Mathematical Expressions

Subsection 1.7.1 Mathematical Expressions

Essential Knowledge 1.7.1 1.7.A.1.

Apps use mathematical expressions to calculate values that help users make decisions, such as prices, time estimates, or progress toward a goal.

In the last lesson, we assigned values to variables like score = 0, but the right side of an assignment statement can be more than just a simple value like 0. We can use expressions which combine numbers, variables, and operators to calculate a value and assign it to the variable in an assignment statement like below.

This looks like a mathematical formula, but in coding, an assignment statement always has a single variable on the left side and an expression on the right side. The expression is evaluated and then assigned to the variable.

Activity 1.7.1.

Subsection 1.7.2 Updating State with Expressions

Essential Knowledge 1.7.2 1.7.A.2.

Mathematical expressions can be used to update the state of a program by changing values stored in variables, such as scores, timers, levels, and resource usage.

If you use a variable to keep score, you would probably increment the score (add one to the current value) as the game progresses. You can do this by setting the variable to the current value of the variable plus one (score = score + 1) as shown below. The formula would look strange in math class, but it makes sense in coding because it is assigning a new value to the variable on the left that comes from evaluating the arithmetic expression on the right. So, the score variable is set to the previous value of score plus 1.

Activity 1.7.2.

Try the code below to see how score is incremented by 1. Try substituting 2 instead of 1 to increment by 2.

Essential Knowledge 1.7.3 1.7.A.3.

An integer is a numeric data type that represents positive and negative whole numbers, including zero. Integers are often used for counts or scores in a program.

In Python, you do not need to indicate the data type of a variable, but variable that are assigned whole number values like score = 5 are automatically type int for integer.

Subsection 1.7.3 Arithmetic Operators

Essential Knowledge 1.7.4 1.7.A.4.

Arithmetic operators appear in most programming languages, but the notation used for those operations is determined by the language. Common arithmetic operators include addition +, subtraction -, multiplication *, division / and modulo %.

Python uses the following basic arithmetic operators:

Table 1.7.5. Python Basic Arithmetic Operators

Operator	Meaning
+	Addition
-	Subtraction
*	Multiplication
/	Division

Activity 1.7.3.

Try the code below to see how the basic arithmetic operators work in Python. Add two more lines to the code as described.

Python also has the following non-standard operators:

Table 1.7.6. Python Special Operators

Operator	Meaning
//	Integer Division (everything after the decimal point is removed)
%	Modulo (integer remainder after integer division)
**	Exponent (raises a number to a power)

Essential Knowledge 1.7.7 1.7.A.5.

The modulo operation % evaluates to the remainder when one integer is divided by another: a % b is the remainder when a is divided by b. Modulo can be used to identify patterns in numbers, such as whether a number is even or odd, and to help format values in user-friendly ways, such as converting total minutes into hours and minutes.

The modulo (mod) or remainder operator is very useful in coding. For example, it can be used to check if a number is even (divisible by 2) if it has no remainders when divided by 2. The expression 5 % 2 is equal to 1, because 5 divided by 2 is equal to 2 with a remainder of 1 (since 2 * 2 = 4).

Activity 1.7.4.

Try the code below to see how the special arithmetic operators work in Python. Compare the result of 5/2 and 5//2 and the remainder 5%2. Add 1 more line to the code as described.

Activity 1.7.5.

What is the result of 2 % 6?

0
This would be correct if it was 6 % 2.
1
This would be correct if it was some odd number divided by 2, but it is not.
2
6 goes into 2 zero times with 2 left over. If you have a smaller number divided by a larger number the remainder is always the smaller number.
6
If you have a smaller number divided by a larger number the remainder is always the smaller number.

If you have a smaller number divided by a larger number the remainder is always the smaller number.

Essential Knowledge 1.7.8 1.7.A.6.

Expressions enclosed in parentheses are evaluated first. Then, multiplication *, division /, and modulo % operations are evaluated from left to right. Then, addition + and subtraction - operations are evaluated from left to right.

