Student Learning Extension Opportunities Grade 6-Grade 8 · 2020-03-24 · Student Learning Extension Opportunities Grade 6-Grade 8 Directions: These learning activities are provided

Student Learning Extension Opportunities Grade 6-Grade 8

Directions: These learning activities are provided for practice opportunities. Refreshing your memory of the concepts learned and keeping your mind engaged will help you maintain the skills you have learned. These learning activities are designed to provide practice over the course of the week, so spread out the work. We look forward to seeing you back in class soon.

WEEK TWO Reading and Writing (Science and Social Studies Integration):

Week 2, Day 1 ● Read a book at your reading level for thirty minutes. Keep track of your daily reading on the

reading log below.● Read the text, “Why Do Volcanoes Erupt?”● Complete the “Why Do Volcanoes Erupt?” comprehension questions.


reading log below.● Read the text, “Teens Health - Hand Washing: Why It’s So Important.”● Complete the “Teens Health - Hand Washing: Why It’s So Important” comprehension

questions.


reading log below.● Read the text, “Early Villages and the Social Networks.”● Complete the “Early Villages and the Social Networks” comprehension questions.


reading log below.● Read the text, “Teens Health - How Much Sleep Do I Need.”● Complete the “Teens Health - How Much Sleep Do I Need” comprehension questions.


reading log below.● Read the text, “Raymond’s Run.”● Complete the “Raymond’s Run” comprehension questions.

Mathematics:

Week 2, Day 1 ● Complete your grade-level worksheet labeled Grade 6, 7, or 8.


Student Learning Extension Opportunities Grade 6-Grade 8


Week 2, Day 4

● Complete your grade-level worksheet labeled Grade 6, 7, or 8.


Reading Log

Keep track of your daily reading.

Beginning Page

Ending Page

Title

Name: Class:

"untitled" by Marc Szeglat is licensed under CC0

Why do volcanoes erupt?By Heather Handley

2018

Volcanic eruptions are amazing events in nature, but what causes a volcano to erupt? In this informationtext, Heather Handley provides valuable information on volcanoes. As you read, take notes on the detailsprovided about magma.

The rock inside the planet we live on can melt toform molten1 rock called magma. This magma islighter than the rocks around it and so it risesupwards. Where the magma eventually reachesthe surface we get an eruption and volcanoesform.

The top part of the Earth is made up of a numberof hard pieces called tectonic plates. Magma andvolcanoes often form where the plates are pulledapart or pushed together but we also find somevolcanoes in the middle of tectonic plates.

Volcanoes have many different shapes and sizes,some look like steep mountains(stratovolcanoes), others look like bumps (shield volcanoes) and some are flat with a hole (a crater orcaldera) in the centre that is often filled with water.

The shape of the volcano and how explosively it erupts depend largely on how “sticky” and how “fizzy”(how much gas) the magma is that is erupted.

For example, if you try to blow bubbles in cooking oil through a straw, the bubbles can escape quiteeasily because the cooking oil is runny.

If you try to blow bubbles in jam or peanut butter you would find it very difficult because the jam andpeanut butter are very sticky, they wouldn’t move much at all if you tried to pour them out of the jar.

It is the same with volcanoes. When magma rises towards the surface gas bubbles start to form.Whether or not they can escape as the magma is rising affects how explosive the eruption will be.

Where the magma is runny like cooking oil and doesn’t have much bubbly gas mixed in it, such asplaces like Hawaii, then we see lots of slow-moving lava flows and shield volcanoes. Lava is what we callmagma when it reaches the surface.

[1]

[5]

1. turned to liquid by heat

1

https://unsplash.com/photos/I1MGVZ42wnU

“Why do volcanoes erupt?” by Heather Handley, Macquarie University, July 22, 2018. Copyright © The Conversation 2018, CC-BY-ND.

However, where the magma is very sticky, like jam or peanut butter, and if it contains a lot of bubblygas then the gas can get stuck and eruptions can be very powerful and explosive, like the recenteruptions at Fuego volcano in Guatemala.

