Unit 7 Rational Numbers — Unit Plan

Title

Assessment

Lesson 1

Positive and Negative Numbers

Agree or Disagree?

State whether you agree with each of the following statements. Explain your reasoning.

A temperature of 35 degrees Fahrenheit is as cold as a temperature of -35 degrees Fahrenheit.
A city that has an elevation of 15 meters is closer to sea level than a city that has an elevation of -10 meters.
A city that has an elevation of -17 meters is closer to sea level than a city that has an elevation of -40 meters.

Show Solution

Disagree. Sample reasoning: 35 degrees Fahrenheit is above 0 degrees Fahrenheit, and -35 degrees Fahrenheit is below 0 degrees Fahrenheit. -35 degrees is 70 degrees colder than 35 degrees.
Disagree. Sample reasoning: -10 meters is 10 meters from sea level. 15 meters is 15 meters from sea level. -10 meters is closer to sea level.
Agree. Sample reasoning: -17 meters is 17 meters from sea level. -40 meters is 40 meters from sea level. -17 meters is closer to sea level.

Lesson 2

Points on the Number Line

Positive, Negative, and Opposite

Plot a point on the number line to represent the value -3.2.
What is the opposite of -3.2?
What is the opposite of the opposite of -3.2?

Show Solution

3.2
-3.2

Lesson 3

Comparing Positive and Negative Numbers

Making More Comparisons

The elevation of Death Valley, California, is -282 feet. The elevation of Tallahassee, Florida, is 203 feet. The elevation of Westmorland, California, is -157 feet.

Label each point on the number line with the name of the city whose elevation is represented by the point.
Use the symbol < or > to compare the elevations of Death Valley and Tallahassee.
Use the symbol < or > to compare the elevations of Death Valley and Westmorland.

Show Solution

A: Death Valley, California
B: Westmorland, California
C: Tallahassee, Florida
$\text-282 < 203$ or $203>\text-282$
$\text-282 < \text-157$ or $\text-157>\text-282$

Lesson 4

Ordering Rational Numbers

Getting Them in Order

Place these numbers in order from least to greatest:

$\frac{16}{5}$

-3

6

3.1

-2.5

$\frac14$

$\text-\frac{3}{4}$

$\text-\frac{3}{8}$
Write a sentence to compare the two points shown on the number line.

Show Solution

-3, -2.5, $\text-\frac34$ , $\text-\frac38$ , $\frac14$ , 3.1, $\frac{16}{5}$ , 6
-2.7 is less than 4.5, or 4.5 is greater than -2.7.

Lesson 5

Using Negative Numbers to Make Sense of Contexts

Event Planner

The table shows records of the money-related activities of an event planner over the period of a week.

date	items	amount in dollars
May 1	rent	-850.00
May 2	birthday party decorations	106.75
May 3	utilities (electricity, gas, phone)	-294.50
May 5	wedding invitations	240.55
May 5	office supplies	-147.95
May 6	anniversary party catering	158.20
May 7	conference event scheduling	482.30

For which items did the event planner receive money?
What does the number -147.95 mean in this context?
Did the event planner receive more or spend more money on May 5? Explain your reasoning.

Show Solution

birthday party decorations, wedding invitations, anniversary party catering, conference event scheduling
$147.95 was spent on office supplies.
Received more money. Sample reasoning: Over $240 was received, and less than $150 was spent.

Lesson 6

Absolute Value of Numbers

Greater, Less, the Same

Write a number that has the same value as each expression:
1. $|5|$
2. $|\text-12.9|$
Write a number that has a value less than $|4.7|$ .
Write a number that has a value greater than $|\text-2.6|$ .

Show Solution

1. 5 or $|\text-5|$
2. 12.9 or $|12.9|$
Sample responses: 4.5, $|\text-4.5|$ , or -10
Sample responses: 2.7 or $|\text-2.7|$

Lesson 7

Comparing Numbers and Distance from Zero

True or False?

Mark each statement as true or false, and explain your reasoning.

$\text-5 < 3$
$\text-5 > 3$
$|\text-5| < 3$
$|\text-5| > 3$

Show Solution

True. Sample reasoning: -5 is farther to the left on the number line than 3.
False. Sample reasoning: -5 is farther to the left on the number line than 3.
False. Sample reasoning: $|\text- 5| = 5$ , and $5>3$ .
True. Sample reasoning: $|\text- 5| = 5$ , and $5>3$ .

