Math 6 – Bid Is Non Negotiable (Read first before bidding!!!)
Read first before bidding. Accept Bid as is or don’t Bid at all. I’m not increasing so don’t ask and don’t waste my time
Lesson
3
.
1
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 11. Distinguish between variables and constants. Represent realworld problem situations using variables and constants. Construct equations to represent relationships between unknown quantities.
·
2
5. Use functional models to make predictions and solve problems.
Specific Objectives
Students will understand
· There are multiple ways to “see” and describe a pattern
Students will be able to
· Form an expression to describe a pattern
· Use that expression to evaluate and solve
In this lesson, you are going to work on seeing patterns and representing them algebraically.
To do this, we’re going to look at a progression of “steps” of pattern, and try to write down what we see, then find an expression that explains it.
Example: Consider the three steps to the right. How many blocks would be in
Step
4
? Step
10
? Step n?
We’ll look at two different student’s approaches.
Student 1 notices that all three steps shown have a single dot on the far left and far right, so that’s 2 dots. There’s a top row and bottom row of dots, each of which is increasing by 1 each time.
So in step 1, we have 2 dots + 2 rows of 1 dot each: 2 + 2·1. In step 2, we have 2 dots + 2 rows of 2 dots each: 2 + 2·2
We jot this down, and note the pattern, which we can then extend:
4 10Step 
What I See Here 
Number of dots 

1 
2 + 2 · 1 
4  
2 
2 + 2 · 2 
6 

3 
2 + 2 · 3 
8 

2 + 2 · 4 
10  
2 + 2 · 10 
22 

n 
2 + 2 · 
2 + 2n 
Student 2 notices that we start with 4 dots, and add 2 dots each time. So, in Step 2, we have 4 dots + 2 more. In step 3 we have 4 dots + 4 more, which is 2 more twice: 2·2, or 4 + 2·2
We jot this down, and note the pattern, which we can then extend:
Step
What I See Here
Number of dots
14, or 4 + 2 · 0
4
2
4 + 2 · 1
6
3
4 + 2 · 2
8
4
4 + 2 · 3
10
10
4 + 2 · 9
22
n
4 + 2 · (
n – 1)
4 + 2(n – 1)
Is one of the students wrong? Or are their answers the same?
We can check by simplifying Student 2’s answer:
4 + 2(n – 1)
Distributing
4 + 2n – 2 Combining like terms, 4 – 2 = 2
2 + 2n
The answers are the same, just written differently.
#1 Points possible: 10. Total attempts: 5
Watch this video to see 4 different students’ approaches to finding a formula for a pattern.
Which of the approaches/formulas in the video are valid?
· Student 1: 3(n+1)+2n3(n+1)+2n
· Student 2: (2n+1)⋅2+(n+1)(2n+1)⋅2+(n+1)
· Student 3: 8+5(n−1)8+5(n1)
· Student 4: 3(3+2(n−1))−n3(3+2(n1))n
Start with one of the valid expressions, and simplify it was much as possible. Your answer should be an expression involving n.
Set 1
Now it’s your turn.
Use the pattern shown for the next set of questions.
#2 Points possible: 5. Total attempts: 5
Complete the table.
Number of dots
Stage 

#3 Points possible: 5. Total attempts: 5
How many dots will there be in stage n? Write an expression involving n.
#4 Points possible: 5. Total attempts: 5
How many dots will there be in stage 20?
dots
#5 Points possible: 5. Total attempts: 5
What stage will have
13
7
dots?
Stage
#6 Points possible: 5. Total attempts: 5
Which best describes this pattern?
· Linear
· Quadratic
· Exponential
· Other
Set 2
Use the pattern shown to answer the next set of questions:
#7 Points possible: 5. Total attempts: 5
Complete the table.
Stage
Number of dots
1
2
3
4
10
#8 Points possible: 5. Total attempts: 5
How many dots will there be in stage n? Write an expression involving n.
#9 Points possible: 5. Total attempts: 5
How many dots will there be in stage 20?
dots
#10 Points possible: 5. Total attempts: 5
What stage will have 144 boxes?
Stage
#11 Points possible: 5. Total attempts: 5
Which best describes this pattern?
· Linear
· Quadratic
· Exponential
· Other
Set 3
One more. Use the pattern shown to answer the next set of questions:
#12 Points possible: 5. Total attempts: 5
Complete the table.
Stage
Number of dots
1
2
3
4
10
#13 Points possible: 5. Total attempts: 5
How many dots will there be in stage n? Write an expression involving n.
#14 Points possible: 5. Total attempts: 5
Which best describes this pattern?
· Linear
· Quadratic
· Exponential
· Other
HW 3.1
#1 Points possible: 8. Total attempts: 5
The first four stages of a pattern are shown in the diagram above. If the pattern continues,
a) How many red boxes will there be in stage 10 of the pattern?
b) Write an expression for the number of red boxes in stage nn of the pattern.