Damage caused by eruptions

In explosive eruptions the frothy, bubbly magma can be ripped apart into tiny bits called volcanic ash.This is not ash like you get after a barbecue or fire, it does not crumble away in your fingers. It is verysharp and is dangerous to breathe in.

Some explosive volcanoes can send ash high up into the sky and it can travel around the world overdifferent countries. If aeroplanes travel through an ash cloud from a volcano it can cause a lot ofdamage to the engine.

Other explosive eruptions create fast-moving, hot clouds of volcanic ash, gas and rocks that traveldown the sides of the volcanoes and destroy pretty much everything in their path.

The benefits of volcanoes

Despite the great damage they can cause, volcanoes also help us to live. Volcanic ash provides food forthe soil around volcanoes which helps us grow plants to eat. The heat from some volcanoes is used tomake energy to power lights, fridges, televisions and computers in people’s houses.

[10]

2

https://theconversation.com/curious-kids-why-do-volcanoes-erupt-98251

“Why Do Volcanoes Erupt?” Comprehension Questions

Answer the following questions, citing evidence from the text.

1) Describe the different shapes and sizes of volcanoes.

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2) Explain the connections the author makes between magma and volcanoes.

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Describe the damage that volcanoes cause.

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“Teens Health - Hand Washing: Why it’s So Important” Comprehension Questions


1) Describe the steps you should use to wash your hands.

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2) Explain how clean hands can help your health.

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In the context of this text, explain how washing your hands is related to limiting the spread of germs.

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This article is available at 5 reading levels at https://newsela.com.

Early Villages and the Social Networks TheyCreated

One of the most important moments in human history was the beginning of farming. It didn'thappen at the same time across all societies. However, the switch from looking for food to growingit changed everything. Farming allowed humans to store extra food, leaving more time for otheractivities. In this way, farming made it possible for humans to build complex civilizations.

As with most big changes, these civilizations did not appear overnight. Cities and states evolvedover a long period of human history. Most people from this period lived in villages and smalltowns. Even the Roman Empire was mostly people living in small settlements.

These early villages grew in different ways. Some eventually developed into large city centers.Others remained independent villages, exchanging with other villages in networks. We don't havewritten records from most villages, but archaeological studies have revealed a lot ofinformation. Archaeologists have been able to reconstruct a story about early village life.

Social life before the city

By Eman M. Elshaikh, Big History Project on 06.17.19Word Count 890Level 850L


What was life like before farming gave us all this freetime? Many scholars consider earlier hunter-gatherersocieties to be largely egalitarian. That means that allpeople were treated equally and no one had morepower than anyone else. This continued to be the casein early farming villages. People in these communitieswere relatively equal. Most people living in villagesspent the majority of their time producing food. Laborwas mostly divided by gender. Women spent moretime taking care of small children but alsoparticipated in food production. However, thesegender divisions did not necessarily mean gender inequalities.

How do historians know this? Evidence shows that people in villages probably shared tools andworkspaces. Excavations of ancient sites in Ukraine and and the town of Çatalhüyük in Turkeyshow that houses were mostly about the same size. Objects in homes and graves were of relativelyequal value, too.

People living in farming communities had very different lives from hunter-gatherers. They lived insmaller areas, which allowed diseases to spread much faster. Some studies suggest that peopleworried more about diseases, too. Farming communities were also more dependent on favorableenvironmental conditions. For this reason, they worried more about the weather.

Farming communities were not models of gender equality, though. As these villages grew, theyintroduced new concepts such as permanent homes and the idea of owning things. A more specificdefinition of the family developed, too. As a result, gender hierarchies tended to intensify. Familysystems became more complicated and rigid. Hunter-gatherer societies had needed full-time effortfrom all men and women to stay fed. However, the new farming communities could build up storesof food. This allowed women to have more children. Over time, most women's lives became morefocused on children and maintaining small family homes.

