Revisiting numbers grade 8

Use the data from the activity to calculate the average speed of the wave in your class.. Calculate the average speed of the wave in seats per second.. Calculate Maddie’s average speed

Revisiting Numbers

Number

Mathematics in Context is a comprehensive curriculum for the middle grades

It was developed in 1991 through 1997 in collaboration with the Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison and the Freudenthal Institute at the University of Utrecht, The Netherlands, with the support of the National Science Foundation Grant No 9054928.

This unit is a new unit prepared as a part of the revision of the curriculum carried out in 2003 through 2005, with the support of the National Science Foundation Grant No ESI 0137414.

National Science Foundation

Opinions expressed are those of the authors and not necessarily those of the Foundation.

Abels, M., Wijers, M., and Pligge, M (2006) Revisiting numbers In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics

in context Chicago: Encyclopædia Britannica, Inc.

Copyright © 2006 Encyclopædia Britannica, Inc.

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Printed in the United States of America.

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The Mathematics in Context Development Team

Development 2003–2005

Revisiting Numbers was developed by Mieke Abels and Monica Wijers

It was adapted for use in American Schools by Margaret A Pligge.

Wisconsin Center for Education Freudenthal Institute Staff

Research Staff

Thomas A Romberg David C Webb Jan de Lange Truus Dekker

Gail Burrill Margaret A Pligge Mieke Abels Monica Wijers

Editorial Coordinator Editorial Coordinator Content Coordinator Content Coordinator

Project Staff

Sarah Ailts Margaret R Meyer Arthur Bakker Nathalie Kuijpers

Teri Hedges Kathleen A Steele Dédé de Haan Nanda Querelle Karen Hoiberg Ana C Stephens Martin Kindt Martin van Reeuwijk Carrie Johnson Candace Ulmer

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Elaine McGrath

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Cover photo credits: (left to right) © William Whitehurst/Corbis;

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Section C Investigating Algorithms

Section E Reflections on Numbers

Multiplication and Division 46

Dear Student,

Have you ever been at a competitive event where

the race was too close to call? Electronic timers

today are very precise and can split a second into

a million parts Precision is extremely important to

scientists as they map out unknown territories in

outer space and inside the human body

In the unit, Revisiting Numbers, you use will learn to use numbers

more precisely You will further investigate ways to represent verylarge and very small numbers You will reflect on all number

operations You will improve your precision working with numberoperations, by looking at related operations

We hope you enjoy this unit

Sincerely,

T

Th hee M Ma atth heem ma attiiccss iin n C Co on ntteex xtt D Deevveello op pm meen ntt T Teea am m

The Cleveland Browns Stadium canseat over 73,200 people Clevelandfans often create a wave—peoplestand up lifting their arms and thenquickly sit down When one groupsits down, an adjacent group takesover The wave moves around theentire stadium

R

P

1 a Estimate how many feet the wave travels one time around the

stadium Describe how you made your estimate The stadiumdimensions are 933 feet by 695 feet

b How much time will it take the wave to go once around the

stadium? Describe how you made your estimate

c Use your estimates to find the average distance the wave

travels in one second You may want to use a ratio table foryour calculations

If a stadium wave travels 60 feet (ft) in 2 seconds (sec), it travels with

Distance (in ft) Time (in sec)

For this activity, you will need:

2 a Use the data from the activity to calculate the average

speed of the wave in your class

b Compare your class’s wave with the Cleveland Stadium

wave Describe your findings

A rateis the ratio of two different measuring units For example, you can express the rate of speed in miles per hour (mi/h), or in feetper second (ft/s) Using metric units, speed is usually expressed inkilometers per hour (km/h), or meters per second (m/s)

Make Some Waves

Rates and Units

Section A: Speed 3

A

Speed

3 What other rates do you know? Copy this table and complete it.

You may remember from the unit Ratio and Rates how you used ratio

tables to express rates as a single number

Stuart and Lexa are researchers who analyze average speeds of largecrowds In a soccer stadium, they timed one wave taking 22 sec totravel 440 seats Each seat had a total width of 2 ft

4 a Calculate the average speed of the wave in seats per second.

b Compare the average speed from this research to the speeds

you found in problem 2b How do they compare?