Section A Check

Section A Checkpoint

Problem 1

Plot and label each of the points on the number line.

-6.5
The opposite of -8
All numbers whose absolute value equals 2

Show Solution

Problem 2

Use <, >, or = to compare each pair of values.

$12\text{ }\underline{\hspace{.5in}}\text{ }\text-30$
$|12|\text{ }\underline{\hspace{.5in}}\text{ }|\text-30|$
$\text-11\text{ }\underline{\hspace{.5in}}\text{ }\text-22$
$|\text-8|\text{ }\underline{\hspace{.5in}}\text{ }|8|$

Show Solution

Problem 3

Tyler is visiting a city that has some locations below sea level. Here are his elevations at different times of the day:

Morning: 7 feet below sea level
Lunch: 11 feet above sea level
Afternoon: -4 feet

At what time of day was Tyler the lowest in elevation?
At what time of day was Tyler farthest from sea level?
What would an elevation of 0 represent in this situation?

Show Solution

Morning
Lunch
Sea level

Lesson 8

Writing and Graphing Inequalities

A Box of Paper Clips

Andre looks at a box of paper clips. He says: “I think the number of paper clips in the box is less than 1,000.”

Lin also looks at the box. She says: “I think the number of paper clips in the box is more than 500.”

Write an inequality to show Andre's statement, using $p$ for the number of paper clips.
Write another inequality to show Lin's statement, also using $p$ for the number of paper clips.
Do you think both Lin and Andre would agree that there could be 487 paperclips in the box? Explain your reasoning.
Do you think both Lin and Andre would agree that there could be 742 paperclips in the box? Explain your reasoning.

Show Solution

$p < 1,000$
$p > 500$
No. Sample reasoning: Andre would agree because the inequality, $487<1,000$ is a true statement. However, Lin would not agree because the inequality $487>500$ is a false statement.
Yes. Sample reasoning: Both inequalities are true for 742 paper clips: $742<1,000$ , and $742>500$ . This means that according to Lin and Andre, there could be 742 paperclips in the box.

Lesson 9

Solutions of Inequalities

1. Select all numbers that are solutions to the inequality $w<1$ .
  
  5
  
  -5
  
  0
  
  0.9
  
  -1.3
2. Draw a number line to represent this inequality.
1. Write an inequality for which 3, -4, 0, and 2,300 are solutions.
2. How many total solutions are there to your inequality?

Show Solution

1. -5, 0, 0.9, -1.3
1. Sample response: $x>\text -5$ .
2. There are infinitely many solutions.

Lesson 10

Interpreting Inequalities

Lin and Andre’s Heights

Lin says that the inequalities $h >150$ and $h <160$ describe her height in centimeters. What do the inequalities tell us about her height?
Andre notices that he is a little taller than Lin but is shorter than their math teacher, who is 164 centimeters tall. Write two inequalities to describe Andre's height. Let $a$ be Andre's height in centimeters.
Select all heights in centimeters that could be Andre's height. If you get stuck, consider drawing a number line to help you.
1. 150
2. 154.5
3. 160
4. 162.5
5. 164

Show Solution

These inequalities tell us that Lin is between 150 and 160 cm tall.
$a < 164$ and $a > h$ (or $h < a$ ).
B, C, D

Section B Check

Section B Checkpoint

Problem 1

A medication has to be stored at a temperature less than 10 degrees Celsius.

Let $t$ be the temperature. Write an inequality that describes the safe storage temperature for the medication.
Draw a number line to represent solutions to the inequality.

Show Solution

$t<10$

Problem 2

Select all the numbers that are solutions to

n >\text -3

A.-11

B.-3.1

C.-3

D.-2.9

E.0

F.11

Show Solution

D, E, F

Lesson 11

Points in the Coordinate Plane

Target Practice

Here are the scores for landing an arrow in the different regions of the archery target.

A: 10 points
B: 8 points
C: 6 points
D: 4 points
E: 2 points

Andre shot three arrows and they landed at $(\text-5, 4),\, (\text-8, 7)$ and $(1, 6)$ . What is his total score? Explain or show your reasoning.
Jada shot an arrow and scored 10 points. She shot a second arrow that landed directly below the first one but scored only 2 points. Name two coordinates that could be the landing points of her two arrows.