If there is more than one math operator in an expression, operator precedence rules are used, just like in math, so that multiplication *, division /, and remainder % are done before addition + and subtraction -. Anything in parentheses is done first. The PEMDAS acronym can help you remember the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (from left to right). It doesn’t hurt to put in extra parentheses if you are unsure as to what will be done first or just to make it more clear.

Activity 1.7.6.

What is the result of 1 + 3 * (2 + 1)?

12
Remember PEMDAS! Don’t do the addition first.
9
Remember to do the parentheses first.
15
Remember PEMDAS
10
Yes, 1 + 3 * (2 + 1) = 1 + 3 * 3 = 1 + 9 = 10

Compound assignment operators +=, -=, *=, /=, and %= are shortcuts that do a math operation and assignment in one step. For example, x += 1 adds 1 to the current value of x and assigns the result back to x. It is the same as x = x + 1. This pattern is possible with any operator put in front of the = sign.

Table 1.7.9. Compound Assignment Operators

Operator	Example	Equivalent to
+=	x += 1	x = x + 1
-=	x -= 1	x = x - 1
*=	x *= 2	x = x * 2
/=	x /= 2	x = x / 2
%=	x %= 2	x = x % 2

Essential Knowledge 1.7.10 1.7.A.7.

The result of a mathematical expression can be determined by evaluating it step by step using the order of operations and operators.

Essential Knowledge 1.7.11 1.7.A.8.

Mathematical expressions can include variables. To determine the result, substitute the current values of variables and then evaluate the expression using the order of operations.

When tracing through code with expressions, always remember to evaluate the right-hand side of the assignment first, then assign the result to the variable on the left-hand side. You can practice and check your answers by adding print statements to show the variables or click on the "Show CodeLens" button on any Active Code exercise to turn on the interactive tracing tool called CodeLens as seen below. This is a tool called Python Tutor that is built into Runestone.

Activity 1.7.7.

Keep clicking on the Next button at the bottom of the code to see how the values of the variables change as you step through the running program. You can see interactive tracing in any Active Code exercise by clicking the CodeLens button.

Activity 1.7.8.

What are the values of x and y after the following code executes? Remember to evaluate the right hand side of = first and then assign the result to the variable on the left side. You can step through this code by clicking on this visualizer link.

a = 2
b = 10
a = b
a = a + 1
b = b * 3

a = 10, b = 30
Remember that a is incremented.
a = 11, b = 33
No, b’s value is copied into a but they are not connected after that.
a = 11, b = 30
Correct!
a = 10, b = 33
Try the visualizer to see the step-by-step evaluation.

Subsection 1.7.4 Input and Math

Many algorithms that use calculations follow a simple pattern:

Input: Get input from the user and save it into variables.
Calculate: Calculate the result based on the input.
Output: Print the result.

Activity 1.7.9.

Put the following code into the correct order. The code should compute and print the salary based on the hours worked and the hourly pay rate provided by the user. Notice how the input is converted to the int or float data type so that it can be used in the calculation.

hoursWorked = int(input("How many hours did you work? "))
hourlyRate = float(input("What is your hourly rate? "))
---
salary = hoursWorked * hourlyRate
---
print("Your salary is:", salary)

When you use the input() command to get input from a user, the value is always treated as a string, even if the user types in a number. If you want to use that input in a mathematical expression, you need to convert it to a number first using the int() command for integers or float() command for decimal numbers. For example:

ageStr = input("How old are you? ")
age = int(ageStr)
# Now we can do math with the age variable since it is an int instead of a string
print("Next year you will be", age + 1, "years old.")

Or you can get the input and convert it in one line by embedding the input in the conversion function:

# Get the age from the user and convert it to an int in one line
age = int(input("How old are you? "))

Activity 1.7.10.

Try the code below to see that there is an error when you try to do math with a string. Then, fix the code by converting the input to an int before doing the math.

Let’s practice getting input, converting it, and using it in a calculation below. The code below also demonstrates how to format a number to 2 decimal places using the round(num, 2) function. This function rounds the number to 2 decimal places, for example 2.555555 will be rounded to 2.56. Another option is to use format(num, ".2f").