Rita found 30 ft/s as the average speed of the wave in her class Shewants to know how fast this is in miles per hour Here is how shestarted to solve the problem

5 a Explain Rita’s second step.

b Copy Rita’s ratio table and

calculate the missing numbers

c Use this information to find the

average speed of the wave intomiles per hour (mi/h)

Did you know? 5,280 ft are in 1 mi?

Units Example

Heart Rate Heartbeats/minute (bmp) Data Transfer Kilobytes/second (kB/s)

My heart beats at a rate of

65 beats per minute.

The speed of the download stream is 1,024 kilobytes per second.

Population Density

Kenny ran 60 m in 15 sec.

6 a Copy this ratio table and use the arrows to

show the steps you have to make to findKenny’s distance in one hour

b What is Kenny’s average speed in meters

per hour? What is Kenny’s average speed

in km/h?

c Will Kenny really be able to run that

distance in one hour? Explain

d Henri ran 50 m in 12 sec Is Henri’s

average speed higher or lower thanKenny’s? Show your work

Distance (in m) 60

15

Time (in sec)

Maddie: “I did the 5K run in 25 minutes

I wonder how fast I ran in kilometers per hour.”

7 Calculate Maddie’s average speed for the

5K run in km/h

To compare speeds, you may have to change theunits Changing kilometers per hour into miles per hour is easy if you have an speedometer like

In most European countries, the speed limit on highways is 120 km/h,

on county roads 80 km/h, and in towns 50 km/h

9 How do these speed limits compare to those in the United States?

Here are some speed records for a variety of creatures

A cheetah, or hunting leopard, is timed at 70 mi/h

The ostrich is a special sort of “sprinter.” It is a bird, but it does not fly It can run 20 mi in 40 min

American quarter horses are the fastest horses in the world They can cover a quarter-mile in less than 21 sec

In the water, the speed record is held by the sailfish, which

in a calm sea, can reach a speed of 100 m per 3.3 sec

The Indian spine-tailed swift bird has repeatedly been clocked in levelflight over a carefully measured two-mile track at 32.8 sec

On September 14, 2002, Tim Montgomery of the United Statesset a world record in the 100 m, clocking 9.78 sec at the IAAFGrand Prix Final

10 Order these speed records on the odometer on Student Activity Sheet 1 Show your work Note that one of the speeds will not fit

Digital timepieces improve accuracy

12 a What time is displayed on this timepiece?

b What does the 04 mean?

c How much more time needs to pass for the

timepiece to read 1 hour?

These hand-held timepieces are accurate; however, theydepend on the reaction time of the humans using thetimepiece From the moment you see or hear a signal, ittakes time for the signal to go to your brain, time foryour brain to react, and finally time for your nerves andthe muscles in your fingers to react You will determineyour personal reaction time using data collected fromthe following activity

Without any signal, the person holding the ruler lets go.The catcher tries to react as quickly as possible andcatches the ruler Record the number of centimeterscaught

The number of centimeters caught is a distance thatwill be used to calculate the catcher’s reaction time

Do this experiment five times per person and recordthe distances in centimeters

b Use one of the correct formulas to calculate the five reaction

times you recorded in the previous activity Calculate youraverage reaction time

Australia’s Cathy Freeman, a world-class athlete, had a reaction

time of 0.223 sec in the women’s 400 m final at the 1995 World

Championships Her reaction time was measured with an electronicdevice inside the starting block This device recorded the intervalbetween the starting shot and the first athlete leaving the blocks

14 a On Student Activity Sheet 1 fill in the missing times indicated

by the blanks under the number line

b How much longer does this number line need to be in order to

locate 1 sec?

c Place Cathy Freeman’s reaction time on the number line Use

an arrow to point to the location

d Tests have confirmed that nobody can react in less than 0.110 of

a second Place this minimum reaction time and your reactiontime from problem 13b on the number line If necessary, extendthe line

e Explain why a false start is declared if the interval between the

starting shot and the athlete leaving the block is less than0.110 of a second

d is the distance you caught, in centimeters.