Before long-distance trade

When we talk about trade, we usually think of shipscrossing oceans. Long before any of that happened,though, villages began trading with each other in localnetworks. That change was also important. It allowedfarming villages to get their hands on things likeobsidian. This is a black glass that comes from cooledvolcanic lava. Villagers used it to make sharp cuttingtools. Parts of obsidian tools weren't just found inTurkey. Archeologists have also found evidence ofobsidian among the Pacific Ocean islands.Trade between islands occurred way before long-distance trade routes emerged. The Lapita cultureexisted from about 1600 to 500 BCE in the Pacific Islands. They left behind plenty of obsidianartifacts, as well as ceramics, marine shells, and plants. The Lapita were the ancestors of historiccultures in Polynesia, Micronesia, and Melanesia.


In the Americas, coastal villages traded fish, mollusks, and shells with inland villages. The inlandvillages grew corn and potatoes and raised llamas.

Once you start trading potatoes and llamas, it isn't long before you are building roads and bridges.This kind of trade also increased specialization. That means that people had different kinds ofjobs. As villages grew, people took on new social roles. Labor became increasingly divided, andlarger projects needed to be managed by leaders. Greater social hierarchies emerged as people hadmore defined jobs.

Gender roles also changed. For example, male heads of family tended to gain control over wealth,leaving women with less power. However, this didn't happen everywhere in quite the same way.There was still plenty of variety in the way people understood gender and family in differentregions of the world.

Trade helped villages to grow, but village networks also boosted trade in a big way. As trade routesgrew, villages located in key areas were able to grow even faster. This cycle reinforced itself overtime. Eventually, many villages would come together into large city centers. This produced a typeof social life that looked very different from village life.

Eman M. Elshaikh is a writer and researcher. She teaches writing at the University of Chicago,where she also completed her master's degree in social sciences. She was previously a WorldHistory Fellow at Khan Academy.

“Early Villages and the Social Networks” Comprehension Questions


1) How did farming impact human history?

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2) What did the excavation of ancient sites discover?

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Explain the mass impact of social networks on daily living in early villages.

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“Teens Health - How Much Sleep Do I Need?” Comprehension Questions


1) What are some of the reasons that teens don’t get enough sleep?

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2) Describe problems people with sleep deficits may have.

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Create a plan for how you can get more sleep. Cite evidence from the text to support your plan.

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NOTES

This version of the selection alternates original text with summarized passages. Dotted lines appear next to the summarized passages.

Squeaky is a confident, sassy young girl who lives in Harlem in New York City. Squeaky has to take care of her brother Raymond, who is “not quite right.” She boldly protects Raymond from kids who try to tease him. Squeaky loves to run races, and she is the fastest runner in her neighborhood.

There is no track meet that I don’t win the first place medal. I used to win the twenty-yard dash when I was a little kid in kindergarten. Nowadays, it’s the fifty-yard dash. And tomorrow I’m subject to run the quarter-meter relay all by myself and come in first, second, and third.

This year, for the first time, Squeaky has some serious competition in the race, a new girl named Gretchen.

So as far as everyone’s concerned, I’m the fastest and that goes for Gretchen, too, who has put out the tale that she is going to win the first-place medal this year. Ridiculous. In the second place, she’s got short legs. In the third place, she’s got freckles. In the first place, no one can beat me and that’s all there is to it.

Squeaky takes a walk down Broadway with Raymond. She is practicing her breathing exercises to get in shape for the race. Raymond is pretending to drive a stagecoach. Squeaky works hard to be a good runner. She dislikes people who pretend that they do not need to work hard to be good at something.

Squeaky sees Gretchen and two of her friends coming toward her and Raymond. One of the girls, Mary Louise, used to be Squeaky’s friend. Now she hangs out with Gretchen instead. Rosie, the other girl, always teases Raymond. Squeaky considers going into a store to avoid the girls, but she decides to face them.

“You signing up for the May Day races?” smiles Mary Louise, only it’s not a smile at all. A dumb question like that doesn’t deserve an answer. Besides, there’s just me and Gretchen standing there really, so no use wasting my breath talking to shadows.

“I don’t think you’re going to win this time,” says Rosie, trying to signify with her hands on her hips all salty, completely forgetting that I have whupped her many times for less salt than that.

“I always win cause I’m the best,” I say straight at Gretchen who is, as far as I’m concerned, the only one talking in this ventriloquist-dummy1 routine. Gretchen smiles, but it’s not a smile, and I’m thinking that girls never really smile at each other because they don’t know how and don’t

1. ventriloquist-dummy (vehn TRIHL uh kwihst) a performer who speaks through a puppet called a“dummy” in a comic performance.

