1.1 Mathematical Thinking: Problem
Solving
Read through the problem below. Take a few minutes and start the
solution process (there are many ways to do this problem). Share your
thinking with the person sitting next to you. Solve the problem together. Be
prepared to share your solution process. Exploration with Ikebana: The visible lengths of the
primary, secondary and tertiary stems add up to approximately 12 inches (the
length of the paper strip). Determine (approximately) the height of a vase to
be used if: (a) the visible length of the
primary stem is 1 ½ times the height of the vase (b) the visible length of the
secondary stem is 2/3 the visible length of the primary stem (c) the visible length of the
tertiary stem is ½ of the visible length of the primary stem |
1.2 Number
Sense
1. (p.17
#25) Simplify mentally using number properties as shortcuts (be ready to
explain your process): (d) 107 × 29 – 7 × 29
(e) 8 (41/4) 2. (p.17
#29) Simplify (round to the nearest tenth): (a)
At a range of 90 meters, the area of a circular archery target, in
square centimeters:
A = p(122 ¸ 2)2 (d) A Puzzle: x = 9 – 4(9 – 2) + 8
¸ 4 – 2 × 5 3. (p.17
#33) Simplify: (c) (d) |
1.3 Numeric and Symbolic
Representations
1.4 Problem
Solving and Verbal Representations
Group 1 a) b) Create an expression that allows you to determine the number of squares
in the nth design in the pattern. In the expression, explain each number and
variable in the context of the problem. a) Create an input/output table. Where the input is the design number and
the output is the expression describing the nth design in the pattern. Group 2: Let x be the input variable and y be the output variable. Write the sentence “the sum of the input and six is the
output decreased by three” as an equation, and then create an
input/output table for this equation. Group 3: Write an equation, letting x represent the
independent (input) variable and y represent the dependent (output)
variable for this situation: The total amount paid for a meal includes 15% of the
cost for tip and 8% of the cost for sales tax. Be prepared to
explain your equation. What would be the simplest equation to model this
situation? Explain.
|
1.5 Rectangular Coordinate Graphs
For
each of the following: (a) suggest axes labels, (b) create an output
expression, (c) create an input/output table, (d) use the table to graph the
relationship on graph paper, then (e) graph the expression on your calculator
adjusting the window to match your axes labels. Group 1: Input is daily sales up to $1000, and
output is sales tax at 8½%. Group 2: Input is meal cost up to $100, and
output is the tip at 18%. Group 3: Input is the number of miles a Boeing
747 can fly without refueling up to 8,380
miles, and output is the number of gallons consumed for each flight at an
average of 5 gallons/mile. |
1.6 Solving Equations with Tables and Graphs
1.7 Solving
Equations and Formulas
Learning to use the graphing Calculator together 1. Use a graph to determine the
solution(s) (the x values that make
the equation true) of the following equations. (TRACE) a: b: c: 2. Find the solution(s) for the equation using
a calculator table. Approximate the solution(s) to the nearest tenth. 3. Plotting the equation as a system
(plot each side of the equation on the same set of axes) can help you to
determine the number of solutions. |
|
Solve for the
indicated variable (solution formula). Group 1: Area of a trapezoid Group 2: Cost of a
skating party for 10 or more people Group 3: Sound Absorption All groups: Check your solution formula by
substituting it back into the original formula. Using this checking method,
how do you know your solution formula is correct? |
2.1 Inequalities and Functions