Other formulas to find the reaction time are:

or

Suppose both athletes ran in the same 100-m race, at theirown world record pace Would the race be too close tocall? To investigate, you will need to zoom in on the finish,the moment when Tim reached the finish line.

15 a How much time (in sec) elapses between Tim’s win

and Maurice’s finish?

b Make a guess What is Maurice’s distance from the

finish line when Tim wins the race?

Instead of guessing, you can calculate this distance in centimeters This ratio table will help you with your calculations

c Explain the numbers 10,000 and 9.79 that are in the

ratio table

d Calculate the distance in the last column What do

you know now?

e If you were at the race, would you be able to tell

who finished first? Explain your answer

The speed of light is 299,792,458 m/s

16 a How many kilometers does light travel per second?

b What is the speed of light in kilometers per hour?

The speed of light is often rounded to 300,000,000 m/s

This table shows the many different ways to write this number

17 Copy the last row of the table in your notebook and fill in the

missing numbers

You may remember writing very large numbers in scientific notation

in the unit Facts and Factors.

A number written in scientific notation is the product of a numberbetween 1 and 10 and a power of 10 The first number is called the

1,680,900, written in scientific notation is 1.68  106

Notice that the mantissa is rounded to two decimal places

A calculator may display this number as:

18 a Write 43,986,000,000,000 in scientific notation Round the

mantissa to one decimal place

b Write the speed of light in km/h from 16b in scientific notation.

Round the mantissa to two decimal places

The average distance from Earth to the sun

is about 1.5  108kilometers Neptune is 30times farther away from the sun

19 Write the distance from Neptune to the

sun in scientific notation

The distance from the sun to Venus is 0.72times the distance from the sun to Earth

20 Write the distance from Venus to the

sun in scientific notation

21 a Who is right, Samantha or Jennifer? How would you explain it

to the person who did the problem incorrectly?

b Explain why 104 104 104 104 104 104 104 104

104 104 105.Samantha and Jennifer disagree on this problem:

106 103

Samantha claims the answer is 102, while Jennifer thinks it has

to be 103

c Who is right, Samantha or Jennifer? How would you explain it

to the person who did the problem incorrectly?

Calculate the following problems without the use of a calculator.Write your answers in scientific notation

Different Number Systems

Our number system uses ten digits, which is the same as the number

of fingers we have

So we use a base-10 number system

Why do we have 24 hours in a day? Why are there 60 minutes in anhour and 60 seconds in a minute?

To explain this, we have to go back in history4,000 years, to the land between the Tigris andEuphrates Rivers, where the Sumerians lived.They also used their fingers to count things, butthey did it differently

The Sumerians counted finger joints instead offingers, and they used their thumb to do thecounting Their number system was a base-12 number system, which

is why they divided a day into twelve parts

A thousand years later, the Babylonians, who lived in the same area asthe Sumerians, used a base-60 number

system It is not sure why they chose

60 One reason might be becausebase-60 makes divisional operationseasy since 60 is divisible by 2, 3, 4, 5,

6, 10, and 12

They divided a day into two times 12 hours because twelve fits nicely

in their system (12  3  4, and 60  3  4  5) Hours were furtherdivided into 60 minutes, and the minutes were divided into 60

seconds This system for time is still used today

For other divisions, they used their base-60 system In our decimalsystem, the first decimal is tenths In the base-60 system, the first frac-tional place is sixtieths The first fractional place is called a minute,the second place is called a second So that is why an hour has 60minutes, and a minute has 60 seconds