#2 Points possible: 8. Total attempts: 5
The table below shows the number of boxes in each stage of a pattern
14
10
Stage 
Boxes 
7  
13 
a) How many boxes will there be in stage 10 of the pattern?
b) Write an expression for the number of boxes in stage nn of the pattern.
Lesson
3
.2
Introduction
Course Objectives
This lesson will address the following course outcomes:
·
1
. Demonstrate operation sense by communicating in words and symbols the effects of operations on numbers. Apply the correct order of operations in evaluating expressions and formulas.
· 13. Evaluate formulas with multiple variables in a variety of contexts, including science, statistics, geometry, and financial math. Solve simple formulas for a specified variable.
· 1
5
. Solve linear equations in one variable, including problems involving the distributive property and fractions.
Specific Objectives
Students will understand that
· t
he behavior of a formula can be explored using a table and graph.
Students will be able to
· simplify a formula given values for some parameters.
· solve for a variable in a linear equation, and evaluate an equation.
Problem Situation: Calculating Blood Alcohol Content
Blood alcohol content (BAC) is a measurement of how much alcohol is in someone’s blood. It is usually measured as a percentage. So, a BAC of 0.3
%
is three
–
tenths of 1%. That is, there are 3 grams of alcohol for every 1,000 grams of blood. A BAC of 0.05% impairs reasoning and the ability to concentrate. A BAC of 0.30% can lead to a blackout, shortness of breath, and loss of bladder control. In most states, the legal limit for driving is a BAC of 0.08%.
#1 Points possible: 12. Total attempts: 5
Think about the variables that might influence BAC, and how they would influence it. Use your intuition to answer these questions:
If you drink more drinks, your BAC would
If the time since your first drink was longer, your BAC would
If you weighed more, your BAC would
Estimating BAC
BAC is usually determined by a breathalyzer, urinalysis, or blood test. However, Swedish physician, E.M.P. Widmark developed the following equation for estimating an individual’s BAC. This formula is widely used by forensic scientists:
B=−0.015t+2.84NW⋅gB=0.015t+2.84NW⋅g
Where
B = percentage of BAC
N = number of “standard drinks” (A standard drink is one 12ounce beer, one 5ounce glass of wine, or one 1.5ounce shot of liquor.) N should be at least 1.
W = weight in pounds
g = gender constant, 0.68 for men and 0.55 for women
t = number of hours since the first drink
The variables B, N, W, g, and t change depending on the person and situation. The units of B are a percentage. So, 0.08 is not 8%, but eighthundredths of a percent.
This equation has been simplified from the one found in the reference. The numbers 0.015 and 2.84 are constants based on the average person. It’s also worth noting many highend beers have twice as much alcohol as the “standard beer,” which assumes 4% alcohol.
#2 Points possible: 5. Total attempts: 5
Consider the case of a male student who has three beers and weighs 120 pounds. For this case, these values can replace the appropriate variables in the formula. What variables are still unknown in the equation? (select all that are still unknown)
· B
· N
· W
· g
· t
#3 Points possible: 5. Total attempts: 5
Consider the case of a male student who has three beers and weighs 120 pounds. Simplify the Widmark equation as much as possible for this case, rounding any constants to 3 decimal places. Hint
#4 Points possible: 12. Total attempts: 5
Using your simplified equation, find the estimated BAC for this student one, three, and five hours after his first drink. Look for patterns in the data.
Hours
BAC
1 % 3
%
5%
#5 Points possible: 6. Total attempts: 5
Complete the following contextual sentence:
Every hour, this student’s BAC by %.
#6 Points possible: 5. Total attempts: 5
Create a graph of the BAC over time for this student. Your graph doesn’t have to be perfect.
Clear All Draw:
#7 Points possible: 10. Total attempts: 5
Using your equation, what would this student’s BAC be after 8 hours?
%
Is this reasonable?
#8 Points possible: 10. Total attempts: 5
How long will it take for this student’s BAC to be 0.08%, the legal limit? Give your answer to 1 decimal place.
hours
How long will it take for the alcohol to be completely metabolized resulting in a BAC of 0.0? Give your answer to 1 decimal place.
hours
Hint: Try this problem on your own. If you’re having trouble after two tries, we’ll give some hints
Another Case
A female student, weighing 110 pounds, plans on going home in two hours. Using the Widmark formula, the simplified equation for this case is
B=−0.03+2.84N60.5B=0.03+2.84N60.5
#9 Points possible: 8. Total attempts: 5
Compare her BAC for one glass of wine versus three glasses of wine at the time she will leave. Give BAC to 3 decimal places.
BAC
1
%
3
%
Number of Drinks 
#10 Points possible: 5. Total attempts: 5
In this scenario, determine how many drinks she can have so that her BAC remains less than 0.08%. Since she could finish only part of a drink, give your answer to one decimal place.
drinks
#11 Points possible: 5. Total attempts: 5
Create a graph of the BAC given number of drinks for this student. Your graph doesn’t have to be perfect.
Clear All Draw:
HW 3.2
#1 Points possible: 5. Total attempts: 5
Which of the following was one of the main mathematical ideas of the lesson?
· Blood Alcohol Content (BAC) is affected by many different variables.
