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| 15 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Vincenzo and Magdalena are two friends eager to explore the world of gaming and programming. They are developing their own mobile game by coding and testing together. As part of their development process, they are monitoring the temperature of their computers. The diagram shows how the temperature of each changes over time.
A linear function is a function with a constant rate of change. A linear function can be represented by a linear equation in two variables. Graphically, a linear function is a nonvertical line.
What is the condition for being a function? What happens if the x-coordinates of M and N are the same.
Analyze the given graph and determine whether it represents a function. If it is a function then determine if it is a linear function or a nonlinear function.
Numerical data, also called quantitative data, is measurable data expressed using numbers. Some examples of numerical data are age, height, and weight of a person. Consider the following personal information.
Person | Age | Weight | Shoe Size | Number of Siblings | Eye Color |
---|---|---|---|---|---|
Dejen | 15 | 43.4 | 6 | 0 | Blue |
Madge | 16 | 56.7 | 7 | 3 | Green |
Yulia | 20 | 64.8 | 7.5 | 1 | Brown |
A person's age, weight, shoe size, and number of siblings are examples of numerical data. In contrast, a person's eye color cannot be expressed in numbers and is an example of categorical data. Additionally, numerical data can be either discrete data that can only take specific values or continuous data that can take any value within an interval.
Data | Numerical? | Type |
---|---|---|
Age | Yes ✓ | Continous |
Weight | Yes ✓ | Continous |
Shoe Size | Yes ✓ | Discrete |
Number of Siblings | Yes ✓ | Discrete |
Eye Color | No * | Non-applicable |
Shoe sizes, for example, can include half-sizes such as 7, 7.5, 8, 8.5, and 9. Shoe Sizes { 7,7.5,8,8.5,9} This provides more options for people to find a better fit. Another example can be a grading system where students can receive scores that include half-points. Scores in a Specific Grading System { 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5} The scores are still discrete because they are specific and separate values. Each score is distinct from the others and there are no values between them. Therefore, in the context of shoe sizes and grading systems, discrete quantities can indeed be fractions or decimal numbers.
A continuous quantity can take any value in a certain interval. A discrete quantity can only take distinct values, so there is a gap between possible values.
To determine if a variable is continuous or discrete, check if there is a gap between possible values. Analyze each given variable separately to reach an answer.
The number of levels is typically counted in whole numbers. It cannot take a real number value, so it is discrete.
The time it takes for a player to complete a level can be any real number. Time can be measured with high precision, including fractions of seconds. Therefore, T is continuous.
A player's score can typically take whole numbers. Scores are limited to whole numbers, so it is discrete.
The speed at which the character moves can be measured on a continuous scale. That is, it can take any real value, including fractions. Therefore, V is continuous.
In conclusion, discrete variables take on distinct values, often whole numbers. Continuous variables, on the other hand, can take any value within a given interval.
Variable | Can it take all values in a certain interval? | Type |
---|---|---|
L | No | Discrete |
T | Yes | Continuous |
S | No | Discrete |
V | Yes | Continuous |
A function is said to have a continuous domain when the independent variable of the function is a continuous quantity. The graph of such a function is typically a curve or a line. Consider the graph of the function, which shows the relationship between a person's age and weight.
A function is said to have a discrete domain when the independent variable of the function is a discrete quantity. A function of this type can be identified by its graph, which is composed of any number of unconnected points. Consider the function that models the costs of concert tickets, specifically for 0, 1, 2, 3, and 4 tickets.
Vincenzo and Magdalena are continuing the development of their own mobile game. In their game, players start with 100 virtual coins and spend 5 coins per day on in-game activities.
Days, x | Number of coins, y |
---|---|
2 | 90 |
4 | 80 |
6 | 70 |
8 | 60 |
10 | 50 |
The function have a discrete domain.
y = mx + b In this form, m is the slope and b is the y-intercept. In this case, the initial amount of virtual coins is 100, so b = 100. The spending rate is 5 coins per day, so the slope is - 5 because the coins are decreasing. Therefore, the linear function is as follows. y = - 5x + 100
Days x | - 5x+100 | Number of Coins y |
---|---|---|
2 | ||
4 | ||
6 | ||
8 | ||
10 |
How about finding the remaining number of coins for a player after 2, 4, 6, 8, and 10 days from starting the game? Try substituting the values of days into the formula and evaluate.
Days x | - 5x+100 | Number of Coins y |
---|---|---|
2 | - 5( 2) + 100 | 90 |
4 | - 5( 4) + 100 | 80 |
6 | - 5( 6) + 100 | 70 |
8 | - 5( 8) + 100 | 60 |
10 | - 5( 10) + 100 | 50 |
Great work so far!
Days x | Number of Coins y |
---|---|
2 | 90 |
4 | 80 |
6 | 70 |
8 | 60 |
10 | 50 |
Plot the ordered pairs (x,y) on a coordinate plane where the x-axis represents the number of days and the y-axis represents the remaining number of coins.
Since the function has a discrete domain, the points are not connected with a straight line. There are missing points in this graph. When other points are included, the graph of the function will be as follows.
As part of their development process, Vincenzo and Magdalena monitor the battery power consumption of a test laptop.
The table shows the percentage, y in decimal form, of battery power remaining x hours after the laptop is turned on.