# rounding a variable to 2 float or decimal places
cost = 10.666666
print("Cost is $", round(cost, 2))

Activity 1.7.11.

In the code below, ask the user for how many pieces of candy they want to buy. Assume that each piece of candy costs $0.25. Then, calculate the total cost and print it out.

In order to compute the number of hours in a given number of minutes, you can divide the number of minutes by 60, since there are 60 minutes in an hour. The remainder operator % can be used to calculate the remaining minutes after the hours are calculated. For example, if there are 150 minutes, then that is 2 hours and a remainder of 30 minutes.

Activity 1.7.12.

The following program segment should calculate the number of hours and then the remaining minutes in 150 minutes, but the blocks are mixed up. For example, if there are 150 minutes, then that is 2 hours and a remainder of 30 minutes. Remember that // in Python is integer division that returns a whole number, / is float division, and % is the remainder operator. Drag the blocks from the left and put them in the correct order on the right. There will be 2 extra blocks that you will not use. Click the Check Me button to check your solution.

minutes = 150
---
hours = minutes // 60
---
hours = minutes / 60 #paired
---
remainingMinutes = minutes % 60
---
remainingMinutes = minutes / 60 #paired

Activity 1.7.13.

Write a program that asks the user for the number of minutes they have spent in class and then calculate the number of hours, remaining minutes, and seconds. Fill in the formulas for 1, 2, and 3 below.

# 1. Ask the user for the number of minutes they have spent in class
minutesStr = input("How many minutes have you spent in class? ")
# convert the input to an int
minutes = int(minutesStr)

# 1. calculate the hours in minutes by using integer division // (there are 60 minutes in an hour)
hours = 
# 2. calculate the remaining minutes in minutes using the mod % operator and 60
remainingMinutes = 
# 3. calculate the number of seconds in minutes with multiplication (there are 60 seconds in a minute)
seconds =

# print the results
print("You have spent", hours, "hour(s) and", remainingMinutes, "minute(s), and a total of", seconds, "second(s) in class.")
====
from unittest.gui import TestCaseGui

class myTests(TestCaseGui):
  def test1(self):
    self.assertIn("minutes//60",self.getEditorText().replace(" ", ""),"Use minutes//60 to calculate hours")
    self.assertIn("minutes%60",self.getEditorText().replace(" ", ""),"Use minutes%60 to calculate remaining minutes")
    self.assertIn("minutes*60",self.getEditorText().replace(" ", ""),"Use minutes*60 to calculate seconds")

myTests().main()

Subsection 1.7.5 Random Numbers

Essential Knowledge 1.7.12 1.7.B.1.

Apps use random number generation to create variety and unpredictability, such as in games, simulations, or randomly generated prompts.

Games would be boring if the same thing happened each time you played the game. Random numbers can be used in games to generate different actions or outcomes each time the game is played. Try a game app on your phone or computer and try to identify where random numbers are used. If you roll dice or flip a coin or if the enemies in the game do not always act the same way, then random numbers are likely being used.

Essential Knowledge 1.7.13 1.7.B.2.

Random values can support variation in app behavior, such as shuffling a playlist, selecting a random item, or assigning outcomes like rolling a die.

Essential Knowledge 1.7.14 1.7.B.3.

Programmers can determine the set of possible values produced by a random command by identifying the endpoints and whether they are included. When the following code runs, a random integer from a to b, including a and b, is generated and returned. Each result is equally likely to occur. The possible results of random(a, b) are all integers from a through b.

In Python, there is a library called random that can be imported and used to generate random values. Use random.randint(start, end) to generate a random integer from start to end, including both endpoints. For example:

import random
# pick a random number from 0 to 10
r = random.randint(0, 10)

Try running the program below a few times to see that it generates different random numbers each time it runs.

Activity 1.7.14.

Try the code below to see how to generate random numbers in Python. Run it a few times to see that it generates different random numbers each time. Then, generate another random number from -10 to 10.

