. Understand and use rational and irrational numbers. |
|
See Assessment Example 2A. |
| 2B. Recognize the order of real numbers. |
|
See Assessment Example 2B. |
| 2C. Apply the properties of real numbers to various subsets of numbers. |
|
See Classroom Idea 2C. |
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 3A. Use addition, subtraction, multiplication, division, and exponentiation with real numbers and algebraic expressions. |
|
See Assessment Example 3A. |
| 3B. Use integral exponents on integers and algebraic expressions. |
|
See Assessment Example 3B. |
| 3C. Recognize and identify symmetry and transformations on figures. |
|
See Assessment Example 3C. |
| 3D. Use field properties to justify mathematical procedures. |
|
See Classroom Idea 3D. |
Key Idea 4
Modeling/Multiple Representation
Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 4A. Represent problem situations symbolically by using algebraic expressions, sequences, tree diagrams, geometric figures, and graphs. |
|
See Assessment Example 4A. |
| 4B. Justify the procedures for basic geometric constructions. |
|
See Classroom Idea 4B. |
| 4C. Use transformations in the coordinate plane. |
|
See Assessment Example 4C. |
| 4D. Develop and apply the concept of basic loci to compound loci. |
|
See Assessment Example 4D. |
| 4E. Model real-world problems with systems of equations and inequalities. |
|
See Assessment Example 4E. |
Key Idea 5
Measurement
Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 5A. Apply formulas to find measures such as length, area, volume, weight, time, and angle in real-world contexts. |
|
See Assessment Example 5A. |
| 5B. Choose and apply appropriate units and tools in measurement situations. |
|
See Classroom Idea 5B. |
| 5C. Use dimensional analysis techniques. |
|
See Assessment Example 5C. |
| 5D. Use statistical methods including the measures of central tendency to describe and compare data. |
|
See Assessment Example 5D. |
| 5E. Use trigonometry as a method to measure indirectly. |
|
See Assessment Example 5E. |
| 5F. Apply proportions to scale drawings and direct variation. |
|
See Assessment Example 5F. |
| 5G. Relate absolute value, distance between two points, and the slope of a line to the coordinate plane. |
|
See Assessment Example 5G. |
| 5H. Explain the role of error in measurement and its consequence on subsequent calculations. |
|
See Classroom Idea 5H. |
| 5I. Use geometric relationships in relevant measurement problems involving geometric concepts. |
|
See Assessment Example 5I. |
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 6A. Judge the reasonableness of results obtained from applications in algebra, geometry, trigonometry, probability, and statistics. |
|
See Classroom Idea 6A. |
| 6B. Use experimental and theoretical probability to represent and solve problems involving uncertainty. |
|
See Assessment Example 6B. |
| 6C. Use the concept of random variable in computing probabilities. |
|
See Assessment Example 6C. |
| 6D. Determine probabilities, using permutations and combinations. |
|
See Assessment Example 6D. |
Key Idea 7
Patterns/Functions
Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 7A. Represent and analyze functions, using verbal descriptions, tables, equations, and graphs. |
|
See Assessment Example 7A. |
| 7B. Apply linear and quadratic functions in the solution of problems. |
|
See Assessment Example 7B. |
| 7C. Translate among the verbal descriptions, tables, equations, and graphic forms of functions. |
|
See Assessment Example 7C. |
| 7D. Model real-world situations with the appropriate function. |
|
See Assessment Example 7D. |
| 7E. Apply axiomatic structure to algebra. |
|
See Assessment Example 7E. |
1A. (back to Key Idea table)
In a school of 320 students, 85 students are in the band, 200 students are on sports teams, and 60 students participate in both activities. How many students are involved in either band or sports?
Show how you arrived at your answer.
1B. (back to Key Idea table)
“If Mary and Tom are classmates, then they go to the same school.”
Which statement below is logically equivalent?
A. If Mary and Tom do not go to the same school, then they are not classmates.
B. If Mary and Tom are not classmates, then they do not go to the same school.
C. If Mary and Tom go to the same school, then they are classmates.
D. If Mary and Tom go to the same school, then they are not classmates.
2A. (back to Key Idea table)
A clothing store offers a 50% discount at the end of each week that an item remains unsold. Patrick wants to buy a shirt at the store and he says, “I’ve got a great idea! I’ll wait two weeks, have 100% off, and get it for free!” Explain to your friend Patrick why he is incorrect, and find the correct percent of discount on the original price of a shirt.
