Unit C Lesson 8-1 Integers and the Number Line

Focus Question: What are integers? Why do we need integers? What do integers allow you to do that whole numbers do not?

Key ConceptPositive Numbers Numbers to the right of zero are positive numbers. Numbers without a sign symbol are assumed to be positive.

Negative Numbers Numbers to the left of zero are negative numbers. Negative numbers are written with a negative sign before the number.

Zero Zero is neither positive nor negative.

Opposites Two numbes that are the same distance from zero on a number line but in opposite directions are called opposites. The opposite of 0 is 0. For example, – 4 and 4 are opposites.

Integers The integers are the set of positive whole numbers, their opposites, and zero.

Part 1 ExampleUse an integer to name the location of each point. Then find the opposite of each integer represented by each point.

Point A Point B Point C Opposite of Point A Opposite of Point B Opposite of Point C

Part 2 ExampleWhich integer represents each situation?a. 5 degrees below zero b. a gain of 8 yards

c. a loss of 6 dollars d. 3 steps backwards

Part 3 ExampleThe highest point on land on Earth is the peak of Mount Everest. Its elevation is 29,028 ft above sea level. The lowest point on land on Earth is the Dead Sea. Its elevation is 1,312 ft below sea level. Express each elevation as an integer. Explain the meaning of zero in each situation.Mount Everest

Dead Sea

Additional Practice1. A meteorologist forecasts the temperature to be 18°C above zero. The integer 18 represents this situation. Find the opposite of this integer. Then give the situation this opposite represents.

The opposite of 18°C is _______˚C.

Which of the following represents the opposite of this situation?o A. The meteorologist forecasts the temperature to be 17°C below zero.o B. The meteorologist forecasts the temperature to be 18°C above zero.o C. The meteorologist forecasts the temperature to be 19°C above zero.o D. The meteorologist forecasts the temperature to be 18°C below zero.

2. What integer does point A represent?

What is the opposite of the integer point A represents?

3. Think About the Process A contestant in a game show has 6,000 points. The contestant answers the next question incorrectly and loses 350 points.

Does the word “lose” suggest a positive integer or negative integer?

What integer represents a loss of 350 points?

4. The news reports that the temperature for today is 11°F colder than yesterday. You dress for the weather and go to meet a friend at the cinema. You forgot your wallet, so your friend gives you $10 to buy your ticket. What integer best represents the change in temperature?

Unit C Lesson 8-2 Comparing and Ordering Integers

Focus Question: What does it mean for one negative number to be greater than another negative number? How can you use that information?

Part 1 IntroRemember, the symbol > means “greater than” and the symbol < means “less than.” On a number line, the numbers increase from left to right.

If a > b, then a is to the right of b on the number line.

If c < b, then c is to the left of b on the number line.

Part 1 ExampleComplete each statement using the < or > symbol.

Part 2 ExampleDuring January, in Montréal, Canada, the temperature at 1PM was –8°C. The temperature at 11PM was –12°C. Write an inequality to compare the two temperatures. When was it warmer? Explain how you know it was warmer.

Part 3 ExampleThe table shows the number of yards gained or lost by a high school football team during the first play in each game. Order the results of the plays from least yards to most yards gained.

Play ResultGame 1 Sack –10 yardsGame 2 Run 5 yardsGame 3 Pass 3 yardsGame 4 Run –4 yardsGame 5 Pass –1 yards

Additional Practice1. Last night, some friends played a board game. They played two rounds. In the first round, one player lost 8 points. In the second round, the same played gained 5 points. Write an inequality to compare this player’s points in these two rounds.

o A. –8 > 5o B. 8 > –5 o C. 8 < 5o D. –8 < 5

o E. 8 < –5 o F. –8 < –5 o G. 8 > 5o H. –8 > –5

2. Five friends played a board game. The table shows their scores at the end of the game. Order the numbers from least to greatest. If the person with the highest score wins, who finished in third place?ScoresPerson Adam Brenda Carrie Daniel EmilyScore –4 points –6 points 2 points 5 points 3 points