· A formula can be explored by evaluating it and solving for specific values.
· When solving an equation, an operation that changes the value of one side must also be done to the other side of the equation.
· Sometimes letters can represent a constant.
#2 Points possible: 20. Total attempts: 5
Find the solution to each of the following:
a. 5+3x=145+3x=14
x =
b. 6x−5=106x5=10
x =
c. 1=7+2×1=7+2x
x =
d. x4+3=8×4+3=8
x =
#3 Points possible: 15. Total attempts: 5
Find the solution to each of the following:
a. 2.5x+1=8.52.5x+1=8.5
x =
b. 3.6+4.2x=9.93.6+4.2x=9.9
x =
c. 14.92=5+1.6×14.92=5+1.6x
x =
#4 Points possible: 15. Total attempts: 5
Find the solution to each of the following, giving the answer in reduced form as proper or improper fractions (no mixed numbers).
a. 13x+38=5813x+38=58
x =
b. 12x+14=1612x+14=16
x =
c. 2x+23=82x+23=8
x =
#5 Points possible: 8. Total attempts: 5
Recall that Blood Alcohol Content (BAC) is a measurement of how much alcohol is in someone’s blood as a percentage. However, police and the public typically omit the language for % when quoting the BAC and simply say, “BAC is 0.04.”
Write an interpretation of what each of the following BAC values means in terms of how much alcohol is in the bloodstream in the form of the amount of alcohol per 1,000 grams of blood. You may want to refer back to the example in the lesson.
a. BAC = 0.1
gram of alcohol for every 1,000 grams of blood.
b. BAC = 0.02
gram of alcohol for every 1,000 grams of blood.
#6 Points possible: 1. Total attempts: 5
Use information from the website
http://en.wikipedia.org/wiki/Blood_alcohol_content
to list effects on an individual having a BAC as given. Give at least one effect for each.
a. BAC = 0.1
b. BAC = 0.5
c. BAC = 0.05
Use the Widmark Equation, B=−0.015t+2.84NW⋅gB=0.015t+2.84NW⋅g to solve the next two questions. Recall that g = 0.68 for men and g = 0.55 for women.
#7 Points possible: 10. Total attempts: 5
A male student had five glasses of wine at a party. He weighs 155 pounds. Using the Widmark equation,
a) Write the equation you would solve to figure out how long it will take before his BAC is 0.08. Is it okay to round any constants to 3 decimal places.
b) Solve that equation. Round to 2 decimal places.
hours
#8 Points possible: 5. Total attempts: 5
This question will apply the Widmark Equation to you.
Enter your weight in pounds: pounds. Your gender:
Answer the next two questions based on the Widmark Equation using the weight and gender you specified.
If you drink alcohol over a period of 5 hours, how many whole drinks would you be able to consume and still ensure that your BAC is less than the legal limit in Washington?
drinks
#9 Points possible: 10. Total attempts: 5
Indicate if each of the following is an expression or an equation:
· 4x+34x+3
· x2−16x−4×216x4
· (x−4)(x+3)=6(x4)(x+3)=6
· (x−4)(x−3)(x4)(x3)
· 4y=34y=3
#10 Points possible: 9. Total attempts: 5
The percentage of Americans who are retired has been increasing over the last decade. This is causing some concern because health care, social security, and other costs will be the responsibility of a smaller group of people. That is, as the percentage of retired people increases, the percentage of working age people decreases. The following model predicts the percentage of retired people based on demographic data
1
:
R=t873.36−2.15R=t873.362.15
where R is the percentage (as a decimal) of Americans who are retired in the year t (for example, the year 1995 would be t = 1995). Use this model to complete the table below by solving for the year.
Year 
% of Retired People 
10% 

15% 

20% 
#11 Points possible: 5. Total attempts: 5
Andy’s house is on a large lot. He got 100 yards of chainlink fence on sale. He wants to use all of the material to completely enclose a rectangular area in his backyard. He wants to make the fenced area 60 feet wide and as long as possible. What is the longest length possible for the sides?
ft
–
Lesson
3
.3
Introduction
Course Objectives
This lesson will address the following course outcomes:
·
2
0
. Translate problems from a variety of contexts into mathematical representation and vice versa (linear, exponential, simple quadratics).
· 22. Identify when a linear model is reasonable for a given situation and, when appropriate, formulate a linear model. In the context of the situation interpret the slope and intercepts and determine the reasonable domain and range.
· 25. Use functional models to make predictions and solve problems.
Specific Objectives
Students will understand that
· linear models are appropriate when the situation has a constant increase/decrease.
· slope is the rate of change.
· the rate of change (slope) has units in context.
· different representations of a linear model can be used interchangeably.
Students will be able to
· label units on variables used in a linear model.
· make a linear model when given data or information in context.
· make a graphical representation of a linear model.
· make a table of values based on a linear relationship.
· identify and interpret the vertical intercept in context.
In this lesson, you will learn about how linear models (linear equations in context) can be useful in examining some situations encountered in real life. A model is a mathematical description of an authentic situation. You can also say that the mathematical description “models” the situation. There are four common representations of a mathematical relationship.