2B. (back to Key Idea table)
A. 1
B. 0
C. -1
D. 4
3A. (back to Key Idea table)
Mr. Cash bought d dollars worth of stock. During the first year, the value of the stock tripled. The next year, the value of the stock decreased by $1,200.
Part A
Write an expression in terms of d to represent the value of the stock after two years.
Part B
If an initial investment is $1,000, determine its value at the end of 2 years.
3B. (back to Key Idea table)
If 0.0154 is expressed in the form 1.54 x 10n, n is equal to
A. -2 C. 3
B. 2 D. -3
3C. (back to Key Idea table)
Pentagon RSTUV has coordinates R (1,4), S (5,0), T (3,-4), U (-1,-4), and V (-3,0).
• On graph paper, plot pentagon RSTUV.
• Draw the line of symmetry of pentagon RSTUV.
• Write the coordinates of a point on the line of symmetry.
4A. (back to Key Idea table)
A 10-foot ladder is placed against the side of a building as shown in Figure 1 below. The bottom of the ladder is 8 feet from the base of the building. In order to increase the reach of the ladder against the building, it is moved 4 feet closer to the base of the building as shown in Figure 2.
To the nearest foot, how much farther up the building does the ladder now reach?
Show how you arrived at your answer.
4C. (back to Key Idea table)
A design was constructed by using two rectangles ABDC and A’B’D’C’. Rectangle A’B’D’C’ is the result of a translation of rectangle ABDC. The table of translations is shown below. Find the coordinates of points B and D’.
4D. (back to Key Idea table)
The distance between points P and Q is 8 units. How many points are equidistant from P and Q and also 3 units from P?
A. 1 C. 0
B. 2 D. 4
4E. (back to Key Idea table)
Mary purchased 12 pens and 14 notebooks for $20. Carlos bought 7 pens and 4 notebooks for $7.50. Find the price of one pen and the price of one notebook, algebraically.
5A. (back to Key Idea table)
Ms. Brown plans to carpet part of her living room floor. The living room floor is a square 20 feet by 20 feet. She wants to carpet a quarter-circle as shown below.
Find, to the nearest square foot, what part of the floor will remain uncarpeted.
Show how you arrived at your answer.
5C. (back to Key Idea table)
Jed bought a generator that will run for 2 hours on a liter of gas. The gas tank on the generator is a rectangular prism with dimensions 20 centimeters by 15 centimeters by 10 centimeters as shown below.
If Jed fills the tank with gas, how long will the generator run?
Show how you arrived at your answer.
5D. (back to Key Idea table)
On his first 5 biology tests, Bob received the following scores: 72, 86, 92, 63, and 77. What test score must Bob earn on his sixth test so that his average (mean) for all six tests will be 80% ?
Show how you arrived at your answer.
5E. (back to Key Idea table)
The tailgate of a truck is 2 feet above the ground. The incline of a ramp used for loading the truck is 11o, as shown.
Find, to the nearest tenth of a foot, the length of the ramp.
5F (back to Key Idea table).
Joan has two square garden plots. The ratio of the lengths of the sides of the two squares is 2:3. What is the ratio of their areas?
A. 2:3
B. 3:2
C. 4:9
D. 9:4
5G (back to Key Idea table).
What is the distance between points A (7,3) and B (5,-1)?
5I. (back to Key Idea table)
In the figure shown below, each dot is one unit from an adjacent horizontal or vertical dot.
Find the number of square units in the area of quadrilateral ABCD.
Show how you arrived at your answer.
6B. (back to Key Idea table)
Paul is playing a game in which he rolls two regular six-sided dice.
What is the probability that he will roll two doubles in a row?
6C. (back to Key Idea table)
The graph below shows the hair colors of all the students in a class.
What is the probability that a student chosen at random from this class has black hair?
6D. (back to Key Idea table)
Erica cannot remember the correct order of the four digits in her ID number. She does remember that the ID number contains the digits 1, 2, 5, and 9. What is the probability that the first three digits of Erica’s ID number will all be odd numbers?
A. 1/4
B. 1/3
C. 1/2
D. 3/4
7A. (back to Key Idea table)
Which of the following tables represents a linear relationship between the two variables x and y?
7B. (back to Key Idea table)
Write an equation to represent the price (P) of mailing a letter that weighs a certain number of ounces (x) if the cost is $0.32 for the first ounce and $0.23 for each additional ounce. Show how that equation would be used to determine the cost of mailing a 4-ounce letter.
7C. (back to Key Idea table)
Two video rental clubs offer two different rental fee plans:
Club A charges $12 for membership and $2 for each rented video.
Club B has a $3 membership fee and charges $3 for each rented video.
