www.everydaymathonline.com 132 Unit 2 Estimation and Computation Advance Preparation For Part 1, use the Class Data Pad to record and display the Mental Math and Reflexes problems and responses. For the optional Readiness activity in Part 3, make one game mat for each partnership by copying, cutting, and taping together Math Masters, pages 491 and 492. For a mathematics and literary connection, obtain a copy of How Much Is a Million? by David M. Schwartz (HarperCollins Publishers, 1985). Teacher’s Reference Manual, Grades 4–6 pp. 256–264 Key Concepts and Skills • Read and write large numbers. [Number and Numeration Goal 1] • Compare order of magnitude for large numbers. [Number and Numeration Goal 6] • Make reasonable estimates for whole number multiplication problems. [Operations and Computation Goal 6] Key Activities Students review time conversion factors. They count the number of times they can tap their desks in 10 seconds and estimate how long it would take to tap 1 million times. Students then estimate how long it would take to tap 1 billion and 1 trillion times. Ongoing Assessment: Informing Instruction See page 134. Key Vocabulary sample Materials Math Journal 1, p. 57 Study Link 2 9 Class Data Pad blank paper or construction paper markers or crayons watch or timer with second hand calculator Playing High-Number Toss Student Reference Book, p. 320 Math Masters, p. 487 per partnership: 1 die 1 sheet of paper Students practice concepts of place value and standard notation by writing and comparing large numbers. Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 487. [Numbers and Numeration Goals 1 and 6] Math Boxes 2 10 Math Journal 1, p. 58 Students practice and maintain skills through Math Box problems. Study Link 2 10 Math Masters, p. 61 Students practice and maintain skills through Study Link activities. READINESS Playing Number Top-It (7-Digit Numbers) Student Reference Book, p. 326 Math Masters, pp. 491 and 492 per partnership: 4 each of number cards 0–9 (from the Everything Math Deck, if available) Students apply place-value concepts to form, read, and compare large numbers. EXTRA PRACTICE Comparing Powers of 10 Using Place Value Math Masters, p. 66B Students apply place value for powers of 10. ENRICHMENT Applying Estimation Strategies Math Masters, p. 62 Students make time estimates and identify the number models used for their estimation strategies. Teaching the Lesson Ongoing Learning & Practice 1 3 2 4 Differentiation Options Comparing Millions, Billions, and Trillions Objectives To provide experience with comparing the relative sizes of 1 million, 1 billion, and 1 trillion and using a sample to make an estimate. s eToolkit ePresentations Interactive Teacher’s Lesson Guide Algorithms Practice EM Facts Workshop Game™ Assessment Management Family Letters Curriculum Focal Points Common Core State Standards
Transcript
www.everydaymathonline.com
132 Unit 2 Estimation and Computation
Advance PreparationFor Part 1, use the Class Data Pad to record and display the Mental Math and Reflexes problems and responses.
For the optional Readiness activity in Part 3, make one game mat for each partnership by copying, cutting, and
taping together Math Masters, pages 491 and 492.
For a mathematics and literary connection, obtain a copy of How Much Is a Million? by David M. Schwartz
(HarperCollins Publishers, 1985).
Teacher’s Reference Manual, Grades 4–6 pp. 256–264
Key Concepts and Skills• Read and write large numbers.
[Number and Numeration Goal 1]
• Compare order of magnitude for large
numbers.
[Number and Numeration Goal 6]
• Make reasonable estimates for whole
number multiplication problems.
[Operations and Computation Goal 6]
Key ActivitiesStudents review time conversion factors.
They count the number of times they can
tap their desks in 10 seconds and estimate
how long it would take to tap 1 million times.
Students then estimate how long it would
take to tap 1 billion and 1 trillion times.
Ongoing Assessment: Informing Instruction See page 134.
Key Vocabularysample
MaterialsMath Journal 1, p. 57
Study Link 2�9
Class Data Pad � blank paper or construction
paper � markers or crayons � watch or timer
with second hand � calculator
Playing High-Number TossStudent Reference Book, p. 320
Math Masters, p. 487
per partnership: 1 die � 1 sheet of
paper
Students practice concepts of place
value and standard notation by writing
and comparing large numbers.
Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 487. [Numbers and Numeration Goals 1 and 6]
Math Boxes 2�10Math Journal 1, p. 58
Students practice and maintain skills
through Math Box problems.
Study Link 2�10Math Masters, p. 61
Students practice and maintain skills
through Study Link activities.
READINESS
Playing Number Top-It (7-Digit Numbers)Student Reference Book, p. 326
Math Masters, pp. 491 and 492
per partnership: 4 each of number cards 0–9
(from the Everything Math Deck, if available)
Students apply place-value concepts to form,
read, and compare large numbers.
EXTRA PRACTICE
Comparing Powers of 10 Using Place ValueMath Masters, p. 66B
Students apply place value for powers of 10.
ENRICHMENTApplying Estimation StrategiesMath Masters, p. 62
Students make time estimates and identify
the number models used for their estimation
strategies.
Teaching the Lesson Ongoing Learning & Practice
132
4
Differentiation Options
���������
Comparing Millions,Billions, and Trillions
Objectives To provide experience with comparing the relative
sizes of 1 million, 1 billion, and 1 trillion and using a sample to
Make a guess: How long do you think it would take you to tap your desk 1 million times,
without any interruptions?
Check your guess by doing the following experiment.
1. Take a sample count.
Record your count of taps made in 10 seconds.
2. Calculate from the sample count.
At the rate of my sample count, I expect to tap my desk:
a. times in 1 minute.
(Hint: How many 10-second intervals are there in 1 minute?)
b. times in 1 hour.
c. times in 1 day (24 hours).
d. At this rate it would take me about full 24-hour days to tap my
desk 1 million times.
3. Suppose that you work 24 hours per day tapping your desk. Estimate how long it
would take you to tap 1 billion times and 1 trillion times.
a. It would take me about to tap my desk 1 billion times.
b. It would take me about to tap my desk 1 trillion times.8,000 years
8 years
3
345,600
14,400
240
40 taps
Answers vary.
(unit)
(unit)
Useful Information1 billion is 1,000 times 1 million. 1 trillion is 1,000 times 1 billion.1 million � 1 thousand � 1 billion 1 billion � 1 thousand � 1 trillion1,000,000 � 1,000 � 1,000,000,000 1,000,000,000 � 1,000 � 1,000,000,000,000
1 minute � 60 seconds 1 hour � 60 minutes 1 day � 24 hours1 year � 365 days (366 days in a leap year)
Sample answers:
Math Journal 1, p. 57
Student Page
Lesson 2�10 133
Getting Started
How many seconds are in...
1 minute? 60
3 minutes? 180
100 minutes? 6,000
How many minutes are in...
1 hour? 60
5 hours? 300
50 hours? 3,000
How many hours are in...
1 day? 24
2 days? 48
200 days? 4,800
How many days are in...
1 year? 365, except 366 in leap years
10 years? about 3,650
100 years? about 36,500
Math MessageExplain the strategy you would use to find the number of minutes in one year.
Study Link 2�9 Follow-Up Have partners share answers and resolve any differences.
Mental Math and Reflexes Have students practice conversions between units of time. Use the Class Data Pad to record the correct responses. Keep this display up for students to refer to during the lesson.
1 Teaching the Lesson
▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION
Have students in small groups discuss their individual strategies. The group then decides which steps they would use to find the number of minutes in a year. Each group should make a poster, using construction paper and markers or crayons, to list their steps and place the poster on display. Allow time for students to read the displayed group posters. Ask volunteers to identify the similarities and differences in the strategies. Guide the class discussion to focus on summarizing the strategies. Sample answers: Convert a year into a unit of time that can be converted to minutes; convert a year into days; convert these days into hours and the hours into minutes.
NOTE Include a walkabout in the follow-up to this Math Message: Display the
group posters in separate areas of your classroom, and allow students time to
browse until they have read all the posters.