The Babylonians not only measured time, they also studied astronomyand measured angles They divided the heavens into twelve sectors,the time it takes the earth to complete one revolution around the sun.Each sector was 30°, so a complete year took 360° (12  30) Eachdegree was further divided into 60 minutes, and each minute wasdivided into 60 seconds

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Persian Gulf

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ARABIAN DESERT

H A W R S

S U M E R

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B A B Y L O N I A

1 2 3

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A rate is the ratio of two different measuring units, written as a singlenumber Examples of rates are:

• Resting heart rate, such as 85 bpm

• Speed of a car on a highway, such as 55 mi/h or 88 km/h

• Data transfer rate, such as 1,024 kB/s

A ratio table is a helpful tool for finding a rate

Suppose you traveled 150 km in 2.5 hours A ratio table can help youdetermine your average speed in km/h

Units

Sometimes you have to convert the measuring units of a rate

• How fast is 5 m/s in kilometers per hour?

1

Time (in sec)

Traveling 150 km in 2.5 hours, what is the average rate of speed? Answer: 60 km/h

How fast is 5 m/s in kilometers per hour? Answer: 18 km/h

Any positive number written in scientific notation is a product of

two factors: a number between 1 and 10 (the mantissa) and a

power of 10

28,600,000 written in scientific notation is 2.86  107

A calculator may display this number as:

You may round the mantissa to one decimal place: 2.9  107

2 a What is the meaning of the six in this number?

b Write this number in scientific notation rounding the mantissa to

one decimal place

How fast is 60 km/h in meters per minute?

Answer: 1,000 m/min

3 Calculate the earth’s rotational speed at the equator in miles per

hour The circumference of the earth at the equator is about 2.5  104miles

A

In one year, the earth travels about 5.8  108miles as it orbits around the sun

4 Calculate the earth’s average orbital speed around the sun in

miles per hour

5 Mercury’s average orbital speed around the sun is about 48 km/sec.

Is Mercury’s orbital speed faster or slower than the orbital speed ofthe earth? Explain your answer

6 Calculate the following problems without the use of a calculator.

Write your final answer using scientific notation

a Multiply 3  104by 2  102

b Divide 2  1010by 106

c Add 2  103and 4.5  102

d Subtract 6  102from 2  103

The earth makes one complete revolution

on its axis in 24 hours The rotational speedbegins at zero at either geographical poleand increases as you head toward theequator

Section A: Speed 15

When timekeepers used hand-held stopwatches, it was very difficult

to rank evenly matched competitors At the 1960 Olympic games in

Rome, Australia’s John Devitt and America’s Lance Larson finished

neck-and-neck in the final of the 100-m freestyle swimming events

All three timekeepers for Devitt’s lane clocked him at 55.2 sec

Larson was clocked at 55.0, 55.1, and 55.1 sec The judges placed

Devitt as the winner The official time for both swimmers was

recorded as 55.2 sec

7 a Is this fair? Explain your reasoning using your knowledge

about reaction time

b Suppose Larson swam 100 m in 55.2 sec and Devitt finished

0.1 sec before Larson What is Larson’s distance (in cm) from

the wall when Devitt finished the race? Would this have been

visible?

Write 236.7  104as a number

Why is 236.7  104not written in scientific notation?

In the previous section, you worked with small numbers with one ormore decimals You will now further investigate decimal numbers,such as 9.78.

1 a What is the value of each digit in the number 9.78?

b What is the value in each digit in the number 97.8?

c How does the number 9.78 compare to the number 97.8?

Describe how they are the same and how they are different

In the unit Facts and Factors, you learned that when you multiply a

number by 10, you multiply the value of each digit by 10

Here are two schemas formultiplying by ten anddividing by ten.