The graph drawn below represents the total cost of renting videos from Club A.
Part A
On the same set of xy-axes, draw a line to represent the total cost of renting videos from Club B.
Part B
For what number of video rentals is it less expensive to belong to club A? Explain how you arrived at your answer.
7D. (back to Key Idea table)
The figure below represents the distances traveled by car A and car B in 6 hours.
Which car is going faster and by how much? Explain how you arrived at your answer.
7E. (back to Key Idea table)
A corner is cut off a 5-inch by 5-inch square piece of paper. The cut is x inches from a corner as shown below.
Part A
Write an equation, in terms of x, that represents the area, A, of the paper after the corner is removed.
Part B
What value of x will result in an area that is 7/8 of the area of the original square piece of paper?
Show how you arrived at your answer.
The following ideas for lessons and activities are provided to illustrate examples of each performance indicator. It is not intended that teachers use these specific ideas in their classrooms; rather, they should feel free to use them or adapt them if they so desire. Some of the ideas incorporate topics in science and technology. In those instances the appropriate standard will be identified. Some classroom ideas exemplify more than one performance indicator. Additional relevant performance indicators are given in brackets at the end of the description of the classroom idea.
2C. (back to Key Idea table)
• Have students make multiplication and addition charts for a 12-hour clock, using only the numbers 1-12.
• Have students determine if the system is closed under addition and multiplication. If not, they should give a counterexample.
• Have students determine if multiplication and addition are commutative under the system, and if not, give a counterexample.
• Have students determine if there is an identity element for addition and multiplication, and if so, what are they?
• Have students determine if addition and multiplication are associative under the system, and if not, give a counterexample.
• Does each element have an additive and multiplicative inverse?
• Determine if multiplication is distributive over addition (if not give a counterexample) and if addition is distributive over multiplication (if not, give a counterexample). [Also 3D.]
3D. (back to Key Idea table)
Identify the field properties used in solving the equation 2(x - 5) + 3 = x + 7.
4B. (back to Key Idea table)
Explain why the basic construction of bisecting a line segment is valid.
5B. (back to Key Idea table)
While watching a TV detective show, you see a crook running out of a bank carrying an attaché case. You deduce from the conversation of the two stars in the show that the robber has stolen $1 million in small bills. Could this happen? Why or why not?
Hints: 1. An average attaché case is a rectangular prism (18” x 5” x 13”).
2. You might want to decide the smallest denomination of bill that will work.
[Also 5A.]
5H. (back to Key Idea table)
An odometer is a device that measures how far a bicycle (or a car) travels. Sometimes an odometer is not adjusted accurately and gives readings which are consistently too high or too low.
Paul did an experiment to check his bicycle odometer. He cycled 10 laps around a race track. One lap of the track is
0.4 kilometers long. When he started, his odometer read 1945.68 and after the 10 laps his odometer read 1949.88. Compare how far Paul really traveled with what his odometer read.
Make a table that shows numbers of laps in multiples of 10 up to 60 laps, the distance Paul really travels, and the distance the odometer would say he traveled.
Draw a graph to show how the distance shown by the odometer is related to the real distance traveled.
Find a rule or formula that Paul can use to change his incorrect odometer readings into accurate distances he has gone from the start of his ride.
An odometer measures how far a bicycle travels by counting the number of times the wheel turns around. It then multiplies this number by the circumference of the wheel. To do this right, the odometer has to be set for the right wheel circumference. If it is set for the wrong circumference, its readings are consistently too high or too low. Before Paul’s experiment, he estimated that his wheel circumference was 210 cm. Then he set his odometer for this circumference. Use the results of his experiment to find a more accurate estimate for the circumference.
6A. (back to Key Idea table)
A box contains 20 slips of paper. Five of the slips are marked with a “X,” seven are marked with a “Y,” and the rest are blank. The slips are well mixed. Determine the probability that a blank slip will be drawn without looking in the bag on the first draw. Have students determine the probability theoretically and then have each conduct the experiment with ten trials and see how close the empirical probability was to the theoretical probability. Combine data from all students in the class to see if a larger number of trials will result in an empirical probability that more closely resembles the theoretical probability. [Also 6B.]
