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| 13 Theory slides |
| 9 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.
Integers are whole numbers and negative numbers without fractions or decimal parts. The following are some examples of integers and non-integers.
Integers | Non-integers |
---|---|
17, 0, -1, 121, -67 | π, -43, 1.5, -12.9, 21 |
An integer can be positive, negative or zero. The set of all integers is denoted by Z.
Z={…-2,-1,0,1,2…}
Similar to natural and whole numbers, integer numbers can be displayed on a number line. This helps to compare numbers and visualize which are greater and smaller.
A number line can display integer numbers by extending it on the left- and right-hand side of zero. The positive integers or natural numbers will be on the right-hand side, and the negative integers will be on the left-hand side of zero.
Determine the integer represented by the point on the number line.
Maya is attending a teen-based summer camp. She is now taking part in a nature scavenger hunt. In this game, players are divided into teams; each has a map, a list of items to find, and tasks to complete. Each action, not only completed task, is worth points.
Action or Task | Points |
---|---|
Find a hidden item (I) | 2 |
Cheating (C) | -7 |
Complete a physical challenge (P) | 4 |
Solve a puzzle (S) | 7 |
Take a wrong turn (W) | -3 |
Run out of time before finishing a task (T) | -5 |
The team that gets more points will be the winner. Maya wants a better idea of the meaning of these points to help her team win. She thinks that plotting them on a number line should help. Which of the following graphs matches Maya's?
Identify if the points given by each action or task are negative or positive. When the number is positive, count that number of units going to the right of 0 on the number line. On the other hand, if the number of points is negative, count that number of units to the left of 0.
Maya is loving the summer camp. The staff even gave her what they call an explorer's box. One of the items in it is a thermometer. Her grandma used one just like it! Maya is excited to record temperatures all week wherever they go.
She created a table with the temperatures she recorded.
Temperatures Measured |
---|
-1∘C |
9∘C |
-4∘C |
3∘C |
12∘C |
A number line helps to compare numbers. The smaller numbers are further left and the greater ones are further right. Now, because the list contains positive and negative integers, the number line must contain 0.
The first recorded temperature is -1∘C. Move one unit to the left of zero and draw a point over the number line to draw this temperature.
The second recorded temperature is 9∘C. Plot this temperature by following a similar method. In this case, move 9 units to the right-hand side of 0.
The graph shows that the first temperature Maya recorded is less than the second. Now, plot the remaining temperatures following a similar reasoning.
The number line shows that the coldest recorded temperature was -4∘C and the warmest was 12∘C. The temperatures can now be written from the coldest to the warmest.The additive inverse of a number is another number such that their sum equals 0. If y is the additive inverse of x, then the following equation holds true.
x+y=0
Given a number, its additive inverse — also called opposite number — can be found by changing the sign. Some examples of additive inverses are listed in the table.
Number | Additive Inverse | Sum |
---|---|---|
5 | -5 | 5+(-5)=0 |
-21 | 21 | -21+21=0 |
0 | -0 | 0+(-0)=0 |
a | -a | a+(-a)=0 |
-b | b | -b+b=0 |
The additive inverse of a number is another number such that the sum of these two numbers equals 0. Use this information to find the additive inverse of the indicated number.
It is common to compare quantities that express distances. They help identify which quantity is further than another. Knowing which number represents a further distance is simple if the numbers are positive. What if they are negative? No fears — the absolute value comes to the rescue.
For any integer number a, these two properties hold true.
Property | Algebraic Representation | Example |
---|---|---|
Non-negativity | ∣a∣≥0 | ∣7∣=7 and 7≥0 |
Symmetry | ∣-a∣=∣a∣ | ∣-7∣=7 and ∣7∣=7∣-7∣=∣7∣ |
Time to identify and compare numbers. See the given statement and evaluate if it is true or false. Some statements involve absolute values.
Near the camp is an animal rehabilitation center. There, the animals have space to roam. Maya gets the chance to visit! She takes photos of the animals and notes their elevation compared to her standing position.
Animal | Elevation (In Feet) |
---|---|
Kingfisher | 10 feet |
River otter | -8 feet |
Porcupine | 7 feet |
Sturgeon | -12 feet |
Painted turtle | -3 feet |
Use the absolute values to order the elevations of the animals from Maya's position.
The elevations contain negative and positive integers. Apply the absolute value of the elevations to get the distances the animals are from Maya's position.
Animal | Elevation (In Feet) |
---|---|
Kingfisher | 10 feet |
River otter | -8 feet |
Porcupine | 7 feet |
Sturgeon | -12 feet |
Painted turtle | -3 feet |
The absolute value of a negative number is its additive inverse. In contrast, the absolute value of a positive number is equal to itself. Now, move 10 units up from 0 in a vertical number line to display the distance of the Kingfisher.
Next, the river otter's elevation is -8 feet. Its absolute value is ∣-8∣=8. That means the river otter's distance is 8 units from 0 in the positive direction.