Show Solution

14 points. Sample reasoning: $(\text-5, 4)$ is 6 points, $(\text-8, 7)$ is 8 points, and $(1, 6)$ is 0 points.
Sample responses:
- $(\text-7, 5)$ and $(\text-7, 1)$
- $(\text-6.5, 5.5)$ and $(\text-6.5, 1.1)$

Lesson 12

Constructing the Coordinate Plane

What Went Wrong: Graphing Edition

Lin labeled this set of axes and plotted the points $A\,(1,2)$ , $B\,(\text-3,\text-5)$ , $C\,(5, 7)$ , $D\,(\text-4, \text-3)$ , and $E\, (\text-4,6)$ in the coordinate plane.

Identify as many mistakes as you notice in Lin's graph.

Show Solution

Point $C$ is plotted at $(5,8)$ instead of $(5,7)$ . Point $D$ is plotted at $(\text-3,\text-4)$ instead of $(\text-4,\text-3)$ .

Lesson 13

Interpreting Points in a Coordinate Plane

Time and Temperature

The temperature in Princeton, MA, was recorded at various times during the day. The times and temperatures are shown in the table.

time (hours after midnight)	temperature (degrees C)
-5	1.2
-2	-1.6
0	-3.5
8	-6.7

Plot points that represent the data. Be sure to label the axes.
In the town of New Haven, CT, the temperature at midnight was $1.2^\circ \text{C}$ . Plot and label this point.
Which town was warmer at midnight, Princeton or New Haven? How many degrees warmer was it?
If the point $(3, \text-2.5)$ were also plotted on the diagram, what would it mean?

Show Solution

See graph.
New Haven is warmer by 4.7 degrees Celsius
3,-2.5 means that 3 hours after midnight, the temperature was -2.5 degrees.

Lesson 14

Distances in the Coordinate Plane

Points and Distances

Here are four points in a coordinate plane.

What is the distance between points $A$ and $B$ ?
What is the distance between points $C$ and $D$ ?
Plot the point $(\text-3, 2)$ . Label it $E$ .
Plot the point $(\text-4.5, \text-4.5)$ . Label it $F$ .

Coordinate plane, axes labeled by ones. Point A, (negative 3 comma negative 2), Point B, (6 point 5 comma negative 2), Point C, (6 point 5 comma 2 point 5), Point D, (6 point 5 comma negative 4).

Show Solution

About 9.5 units
About 6.5 units
See image.

Lesson 15

Shapes in the Coordinate Plane

Perimeter of a Polygon

Plot the following points in the coordinate plane, and connect them in the order listed to create a polygon.

$A~(1, 3)$

$B~(3, 3)$

$C~(3, \text-2)$

$D~(\text-2, \text-2)$

$E~(\text-2, 0)$

$F~(0, 0)$

$G~(0, 2)$

$H~(1, 2)$

$I~(1, 3)$
Find the perimeter of the polygon.

Show Solution

The perimeter is 20 units.

Section C Check

Section C Checkpoint

Problem 1

Write the coordinates of each point:
1. $A$ :
2. $B$ :
3. $C$ :
Plot each point:
1. $D (\text-4, \text-1)$
2. $E (0, \text-4)$
3. $F (\text-3, 4)$

Show Solution

1. $A: (5,2)$
2. $B: (5,\text-2)$
3. $C: (\text-3,0)$

Problem 2

The points $R$ , $E$ , $C$ , and $T$ form a rectangle.

What are the coordinates of point $T$ ?

$R \text{ }(\text-7, 10)$
$E \text{ }(\text-7, \text-10)$
$C \text{ } (7, 10)$

Show Solution

T \text{ } (7,\text -10)

Problem 3

Each pair of points are connected to form a line segment. What is the length of each?

$A (2, 9)$ and $B (2, 6)$
$C (\text{-}5, \text{-}1)$ and $D (\text{-}7, \text{-}1)$
$E (3, 4)$ and $F (3, \text{-}4)$

A blank coordinate plane with 16 evenly spaced horizontal units and 12 evenly spaced vertical units.

Show Solution

3 units
2 units
8 units

Lesson 16

Common Factors

In Your Own Words

What is the greatest common factor of 24 and 64? Explain or show your reasoning.
In your own words, what is the greatest common factor of two whole numbers? How can you find it?