Math B(to the top)
Key Idea 1
Mathematical Reasoning
Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 1A. Construct proofs based on deductive reasoning. |
|
See Classroom Activity 1A. |
| 1B. Construct indirect proofs. |
|
See Classroom Activity 1B. |
Key Idea 2
Number and Numeration
Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 2A. Understand and use rational and irrational numbers. |
|
See Classroom Activity 2A. |
| 2B. Recognize the order of the real numbers. |
|
See Classroom Activity 2B. |
| 2C. Apply the properties of the real numbers to various subsets of numbers. |
|
See Classroom Activity 2C. |
| 2D. Recognize the hierarchy of the complex number system. |
|
See Classroom Activity 2D. |
| 2E. Model the structure of the complex number system. |
|
See Classroom Activity 2E. |
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 3A. Use addition, subtraction, multiplication, division, and exponentiation with real numbers and algebraic expressions. |
|
See Classroom Activity 3A |
| 3B. Develop an understanding of and use the composition of functions and transformations. |
|
See Classroom Activity 3B. |
| 3C. Use transformations on figures and functions in the coordinate plane. |
|
See Classroom Activity 3C. |
| 3D. Use rational exponents on real numbers and all operations on complex numbers. |
|
See Classroom Activity 3D. |
| 3E. Combine functions, using the basic operations and the composition of two functions. |
|
See Classroom Activity 3E. |
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 4A. Represent problem situations symbolically by using algebraic expressions, sequences, tree diagrams, geometric figures, and graphs. |
|
See Classroom Activity 4A. |
| 4B. Manipulate symbolic representations to explore concepts at an abstract level. |
|
See Classroom Activity 4B. |
| 4C. Choose appropriate representations to facilitate the solving of a problem. |
|
See Classroom Activity 4C. |
| 4D. Develop meaning for basic conic sections. |
|
See Classroom Activity 4D. |
| 4E. Model real-world problems with systems of equations and inequalities. |
|
See Classroom Activity 4E. |
| 4F. Model vector quantities both algebraically and geometrically. |
|
See Classroom Activity 4F. |
| 4G. Represent graphically the sum and difference of two complex numbers. |
|
See Classroom Activity 4G. |
| 4H. Model quadratic inequalities both algebraically and graphically. |
|
See Classroom Activity 4H. |
| 4I. Model the composition of transformations. |
|
See Classroom Activity 4I. |
| 4J. Determine the effects of changing parameters of the graphs of functions. |
|
See Classroom Activity 4J. |
| 4K. Use polynomial, trigonometric, and exponential functions to model real-world relationships. |
|
See Classroom Activity 4K. |
| 4L. Use algebraic relationships to analyze the conic sections. |
|
See Classroom Activity 4L. |
| 4M. Use circular functions to study and model periodic real-world phenomena. |
• Use the concept of the unit circle to solve real-world problems involving: -radian measure -sine -cosine -tangent -reciprocal trigonometric functions. • Relate reference angles, amplitude, period, and translations to the solution of real-world problems. |
See Classroom Activity 4M. |
| 4N. Use graphing utilities to create and explore geometric and algebraic models. |
• Graph quadratic equations and observe where the graph crosses the x-axis, or note that it does not. |
See Classroom Activity 4N. |
Key Idea 5
Measurement
Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 5A. Use trigonometry as a method to measure indirectly. |
|
See Classroom Activity 5A. |
| 5B. Understand error in measurement and its consequence on subsequent calculations. |
|
See Classroom Activity 5B. |
| 5C. Derive and apply formulas relating angle measure and arc degree measure in a circle. |
|
See Classroom Activity 5C. |
| 5D. Prove and apply theorems related to lengths of segments in a circle. |
|
See Classroom Activity 5D. |
| 5E. Define the trigonometric functions in terms of the unit circle. |
|
See Classroom Activity 5E. |
| 5F. Relate trigonometric relationships to the area of a triangle and to general solutions of triangles. |
|
See Classroom Activity 5F. |
| 5G. Apply the normal curve and its properties to familiar contexts. |
|
See Classroom Activity 5G. |
| 5H. Derive formulas to find measures such as length, area, and volume in real-world context. |
|
See Classroom Activity 5H. |
| 5I. Design a statistical experiment to study a problem and communicate the outcome, including dispersion |
|
See Classroom Activity 5I. |
| 5J. Use statistical methods, including scatter plots and lines of best fit, to make predictions. |
|
See Classroom Activity 5J. |
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 6A. Judge the reasonableness of results obtained from applications in algebra, geometry, trigonometry, probability, and statistics. |
|
See Classroom Activity 6A. |