Essential Knowledge 1.7.15 1.7.B.4.

When a program uses random number generation, the same code can produce different results each time it runs.

Computers can’t usually generate truly random numbers without measuring something that is random (like static noise on radio frequencies or lava lamps). If they don’t have access to something like that, they must rely on generating pseudorandom numbers that look random even though they are created using a mathematical formula. Python takes the system time as the seed or starting point (since it is always changing) and then uses a mathematical formula to generate the next random number each time we ask for one.

Subsection 1.7.6 Coding Challenge: Random Turtle Project

Let’s practice with turtles drawing with random colors and random movements. Colors on computers and on the web can be set using RGB (red, green, blue) values that range from 0 to 255 (which can be saved in 8 bits). Try the Color Picker and look at the RGB values for different colors below the selected color box. Any color can be made by mixing different amounts of red, green, and blue light. Mixing light is different than mixing paint, so when you mix red, green, and blue light, you get white light.

Activity 1.7.15.

What RGB values do you need to make black? Try the Google color picker to test.

0, 0, 0
Correct! Black light is the default state of no light. For example at nighttime, all is black because of there is no sunlight.
255, 255, 255
This is white, not black. If you mix all three primary RGB colors of light, you get white.
255, 0, 0
This is red light, not black. Red light is made with all red and no green or blue.
0, 255, 255
This is cyan light, not black. Cyan light is made with all green and blue and no red.

Activity 1.7.16.

The assignment statement x = random.randint(-200, 200) can be used to generate a random x-coordinate for a turtle, because it is initially placed at coordinates (0, 0) and can move 200 pixels to the right to 200 or left to -200. Which of the following numbers cannot be generated by this expression?

0
0 is between -200 and 200, so it can be generated.
-200
-200 is between -200 and 200, so it can be generated.
200
200 is between -200 and 200, so it can be generated.
-255
-255 is not between -200 and 200, so it cannot be generated by random.randint(-200, 200).

Project 1.7.17 Random Turtle Project.

In the code below, you will use random numbers to set the background color, pen color, and turtle coordinates. Right now it uses default colors. Change the lines of code that say "# Change these to use random.randint". Also calculate the distance that the turtle has traveled. Run the code a few times to see that it generates different random colors and movements each time it runs.

import turtle
import random
screen = turtle.Screen()
turt = turtle.Turtle()
turt.shape("turtle")
# total_distance keeps track of the total distance the turtle has traveled
total_distance = 0
# Do not repeat the code above

# Repeat code from here
# Choose random RGB values for a light background color from 150 to 255
# You can choose different ranges within that for r, g, and b.
r = 255    # TODO: Change these to use random.randint(start,end)
g = 255
b = 255
# Set the background color using the random RGB values (scaled by /255)
screen.bgcolor(r/255, g/255, b/255)

# Choose random RGB values for a dark pen color from 0 to 255 for 2 of the colors
# and 0 to 100 for the other color to make it darker.
r = 0    # TODO: Change these to use random.randint(start,end)
g = 0
b = 0
# Set the pen color using the random RGB values (scaled by / 255)
turt.color(r/255, g/255, b/255)

# Choose a random x and y coordinate from -200 to 200
x = 100   # TODO: Change these to use random.randint(start,end)
y = 100
turt.begin_fill()  # Start filling the shape with the pen color
turt.goto(x, y)  # Move the turtle to the random (x, y) coordinates
# distance from the origin (0, 0) to the random (x, y) coordinates
distance_from_origin = turt.distance(0, 0)

# Choose a random angle from 80 to 100 and a random distance from 50 to 100
angle = 45   # TODO: Change these to use random.randint(start,end)
distance = 50
turt.right(angle)       # Turn the turtle right by the random angle
turt.forward(distance)  # Move the turtle forward by the random distance
turt.goto(0, 0)         # Go back to the origin (0, 0)
turt.end_fill()         # Stop filling the shape with the pen color

# TODO: Add to total_distance, 2 times distance_from_origin + distance

print("Total distance: ", format(total_distance, ".2f"))