Raymond’s RunToni Cade Bambara

SHORT STORY

GRADE 6 • UNIT 1 • Accessible Leveled Text • Raymond’s Run 1

MPELA17_SE06_U1C_LIT_Bambara.indd 1 2/9/17 3:04 PM

© P

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NOTESwant to know how and there’s probably no one to teach us how cause grown-up girls don’t know either. Then they all look at Raymond who has just brought his mule team to a standstill. And they’re about to see what trouble they can get into through him.

Mary Louise starts to tease Raymond, but Squeaky defends him. Gretchen and her friends leave, and Squeaky smiles at her brother. The next day, Squeaky arrives late at the May Day program because she does not want to see the May Pole dancing. She thinks it is silly. She arrives just as the races are starting. She puts Raymond on the swings and finds Mr. Pearson, a tall man who gives the racers their numbers.

“Well, Squeaky,” he says, checking my name off the list and handing me number seven and two pins. And I’m thinking he’s got no right to call me Squeaky, if I can’t call him Beanstalk.

“Hazel Elizabeth Deborah Parker,” I correct him and tell him to write it down on his board.

“Well, Hazel Elizabeth Deborah Parker, going to give someone else a break this year?” I squint at him real hard to see if he is seriously thinking I should lose the race on purpose just to give someone else a break.

Mr. Pearson suggests that Squeaky let Gretchen, the new girl, win the race. Squeaky gets mad and walks away. When it is time for the fifty-yard dash, Squeaky and Gretchen join the other runners at the starting line. Squeaky sees that Raymond has left the swings and is getting ready to run on the other side of the fence. Squeaky mentally prepares herself to win and takes off like a shot, zipping past the other runners.

I glance to my left and there is no one. To the right, a blurred Gretchen, who’s got her chin jutting out as if it would win the race all by itself. And on the other side of the fence is Raymond with his arms down to his side and the palms tucked up behind him, running in his very own style, and it’s the first time I ever saw that and I almost stop to watch my brother Raymond on his first run. But the white ribbon is bouncing toward me and I tear past it, racing into the distance till my feet with a mind of their own start digging up footfuls of dirt and brake me short.

Squeaky believes that she has won the race, but it turns out that she and Gretchen crossed the finish line at almost the same time. The judges are not sure which girl is the winner.

And I lean down to catch my breath and here comes Gretchen walking back, for she’s overshot the finish line too, huffing and puffing with her hands on her hips taking it slow, breathing in steady time like a real pro and I sort of like her a little for the first time. “ In first place ... ” and then three or four voices get all mixed up on the loudspeaker and I dig my sneaker into the grass and stare at Gretchen who’s staring back, we both wondering just who did win.

As Squeaky waits to find out whether she has won, Raymond calls out to her. He starts climbing up the fence. Suddenly, Squeaky remembers that Raymond ran the race too, on the other side of the fence.



© P

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NOTESAnd it occurs to me, watching how smoothly he climbs hand over hand and remembering how he looked running with his arms down to his side and with the wind pulling his mouth back and his teeth showing and all, it occurred to me that Raymond would make a very fine runner. Doesn’t he always keep up with me on my trots? And he surely knows how to breathe in counts of seven cause he’s always doing it at the dinner table, which drives my brother George up the wall. And I’m smiling to beat the band cause if I’ve lost this race, or if me and Gretchen tied, or even if I’ve won, I can always retire as a runner and begin a whole new career as a coach with Raymond as my champion.

Squeaky gets very excited about the idea of teaching Raymond to be a champion runner. She wants him to have something to be proud of. Raymond runs over to her, and she jumps up and down with happiness because of her plans to help him.

But of course everyone thinks I’m jumping up and down because the men on the loudspeaker have finally gotten themselves together and compared notes and are announcing “In first place—Miss Hazel Elizabeth Deborah Parker.” (Dig that.) “In second place—Miss Gretchen P. Lewis.” And I look over at Gretchen wondering what the “P” stands for. And I smile. Cause she’s good, no doubt about it. Maybe she’d like to help me coach Raymond; she obviously is serious about running, as any fool can see. And she nods to congratulate me and then she smiles. And I smile. We stand there with this big smile of respect between us. ...