| 6B. Judge the reasonableness of a graph produced by a calculator or computer. |
|
See Classroom Activity 6B. |
| 6C. Interpret probabilities in real-world situations. |
|
See Classroom Activity 6C. |
| 6D. Use a Bernoulli experiment to determine probabilities for experiments with exactly two outcomes. |
|
See Classroom Activity 6D. |
| 6E. Use curve fitting to fit data. |
|
See Classroom Activity 6E. |
| 6F. Create and interpret applications of discrete and continuous probability distributions. |
|
See Classroom Activity 6F |
| 6G. Make predictions based on interpolations and extrapolations from data. |
|
See Classroom Activity 6G. |
| PERFORMANCE INDICATORS |
MAY INCLUDE | EXAMPLES |
| 7A. Use function vocabulary and notation. |
|
See Classroom Activity 7A. |
| 7B. Represent and analyze functions, using verbal descriptions, tables, equations, and graphs. |
|
See Classroom Activity 7B. |
| 7C. Translate among the verbal descriptions, tables, equations, and graphic forms of functions. |
|
See Classroom Activity 7C. |
| 7D. Analyze the effect of parametric changes on the graphs of functions. |
|
See Classroom Activity 7D. |
| 7E. Apply linear, exponential, and quadratic functions in the solution of problems. |
|
See Classroom Activity 7E. |
| 7F. Apply and interpret transformations to functions. |
|
See Classroom Activity 7F. |
| 7G. Model real-world situations with the appropriate function. |
|
See Classroom Activity 7G. |
| 7H. Apply axiomatic structure to algebra and geometry. |
|
See Classroom Activity 7H. |
| 7I. Solve equations with complex roots, using a variety of algebraic and graphical methods with appropriate tools. |
|
See Classroom Activity 7I. |
| 7J. Evaluate and form the composition of functions. |
|
See Classroom Activity 7J. |
| 7K. Solve equations, using fractions, absolute values, and radicals. |
|
See Classroom Activity 7K |
| 7L. Use basic transformations to demonstrate similarity and congruence of figures. |
|
See Classroom Activity 7L. |
| 7M. Identify and differentiate between direct and indirect isometries. |
|
See Classroom Activity 7M. |
| 7N. Analyze inverse functions, using transformations. |
|
See Classroom Activity 7N. |
| 7O. Apply the ideas of symmetries in sketching and analyzing graphs of functions. |
|
See Classroom Activity 7O. |
| 7P. Use the normal curve to answer questions about data. |
|
See Classroom Activity 7P. |
| 7Q. Develop methods to solve trigonometric equations and verify trigonometric functions. |
|
See Classroom Activity 7Q. |
4A. (back to Key Idea table)
Draw the graph y = 48/x (x&Mac173; 0). Make a table, using some integral values of x from x = -16 to x = 16.
Identify the graph.
4B. (back to Key Idea table)
Prove: If x and n are real numbers, and x > 0, then loga (xn) = n logax a > 0, a&Mac173; 1.
4C. (back to Key Idea table)
Students model population growth and decline of people, animals, bacteria, and decay of radioactive materials, using the appropriate exponential functions. [Also 4B.]
4D. (back to Key Idea table)
On your graphing calculator, graph the two conic sections on the
same set of axes. Determine the number of points of intersection and estimate their value from the graph. Check your estimates by substitution. Discuss an allowable margin of error in the check.
4E. (back to Key Idea table)
Two toy rockets are launched, one ten seconds after the other. The height in feet of the first rocket after 0 < t < 16 seconds is given by h(t) = -16t2 + 256t. The height of the second one after 10 < t < 20 seconds is given by g(t) = -16t2 + 480t - 3200. How many seconds after the first rocket is launched are the rockets at the same height?
4F. (back to Key Idea table)
The lift of an airplane wing is 750 lb. The drag is 300 lb. What is the magnitude and the direction of the resulting force? Draw a picture of the wing showing the lift and drag forces. Represent the problem geometrically and find the resultant force algebraically.
4G. (back to Key Idea table)
Have students investigate whether or not the difference of two complex conjugates can be a real number.
4H. (back to Key Idea table)
Solve the following inequality algebraically and graphically: x2 - 5x - 6 < 0.
4I. (back to Key Idea table)
Give students cut-out triangles. Have them draw a line and put a point on it for a vertex (straight angle). By doing translations with the triangle, students are to show that the sum of the measures of the angles of a triangle is 180o. Have students list their translations in order. (The translation for the first two angles can be done with a slide. The third angle can be done with a composition of line rotation and slide.) Have students prove that the translations are legitimate, using rules of transformations and parallel lines.
4J. (back to Key Idea table)
The graph of a function can be transformed in a number of ways. We will consider three:
vertical shift, horizontal shift, and vertical stretch. The function we will use is f(x) = x2.