All groups: Write equations with conditions on the inputs. Group 1: (p.86 #45) Rental of a pressure washer costs $30 for the first 4 hours
and $10 per additional hour or part of an hour. If x is the total hours
rented, what is the total cost, y ? Group 2: (p.86 #46) A taxi cab costs $5 to hire and ride the first ½ mile. Each
additional 1/10 of a mile cost $0.75. If x is the total miles traveled,
what is the total cost, y ? Group 3: (p.86 #48) A former e-mail account
cost $9.95 each month for the first ten hours and $2 for each additional hour
or part of an hour. What is the total cost, y, for x hours a month? Why was this a disaster when the user
forgot to sign off before going to bed? |
2.2 Functions
Group 1: (p.98 #60) What is the
relevant domain for the area of a square, A(s) = s2, as
a function of the length of its side. Explain your answer. Group 2: (p.99 #76a) If an example of
the input is 3, -2, 4 and the respective output is 15, -10, 20; what is the function? Write the function as an equation and then
write it in function notation. What part of the function represents the
independent variable and what part represents the dependent variable? Group 3: (p. 99 #76b) If an example of
the input is 3, -2, 4 and the respective output is -1, -6, 0; what is the
function? Write the function as an
equation and then write it in function notation. What part of the function
represents the independent variable and what part represents the dependent
variable? |
2.3 Linear Functions
2.4 Modeling
with a Linear Function
Group 1: (p.122 #58e) Find the linear function that represents
this arithmetic sequence: 12, 5, -2, -9, -16 . . . Explain your process for
finding this equation. Group 2: (p.122 #41) Find the linear function that models the
relationship between the price and capacity of Pyrex measuring cups: 1 cup,
$3.19; 2 cups, $3.99; 4 cups, $4.99; 8 cups. $7.49 Group 3: (p.122 #40) Assume that the relationship between the
date and the swim times for the women’s 100-meter freestyle swim Olympic
record can be modeled with a linear function. Use this data: 1912, 82.2 sec.;
2004, 53.84 sec. to create a linear model and make a prediction for the year
2008. |
2.5 Special Lines
Group 1: (p.130 #22) Different-sized
rental trucks cost $19.99, $29.99, or $39.99 plus $0.59 per mile. Write an
equation (in function notation) that will model the rental cost, C, in dollars, for x miles driven for each size of truck.
Compare the equations. Would their graphs be parallel lines? Explain your
thinking. Group 2: (p.131 #38) Points (1, 2), (2,
5), (5, 4), and (4, 1) are connected to one another with line segments to
create a quadrilateral figure. What can you conclude about the line segments
and their relationship to one another? Explain your thinking. Group 3: (p.131 #47) Hand-packed ice
cream costs: $2.75 for 12 ounces, $4.75 for 24 ounces, and $7.95 for 50
ounces. Draw the graph of this data. Then determine the equation (in function
notation) for the line of best-fit for this data. Be prepared to share your
reasoning and process. |
2.6 Special Functions
Group 1:
(p.145 #61) A square is drawn with its corners at (0, 0), (4, 0), (4, 4), and
(0, 4). Write the equations for as many lines as possible that divide the area
of the square in half. Group 2:
(p.145 #62) Create a table that records all the possible sums when two dice
are added together. Look on page 145 to find all the possible outcomes (36)
for rolling a set of dice. Make a graph showing the possible sums as inputs, n, and the number of times this sum
appears among the 36 outcomes as the output or frequency of n, f(n). What is the domain and range of the function? Group 3:
(p.144 #54) Rental of a pressure washer costs $30 for the first 4 hours and
$10 for each additional hour or part of an hour. The total cost, y, is for an x-hour rental.