▶ Solving a Tapping Problem WHOLE-CLASSDISCUSSION
with Sampling Strategies(Math Journal 1, p. 57)
Ask students to refer to the Useful Information chart on journal page 57. Pose several questions from the first row of information to highlight these large number relationships for students. What is 1,000 times 1 million? 1 billion One billion has what relationship to 1 trillion? One trillion is 1,000 times 1 billion. Ask a volunteer to read the question labeled Make a guess. Ask students to guess how long it would take them to tap their desks 1 million times without any interruptions. Have students
Materials � 1 six-sided die� 1 sheet of paper for each player
Players 2Skill Place value, exponential notation Object of the game To make the largest numbers possible. Directions
1. Each player draws 4 blank lines on a sheet of paper to record the numbers that come up on the rolls of the die.
2. Player 1 rolls the die and writes the number on any of his orher 4 blank lines. It does not have to be the first blank—it canbe any of them. Keep in mind that the larger number wins!
3. Player 2 rolls the die and writes the number on one of his orher blank lines.
4. Players take turns rolling the die and writing the number 3 more times each.
5. Each player then uses the 4 numbers on his or her blanks tobuild a number.
♦ The numbers on the first 3 blanks are the first 3 digits ofthe number the player builds.
♦ The number on the last blank tells the number of zerosthat come after the first 3 digits.
6. Each player reads his or her number. (See the place-valuechart below.) The player with the larger number wins theround. The first player to win 4 rounds wins the game.
Games
Hundred Ten Hundred TenMillions Millions Millions , Thousands Thousands Thousands , Hundreds Tens Ones
Note
If you don’t have a die,you can use a deck ofnumber cards. Use allcards with the numbers 1 through 6. Instead ofrolling the die, draw thetop card from thefacedown deck.
record their responses on the journal page. Conduct a quick survey of the class for their responses.
Discuss the difference between a guess and an estimate. Use the discussion to clarify for students that a guess is an opinion that you might state without the support of other information. An estimate is based on some knowledge about the subject and is often called an educated guess. Students can only guess the amount of time it would take to tap 1 million times until they collect additional information with which to make an estimate.
Encourage students to think about what information they would need to make a more educated guess, and then ask volunteers to explain strategies that could be used to gather additional information.
Ongoing Assessment: Informing Instruction
Watch for students who are having difficulty developing a strategy. Explain that
this is another situation for which obtaining the exact answer is impossible, such
as the Estimation Challenge from Lesson 2-1. Recall the strategies students
used in that lesson and have them use similar approaches here.
Students might suggest strategies such as the following:
� Count how many times you can tap your desk in a set amount of time, such as 10 seconds.
� Time how long it takes you to tap a certain number of times, such as 100 times.
� Pick a reasonable number of taps for a set amount of time, and make an estimate based on that rate, such as 3 taps per second.
When the class found and used the median step length, they were using a sample. Ask whether students know of any other situations where samples are used.
NOTE A sample of anything is a small piece or part that is intended to give
information about the whole thing. Consumers use product samples to decide
whether products suit their needs. Pollsters use population samples to estimate
information for the whole population. The finger-tapping samples here are time
samples: The count of taps in a 10-second period is a sample used to determine
a tapping rate, and the rate is then used to estimate how long it would take to
make large numbers of taps.
Each student will take a 10-second sample count of their own finger tapping. Practice taking sample counts by timing students as they tap and count for 10 seconds.
3. Use the map on page 354 of your Student Reference Book to answer the questions.
Choose the best answer.
a. What is the shortest distance between San Francisco and San Antonio? About:
500 miles 1,000 miles 1,500 miles 2,000 miles
b. What is the shortest distance between New York City and Chicago? About:
700 miles 800 miles 900 miles 1,000 miles
139
4. a. Circle the times for which the hands on a clock form an acute angle.
2:00 6:40 1:30 12:50
b. Circle the times for which the hands on a clock form
an obtuse angle.