Four Representations
Four Representations of a Relationship
In the last lesson, you explored how BAC could be estimated. If we were considering a
1
50 pound male who has had 3 drinks, the equation would simplify to
B=−0.015t+
0.0835
B=0.015t+0.0835
This is an equation for a model. This equation is useful because it can be used to calculate BAC given a number of hours (for a 150 pound man who’s had 3 drinks). Equations are useful for communicating complex relationships. In writing equations, it is always important to define what the variables represent, including units. For example, in the equation above, B = Blood alcohol content percentage, t = number of hours since the first drink.
Another way that you could have represented this relationship is in a table that shows values of t and B as ordered pairs. An ordered pair is two values that are matched together in a given relationship. You used a table in the last lesson. Tables are helpful for recognizing patterns and general relationships or for giving information about specific values. A table should always have labels for each column. The labels should include units when appropriate.
Time since first drink (hours) 
BAC (%) 

0  0.0835  
1 
0.0 6 85 

2 
0.0535 

3 
0.0385 
A graph provides a visual representation of the situation. It helps you see how the variables are related to each other and make predictions about future values or values in between those in your table. The horizontal and vertical axis of the graph should be labeled, including units.
A verbal description explains the relationship in words. In this case, we can do that by looking at how the values in the formula affect the result. Notice that the person’s BAC begins at 0.0835% (at the time he first consumes the 3 drinks), and whenever t increases by 1, the BAC drops by 0.015%. So we could verbally describe the relationship as: The persons BAC starts at 0.0835%, and drops by 0.015% each hour.
Slope
Linear Equations and Slope
The type of equation we looked at for BAC is a linear equation, or a linear model. Recall that the graph of a linear equation is a line. The primary characteristic of a linear equation is that it has a constant rate of change, meaning that each time the input increases by one, the output changes by a fixed amount. In the example above, each time t increased by 1, the BAC dropped by 0.015%. This constant rate of change is also called slope.
Example: If I were walking at a constant speed, my distance would change 6 miles in 2 hours. We could compute the slope or rate of change as 6 miles2 hours=3 miles1 hour6 miles2 hours=3 miles1 hour = 3 miles per hour.
In general, we can compute slope as change of outputchange of inputchange of outputchange of input .
In the graph shown, we can see that the graph is a line, so the equation is linear. We can compute the slope using any two pairs of points by counting how much the input and output change, and divide them. Notice that we get the same slope regardless of which points we use.
If the output increases as the input increases, we consider that a positive change. If the output decreases as the input increases, that is a negative change, and the slope will be negative.
Some people call the calculation of slope “rise over run“, where “rise” refers to the vertical change in output, and “run” refers to the horizontal change of input.
The units on slope will be a rate based on the units of the output and input variables. It will have units of “output units per input units”. For example,
· Input: hours. Output: miles. Slope: miles per hour
· Input: number of cats. Output: pounds of litter.
Slope: pounds of litter per cat
Slopeintercept Equation
Slope Intercept Equation of a Line
The slopeintercept form of a line, the most common way you’ll see linear equations written, is
y=mx+by=mx+b
where m is the slope, and b is the vertical intercept (called y intercept when the output variable is y). In the equation, x is the input variable, and y is the output variable.
Notice our BAC equation from earlier, B=−0.015t+0.0835B=0.015t+0.0835 , fits this form where the slope is 0.015 and the vertical intercept is 0.0835. The input variable is t and the output variable is B.
Sometimes you’ll see the equation written instead in the form y=b+mxy=b+mx .
The slope tells us a rate of change. As we interpreted earlier, in this equation the slope tells us the person’s BAC drops by 0.015% each hour. Or we could say that the rate of change is 0.015% per hour.
The vertical intercept tells us the initial value of the equation – the value of the output when the input is zero. For the BAC equation, the vertical intercept tells us the persons BAC starts at 0.0835%.
#1 Points possible: 5. Total attempts: 5
Write the equation of a line with slope 3 and yintercept of (0, 7). Use x as the input variable, and y as the output variable.
#2 Points possible: 1
4
. Total attempts: 5
Terry is skiing down a steep hill. Terry’s elevation, EE, in feet after tt seconds is given by E=3300−40tE=330040t.
The equation tells us that Terry started skiing and his is by
Graphing
Graphing using Slope and Intercept
In earlier lessons, you graphed a formula by calculating points. You certainly can continue to do that, and knowing that an equation is linear just makes that easier by only requiring two points. But it can be helpful to think about how you can use the slope to graph as well.
Suppose we want to graph y=−23x+5y=23x+5 using the vertical intercept and slope.
The vertical intercept of the function is (0, 5), giving us a point on the graph of the line.
The slope is −2323 . This tells us that every time the input increases by 3, the output decreases by 2. In graphing, we can use this by first plotting our vertical intercept on the graph, then using the slope to find a second point. From the initial value (0, 5) the slope tells us that if we move to the right 3, we will move down 2, moving us to the point (3, 3). We can continue this again to find a third point at (6, 1). Finally, extend the line to the left and right, containing these points.