Which list below shows the numbers ordered from least to greatest?o A. –6, –4, 2, 3, 5o B. 5, 2, –6, –4, 3o C. 5, 3, 2, –4, –6 o D. –4, 2, 5, –6, 3

Who came in third place? _______________________

3. Error Analysis Marco and Lucinda compare 1 and –7. Marco claims that 1 is to the left of –7 on a number line, and so 1 < –7. Lucinda claims that 1 is to the right of –7 on a number line, and so 1 > –7. Who is incorrect, and what is the error?

o A. Marco is incorrect. On a number line, 1 is not to the left of –7.o B. Lucinda is incorrect. On a number line, 1 is not to the right of –7.o C. Lucinda is incorrect. Because 1 is to the right of –7 on a number line, the inequality

should be 1 < –7.o D. Marco is incorrect. Because 1 is to the left of –7 on a number line, the inequality should

be 1 > –7.

4. Mental Math Five people started an exercise program. Each person ran 10 laps around a track and recorded the time. After doing this daily for two months, each person ran another 10 laps around the same track and recorded the time. The table shows the change from the first time to the second time for each person. Order the numbers from least to greatest. Which person’s time decreased the most?

Changes in TimesPerson Change (seconds) Person Change (seconds)

A 20 D 30B –10 E –30C 10

Unit C Lesson 8-3 Absolute Value

Focus Question: How can you express the distance of any number from 0? When might you need to know the distance from 0?

Key ConceptThe absolute value of a number is its distance from zero on a number line. Since absolute value is a distance, an absolute value is never negative. The symbol for the absolute value of a number is |n|.

The absolute value of –5 is the distance from –5 to 0.

There are 5 units from –5 to 0 so | –5 | = 5.Part 1 ExampleFind each absolute value.a. |9| b. |–3| c. |0|

Part 2 ExampleOrder the values from least to greatest.

Part 3 IntroYou can represent the positions of the divers relative to sea level using the integers – 2 and – 5.

However, depth is a positive quantity, so the depths of the divers are given by |–2| and |–5|.

Part 3 ExampleA college student is working during the summer to save money for the upcoming school year. She deposits the money she earns working. The money she withdraws she uses to pay for her living expenses.Month Deposits WithdrawalsMay $925 –$400 June $875 –$325 July $1,300 –$450August $1,250 –$550

In which month did the student spend the most money?

Additional Practice1. Find |18|.

2. Order these numbers from least to greatest. |–12|, –12, |–8|, –8, 0, |4|, –4

3. Two goldfish swim in a tank. The position of fish A, relative to the top of the tank, is –97 cm. The position of fish B is –92 cm. Use |–97| and |–92| to represent the depths of the fish.

Compare the depths. |–97| _______ |–92|

Goldfish _____ is closer to the top of the tank.

4. Find –|719|.

5. Challenge You catch and measure 5 fish. The table shows the difference between the length of each fish and the legal minimum length. Which fish has a length that is farthest from the legal minimum? Can you legally keep that fish?

Fish You CaughtFish A B C D EDifference from the legal minimum (mm) –

48–41 40 –39 55

Fish ______ has length farthest from the legal length.

Can you legally keep that fish? Explain your reasoning.

Unit C Lesson 8-4 Integers and the Coordinate Plane

Focus Question: How is a number line related to the coordinate plane? How are integers used within the coordinate plane?

Key ConceptThe coordinate plane is a surface formed by the intersection of two number lines.

The axes divide the coordinate plane into four regions, called quadrants.

The points on the axes do not lie in any of the quadrants.

An ordered pair is a pair of numbers that decsribe the location of a point in the coordinate plane.

Part 1 IntroYou can graph points if you know their coordinates. (x,y) = (x-coordinate, y-coordinate) The x-coordinate tells you how far to move right or left from the origin.

When the x-coordinate is positive, move to the right of the origin. When the x-coordinate is negative, move to the left of the origin.

The y-coordinate tells you how far to move up or down from the origin. When the y-coordinate is positive, move up from the origin. When the y-coordinate is negative, move down from the origin.