“Raymond’s Run,” copyright © 1971 by Toni Cade Bambara; from Gorilla, My Love by Toni Cade Bambara. Used by permission of Random House, an imprint and division of Penguin Random House LLC. All rights reserved. Any third party use of this material, outside of this publication, is prohibited. Interested parties must apply directly to Penguin Random House LLC for permission.



“Raymond’s Run” Comprehension Questions


1) Explain how Squeaky feels about her brother and her role in his life.

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2) How do Squeaky's feelings change about Gretchen at the end of the story?Provide evidence from the story to support your answer.

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In the story, Squeaky’s running is an important part of her identity. Describe special skills, abilities, or interests that are an important part of your identity.

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Examples:

40% of 50 is what? 66% of 400 is what?

40100 50

20

x

x

=

=

1. 50% of 36 is what? 2. 75% of 48 is what 3. 60% of 800 is what?

4. 25% of 60 is what? 5. 10% of 70 is what? 6. 20% of 50 is what?

7. 100% of 75 is what? 8. 15% of 80 is what? 9. 40% of 200 is what?

10. 33 13

% of 99 is what? 11. 22% of 50 is what? 12. 1% of 70 is what?

Percent Problems – Finding the Part

66100 400

264

x

x

=

=

Grade 6, Week 2, Day 1

Examples:

What % of 48 is 12? What % of 99 is 9?

12100 48

25%

x

x

=

=

1. What % of 28 is 7? 2. What % of 60 is 12? 3. What % of 90 is 45?



10. What % of 32 is 24? 11. What % of 80 is 12? 12. What % 40 is 16?


Percent Problems – Finding the Percent

1

10

1

4

Grade 6, Week 2, Day 2Name _________________________________ Period_________ Date___________________

Examples:

40% of what is 48? 66% of what is 99?

40 48100

120

x

x

=

=

1. 50% of what is 45? 2. 70% of what is 42? 3. 60% of what is 18?





Percent Problems – Finding the Total

66 99100

150

x

x

=

=

50

4

10


1. What % of 22 is 11? 2. 50% of 36 is what? 3. 25% of what is 13?

4. 20% of what is 4? 5. What % of 40 is 4? 6. 10% of 11 is what?

7. 15% of 60 is what? 8. 75% of what is 12? 9. What % of 37 is 37?

10. 25% of 14 is what? 11. What % of 16 is 2? 12. 30% of what is 9?

13. What % of 36 is 6 14. 80% of 25 is what? 15. 12% of 100 is what?

Percent Problems - Mixed


1. 20% of the girls in class have blonde hair. 2. 40% of the farm animals are cows.72 animals are not cows.16 girls do not have blonde hair. How

many total girls are in the class? How many animals in all?

3. Of the 80 shirts sold for the fundraiser 4. 600 students attend Walker Middle School.44 were size medium. 240 are 6th graders.What percent were not size medium? What percent of the students are not 6th graders?

5. Molly read 30% of a novel for her English class. 6. A glucose molecule is composed of 24 atoms.She still needs to read 105 pages. Oxygen atoms make up 25% of the molecule. How many pages are in the novel? How many atoms in the molecule are not oxygen

atoms?

7. Victoria’s trip requires a 200 mile drive. 8. Kalecka charged a computer for $600 on her card.She has driven 80 miles so far. She has paid 40% so far.What percent of the trip has not been How much does Kalecka still owe for thedriven yet? computer?

More Percent Word Problems


Name _________________________________ Period_________ Date___________________

1714

17:14 17 to 14

All of these expressions are read seventeen to fourteen. A ratio is said to be in lowest terms if the two numbers are relatively prime. You do not change an improper fraction to a mixed number if the improper fraction represents a ratio.

Example: There are 9 players on a baseball team. Four of these are infielders and 3 are outfielders. Find each ratio in lowest terms.