3 On Student Activity Sheet 2, complete the

schema that showsdividing by ten

4 Calculate without using a calculator.

5 What portion of Shelia’s solution

is food coloring? What portion iswater?

6 What portion of Shelia’s second solution is food coloring?

9 Copy and complete the table

for six dilutions

Suppose the table extended up

to 20 dilutions

10 After 12 dilutions, what

portion of the 10-ounce solution is food coloring?

What about after

20 dilutions?

Shelia loves the new color She decides to do it again and poursone ounce of this second solution into another empty measuringcontainer She adds enough water to make a 10-ounce solution

7 What portion of Shelia’s third solution is food coloring?

8 Suppose Shelia repeats this process again What portion of

Shelia’s fourth 10-ounce solution would be food coloring?

Notation

B

Number of Dilutions

Portion of the 10-oz Solution That Is Food Coloring

1 2 3 4 5 6

0.1 or 1010.01 or 1001

You have probably found that it is just as tedious to write very smallnumbers as it is to write very large numbers An abbreviation systemlike the one that you use for large numbers can be developed andused for very small numbers as well

When you were diluting food coloring, you divided the portion offood coloring in each solution by 10

Here is a way of describing this dilution process using arrow language

11 Copy and complete the arrow string until you reach the

You can display this pattern of dividing by 10 vertically.

13 a Copy the vertical display on the left in your

notebook Write each result also as a power

of 10 Do this only with the numbers for whichyou know the power of 10

b Describe any patterns you notice in the powers

of 10 How would you continue this pattern?

c How would you explain to someone that 100 1?

14 a Look back at your chart of Shelia’s food coloring

dilutions Suppose you have a solution that has a

104portion of food coloring How many dilutionshave taken place? How do you know?

b How many dilutions have taken place if the

solution is 106food coloring?

c What portion of the solution is food coloring if

you have done five dilutions? Write this number

as a power of 10

d Describe the relationship between the number

of dilutions and the portion of the solution that

15 a What is 107 10 written as a power of 10?

b What is 106 100? (Hint: Dividing by 100 is two dilutions.)

16 Show that you can write the result of 100  1,000 as 101.

17 a Calculate 106 103

b Calculate 104 107

Small Numbers

Near the end of Section A, you created a general rule for multiplyingpowers of ten

If m and n are natural numbers, then 10 m 10n 10m+n

18 a Give examples to show that this rule is also valid for negative

values for m and n.

b Create a general rule for dividing powers of ten.

19 Find each product or quotient Write your answers both as a

power of ten and as a numeral

Here is a photo of bacteria enlarged 10,000 times

20 a Measure the diameter of the image of one of

the bacterium in the photo in millimeters (mm)

b Find the actual size of this bacterium.

Mycoplasmal pneumoniae is a type of bacteria known

to cause a sore throat, bronchitis, and pneumonia.Mycoplasma bacteria range between 0.0015 mm and0.0025 mm in size

21 Can the bacteria from problem 20 be an example

of the mycoplasmal pneumoniae bacterium?

Justify your reasoning

10,000x

22 Copy these lines in your notebook and fill in the missing

numbers

The last line shows how to write 0.0025 in scientific notation

0.0025 written in scientific notation is 2.5  10 3.

To write 0.00000075 in scientific notation, you can start a pattern.0.00000075  1  0.00000075  100

0.0000075  0.1  0.0000075  101

23 a Continue the pattern until the number is in scientific notation.

b Write 0.00261 in scientific notation.

c Write what your calculator displays after you enter:

20  30,000,000,000,000 

How do you write this answer in scientific notation?

d Write 1.8  105as a numeral.

How would you write this number in scientific notation?





Notational Systems

The number system we use today is a positional notation systembased upon powers of ten Each digit occupies a position connected

to a power of ten

• The position of each digit in a number determines its value

• You can read the number 14.75 as:

“fourteen and seventy-five hundredths.”

• You can write the number 14.75 as:

B

Section B: Notation 23

0.000025  2.5  0.00001, so 0.000025 written in scientific notation is

2.5  105.