Explain whether
this is a dot or step graph. Draw the graph of the equations. Label the graph
and be prepared to explain your reasoning. |
3.1 Solving Systems by Substitution
or Elimination
3.2 Solving Systems of Two Linear
Equations by Graphing
Group 1:
(p.162 #44) A deluxe box of chocolate costs $45. The fancy box costs $40 less
than the chocolates it contains. What is the cost of each? Group 2: (p.162
#50) Two angles are supplementary. One angle is 35° more than the other. Find
the angle measures. Group 3: Solve
the systems by using either substitution or elimination. Explain how you can tell what is happening
graphically from the algebraic methods for solving the system. a: (p.168 #12) b: (p.168 #14) c: (p.168 #16) |
3.3 Solving Equations Involving
Quantity and Rate
All Groups: Set up a quantity-rate table, determine the
two equations, then solve using any method you prefer (substitution,
elimination, or graphing) Group 1: (p.175
#26) A nurse’s aide must prepare 4000 milliliters of a 0.5% potassium
permanganate solution for an astringent. How much distilled water and 4%
potassium permanganate must be blended? Group 2:
(p.175 #18) Polly earns $2343 interest on a total of $29,800 placed in two
investments. The investments earn 3.5% and 8.5%. How much money is in each
investment? Group 3:
(p.175 #20) How many pounds of cat food A, containing 8% protein, need to be
blended with cat food B, containing 13% protein, to obtain 500 pounds of a
blend with 9% protein? |
4.1 Quadratic Functions
4.2 Modeling Quadratic Functions
Group 1: (p.217 #24) Group 2: (p.218 #26) The formula for the amount of money in
an account earning interest compounded yearly for 2 years is a) For what interest
rate will the account contain $1100 at the end of 2 years? b) For what interest
rate will the account contain $1200 at the end of 2 years? c) For what interest
rate will the account contain $1400 at the end of 2 years? Group 3: (p.226
#10) Using first and second differences, find a quadratic function for the
following list of sequential outputs: 36, 44, 50, 54, 56, . . . |
4.3 Polynomial Functions and
Operations
Group 1: (p.237 #72) Write the polynomial, Group 2: (p.238 #98) The equations y = 4 – x2 and y = (2 – x)(2 + x) both describe
the same parabola. Graph the parabola and explain how each equation gives
different information about the graph. Groups
1 & 2 when done with your problem work on Group 3’s
problem Group 3: (p.239 #10) Post Office box rent is a function of the size of
the box. A box front measuring 3 in. by 5.5 in. costs $19 every 6 months; 5
in. by 5.5 in. costs $34; 11 in. by 5.5 in. costs $63; 11 in. by 11 in. costs
$110; and 22.5 in. by 12 in. costs $175. In a remodel, a set of 10 in. by 15
in. boxes is added. What price fits in with the current pricing scheme?
Explain your reasoning. |
4.4 Special Products and Completing
the Square
4.5 Solving Quadratic Equations
Group 1:
(p.246 #72) A square picture frame of outer dimensions N by N evenly surrounds a square mirror of dimensions n by n. Write the expression for the area of the frame in standard
form and in factored form. What is the thickness of the frame? Group 2:
(p.256 #60) If you had a sheet of plywood that is 4 feet by 8
feet, and you cut off an x feet
from each of the 4 sides, what would be the value of x if the area left is 15 square feet? (If it won’t factor nicely,
try using “completing the square” to solve”.) Group 3:
(pp.256-7 #64) While working on a cooling tower, a construction
worker tosses a candy bar to a friend at the base (height = 0 feet) of the
tower. The formula describing the height in feet as a function of time in
seconds is All Groups:
(p.246 #68) Factor the expressions: (a) |
5.1 Square Root Function and
Pythagorean Theorem
Group 1: (p.288 #77) A support wire for a 50-foot tall radio
antenna is to be fastened halfway up the
antenna. The other end of the wire is to be attached to the ground, 16 feet
from the base of the antenna. Draw a picture and label all parts. How long a
wire is needed (answer should be in feet and inches. Do not include the part
of the wire needed to connect to the antenna or the ground pin.) Group 2: (p.288 #79) A ladder is in a safe position if the
height it reaches on the wall is four times the distance of the base to the
wall. Draw a picture and label all parts. Find the length of the ladder that
must reach a 16-foot height (answer should be in feet and inches.)