1. infielders to outfielders 43

4:3 4 to 3

2. outfielders to total players 3 19 3= 1:3 1 to 3

3. outfielders to infielders 34

3:4 3 to 4

4. infielders to total players 49

4:9 4 to 9

Express each ratio as a fraction in lowest terms.

1. 5 to 7 2. 11 to 6 3. 10:30 4. 12:24

5. 8 to 2 6. 32 to 4 7. 68:17 8. 45:18

Express each rate in per unit form.

Example: 15 cans for $3 15 53 1= 5 cans per dollar

200 miles in 4 hours 200 504 1

= 50 miles per hour

7. 120 km in 3 hours 8. 10 walls using 4 gallons of paint

9. 10 gallons in 2 minutes 10. $1,000 in 4 months

11. 300 miles on 6 gallons 12. 75 meals in 3 hours

Ratios and Per Unit Rate

A ratio is a comparison between two quantities. We use them everyday; $2.75 per gallon of gasoline, one Pespi costs 50 cents, the legend says this map tells us one inch is equivalent to 100 miles, five fi ngers per ha nd, etc.

If this class has 17 girls and 14 boys, we can write the ratio of girls to boys in the following ways:


Example: Write each ratio as a fraction in lowest terms.

1. The number of prime numbers between 10 and 25 to the number of whole numbers between 10 and 25.

2. The number of vowels to the number of consonants in the alphabet (consider y a consonant).

3. The numbers of consonants to the total number of letters in the alphabet.

4. The total number of letters in the alphabet to the number of vowels.

5. The number of weekdays to the number of weekend days in a week.

6. The number of weekdays to the number of weekend days in the month of February (not a leap year).

7. The number of diagonals drawn in the figure to the right to the total number of segments.

8. The number of sides on a triangle to the number of sides on a hexagon.

9. The number of even dates in a 30-day month to the number of odd dates in a 31-day month.

10. The number of factors of 18 to the number of factors of 15.

11. In an equilateral triangle, what is the ratio of the length of one side to the perimeter.

Writing Ratios


1. A recipe has a ratio of 2 cups of flour to 3 cups of sugar. Find the per unit rate in terms of eachingredient.

2. If a person jogs 12

mile in 14

hour, compute the unit rates.

3. If a person walks 14

mile in 16

hour, compute the unit rates.

4. An Olympic freesyle swimmer averaged 0.55meters in 14

second. Compute the unit rates.

5. At the Daytona 500, drivers can average 2 12

miles in 45

minute. Compute the unit rates.

Unit Rates Involving Complex Fractions


Solve each proportion. Examples:

5 357 n= 49n = 14

16 40y

= 35y = 3 54 m=

3 4 53 203 203 3

263

mmm

m

==

=

=

1. 29 3n= 2. 3

5 20n

= 3. 2 67 n=

4. 3 610n

= 5. 1211 33n= 6. 21 3

8r=

7. 8 243 m= 8. 3

16 4n= 9. 7 56

8 v=

10. 14 2128 x

= 11. 4 610 y

= 12. 3 275 y

=

13. 5 23t

= 14. 4 75 n= 15. 5

6 9x

=

Solving Proportions

7

8


1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

0 x

y

1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

0 x

y

1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

0 x

y

Proportional Relationships involve collections of pairs of measurements in equivalent ratios. Equivalent ratios have the same unit rate.

Example: Are the following ratios proportional?

23

and 49

2 2 43 2 9→×

≠→×

NO 810

and 2025

8 410 5

= and 20 425 5

= YES

1. Determine whether the ratios 25

and 2050

are proportional. A. are proportional

B. not proportional

2. Determine whether the ratios 812

and 1018

are proportional. A. are proportional

B. not proportional

Determine whether the values in each table represent a proportional relationship.

3. 4. 5.

Determine whether the graphs show a proportional relationship.

6. 7. 8.