Calculators display very small or large numbers using scientific notation

A calculator may display 2.5  105as:

Operating with Powers of 10

If p and q are whole numbers, then 10 p 10q 10p+ q, and

2.5 –05 or as 2.5  E– 05

Here is an enlarged photo of a human hair

1 a What does the 1,000x on the picture mean?

b Measure the width of the hair shown shown

in the photo in centimeters

c What is the actual thickness of this hair?

Write your answer as a numeral and in

4 a Use a calculator to find the answer of 4  250,000,000 Writeyour answer in scientific notation.

b Calculate the product of 3.5  103and 1.2  102.

5 Write in scientific notation.

Your neighbor offers to pay you to be hisson’s math tutor You agree and set upyour first tutoring session with Harvey.Harvey comes over to your house, flopsdown in a chair, and opens his 4th grademath book to read this problem.

• A crate of Lemon Drop lemonadecontains 24 bottles

• If a supermarket manager bought

49 full crates, how many bottles didshe buy?

Section C: Investigating Algorithms 25

C

Investigating Algorithms

Multiplication

You ask Harvey to try the problem, and he reluctantly picks up his pencil

to begin solving it As he works, you do the problem in your head

1 a Describe a way to estimate the answer.

b Adjust your estimate to find an exact answer.

Harvey uses a ratio table to do the problem

When he finishes, he looks unsure and asks, “Did I do it right?”

2 Review Harvey’s ratio table solution and explain each entry.

Crates Bottles

1 24

10 240

5 120

4 96

9 216

40 960

49 1,176

3 How would you explain Sean’s reasoning to Harvey?

Harvey shows you Sandra’s and Hattie’s solutions for the lemonadeproblem, 24  49

Sandra worked the problem like this:

Hattie worked the problem like this:

Harvey says, “Sandra and Hattie say that they did the same thing But Sandra used a ratio table, and Hattie used something that lookedtotally different How can these two things be the same?”

4 How are Sandra’s and Hattie’s ways of multiplying 24  49 thesame? How are they different?

Crates Bottles

1 24

50 1,200

49 1,176

Crates Bottles

1 24

9 216

40 960

49 1,176

24

 49 216 960 1,176

Hattie used a standard algorithmto calculate 24  49 An algorithm is

a predetermined set of rules used to perform computations The wordcomes from the name of an Arabic scientist, al-Khwarizmi, who lived

in the ninth century

Harvey also shows you Clarence’s solution

Clarence lists four different multiplication problems Harvey seems really confused and says, “Look at what Clarence did!

I know this can’t be right!”

5 How would you explain to Harvey that Clarence’s method is a

legitimate way to multiply 24  49?

Section C: Investigating Algorithms 27

800

?

?

problem 5, show how to use the area model

to multiply 24  49

7 a Which of the multiplication methods

presented on pages 25–27 do you prefer?Explain

b Show how to find the product of 28  36using two different methods You can useany two you prefer

Sometime in early March, Harvey comes to the tutoring session veryupset He says, “Just when I figured out multiplication, they changethe problems Now they are making us do division!”

You assure him that he will be able to do division problems as easily

as he can now do multiplication problems You suggest that since helikes to use ratio tables to multiply, he may be able to use them todivide Harvey decides to try it He reads this problem from his book

A stapler factory can pack 32 staplers in a standard box

A large company orders 2,000 staplers How many boxes

do you need to fill the order?

Harvey makes this ratio table

Division

Boxes Staplers

1 32

2 64

20 640

60 1,920

62 1,984

Harvey finishes his work and says, “The factory has to pack and ship

62 boxes and then figure out how to send 16 more staplers Or theycould just call the people at the company and ask them whether theywant 62 or 63 boxes.”