Construct a table for the function f(x) = x2 for -3&Mac178; x&Mac178; 5. Construct similar tables and:
• use the tables to graph f(x) = 2x2, g(x) = x2-3, and h(x) = (x-3)2 on the same coordinate axis
• compare each graph you drew to the graph of f(x) = x2.
• determine which function has a graph that is a vertical shift of the graph of f(x) = x2? Is the shift upward or downward?
• determine which function has a graph that is a horizontal shift of the graph of f(x) = x2? Is the shift right or left?
• determine which function has a graph that is a vertical stretch of the graph of f(x) = x2?
4K. (back to Key Idea table)
Have students make pop rockets from paper and film canisters, using water and baking soda for fuel. (For directions see Rockets: A Teacher’s Guide with Activities in Science, Mathematics and Technology by NASA.) Have students make an astrolabe to measure the angle of altitude of the rockets assent. If students are a known distance from the rocket when they determine the angle of altitude, they can use Tan A = Opposite/adjacent to determine the height the rocket reached by adding that result to the distance of their own eye level from the ground. Fire the rocket and measure its height.
4L. (back to Key Idea table)
Write an equation of a circle with a center T(4,-3) and radius 3, using the distance formula.
4M. (back to Key Idea table)
The brightness of the star MIRA over time is given by y = 2 sin ((&Mac185;x)/4) + 6 where x measures years and y is the brightness factor. A new star has a brightness factor determined by y = 4 sin ((&Mac185;x)/16) + 4.
A. Do the two stars have the same maximum brightness factor?
B. Do the two stars have the same minimum brightness factor?
C. Compare the period of the brigntness factor of the new star to the period of MIRA?
D. Is it possible for the two stars to be equally bright at the same time?
4N. (back to Key Idea table)
Use your graphing calculator to graph y = x2 - 1. Compare the x values of where the graph crosses the axis and the solution to the equation x2 - 1 = 0.
5A. (back to Key Idea table)
Using the formula for the area of a triangle (area equals one-half of the product of any two sides and the sine of the included angle), show that the area of a right triangle is equal to one-half the products of its legs.
5B. (back to Key Idea table)
In<ABC, AC=8, BC=17, AB=20. Find the measure of the largest angle to the nearest degree
A. in one step using the Law of Cosines to find angle C.
B. in three steps using the Law of Cosines to find angles A and B and then the Law of Sines to find angle C.
5C. (back to Key Idea table)
Give students a cone-shaped drinking cup. Have the students cut the side from the brim to the apex of the cone and flatten out the cup. The shape of the flattened surface will be a circle with a sector missing. Ask them to use the shape and the ideas of unit circles to help them find the surface area of the cone.
5D. (back to Key Idea table)
Prove that any trapezoid inscribed in a circle is an isosceles trapezoid; that is, the non-parallel sides are equal.
5E. (back to Key Idea table)
Sketch the six basic trigonometric functions and their inverses on the graphing calculator by superimposing each function with its inverse.
5F. (back to Key Idea table)
Prove that if<ABC is a right triangle, the Law of Cosines reduces to the Pythagorean theorem. [Also 1D.]
5G. (back to Key Idea table)
As one of its admissions criteria, a college requires an SAT math score that is among the top 70% (69.1%) of all scores. The mean score on the math portion of the SAT is 500 and the standard deviation is 100. What is the minimum acceptable score? Justify your answer by drawing a sketch of the normal distribution and shading the region representing acceptable scores. [Also 6G.]
5H. (back to Key Idea table)
Use your knowledge of the area of squares and triangles to derive the Pythagorean Theorem.
5I. (back to Key Idea table)
A business owner pays each of his employees $50,000 per year. His salary is $150,000 per year. He wants to place an ad in the newspaper for more help. What would be the problem with only mentioning the mean with regard to salary? What measures of central tendency are more accurate when discussing salary? Would it be helpful to mention dispersion? Why? Support your answers with calculations.
5J. (back to Key Idea table)
Record, in seconds, the time for each student to run a 100 meter dash. Also record their height in inches. Sketch a scatter plot of the data (Use a minimum of 10 students.).
• Can any conclusions be made concerning height and speed?
• Using a calculator, find the equation of the best fit line.
• Does this equation support your conclusions?
• Make predictions for other students based on their height.
• Discuss the accuracy of these predictions.
6A. (back to Key Idea table)
The following ads for truck rentals appear in the paper.
A. You plan to rent a truck for one day. From which company would you rent? Why? Suppport your answer with a discussion of the factors you need to take into consideration. Use both equations and graphs to help illustrate your solution. Substitute specific values to check your results.