Inches on map

Distance In miles

1 90 2 170 3 250 4 330 5 410

Number of baskets

Total Cost

3 $24 5 $40 6 $48 8 $64 15 $120

Gallons Used

Miles Travelled

1 36 2 72 4 144 5 180 8 288

Proportional Relationships


Name _________________________________ Period_________ Date________________

Slope of a Line Linear equations have a constant slope. Slope is the ratio of a line’s vertical change (rise) to its horizontal change (run). On the coordinate plane, you can find the slope m:

--

rise difference of y coordinatesm orrun difference of x coordiantes

=

Examples: Find the slope of the line through the given points.

(a) (5, 4), (2, 0) (b) (1, 5), (3, 1) (c) (−3, 6), (4, 6) (d) (3,−1), (3, 2) (e)

4 05 243

m −=

−

=

5 11 3422

m −=

−

=−

= −

6 64 ( 3)070

m −=

− −

=

=

2 ( 1)3 3

30

m

undefined

− −=

−

=

-5 5

-5

5

x

y

1 ( 3)3 0

23

m − − −=

−

=


Slope of a Line For problems 1 – 7, find the slope of the line through the given points. Show your work.

2. (4, 1), (12, 8) 2. (−1, 0), (0,− 3) 3. (2,−3), (−5,−3) 4. (5, 7), (5,−2)

-5 5

-5

5

x

y

-5 5

-5

5

x

y

-5 5

-5

5

x

y

5. 7.6.

1 2 3 4 5 6 7

10

20

30

40

50

60

70

x

y

Name _________________________________ Period_________ Date________________

Direct Variation: Constant of Proportionality (Variation), Slope & Unit Rate

Consider the table. Note that the ratio of the two quantities

is constant ( 20 40 602 4 6= = ), indicating a proportional

relationship. This relationship is called a direct variation. This constant ratio is called the constant of proportionality or constant of variation.

Example problems: Determine the unit rates:

1. Bamboo that grows 5 inches in2.5 hours.

Cyclist Ride Hours Miles

3 24 6 48

Babysitting (hours), x

Money Earned ($), y

2 20 4 40 6 60

Consider the graph of a line containing these points.

Determine the slope: 20 102

changein ychangein x

= =

What does this mean? $10 is earned per hour babysitting Recall, this is also called the unit rate (a rate with 1 in the denominator). Therefore, note that the constant of proportionality (variation), the slope, and the unit rate all have the same value.

2.

1 2 3 4 5

1

2

3

4

5

x

y

3.


Direct Variation: Constant of Proportionality (Variation), Slope & Unit Rate

You can use tables, graphs and words to represent proportional relationships. Fill in the missing information; determine the unit rate.

TABLE GRAPH WORDS

A grocery store sells 2 red peppers for $4.

Raffle Tickets

1 2 3 4 5

1

2

3

4

5

x

y

1 2 3 4 5

1

2

3

4

5

x

y

TICKETS

CO

ST ($

)

Deriving y = mx We know that the graphs for direct variation always go through the origin (0, 0). Knowing that, let’s derive the equation for direct variation.

2 1

2 1

1 1 2 2

slope formula

0 ( , ) (0,0) and ( , ) ( , )0

simplify

Multiplication Propertyof Equality

y y mx x

y m x y x y x yx

y mxy mx

−=

−−

= = =−

=

=

So, in a direct variation equation, y mx= , the m represents the constant of proportionality (variation), the slope and the unit rate. Example: Which functions show a proportional relationship? How do you know?

Yes, passes through (0,0)

x 3 6 9 y 1 2 3

(0, 0)

y

x

(x, y)

-5 5

-5

5

x

y

x 0 1 2 y 0 3 6

-5 5

-5

5

x

y

Yes, passes through (0,0)

Yes, y=mx

No, does not pass through (0,0)

No, not y=mx Yes, does pass through (0,0)


Name _________________________________ Period_________ Date________________

Deriving y = mx Which functions show a proportional relationship? How do you know?

x 0 1 2 y 0 4 8

x 2 4 6 y 4 8 12

x -2 -1 0 y 0 3 6

-5 5

-5

5

x

y

-5 5

-5

5

x

y

-5 5

-5

5

x

y

1. 2. 3.

4.

5.

6.

7. 8. 9.

Solving Proportions with Word Problems; y = kx

Example: If a car drives 7 miles (x) in 28 minutes (y), how far will it drive in 64 minutes?