#3 Points possible: 5. Total attempts: 5
Sketch a graph of y=−32x+2y=32x+2 by first placing a point at the yintercept, then use the slope to find a second point.
Clear All Draw:
Fundraiser
Problem Situation 1: Fundraiser
A sports team is planning a fundraiser to help pay for equipment. Their plan is to sell team Tshirts to friends and family, and they hope to raise
$
500. Their local screen printing shop will charge them a $75 setup fee, plus $5 per shirt, and they plan to sell the shirts for $15 each.
#4 Points possible: 12. Total attempts: 5
Explore their profit (the amount they bring in from sales minus costs) by filling in the table below. In all cases (including 0 sold) assume they still paid the $75 setup fee.
0 1$
2$
$
Number of shirts 
Profit 

$  
10 
#5 Points possible: 5. Total attempts: 5
How much additional profit does the team make for each shirt they sell?
$
#6 Points possible: 5. Total attempts: 5
Create a linear model for their profit in terms of the number of shirts they sell. Use P for the profit, and n for the number of shirts they sell.
#7 Points possible: 5. Total attempts: 5
Create a graph of the linear model.
Clear All Draw:
#8 Points possible: 5. Total attempts: 5
How much money will they raise if they can sell 30 shirts?
$
#9 Points possible: 5. Total attempts: 5
How many shirts will they have to sell to raise their goal of $500?
shirts
Lattes
Problem Situation 2: Daily Latte
A local coffee shop offers a Coffee Card that you can preload with any amount of money and use like a debit card each day to purchase coffee. At the beginning of the month (when you get your paycheck), you load it with $50. Each day, your short soy latte costs $2.63.
#10 Points possible: 5. Total attempts: 5
Estimate if the Coffee Card will last until the end of the month if you purchase a latte every weekday.
· No, it will run out
· Yes, it will last
#11 Points possible: 5. Total attempts: 5
If you purchase a latte every weekday, create a formula for a linear model for the balance, B, on your coffee card after t weekdays.
#12 Points possible: 5. Total attempts: 5
Use the linear model to calculate if your $50 Coffee Card will last until the end of the month.
The card will run out of money after weekdays. (Answer to the nearest whole day)
The value you found in the last question is called the horizontal intercept (sometimes called the xintercept). It is the point where the output value is zero and the graph crosses the horizontal axis.
HW 3.3
#1 Points possible: 20. Total attempts: 5
Which of the following representations depict a linear model? Justify your decision based on one of the three characteristics listed above.
a. R=t873.36−2.15R=t873.362.15 where R is the % of Americans who are retired in the year t. The graph is shown below.
Justification:
b. P = 40 + 2L where P = the perimeter of an area in yards and L = the length of the area in yards.
Justification:
c. The graph for braking distance versus speed for the coefficient of friction, f = 0.8, and roadway grade of 5% is shown below.
Justification:
d.
0 2
Time (min) 
Distance (ft) 
3.5 

13.5 

4 
23.5 
6 
33.5 
e.
Justification:
f. The formula for the area of a circle: A = πr2.
Justification:
#2 Points possible: 5. Total attempts: 5
A rental car company charges $45$45 plus 2020 cents per each mile driven.
Part1. Which of the following could be used to model the total cost of the rental where mm represents the miles driven.
· C=20m+45C=20m+45
· C=45m+20C=45m+20
· C=0.2m+45C=0.2m+45
· C=2m+45C=2m+45
· C=45m+0.2C=45m+0.2
Part 2. The total cost of driving 300300 miles is;
$
#3 Points possible: 12. Total attempts: 5
Indicate if each of the following statements is true or false.
a. The slope of a linear model is a ratio that describes how the outputs of the model increase or decrease as the inputs increase.
b. The slope of a linear model changes depending on the values substituted into the model.
c. The horizontal intercept is represented by an ordered pair in which the first value is 0: (0, __).
d. The vertical intercept is sometimes called a starting or initial value.
#4 Points possible: 15. Total attempts: 5
Sheila wants to lose weight for an upcoming wedding. She currently weighs 186 pounds and her goal is to weigh 140 pounds. After consulting with her doctor, she feels she can safely lose 2 pounds per week. The graph tracks the projected weight loss over time.
a. Write an equation for the weight loss trend. Use W = weight (lb) and t = time (weeks).
b. Use your equation to determine how long will it take Sheila to achieve her desired weight goal.
weeks
c. What is the slope (and select the correct units)?
#5 Points possible: 15. Total attempts: 5
Ben has $70 in his savings account. He plans to deposit $40 per week to build his account balance.
a. Complete the following equation to represent the amount of money (A) Ben will have in his account after any number of weeks. Let x represent the number of weeks.
A =
b. Which of the following values could be the value of the variable in this context?