Hours, x | 0 | 2 | 3 |
---|---|---|---|
Power Remaining, y | 1.0 | 0.4 | 0.1 |
Graph:
Hours, x | 0 | 2 | 3 |
---|---|---|---|
Power Remaining, y | 1.0 | 0.4 | 0.1 |
y= 0
Subtract term
.LHS /(- 0.3).=.RHS /(- 0.3).
a/b=a * (- 10)/b * (- 10)
Add fractions
Rewrite 3 +1/3 as 3 13
Rearrange equation
Rewrite 3 13h as 3h + 1/3h
1h=60 min
1/b* a = a/b
Calculate quotient
Rewrite 3h + 20min as 3h and20min
y= 0.25
Subtract term
.LHS /(- 0.3).=.RHS /(- 0.3).
Calculate quotient
LHS+3=RHS+3
Rearrange equation
Finally, the challenge presented at the beginning of the lesson will be solved. In the challenge, Vincenzo and Magdalena are monitoring the temperatures of their computers which tend to increase while running applications.
It is helpful to answer the questions by recalling the concepts and rules presented throughout the lesson.
Vincenzo's Computer | Magdalena's Computer | |
---|---|---|
y-intercept | 32 | 24 |
The initial temperature of the computer is 32^(∘)C. | The initial temperature of the computer is 24^(∘)C. | |
Slope | 6 | 8 |
Its temperature increases by 6^(∘)C per hour. | Its temperature increases by 8^(∘)C per hour. |
The y-intercept of the blue line is 32 and the y-intercept of the red line is 24. Note that the y-axis represents the temperature in Celsius degrees. Since the time is zero at those points, the y-intercepts represent the initial temperatures of the computers.
Vincenzo's Computer | Magdalena's Computer | |
---|---|---|
y-intercept | 32 | 24 |
The initial temperature of the computer is 32^(∘)C. | The initial temperature of the computer is 24^(∘)C. |
Now the slope of each line will be found. The Slope Formula can be used to find the slopes. m = y_2-y_1/x_2-x_1 It is necessary to determine one more point on each line in order to use the formula.
The blue line passes through the points (0,32) and (4,56), while the red line passes through the points (0,24) and (1,32). Substitute these points into the Slope Formula.
Blue Line | Red Line | |
---|---|---|
Points | ( 0, 32) and ( 4, 56) | ( 0, 24) and ( 1, 32) |
m = y_2-y_1/x_2-x_1 | 56- 32/4- 0 | 32- 24/1- 0 |
Evaluate | 24/4 = 6 | 8/1 = 8 |
In this context, the slope of the functions indicate how quickly the temperature of a computer rises within an hour.
Vincenzo's Computer | Magdalena's Computer | |
---|---|---|
Slope | 6 | 8 |
Its temperature increases by 6^(∘)C per hour. | Its temperature increases by 8^(∘)C per hour. |
Magdalena's computer temperature increases faster than Vincenzo's when both are running the applications. Note also that both functions have continuous domains as the inputs can take any value.
Blue Line | Red Line | |
---|---|---|
y-intercept | 32 | 24 |
Slope | 6 | 8 |
Using these values, the equation of each line can be written in slope-intercept form. y= mx+ b In this form, m is the slope and b is the y-intercept.
Blue Line | Red Line | |
---|---|---|
y-intercept | 32 | 24 |
Slope | 6 | 8 |
Equation | y= 6x+ 32 | y= 8x+ 24 |
The blue line corresponds to Vincenzo's computer temperature, whereas the red line corresponds to Magdalena's computer temperature.
The tables show conversions between particular units of weight.
Let's start by writing a function for each given table. The first table relates the number of cups and ounces. We can represent the number of cups with c and the number of ounces with u. Notice that the number of ounces u is 8 times the number of cups u.
Cups, c | 8c | Ounces, u |
---|---|---|
1 | 8( 1) | 8 |
2 | 8( 2) | 16 |
3 | 8( 3) | 24 |
4 | 8( 4) | 32 |
We can conclude that 8c is equal to u. u=8c Therefore, the first statement is false. Let's continue to the next table. The second table relates the number of pints and ounces. Let the number of pints be p and the number of ounces be u. Notice that the number of ounces u is 16 times the number of pints p.
Pints, p | 16p | Ounces, u |
---|---|---|
1 | 16( 1) | 16 |
2 | 16( 2) | 32 |
3 | 16( 3) | 48 |
4 | 16( 4) | 64 |
Thus, the table can be represented as a function as shown. u=16p Let's analyze the last table. In this table, the relation between quarts and ounces is shown. We will represent the number of quarts with q and the number of ounces with u. Notice that if we multiply the number of quarts q by 32, we will get the number of ounces u.
Quarts, q | 32q | Ounces, u |
---|---|---|
1 | 32( 1) | 32 |
2 | 32( 2) | 64 |
3 | 32( 3) | 96 |
4 | 32( 4) | 128 |
Therefore, we can write the following equation. u=32q This equation is the same as the one in Statement II, so it is true. Now, let's find the rate of change of each function. Since the functions we wrote are linear functions, the rate of change is constant and refers to the slope of the function's graph.
We see that the rate of change of the function for Table III has the highest rate of change. 8 < 16 < 32 Therefore, its graph have the steepest slope. As a result, Statement III is false and Statement IV is true. Answer: IIandIV