First, we find the minutes per mile. y = k x, 28 = k (7), 4 = k. Thus, it takes the car 4 minutes per mile. Now, we must find how many miles are driven in 64 minutes. y = k x, 64 = 4 x, 16 = x. Therefore, the car will drive 16 miles in 64 minutes.

Solve the following problems by finding the constant of variation.

1. A speed reader can read a 90,000 word book in 150 minutes. How long would it takethem to read a 10,800 word chapter?

2. If a car drives 390 miles in 6 hours, how long does it take to drive 130 miles?

3. A student can type on average 210 words in 3 minutes. How long would it take for themto type a 1,001 word paper?

4. A girl’s hair is 23 inches long and grows about half an inch every month. How long willher hair be in 2 years?

5. A child jumping rope can jump at a speed of 280 revolutions every 2 minutes. What is thenumber of revolutions the child can jump in 3.5 minutes?


Name _________________________________ Period_________ Date________________

5 10 15

5

10

x

y

The equation that will represent this data is y = 3.50x, where x is the number of gallons of gasoline and y is the total cost (y = mx). Slope is

3.50 as indicated in the table as the unit rate.

The graph is shown. (Note: The equation does extend into the third quadrant because this region does not make sense for the situation. We will not buy negative quantities of gasoline, nor pay for it with negative dollars!) We can find the slope by creating a “slope triangle” which represents

7 3.52

riserun

= = , which confirms the slope we show in the equation.

Either way, the constant of proportionality is the slope, which is 3.5.

Example: Graph the proportional relationship between the two quantities, write the equation representing the relationship, and describe how the unit rate or slope is represented on the graph.

Five Fuji Apples cost $2 Again, we can begin by creating a table relating the number of apples to their cost. We can use this table to plot the points and

determine the slope of the line.

Using the slope triangle, we can see that 25

riserun

= .

Using ymx, the equation for the line is 25

y x= .

For unit rate: if five apples cost $2.00, then one apple costs 2.00 .405

=

or 40 cents per apple. (It is also represented on the graph: for one apple, the graph rises .40.)

Gas (gal) Cost ($) 0 0 1 3.50 2 7 3 10.50

# of apples 0 5 10 15 cost ($) 0 2 4 6

Gallons of Gas 5 10

5

10

x

y

Tota

l Cos

t ($)

rise +7

Run +2


Name _________________________________ Period_________ Date________________

Representing Proportional Relationships and Slope (page 1)

A proportional relationship between two quantities exists if they have a constant ratio and a constant rate of change. This relationship is also called a direct variation. The equations of such relationships are always in the form y = mx. When graphed, they produce a line that passes through the origin. In this equation, m is the slope of the line; it is also called the unit rate, the rate of change, or the constant of proportionality of the function.

Example: Graph the proportional relationship between the two quantities, write the equation representing the relationship, and describe how the unit rate or slope is represented on the graph.

Gasoline cost $3.50 per gallon We can start by creating a table to show how these two quantities, gallons of gas and cost, vary. Two things show us that this is definitely a proportional relationship. First, it contains the origin, (0, 0), and this makes sense: if we buy zero gallons of gas it will cost zero dollars. Second, if the number of gallons is doubled, the cost is doubled; if it is tripled, the cost is tripled.

Representing Proportional Relationships and Slope (page 2)

For each of the problems, graph the proportional relationship between the two quantities, write the equation representing the relationship, and describe how the unit rate or slope is represented on the graph.

1. Mario walks at 3 miles per hour.

2. In a food eating contest, a contestant eats 60 hot dogs in10 minutes.

3. Every six days, Draco receives four boxes of cauldron cakes.

5 10

5

10

x

yEQUATION:

DESCRIPTION:

EQUATION:

DESCRIPTION:

EQUATION:

DESCRIPTION:

5 10 15 20

5

10

15

20

x

y

4 8 12 16 20

20

40

60

80

100

120

x

y


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Student Learning Extension Opportunities Grade 6-Grade 8 · 2020-03-24 · Student Learning Extension Opportunities Grade 6-Grade 8 Directions: These learning activities are provided