All Groups: (p.282 example 11) A right triangle is always formed when the diameter of a circle is one of the triangle’s legs and the other two legs connect on the edge of the circle. Find the missing side (give your answer in exact form.) |
5.2 Solving Quadratic Equations with
Square Roots
Group 1: (p.296 #55) On Earth, the approximate distance in
miles seen to the horizon from a height, h, in feet, is: Group 2:
(p.296 #53) The height of an object is determined using this formula: Group 3: (p.295
#49) The cross-sectional area of a water pipe (the
round opening) is 5026 square inches.
(a) Convert the area of a circle formula, |
5.3 Using the Quadratic Formula
Groups 1 and 2: Vertical motion uses this formula: Group 1: (p.305 #35) (a) What is the meaning of the
y-intercept in the context of the problem? (b) Graph the equation on your
graphing calculators and determine a reasonable window. Sketch the graph on
paper. (c) At what two times is the diver at 32.8 feet? Explain what is
happening at those times. Group 2: (p.305 #37) Suppose another Olympic diver has an
initial velocity Group 3: (p.305
#33) The annual simple interest
rate, r, for a $2000
loan with a $3000 repayment due at the end of two years is found by solving: |
5.4 Solving Minimum and Maximum
Problems
Group 1: (p.315 #39) Juan has 120 feet of fencing with which to make a
movable pen as shown. (a) Find the largest possible fenced area. (b) What is
the length and width of the largest possible area?
h=-0.5gt2+vot+ho where g is acceleration
due to gravity at 32.2 ft/sec2, and vo and ho are the initial velocity and initial height
respectively. If a ball of fireworks is shot vertically into the air from the
ground with an initial velocity of 115 ft/sec and the fuse is set to go off
at the maximum height what is that time and height? Group 3: (p.315
#35) Transition
curves, as described on p.310 Example 4, are used to design roadbeds and to
position the storm drains at the minimum point in the bed. For a highway
roadbed, the transition curve between two hills is given by: |
7.1 Exponents and their Properties
Group 1: (p.444 #25) Name the operation as adding like terms, multiplying
like bases, or neither, and simplify if possible: (a) Group 2: (p.444 #28) The radius of a soccer ball is about 5.3 times the
radius of a golf ball. Compare the ratio of their volumes, Group 3: (p.444 #29) If we double the radius of a sphere, how does the new volume compare with the original volume? |
7.2 Scientific Notation and
Significant Digits
Circular velocity,
Vcirc = Escape velocity, Vesc = Group 1: (p.451 #27b) Find the Circular velocity of a space vehicle
orbiting Jupiter: R = 8.14 x 107 m, M = 1.90 x 1027
kg Group 2: (p. 451 #27c) Find the escape velocity of a space vehicle orbiting
Earth: R = 7.18 x 106 m, M = 5.98 x 1024
kg Group 3: (p. 451 #27e & f) (e) What happens when the velocity drops below circular velocity? (f) What happens to the orbit when the velocity is between circular and escape velocity? (g) How do you know where to round off to (number of significant digits)? |
7.3 Rational Exponents
All groups: Use (original
amount invested), r is the interest rate, t is the number of
years invested, n is the number of times per year the interest is
compounded (calculated). Group 1: (p.460 #41) Find the amount of money in a savings account for
4.75 years at 7% annual interest rate, with quarterly compounding. Start with
$1000 in the account. Group 2: (p.459 #33) Interest on late property taxes is 16% per year
compounded monthly. Find the total to be paid for $650 tax that is 2 months
late. Group 3: (p.459
#29) If you borrow
$200 from a payday lender at 1% per day, what will you owe in 30 days? |
7.4 Roots and Rational Exponents
Group 1: (p.471 #68m, u,
w) (a) Group 2: (p.470 #53) Change Group 3: (p.470
#54) Change |
7.5 More Operations with Radicals
Group 1: (p.479 #49) Rationalize the denominator Group 2: (p.480 #51a-c) Multiply the numerator and denominator by a number or
variable that makes a perfect cube under the radical in the denominator.