Show Solution

8. Sample reasoning: The common factors of 24 and 64 are 1, 2, 4, and 8, and 8 is the greatest.
Sample response: The greatest common factor of two whole numbers is the largest number that divides evenly into both numbers. You can find the greatest common factor by listing the factors of each number and then finding the greatest one that both numbers share.

Lesson 17

Common Multiples

In Your Own Words Again

What is the least common multiple of 6 and 9? Explain or show your reasoning.
In your own words, what is the least common multiple of two whole numbers? How can you find it?

Show Solution

The least common multiple of 6 and 9 is 18. Sample reasoning: The first few multiples of 6 are 6, 12, 18, 24, 30, and 36. The first few multiples of 9 are 9, 18, 27, and 36. The number 18 is the first to appear on both lists.
Sample response: The least common multiple of two numbers is the smallest multiple that the numbers share. You can find the least common multiple by listing the multiples of each number until you find one that is common to both lists. The first multiple that is common to both lists is the least common multiple.

Lesson 18

Using Common Multiples and Common Factors

What Kind of Problem?

For each problem, tell whether finding the answer requires finding a greatest common factor or a least common multiple. You do not need to solve the problems.

Elena has 20 apples and 35 crackers. She wants to make as many snack bags as possible that each contain the same combination of apples and crackers. What is the largest number of snack bags she can make?
A string of lights flashes two colors—red every 5 seconds and blue every 3 seconds. How long before both lights flash at the same time?
A florist orders sunflowers every 6 days and daisies every 4 days. On which day will she order both kinds of flowers on the same day?

Show Solution

Greatest common factor
Least common multiple
Least common multiple

Section D Check

Section D Checkpoint

Problem 1

Jada buys 30 tulip bulbs and 40 daffodil bulbs. Her plan is to plant all the bulbs in groups. She wants the groups to be identical and each group to have both types of flowers.

List three different ways Jada can group the flowers.

Show Solution

Sample responses:

1 group with 30 tulips and 40 daffodils
2 groups, each with 15 tulips and 20 daffodils
5 groups, each with 6 tulips and 8 daffodils
10 groups, each with 3 tulips and 4 daffodils

Problem 2

Pencils are sold in packages of 12, and erasers are sold in packages of 9. Kiran wants to buy the same number of erasers as pencils.

Find a number of pencils and erasers that will meet Kiran’s requirement. How many packages of pencils and how many packages of erasers will Kiran need to buy?
Find a different number of pencils and erasers that also meets Kiran’s requirement. Explain your reasoning.

Show Solution

Sample response: Kiran can have 36 pencils and 36 erasers if he buys 3 packages of pencils and 4 packages of erasers. (Any multiple of 36 pencils and erasers is also correct.)
Sample response: Kiran could also have 72 pencils and 72 erasers if he bought 6 packages of pencils and 8 packages of erasers.

Lesson 19

Drawing in the Coordinate Plane

No cool-down

Unit 7 Assessment

End-of-Unit Assessment

Problem 1

These four numbers are plotted on a number line: $\text{-}\frac {2}{3}, \frac{5}{8}, \text{-}\frac {3}{5}, \text{-}\frac {1}{2}$

From left to right, which is the correct ordering on the number line?

$\text{-}\frac {1}{2}, \text{-}\frac {3}{5}, \text{-}\frac {2}{3}, \frac 5 8$

$\text{-}\frac {1}{2}, \text{-}\frac {3}{5}, \frac 5 8, \text{-}\frac {2}{3}$

$\text{-}\frac {2}{3}, \text{-}\frac {3}{5}, \text{-}\frac {1}{2}, \frac 5 8$

$\text{-}\frac {3}{5}, \text{-}\frac {2}{3}, \text{-}\frac {1}{2}, \frac 5 8$

Show Solution

$\text{-}\frac {2}{3}, \text{-}\frac {3}{5}, \text{-}\frac {1}{2}, \frac 5 8$

Problem 2

Diego’s dog weighs more than 10 kilograms and less than 15 kilograms.

Write two inequalities to describe the minimum and maximum values of $w$ , the weight of Diego’s dog in kilograms.

Show Solution

w>10

(or

10<w></w></span>) and <span class="math">\(w<15

(or

15>w

)

Problem 3

Select all the numbers that are a common multiple of 4 and 6.