# TODO: Copy the code from line 10 to 45 (or up to the line above) 
#       and paste below two more times
====
from unittest.gui import TestCaseGui

class myTests(TestCaseGui):
    def findLine(self, lines, target):
        results = []
        for line in lines:
            if "#" in line:
                index = line.find("#")
                line = line[0:index]
            if target in line:
                #print(line)
                results.append(line)
        return list(set(results))
    
    def remove_comments(self, code):
        lines = code.split("\n")
        new_lines = []
        for line in lines:
            if "#" in line:
                line = line.split("#")[0]
            new_lines.append(line)
        return "\n".join(new_lines)

def test1(self):
        code = self.getEditorText()
        code = self.remove_comments(code)
        code = code.replace(" ", "")

count = code.count("randint")
        self.assertLessEqual(30,count, "at least 30 randint used")

# background color
        count = code.count("=random.randint(150,255)")
        self.assertLessEqual(9,count, "randint(150, 255) used to determine background colors")

# dark pen color
        count = code.count("=random.randint(0,100)")
        count += code.count("=random.randint(0,255)")
        self.assertLessEqual(9,count, "randint(0, 100) used to determine dark pen colors")

# distance
        count = code.count("=random.randint(-200,200)")
        self.assertLessEqual(6,count, "randint(-200, 200) used to determine x and y coordinates")

# angle
        count = code.count("angle=random.randint(80,100)")
        self.assertLessEqual(3,count, "randint(80, 100) used to determine angles")

# distance
        count = code.count("distance=random.randint(50,100)")
        self.assertLessEqual(3,count, "randint(50, 100) used to determine distances")

# distance
        #count = code.count("total_distance+=2*distance_from_origin+distance")
        lines = self.findLine(code.split("\n"), "total_distance")
        for i in range(len(lines)-1, -1, -1):
            line = lines[i]
            if "=0" in line or "print" in line:
                print(lines[i])
                lines.pop(i)

if len(lines) == 0:
            self.assertTrue(False, "total_distance calculated correctly 3 times (total_distance += 2*distance_from_origin + distance)")
        else:
            line = lines[0]

print(line)

hasMult = "*" in line
            hasAdd = line.count("+") >= 1
            hasDistanceFromOrigin = "distance_from_origin" in line
            passes = hasMult and hasAdd and hasDistanceFromOrigin

self.assertTrue(passes, "total_distance calculated correctly 3 times (total_distance += 2*distance_from_origin + distance)")

# printing distance
        output = self.getOutput()
        count = output.count("Total distance:")
        self.assertLessEqual(3,count, "\"Total distance:\" printed three times")
      
myTests().main()

Activity 1.7.18 Project Reflection.

Describe how you used random numbers in your turtle project. What was the range of random numbers you used for the RGB values and the x, y coordinates? How did the random numbers enhance the project?

Subsection 1.7.7 Learning Objectives Review

After this lesson, you should be able to:

Learning Objective 1.7.16 1.7.A.

Develop code using mathematical expressions, and determine the result of these expressions.

Learning Objective 1.7.17 1.7.B.

Develop code to generate random numbers, and determine the possible values that could be produced as a result.

Subsection 1.7.8 Vocabulary Review

Activity 1.7.19.

Review the vocabulary in this lesson. Drag the vocabulary term from the left and drop it on its correct definition on the right. Click the "Check Me" button to see if you are correct.

Find the words in the lesson above and try again.

expression
combination of numbers, variables, and operators that represents a value
increment
to increase a variable’s value, often by adding 1 to it
integer
a numeric data type that represents positive and negative whole numbers, including zero.
modulo (%)
operator that returns the remainder of a division operation
random.randint
Python function that generates a random integer between two specified values (inclusive)

Subsection 1.7.9 App Inventor Project Continued (Optional)

If your class started the Paint Pot app using MIT’s App Inventor, you can now finish the Paint Pot Tutorial part 2, enhancements, and reflections.

Section 1.7 Mathematical Expressions