8:00 1:20 5:15 10:30
Math Journal 1, p. 58
Student Page
Lesson 2�10 135
▶ Using Sampling to Make PARTNER ACTIVITY
an Estimate(Math Journal 1, p. 57)
Have partners complete the journal page. They begin by finding their individual sample counts. Partners take turns. While one partner taps and counts the taps for 10 seconds, the second partner keeps time for 10 seconds, signaling when to start and stop.
Partners then use their 10-second sample counts to estimate the number of taps they could make in 1 minute; in 1 hour; and in 1 day. Encourage students to use their calculators, as needed.
Next students use their estimates to calculate the approximate number of days it would take to tap 1 million times. Encourage students to devise their own solution strategies. One possible approach is to divide 1 million by the number of taps per day.
▶ Making Time Estimates for PARTNER ACTIVITY
1 Billion and 1 Trillion Taps(Math Journal 1, p. 57)
In Problem 3 on journal page 57, students estimate the time it would take to tap 1 billion and 1 trillion times. Remind students that they can use the relationships between 1 million, 1 billion, and 1 trillion found in the Useful Information chart to help them estimate. They will also need to decide whether to report their estimates for 1 billion and 1 trillion taps as days or years.
NOTE Expect that the tapping rate for most students will be about 40 times in 10
seconds. At this rate they will tap about 250 times in 1 minute (6 ∗ 40, rounded up);
15,000 times in 1 hour (60 ∗ 250); 350,000 times in 1 day (24 ∗ 15,000; rounded
down), and about 3 days, without interruptions, to tap 1 million times.
▶ Sharing and Discussing SMALL-GROUPDISCUSSION
the Results(Math Journal 1, p. 57)
When most students have completed the problems, have partners form small groups to discuss their strategies. Then have the groups report on the similarities and differences of the strategies used as well as any notable experiences they encountered. Use the following questions as a guide:
● How does your estimate of the time for 1 million taps compare with your initial guess?
● Did you use your estimate for the number of taps in 1 day to estimate how long it would take to tap 1 million times?
● Did you use the time for 1 million taps to estimate the time for 1 billion and 1 trillion taps?
� The value of the digit in the thousandths place is equal to the sum of the measures of
the angles in a triangle (180°) divided by 30.
� If you multiply the digit in the tens place by 1,000; the answer will be 9,000.
� Double 35. Divide the result by 10. Write the answer in the tenths place.
� The hundreds-place digit is �12
� the value of the digit in the thousandths place.
� When you multiply the digit in the ones place by itself, the answer is 0.
� Write a digit in the hundredths place so that the sum of all six digits in this number is 30.
What is the number? .
Puzzle 2
� Double 12. Divide the result by 8. Write the answer in the thousands place.
� If you multiply the digit in the hundredths place by 10, your answer will be 40.
� The tens-place digit is a prime number. If you multiply it by itself, the answer
is 49.
� Multiply 7 and 3. Subtract 12. Write the answer in the thousandths place.
� Multiply the digit in the hundredths place by the digit in the thousands place. Subtract 7
from the result. Write the digit in the tenths place.
� The digit in the ones place is an odd digit that has not been used yet.
� The value of the digit in the hundreds place is the same as the number of
sides of a quadrilateral.
What is the number? , .
Check: The sum of the answers to both puzzles is 3,862.305.
9451743
657093
Millions Thousands OnesHundred- Ten- Millions Hundred- Ten- Thousands Hundreds Tens Onesmillions millions thousands thousands
3. 7,772 � 1,568 � 4. 472 � 228 �
5. 826 º 54 � 6. 59 / 3 ∑ 19 R244,604
2449,340
Practice
Math Masters, p. 61
Study Link Master
ExampleExample Andy and Barb played 7-digit Number Top-It. Here is the result for onecomplete round of play.
Andy s number is larger than Barb s number. So Andy scores 1 point for thisround, and Barb scores 2 points.