· 3
· 0
· 18
· 5
c. Ben wants to use his savings to buy a computer for $750. Use your algebraic expression to determine the number of weeks it will take to buy the computer.
weeks
#6 Points possible: 6. Total attempts: 5
Suppose you work for a shuttle service. You see the following spreadsheet that gives information about what your company will charge for rides of different lengths. The formula shown is for cell B2. Complete the statement that tells how the fare is calculated based on the number of miles.
The fare is cents per mile plus a $ flat fee.
#7 Points possible: 20. Total attempts: 5
Doctors use intravenous (IV) drips to deliver fluids and medications to patients. Suppose an IV drip starts with 1,000 ml of saline and dispenses fluid at a rate of 2.5 ml per minute.
a. Write an equation for this situation using F to represent the amount of fluid (saline solution) left in milliliters and t to represent the time in minutes. Hint: If you have trouble writing a model, try making a table first.
b. Is this a linear model?
Explain your answer.
c. How long will it take to use all of the saline solution? Give you answer in hours and minutes.
hours and minutes
d. Which of the following can be used to describe the point you found in Part (c)?
· Horizontal intercept
· Vertical intercept
· Slope
· Rate of change
#8 Points possible: 15. Total attempts: 5
The Bureau of Labor and Statistics published the employment projections for the Heathcare industry in 2011.
Healthcare employment in 2008 was 14,336,000, and is projected to rise to 17,561,600 by 2018.
a. On average, how many healthcare workers will be added each year during this period?
workers/year
b. Which of the following can be used to describe the value you found in Part (a)?
· Horizontal intercept
· Vertical intercept
· Slope
c. Write a linear equation for this relationship. Let W = the number of healthcare workers and t = the number of years after 2008.
#9 Points possible: 5. Total attempts: 5
Give the slope and the yintercept of the line y=−8x−9y=8x9. Make sure the yintercept is written as a coordinate.
Slope =
yintercept =
#10 Points possible: 5. Total attempts: 5
Give the equation of the line with a slope of 2727 and a yintercept of 22.
#11 Points possible: 12. Total attempts: 5
For the line sketched below
1234512345246810246810xy
a. What is the value of b, the yintercept?
b =
b. What is the value of m, the slope?
m =
c. What is the equation for the line?
#12 Points possible: 8. Total attempts: 5
Sketch a graph of y=12x−2y=12x2
Clear All Draw:
Lesson 3.4
Introduction
Course Objectives
This lesson will address the following course outcomes:
·
2
0. Translate problems from a variety of contexts into mathematical representation and vice versa (linear, exponential, simple quadratics).
· 22. Identify when a linear model is reasonable for a given situation and, when appropriate, formulate a linear model. In the context of the situation interpret the slope and intercepts and determine the reasonable domain and range.
· 25. Use functional models to make predictions and solve problems.
Specific Objectives
Students will understand
· that linear models are appropriate when the situation has a constant rate of increase/decrease or can be approximated by a constant rate.
· that the rate of change (slope) has units in context.
· the difference between a positive slope and a negative slope.
· that the linear models for authentic situations have limitations in using them to make predictions.
Students will be able to
· make a linear model when given data or information in context.
· calculate a slope given data or information in context.
· estimate and calculate the value that makes two linear models equivalent.
Weight Loss
Problem Situation
1
: Weight Loss Challenge
Four friends decided to do a joint weight loss challenge. They weighed in at the start, and again after 3 or 4 weeks. Their results are shown in the graph below.
#1 Points possible: 5. Total attempts: 5
In this context, what would the rate of change (slope) mean? What does the vertical intercept mean? Give an answer, then compare it to ours.
Recall from the last lesson that we can compute rate of change (slope) as change of outputchange of inputchange of outputchange of input
#2 Points possible: 20. Total attempts: 5
Calculate each person’s rate of change. Give your answers as fractions or integers. Do not use mixed numbers.
Marcel:
Jonas:
Jamie:
Carlie:
The units on these numbers is:
#3 Points possible: 12. Total attempts: 5
Consider how the rate of change relates to the shape of the graph.
If the rate of change is positive, then the line will be
If the rate of change is negative, then the line will be
If the rate of change is zero, then the line will be
#4 Points possible: 5. Total attempts: 5
If Jamie continues losing weight at the same rate, find a formula for a linear model for Jamie’s weight, W, after t
weeks
.
#5 Points possible: 5. Total attempts: 5
Using your equation, predict how much Jamie will weigh after 5 weeks. Give your answer as a decimal, to the nearest pound.
pounds
#6 Points possible: 5. Total attempts: 5
Using your equation, predict how long it will take until she reaches her target weight of 173 pounds. Give your answer to one decimal place.
weeks
Slope From 2 Points
In the previous problem, you were able to calculate the rate of change by counting on the graph how much each person’s weight changed over 3 or 4 weeks. This approach works well with clear graphs or simple numbers, but becomes more problematic in other cases.
For example, suppose that another friend, Raj, decides to join the challenge. Being a mathematician and a technology fan, he has a much more accurate scale and tracks things more closely. Since he joined late, he doesn’t know what his weight was when the others started, but he knows after 0.5 weeks his weight was 193.4 pounds, and after 4 weeks his weight was 187.6.