Study Harvey’s ratio table

8 How did Harvey find that 60 boxes would hold 1,920 staplers?

Harvey asks you how you would do the problem You tell him thatyou would do it mentally Suppose you shared this mental strategywith Harvey

Investigating Algorithms

C

One box holds 32 staplers.

One hundred boxes hold 3,200 staplers.

Fifty boxes hold 1,600 staplers.

Ten boxes hold 320 staplers.

So sixty boxes hold 1,920 staplers.

I still need to pack 80 staplers.

Two more boxes will hold 64 staplers, but

there will still be 16 of the 2,000 staplers left to pack.

Just then, Harvey’s father comes to take Harvey home Harvey is soexcited about being able to do division problems that he gives hisdad the stapler factory problem to see how his father would do it His father picks up a pencil and writes out the following

9 a Compare the mental strategy with Harvey’s ratio table strategy

and his Dad’s strategy Describe any similarities and differencesamong these strategies

b Which method do you usually use to solve this kind of problem?

32 2000  192 80  64 1662

Section C: Investigating Algorithms 29

C

Investigating Algorithms

Multiplication and division are very closely related In fact, for every multiplication problem, there are at least two related divisionproblems You will explore this relationship using the multiplicationproblem Harvey read from his book

A crate of Lemon Drop lemonade contains 24 bottles

If a supermarket manager bought 49 full crates, how many bottles did she buy?

The number sentence that matches this situation is 24  49  1,176

10 Can the number sentence also be 49  24  1,176? Why or why not?

You can also think about the crates and the bottles like this:

“There are 1,176 bottles of lemonade in crates

24 bottles fill up each crate.”

That means that there are 49 crates filled up

The number sentence that matches this situation is 1,176 ÷ 24  49

11 Another related number sentence is 1,176 ÷ 49  24 Write a storyabout the crates and bottles to match this number sentence

In the town where Harvey lives, each city block is about 18of a mile long Here is a double number showing the relationship between cityblocks and miles

12 a Harvey lives forty blocks from the mall How many miles is

Harvey’s home from the mall? How did you figure this out?

b Explain how 40 1 8is related to the home-to-mall situation

c Show how to use the ratio table to calculate 40 1 8

1 1 8

Investigating Algorithms

C

Blocks Miles

1 1 8

13 a Harvey lives 312miles from school How many blocks does

Harvey live from school? How did you figure this out?

b Explain how 312 1 8is related to the school-to-home situation

To calculate 314 1 8, you can make up a context problem like the following

How many city blocks of 18 mile are there in 314 miles? Or how many times does 18 mile fit into 314 miles?

14 a Calculate 314 1 8 using this ratio table

b Calculate 212  1 8

c How would you calculate 212 1 6?

15 a Harvey’s mother made 6 liters of lemonade, and she wants

to store it in34 liter bottles How many bottles can she fill with lemonade? Show your work

b Make up a context problem that fits with 6 2 3 and find theanswer

To calculate 6 2 3, you can use a ratio table, but Chi likes to use a different strategy

“6 2 3 means that I have to figure out how many times23 fits into 6

So I will rewrite 6 into thirds.”

2 3

18 3 18  2  9

Chi

Harvey’s mother made 712liters of ice cream She wants to store

it in 34 liter boxes How many boxes can she fill up?

18 a Write a division problem for this situation.

b How many fourths is 712liters of ice cream?

Copy and complete: 71

2…2

… 4

c How many boxes can Harvey’s mother fill up?

The fractions in the previous problem had different denominators

To find the number of times that 34 liter fits into 712liters, it helps tomake the denominators of the fractions equal

19 a Make up a context problem to fit 31

35 6.

b 313…

3

…6

c Solve your context problem.

Your last answer was a whole number Your answer can also be a fraction See problems 17b and 17c

b Make up a division problem where mixed numbers are

involved and solve your own problem

b Make up three more problems with the same phenomenon.

Solve each one

Section C: Investigating Algorithms 31

C

Investigating Algorithms