Use a similar reasoning to identify the remaining distances.
The graph shows that the distance of the sturgeon is the furthest. That is even with a negative elevation! On the other side of the spectrum, the painted turtle is the closest.
Many campers trust Maya's math skills and entrust their money to her. Vincenzo, has already given her $20. He then goes to the gift shop and orders a custom bear sculpture. Before paying, he runs to Maya to retrieve his $20. Maya sees the ticket and notices that the bear sculpture costs $30. She shows his total balance to Vincenzo.
What does this amount represent? The negative amount in Vincenzo's balance means he has a debt or money owed. The amount he is in debt is given by the absolute value of the total.Balance | Meaning |
---|---|
Negative | Debt or money owed |
0 | No debt and no credit |
Positive | Cash and credit |
Vincenzo needs to pay that $10 because the bear sculpture costs more than he actually has. Scouring for loose change, he finds $10 in a secret pocket in his backpack. That bear sculpture is all his!
Read each situation carefully and determine the integer number that represents it.
Let's consider the given situation.
A student loses 4 points for not doing her homework.
A loss indicates a negative number. This means we must use a negative integer to represent these 4 points the student lost for not doing her homework. The negative integer for this situation is -4.
Let's look at the given situation.
A football team gains 3 yards.
A gain indicates a number greater than 0. We then need a positive integer to represent this situation. This integer is 3.
Let's now analyze the third situation.
The temperature outside is 6 degrees below zero.
This situation describes a temperature below zero. Below zero means a number less than 0 on a number line. We must represent this situation with a negative integer. This integer is -6.
Find the integer that represents each of the given situations.
Let's begin by looking at the given situation.
Tiffaniqua has a debt of $50.
Debt means that we owe money. This represents a negative situation. Therefore, we represent this situation with a negative number. This integer is -50.
Consider the given situation.
A diver is 20 meters below sea level.
In this case, the sea level represents the starting point of 0. The diver is below sea level, a number less than 0. This means that we need to represent this situation with the integer -20.
Finally, let's analyze the third situation.
A hiker climbs 300 meters up a mountain.
For this situation, the bottom or base of the mountain represents the starting point of 0. Because the hiker is above 0, it represents a positive number. This integer is 300.
We can use a number line to plot the numbers. The smaller numbers will be more left and the greater righter. With this in mind, let's look at the first number in the list. |-6| Recall that the absolute value of a number is the distance between that number and 0. In addition, the absolute value of a negative number is its additive inverse. Because the additive inverse of -6 is 6, let's plot |-6| over the number 6.
The following number in the list is 0. Since we do not move any unit to the right or left of 0, we plot a point over the number 0 to represent this number.
We can now follow a similar process to graph the remaining points on the number line.
The numbers are now ordered on the number line. The more left is -10, meaning it is the smallest number in the list. In contrast, |13| is the rightest, meaning it is the greatest. Let's write all the given numbers from the smallest to the greatest. The Numbers From the Smallest to the Greatest -10, -8, 0, |-6|, 10, |13|
The table contains the average winter temperature of some states of the US.
State | Average Winter Temperature |
---|---|
Pennsylvania, PA | -2∘C |
Georgia, GA | 9∘C |
Tennessee, TN | 4∘C |
New York, NY | -5∘C |
Maine, ME | -8∘C |
Let's begin by graphing the temperatures on a number line. Consider that the smallest temperatures will be more left and the higher will be further right. The first temperature is for Pennsylvania, PA. This place has an average winter temperature of -2^(∘) C. Let's draw a point on -2 on the number line.
Georgia, GA, has an average winter temperature of 9^(∘) C. Let's plot a point on the number 9 to represent this temperature.
We can now plot the remaining temperatures following a similar fashion.
Now that we graphed the temperatures, we can see that Maine has the lowest average winter temperature. At the same time, Georgia is the warmest place. Let's write the places according to their temperature in ascending order. Maine, New York, Pennsylvania, Tennessee, Georgia
Simplify the following absolute value expressions.
Let's look at the given expression. |35| We are asked to find the absolute value of 35, which is a positive integer. The absolute value of a positive integer is equal to itself. This means that the absolute value of 35 is 35. |35|=35
Consider the given expression.
|-13|
This expression asks for the absolute value of -13. The absolute value of a negative integer is its additive inverse. Change the sign of a number to find its additive inverse. The additive inverse of -13 is then 13. We can now write the absolute value of -13.
|-13|=13
This expression asks for the absolute value of -7 and then applies to it a negative sign.
-|-7|
We first calculate the absolute value. The opposite of -7 is 7. This means that |-7|=7. Next, this result is transformed to negative, representing the value that simplifies the given expression.
-|-7|=-7
Finally, let's look at the last expression.
-|15|
Again, we first calculate the absolute value. 15 is a positive number, which means its absolute value is itself. The negative sign outside the absolute value transforms it into a negative. We can then write the value to this expression.
-|15|=-15