Place-Value Mat
MillionsHundred
ThousandsTen
Thousands Thousands Hundreds Tens Ones
Number Top-It (7-Digit Numbers)
Materials � number cards 0—9 (4 of each)� one Place-Value Mat
(Math Masters, pp. 491 and 492)Players 2 to 5Skill Place value for whole numbersObject of the game To make the largest 7-digit numbers. Directions
1. Shuffle the cards and place the deck number-side down onthe table.
2. Each player uses one row of boxes on the place-value mat.
3. In each round, players take turns turning over the top cardfrom the deck and placing it on any one of their emptyboxes. Each player takes a total of 7 turns, and places 7cards on his or her row of the game mat.
4. At the end of each round, players read their numbers aloudand compare them to the other players numbers. The playerwith the largest number for the round scores 1 point. Theplayer with the next-largest number scores 2 points, and so on.
5. Players play 5 rounds for a game. Shuffle the deck betweeneach round. The player with the smallest total number ofpoints at the end of 5 rounds wins the game.
Games
7
7
4
4
6
6
9
9
4
4
7
7
5
5
3
3
2
2
5
5
0
0
2
2
1
1
4
4
Andy
Barb
Student Reference Book, p. 326
Student Page
136 Unit 2 Estimation and Computation
To conclude Part 1 ask students: If one person could tap 24 hours per day without stopping, would it be possible to tap 1 trillion times? No Aim follow-up questions at getting students to support their responses. They would still need many more years than are in a normal lifetime. An exit question might be: Do you feel that your informed estimate was more reasonable than your guess?
2 Ongoing Learning & Practice
▶ Playing High-Number Toss PARTNER ACTIVITY
(Student Reference Book, p. 320; Math Masters, p. 487)
High-Number Toss provides students with the opportunity to apply their knowledge of place value and standard notation to create, write, read, and compare large numbers. Provide students with a reminder box on the board noting that < means less than and > means greater than.
Ongoing Assessment: Math Masters
Page 487 �Recognizing Student Achievement
Use the Record Sheet for High-Number Toss (Math Masters, page 487) to
assess students’ knowledge of place value and comparing numbers. Students
are making adequate progress if they correctly insert the relational symbols
between the two numbers in five rounds of the game.
[Numbers and Numeration Goals 1 and 6]
▶ Math Boxes 2�10
INDEPENDENT ACTIVITY
(Math Journal 1, p. 58)
Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 2-8. The skill in Problem 4 previews Unit 3 content.
▶ Study Link 2�10
INDEPENDENT ACTIVITY
(Math Masters, p. 61)
Home Connection Students use their knowledge of place value and number relationships to solve number puzzles.
11. In Problem 10, explain what happens to the value of the digit 5 when you go
from millimeters to centimeters, and then from decimeters to meters.
LESSON
2�10 Using Place Value to Compare Powers of 10
10 times
1
___ 100 of
10 times 1
__ 10 of
1
___ 100 of
Sample answer: You divide by 10 with each
conversion; so each time, you move the decimal
point one position to the left. So with each move, the
digit 5 is worth 1
__ 10 as much as before.
3,0001,750
974.3
Answers vary.
30097.43
3017.5
9.743
1.75
EM3cuG5MM_U02_066A-066B.indd 66B 1/30/11 9:11 AM
Math Master, p. 66B
Teaching Master
Lesson 2�10 137
3 Differentiation Options
READINESS PARTNER ACTIVITY
▶ Playing Number Top-It 15–30 Min
(7-Digit Numbers)(Student Reference Book, p.326; Math Masters,
pp. 491 and 492)
To review place-value concepts, have students play Number Top-It (7-Digit Numbers).
EXTRA PRACTICE SMALL-GROUPACTIVITY
▶ Comparing Powers of 10 5–15 Min
Using Place Value(Math Masters, p. 66B)
To provide additional practice with place value and understanding the relationships between powers of 10, have students complete Math Masters, page 66B.
ENRICHMENT
INDEPENDENT ACTIVITY
▶ Applying Estimation Strategies 5–15 Min
(Math Masters, p. 62)
To apply students’ ability to use estimation strategies, have them solve problems that involve converting situational information into open number sentences. Direct students to focus on making informed estimates.