Further simplify, as needed. (a) Group 3: (p.480
#59) Find slope and
the distance between points (6, 7) and (-3,
-4). Use the distance formula |
7.6 Finding Inverse Functions
Group 1: (p.488 #3) Name the ordered pairs that are in the inverse for
the set: {(1,2),(2,5),(3,8),(4,11)} Group 2: (p.488 #13) Graph the equation, Group 3:
(p.489 #41) Use algebra to find the inverse for |
7.7 Solving Root and Power Equations
Group 1: (pp.498-499 #60a) A pizza has a 5-inch radius. Find the area of a
slice of pizza with a central angle of 44o. Use the proportion: a
circle with radius, r, as the sector angle, q, is to the total degrees in a circle) to help you
solve this problem. Give your answer in exact form, then round it to the
nearest tenth of an inch. Groups 2 & 3: Use the compound interest formula: Group 2: (p.498 #47) In 1985, my house was valued at $55,000. In 1998,
the same house was valued at $119,000. What annual rate of growth describes
the change in value? At this rate, of growth, what will
the value be in 2010, to the nearest thousand? Group 3: (p.498
#53) Microsoft stock
valued at $6.953 per share in 1994 was sold at $52.02 in 2002. What is the
annual rate of growth? All Groups: (p.499
#61a) the volume of a
sphere with radius, r is your answer in
exact form, and then round to the nearest hundredth. |
8.1 Exponential Functions
Group 1: (p.519 #43) For the sequence: 3, 1, 1/3, 1/9,
1/27 (a) give the common ratio, (b) find the expression
for the nth term, (c) find the exponential regression, and the (d) use the
properties of exponents to show that your description of an is the same as that given by Group 2: (p.519 #49d) Find a linear or exponential regression equation
(use your calculator) for Group 3: (p.520
#51) The national debt of the |
8.2 Exponential Equations and Graphs
All Groups: (p.530 #56) Group 1: (p.529 #47) Group 2: (p.529 #51) Group 3: (p.529 #45) |
8.3 Solving Exponential Equations:
Logarithms
Find
the inverse equations: Group 1: (p.538 #19a, 23e) (a) 17=10x (b) logx64=3 Group 2: (p.538 #19b, 23b) (a) 125=10x (b) log33=x Group 3: (p.538 #19d, 23c) (a) 10x=0.05 (b) log100.01=x |
8.4 Exponential and Logarithmic
Function Applications
Group 1: (p.551 #43) Suppose water consumption in a large city increases
by 6% per year. In how many years will the water consumption double? Group 2: (p.551 #45) Suppose a $30,000 car loses 10% of its value each
year. Write an exponential equation and use it to find in how many years (to the
nearest hundredth) the value will be half the original. Group 3:
(p.550 #19) The moment magnitude scale is based on the energy
released by the rupture (source of an earthquake). The magnitude of an
earthquake using the moment magnitude scale can be modeled with the equation:
M=log x, where M is the magnitude and x is the energy
released. Write the energy input in scientific notation then calculate the
magnitude of the quake. (a) |
8.5 Properties
of Logarithms
Group 1: (p.560 #49) For 1000(1 – 0.08)t
= 500, (a) take the log of both sides, simplify and then solve. b) Change original from exponential form to
logarithmic form to solve.
Groups 2
and 3: Fit an equation using exponential regression, use (0,S) and (half life,
½ S) for data points
Group 2: (p.551 #49d) Neptunium-239
decays into plutonium-239, with a half-life of 24,000 years. The amount of
neptunium-239 released decayed into 1.7 curies of plutonium-239. From your
regression line, can you determine how much was released from Hanford in 1970,
if 1.7 curies are what is remaining today?
Group 3: (p.551 #49e) Sodium-24
has a half life of 15 hours. The amount released in 1970 has decayed into
12,000,000 curies. From our regression line, can you determine how much as
originally released?