B. Determine under what conditions, considering both days and milage, the expense of renting a truck from Fast Rent-A-Truck would be less expensive than renting from Easy Rent-A-Truck.
6B. (back to Key Idea table)
A rich philanthropist, who loved mathematics, agreed to sponsor an 18-hole golf tournament at the local country club. In order to enter, a contestant had to pay 2 cents and select either a linear, quadratic, or exponential formula to calculate how many CENTS he/she would receive for a winning hole. In each of the following formulas, x represents the number of the winning hole. linear, y = 2x; quadratic, y = x2, exponential, y = 2x. Why bother entering if the payoff is in pennies? Use your graphing calculator to investigate. Describe numerically how the amounts change from one hole to the next for each method. Which method would you select on your entry form and why?
6C. (back to Key Idea table)
The principal of the local high school was willing to participate in the school fair dunking booth in which students who paid $1 could push a button that operated a light over the booth which was programmed to flash either red or green. If the light flashed green, the principal would fall into the water. If it flashed red, he would not. He was told that the light was set to flash either red or green randomly with a 50% chance of turning green. As it turned out, the principal seemed to be dunked more than 50% of the time. In the first 20 pushes of the button, he was dunked 15 times. He was getting suspicious that probability had been misrepresented to him. Based on the results so far, do you think the principal has justification for being suspicious? What is your reasoning? If you do not think the principal is justified in his suspicions, how many occurrences of 75% dunks would it take to convince you that the light was not set at 50% green? If you think the principal is justified in being suspicious, what are the smallest occurrences of 75% that would be required to convince you? [Also Performance Indicators 6C., 6D., and 6E.]
6D. (back to Key Idea table)
If each problem can be regarded as a Bernoulli experiment, state the values of n, p, q, and r, and give the answer in symbolic form. If the problem cannot be regarded as a Bernoulli experiment, explain why. Four balls are drawn with replacement from an urn containing 4 red balls and 2 white balls.
What is the probability of drawing exactly 2 red balls?
Four balls are drawn without replacement from an urn containing 4 red balls and 2 white balls.
What is the probability of drawing exactly 2 red balls? [Also Performance Indicator 6F.]
6E (back to Key Idea table)
Given x = {10, 20, 30, 40, 6E
Given x = {10, 20, 30, 40, 50} and y = {11.0, 12.1, 13.0, 13.9, 15.1} where x is measured in lbs. force and y measures the length of a spring in inches.
• Find the equation which best fits the data.
• Determine the load when y = 17 inches and determine the length of the spring when x = 62 lbs.
6F. (back to Key Idea table)
In her algebra class, Ms. Goodheart predicts 8 of her 26 students will earn a score of 90 or above on a particular exam with a normal distribution. After taking the exam, the mean score was 84 with a standard deviation of 6. Was her prediction accurate? What should she have predicted to be more precise?
6G. (back to Key Idea table)
The boiling point of water is a function of altitude. The table shows the boiling points at different altitudes.
Location Altitude Boiling point of
h meters water, toC
Halifax, NS 0 100
Banff, Alberta 1383 95
Quito, Ecuador 2850 90
Mt. Logan 5951 80
- Graph the relation between the altitude and the boiling point.
- Use the graph to estimate the boiling point of water at:
a) Lhasa, Tibet, altitude 3680m
b) the summit of the Earth’s highest mountain, Mt. Everest, 8848m.
7A. (back to Key Idea table)
f(x) = x3 + 5
A. Does f(x) have an inverse?
B. If so, find the inverse and decide if it is a function
7B. (back to Key Idea table)
A projector throws an image on a screen. To determine how the width of the image is related to the distance of the screen from the projector, the following measurements were made.
Graph the data and find the equation relating x and y.
Find the width of the image when the projector is 3.0 m from the screen.
Find the distance from the projector to the screen when the image is 3.0 m wide.
What is the domain of the relation?
7C. (back to Key Idea table)
A printer agrees to print a brochure for a sum of $300 plus 15 cents for each copy. Express the cost as a function of the number of copies.
7D. (back to Key Idea table)
Explain the similarities and differences in the equations that might be used for
each of the following graphs.
7E. (back to Key Idea table)
About Decay
Start this experiment with one cupful of M & M’s. Shake the cup and pour the M & M’s onto the napkin. Count the total number of M & M’s. Write this as the number for trial #1. Then remove all M & M’s that have the M showing. Record the total number left in the table below. Using the new total of M & M’s each time, repeat the procedure five more times. Note if the number of M & M’s reaches zero at any trial, the experiment is over at that time and you should not use the zero result as part of your data.