Part 1 ExamplePlot each point on the coordinate plane. Then identify the quadrant.

A (3,4)

B (– 8,– 4)

C (– 6,5)

D (1,– 7)

E (0,2)

Part 2 Intro

Part 2 ExampleIdentify which quadrant each point lies.

(1,–2) (2,–5) (2,1) (3,5) (–3,4)

(4,–1) (–4,–4) (–1,–3) (–5,2) (–2,5)

Part 3 Intro

A transformation of a figure is a change in its position, size, or shape. The new figure is the image of the original figure. A reflection is a type of transformation that flips a figure across a line called the line of reflection. A line of reflection acts as a mirror.

Reflection across the x-axis The reflection of a point (x,y) across the x-axis is located at (x,–y).

When a point is reflected across the x-axis, x-coordinate stays the same. y-coordinate changes signs.

Reflection across the y-axis The reflection of a point (x,y) across the y-axis is located at (–x,y).

When a point is reflected across the y-axis, y-coordinate stays the same. x-coordinate changes signs.

Reflection across the x- and y-axes The reflection of a point (x,y) across the x- and y-axes is located at point (–x,–y).

When a point is reflected across the x- and y-axes, neither coordinates stay the same. both coordinates change sign.

Part 3 ExampleFill in the blank to complete each statement.

_______________ is the reflection of the point (3,2) across the x-axis.

_______________ is the reflection of the point (3,2) across the y-axis.

_______________ is the reflection of the point (–2,3) across the x-axis.

_______________ is the reflection of the point (–2,3) across the y-axis.

Additional Practice

1. What ordered pair could be the coordinates of point E?

o A. (3,5)o B. (–5,–4)o C. (4,–5)o D. (–2, 1)

2. Map Coordinates The coordinate plane shows certain locations in a town. Find the coordinates of the hospital. (Type an ordered pair.)

3. Error Analysis Aaron says the reflection of the point (–7,8) across the x-axis is (7,8). Leah says it is (–7,–8). Who was correct? What was the other student’s error?

Who was correct?

What was the other student’s error?o A. The student switched the x- and y-coordinates.o B. The student reflected the point across both the x- and y-axes.o C. The student reflected the point across the y-axis instead of the x-axis.

4. Think About the Process To plot the point (–4,9) on the coordinate plane, start at the origin (0,0). What are the next steps? Select the correct choice below and fill in the answer boxes to complete your choice.

o A. Move ____ unit(s) to the right and ____ unit(s) down.

o B. Move ____ unit(s) to the left and ____ unit(s) down.

o C. Move ____ unit(s) to the left and ____ unit(s) up.

o D. Move ____ unit(s) to the right and ____ unit(s) up.

Unit C Lesson 8-5 Distance [on a Coordinate Plane]

Focus Question: How are integers used to express distance? What does a negative number mean when you are talking about distance?

Part 1 IntroThere are some measurements, such as temperature, where negative numbers have a meaning. There are other measurements where negative numbers are meaningless. Distance is an example of a measurement that is always non-negative.

Recall that absolute value measures the distance a number is from zero on a number line. Absolute vallue is never negative. Distance cannot be measured in negative units.

Part 1 Example

Use the graph to answer the questions below. Show all work of how you determined your answer.

a. What is the distance between point X and point Y?

b. What is the distance between point Y and point Z?

c. What is the distance between point X and point Z?

Part 2 Example

Your cousin rides the subway to work. His trip is shown on the coordinate plane. Each grid line represents 1 block.

What is the distance between Start and Station 1?

What is the distance between Station 1 and the end?

Part 3 ExampleA city planner is using a coordinate plane to plan a new city park. The four corners of a rectangular dog park will be located at A (-3,1), B (-3, 8), C (5, 8) and D (5,1). What is the area of the dog park?

Additional Practice1. If point A is located at (5,2) and point B is located at (5,18), what is the distance between points A and B?

2. What is the distance between (–4,–1) and (–26,–1)?

3. Challenge The lengths of the sides of a regular figure are equal. This figure is a regular hexagon.

What is the length of side DE?

What is the perimeter of the figure?