Show Solution

D, E, G

Problem 4

Select all the true statements.

A.Point

A

is at the opposite of 5.

B.Point

B

is the opposite of point

E

C.Point

C

is at

|\text-2|

D.Point

D

is at

|1|

E.Point

E

is the opposite of point -2.5.

F.Point

F

is at

|\text-6|

Show Solution

A, D, E, F

Problem 5

Which temperature is warmer: -2 degrees Celsius or -5 degrees Celsius?
Write an inequality to express the relationship between -2 and -5.
On the number line, graph all the temperatures that are warmer than -2 degrees Celsius.

Show Solution

-2 degrees Celsius
$\text -2 > \text -5$ or $\text-5<\text-2$
Graph has an open circle at -2 and an arrow pointing to the right.

Minimal Tier 1 response:

Work is complete and correct.
Acceptable errors: No arrow on the graph, provided that the graph extends all the way to the right on the number line.
Sample:

$\text{-}2$ degrees Celsius
$\text{-}5 < \text{-}2$
Graph includes an open circle at -2 and extends fully to the right.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Graph has a closed circle or no circle at -2; answers to the first two parts are incorrect but consistent with one another.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: No graph; graph does not start or end at -2; graph has multiple errors including a closed circle at -2 and an arrow pointing to the left; work in the first two parts shows a lack of understanding about negative numbers.

Problem 6

Draw polygon $ABCDEF$ in this coordinate plane given its vertices $A (\text -2,\text -3)$ , $B (0,\text -3)$ , $C (0,1)$ , $D (3,1)$ , $E (3,3)$ , and $F (\text -2,3)$ .

A coordinate plane with the origin labeled "O". Both axes have the numbers negative 5 through 5 indicated.

Show Solution

Minimal Tier 1 response:

Work is complete and correct.
Sample: Polygon drawn as above. Vertices do not need to be marked as discrete points or labeled.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: One or two vertices are incorrect; vertices are correct but connected in the wrong order, for example, $A$ to $C$ to $B$ ; the points $A$ and $F$ are not connected, though the rest of the sides of the polygon are present.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Consistent errors, such as reversing the $x$ - and $y$ -coordinates; several missing vertices; vertices are plotted but the sides of the polygon are not.

Problem 7

Lin spent a day hiking. This graph shows her elevation (in meters) at some different times. Negative values of $x$ represent the number of hours before noon, and positive values of $x$ represent the number of hours after noon.

<p>Coordinate plane, origin O, horizontal axis, x, labeled by ones, vertical axis, y labeled by twos. Points marked at (negative 5 comma 10), (0 comma 3), and (2 comma negative 2).</p>

What was Lin’s elevation at noon? Explain your reasoning.
At 10:00 a.m., Lin’s elevation was 70 meters. Add this point to the graph.
At 1:00 p.m., Lin was at sea level. Add this point to the graph.
Did Lin’s elevation increase or decrease between 7:00 a.m. and 2:00 p.m.? Explain your reasoning.
Lin’s elevation decreased from 2:00 p.m. to 3:00 p.m. Add a point to the graph that shows her possible elevation at 3:00 p.m. Explain your reasoning.

Show Solution

30 meters. Sample reasoning: An $x$ -coordinate of 0 represents noon, and the point $(0,30)$ is on the graph.
The graph should include the point $(\text{-}2,70)$ .
The graph should include the point $(1,0)$ .
Lin’s elevation decreased. Her elevation changed from 100 meters to -40 meters.
Sample response: A point at $(3,\text{-}50)$ is graphed; Lin climbed downward, so her elevation will be more negative. A correct point will be $(3,y)$ with $y < \text{-}40$ .

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:

30, because the point $(0, 30)$ is on the graph
$(\text{-}2,70)$ plotted
$(1,0)$ plotted
Decreased. She went down 140 meters.
$(3,\text{-}50)$ plotted. Lin dropped further, so the new point is lower.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Misinterpretation of the y-axis scale so that each grid line is treated as one unit (or ten units); one problem part is incorrect.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Does not understand that the zero mark is noon, but otherwise, interpretation of points shows understanding; $x$ - and $y$ -coordinates are consistently reversed.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Work does not show understanding of the relationship between points on the graph and the situation.