To compute his rate of change, we have to determine the change in output (weight) and change in input (weeks). This is harder to just count from the graph, so instead we can use the values themselves and find the difference:
187.6 − 193.4 = −5.8 He lost 5.8 pounds
4 − 0.5 = 3.5 He lost it over 3.5 weeks
So the slope is change of outputchange of input=187.6−193.44−0.5=−5.83.5≈−1.66change of outputchange of input=187.6193.440.5=5.83.5≈1.66 pounds per week.
In general, if we have two points (x1, y1) and (x2, y2), then we can calculate the slope as
slope=change of outputchange of input=y2−y1x2−x1slope=change of outputchange of input=y2y1x2x1
It doesn’t matter which point you call (x1, y1) and which you call (x2, y2), but it is important that when calculating the differences, both start with values from the same point.
#7 Points possible: 5. Total attempts: 5
Find the slope between the points (50, 600) and (65, 480).
Slope =
Comparing Consumption
Problem Situation 2: Milk and Soft Drink Consumption
Since 1950, the U.S. perperson consumption of milk and soft drinks has changed drastically. For example, in 1950, the number of gallons of milk consumed per person was 36.4 gallons; in 2000 that number had decreased to 22.6 gallons. Meanwhile, the number of gallons of soft drinks consumed per person in 1950 was 10.8 gallons. By 2000, this number had increased to 49.3 gallons per person.
#8 Points possible: 5. Total attempts: 5
Graph the lines for milk consumption and soft drink consumption. Plotting the exact points will not be possible, so just do the best you can. The horizontal axis is in years after 1950 and the vertical axis is in gallons.
Clear All Draw:
#9 Points possible: 5. Total attempts: 5
From the graphs, estimate the year in which the consumption (per person) of milk equaled the consumption (per person) of soft drinks. Estimate to the nearest year.
#10 Points possible: 10. Total attempts: 5
Find equations for the linear models for milk consumption, M, and soda consumption, S, both in terms of t, years since 1950. Keep at least 3 decimal places on any values calculated.
Try the problem on your own first. If you are having trouble after 2 tries, we will break it down.
Equation for Milk consumption:
(this equation should involve M and t)
Equation for Soft drink consumption:
(this equation should involve S and t)
#11 Points possible: 8. Total attempts: 5
Use your equations to solve for the year when the consumption (per person) of milk equaled the consumption (per person) of soft drinks.
Algebraically, the answer is t = (to one decimal place).
This means that the consumption will be equal in the year
This is a good time to mention the limitations of the models. Although the vertical intercepts have meaning (the amounts consumed in 1950 if 1950 is time = 0), the horizontal intercepts do not always have meaning in reality. When the linear model for milk equals 0 (crosses the horizontal axis), this means that the milk consumption per person is none. However, in reality, the consumption will not likely go to 0 any time soon. So, there are limits to the time for which the model is reliable and accurate. There are limitations to using mathematical models (linear or nonlinear), but the models can still be extremely useful – as long as they are used for a period of time.
HW 3.4
#1 Points possible: 5. Total attempts: 5
Which of the following was one of the main mathematical ideas of the lesson?
· A slope can be written as a fraction or a decimal.
· A slope is a ratio with units that describes how two variables change in relationship to each other.
· A slope tells you how much one variable has changed.
· Milk consumption is decreasing as soft drink consumption increases.
#2 Points possible: 9. Total attempts: 5
You have learned about absolute change, relative change, and average rate of change. Identify which type of change best describes each of the following values and explain your answer.
a. 30%
b. $30/year
c. $30
#3 Points possible: 25. Total attempts: 5
The three graphs below model the relationship between time spent in class and time spent studying out of class for three students. Use the graphs to answer the following questions.
a. Which of the graphs has a slope of 2?
b. Which graph shows a relationship of 1 hour studying for each hour in class?
c. Which graph matches the following table?
Hours in class
1
4
7
9
Hours studying
2
8
14
18
d.
e. What is the equation for Graph B? Use C for hours in class and s for hours studying.
f. Which graph would contain the point (20, 10) if it were extended?
#4 Points possible: 15. Total attempts: 5
For each of the following, sketch a line that meets the conditions.
(a) Any line with a positive slope greater than 1
Clear All Draw:
(b) Any line with a positive slope smaller than 1
Clear All Draw:
(c) Any line with a slope of −1
Clear All Draw:
#5 Points possible: 15. Total attempts: 5
A parents’ group is writing a grant to support an afterschool program for their children. They want to make the point that government funding for afterschool programs in their school has decreased over the last decade while the cost of offering programs has increased. They have the data given below.
2000
2010
Government funding for afterschool programs ($/child)
$3,000
$1,800
Cost of offering afterschool programs ($/child)
$3,800
$5,100
a. Which of the following is the best interpretation of the average rate of change in the government funding?
· Government funding has decreased by an average of $1,800 per child each year.
· Government funding has changed by an average of $120 each year.
· Government funding has decreased by an average of $1,200 per child each year.
· Government funding has decreased by an average of $120 per child each year.
b. Complete the interpretation below of the average rate of change in the cost of offering afterschool programs.