Create a scatterplot of Trial for x and total Number for y.
Enter the data in lists using your graphing calculator.
Write the equation:
Graph the exponential function on the grid above.
Use the equation to predict the number of M & M’s you would have had two times before trial #1:
If there were a larger number of M & M’s before trial #1, use the equation to predict the trial when there were 900 M & M’s (a negative number):
Explain the coefficients a and b in the equation.
7F. (back to Key Idea table)
Given f (x) = x2 - 2x
A. Determine an expression for h (x), if h (x) = f (-x).
B. Determine an expression for g (x), if g(x) is represented by the rotation of 180o of f (x) about the origin.
C. Rotate f (x) 90o about the origin. Find the coordinates of the point(s) for which x = -1, under the rotation.
7G. (back to Key Idea table)
Nita Pass is about to study for a mathematics exam. Nita knows that the test grade is a function of the number of hours studied and knows from past experience that 1 hour of studying will result in a grade of 60; 2 hours, in a grade of 74; and 7 hours in a grade of 84.
Show Nita that the grade is not a linear function of the number of hours studied.
Assume that the grade varies quadratically with the number of hours studied. Find the equation for the function, and draw the graph (show important features: vertex and intercepts).
What is the minimum amount of study time needed to pass the test if the lowest passing grade is 70? What is the grade-intercept and what does it represent in the real world?
The quadratic model predicts that Nita could earn zero points on the test. What might happen in the real world that could actually cause her to score zero by studying this long?
Use the graph to show that there is no real value of time for which the grade will be 100.
7H. (back to Key Idea table)
Conjecture:
The angle bisector of the vertex angle of an isosceles triangle is also a median to the base.
Given:
Isosceles ABC with AC = BC and with CD an angle bisector of vertex angle C.
Show:
CD is a median to the base.
Two-column proof:
7I. (back to Key Idea table)
Solve the following equation for x: 2x2 + 5x - 1 = 0. Sketch the graph of the function y = 2x2 + 5x - 1. Explain the relation between the roots of the equation and the x-intercepts of the graph of the function. [Also 3D., 4A., 6A., and 7C.]
7J. (back to Key Idea table)
The area A and perimeter P of a square are functions of its side length S. Express the area as a function of the perimeter.
7K. (back to Key Idea table)
The time it takes for a pendulum to swing back and forth once depends only on the length of the pendulum. This period T seconds is given by the formula, where l is the length of the pendulum in meters. By what factor is the
period increased when the pendulum length is tripled?
7L. (back to Key Idea table)
Provide students with examples of Escher prints and have them identify two congruent shapes and the isometries that provide the congruence. [Also 4J.]
7M. (back to Key Idea table)
Note the tessellations below, using capital block letters T and E. Have students work in groups to:
-determine what transformations were used in these tilings.
-identify those that are direct or indirect isometries.
-determine what other capital block letters would tile a plane.
-use graph paper to create their tessellations and make a list describing their findings.
7N. (back to Key Idea table)
Graph each of the relations below, its inverse, and y = x on the same coordinate system. Which of the four relations are functions? Which of the inverses are functions?
g: y = 2x - 2 f: y = -1/2x + 2
p: y = x2 + 1 q: y = (x+2)2
7O. (back to Key Idea table)
Find, if it exists, a line of symmetry of the graph of each equation. If there is no line of symmetry, write none.
y = x2 + 5 y = x2 + 4x + 1
y = x 7P.
The table below shows the scores on a writing test in an English class:
A. Using the accompanying set of data, find both the mean and the standard deviation to the nearest tenth.
B. What is the number of scores that fall within one standrad deviation of the mean?
C. Find, to the nearest tenth, the percentage of scores in this set of data that are within one standard deviation
of the mean?
D. What is the number of scores that fall within two standard deviations of the mean?
E. Find, the percentage of scores in this set of data that are within two standrad deviations of the mean.
7P. (back to Key Idea table)
The table below shows the scores on a writing test in an English class:
A. Using the accompanying set of data, find both the mean and the standard deviation to the nearest tenth.
B. What is the number of scores that fall within one standrad deviation of the mean?
C. Find, to the nearest tenth, the percentage of scores in this set of data that are within one standard deviation
of the mean?
D. What is the number of scores that fall within two standard deviations of the mean?
E. Find, the percentage of scores in this set of data that are within two standrad deviations of the mean.
7Q. (back to Key Idea table)
Find all positive values of A less than 360o that satisfy the equation 2 cos 2A - 3 sin A = 1. Express your answers to the nearest ten minutes or nearest tenth of a degree.