The cost of offering afterschool programs has by an average of $ per child each year.
c. Sketch a graph showing linear models of the two sets of data, using years since 2000 on the horizontal axis.
$/child
Clear All Draw:
Years since 2000
d.
#6 Points possible: 24. Total attempts: 5
During the summer, Tamir likes to swim at his local pool a few times a week. He is debating if he should buy a “season pass” that allows him to swim as many times as he wants or just pay each time he goes swimming. The pool charges $4.50 each day if he pays each time. The “season pass” is $207, which is good for the entire summer (91 days).
a. Create a table to compare the two options by picking a set of inputs (Number of Swim Days) and calculating the corresponding cost of a season pass and cost of paying per day.
No. Swim Days
Cost of Season Pass ($)
Cost of Paying Per Day ($)
b.
c. How many times would Tamir have to swim to make the “season pass” less expensive than paying for each visit individually?
He would have to swim more than times
d. Which of the graphs below models each situation?
Season Pass: Paying per day:
#7 Points possible: 15. Total attempts: 5
Delilah wants to join a gym, so she shops around to find the one with the lowest overall price (she is not sure how long she will be a member). She finds the Harbor Square Athletic Club is running a special, and only charges a $25 initiation fee plus $87 a month to be a member. The local YMCA charges $100 to join, and has a monthly fee of $72.
a. Find the equations for the linear models for the costs of the Harbor Square Athletic Club and the YMCA. Use C = cost ($) and m = months.
Athletic Club:
YMCA:
b. Under what conditions would it be less expensive to join the YMCA?
The YMCA will be less expensive if she remains a member for months
#8 Points possible: 5. Total attempts: 5
Janey bought a used bicycle for $640. The bike was 3 years old when she bought it, and cost $850 new.
Assuming it decreases in value by the same amount each year, write a linear equation for the value of the bicycle, V, when it is t years old (so t = 0 was when it was new).
#9 Points possible: 20. Total attempts: 5
In Lesson 2.6, you used the following data about population changes in the United States. Suppose state planners in Indiana and Michigan think that the average rate of change will continue through 2020.
State
2000 Population
2010 Population
Michigan
9,938,444
9,883,640
Indiana
6,080,485
6,483,802
a. Write equations to model the populations for each state. Use M for the population of Michigan; I for the population of Indiana and t for time in years after 2000.
Michigan:
Indiana:
b. Use your equations to find the projected population for Michigan and Indiana in 2020.
Michigan:
Indiana:
#10 Points possible: 25. Total attempts: 5
Netflix made headlines in September 2011 when it split its streaming video service and its DVD bymail service into two companies. This decision was based on projections that streaming videos will replace DVDs in the future. In this problem, you will explore these projections.
Total DVD and Bluray disc sales in 2009 were $8.73 billion. For the purposes of this problem, you will combine DVD and Blueray disk sales into one category of DVD sales. A study by InStat predicted that “physical disc sales will decline by $4.6 billion by 2014,” and that “streaming, on the other hand, should grow from its current $2.3 billion to $6.3 billion over the same time period (2009–2014).”
1
2
a. What is the slope, including units, of the model for DVD sales?
Write a statement interpreting the meaning of the slope in the context of the problem.
b. Write an equation that represents the linear model for DVD sales. Let D = the sales of DVDs in billions of dollars, and t = the number of years after 2009.
c. What is the slope, including units, of the model for streaming videos?
Write a statement interpreting the meaning of the slope in the context of the problem.
d. Write an equation that represents the linear model for streaming videos. Let S = the total sales of streamed videos in billions of dollars, and t = the number of years after 2009.
e. In what year do the models predict streaming video sales to exceed DVD sales?
#11 Points possible: 5. Total attempts: 5
Given a line passing through (19,1)(19,1) and (11,17)(11,17), which of the following is the correct slope of the line?
· m=(17)−(−1)(11)−(−19)m=(17)(1)(11)(19)
· m=(11)−(19)(17)−(−1)m=(11)(19)(17)(1)
· m=(17)−(−1)(11)−(19)m=(17)(1)(11)(19)
· m=(17)−(1)(11)−(19)m=(17)(1)(11)(19)
· m=(11)−(19)(17)−(1)m=(11)(19)(17)(1)
· m=(11)−(−19)(17)−(1)m=(11)(19)(17)(1)
· m=(17)−(1)(11)−(−19)m=(17)(1)(11)(19)
· m=(11)−(−19)(17)−(−1)m=(11)(19)(17)(1)
#12 Points possible: 5. Total attempts: 5
Determine the slope of the line passing through the points (9,2)(9,2) and (−3,−1)(3,1).
m=m=
#13 Points possible: 5. Total attempts: 5
Determine the slope of the line passing through the points (7,−3)(7,3) and (−4,6)(4,6).
m=m=
#14 Points possible: 5. Total attempts: 5
Given the points (0,−5)(0,5) and (−9,−2)(9,2) and on a line, find its equation in the form y=mx+by=mx+b.
y=y=
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