lesson 2 skills maintenance lesson planner · skills maintenance working with exponents activity 1...

804 Unit 7 • Lesson 2

250 Unit7•Lesson2

Name Date

SkillsMaintenanceWorkingWithExponents

Activity1

Writetherepeatedmultiplicationasapower.

1. 2 · 2 · 2 · 2 · 2 · 2 · 2 27

2. 3 · 3 · 3 · 3 34

3. 4 · 4 42

4. 2 · 2 · 2 23

5. 3 · 3 32

6. 4 · 4 · 4 43

Activity2

Writethepowersasrepeatedmultiplication.

Model 25 2 · 2 · 2 · 2 · 2

1. 56 5 · 5 · 5 · 5 · 5 · 5

2. 33 3 · 3 · 3

3. 42 4 · 4

4. 21 2

5. 104 10 · 10 · 10 · 10

6. 65 6 · 6 · 6 · 6 · 6

Lesson2 SkillsMaintenance

Skills MaintenanceWorking With Exponents(Interactive Text, page 250)

Activity 1

Students write the repeated multiplication as a power.

Activity 2

Students write the powers as repeated multiplication problems.

Vocabulary Developmentsample space

Skills MaintenanceWorking With Exponents

Building Number Concepts: Powers and Place Value

We use a base number of 10 to show patterns with exponents in the place-value chart. We also introduce and define the powers of 0 and 1. When we look at a place-value chart and various powers with 10 as the base, we see a pattern involving the zeros in the number and the exponent.

The rule for exponents is that we multiply the base number by itself that number of times. We use rules for explaining exponents of 0 and exponents of 1: x0 = 1 and x1 = x.

ObjectiveStudents will demonstrate patterns with exponents with a base number of 10.

Problem Solving: Calculating Chances

We show students how to use probability to understand chances of winning in a lottery.

ObjectiveStudents will use probability to calculate the chances of winning lottery prizes.

HomeworkStudents rewrite powers as numbers and products using exponents, then determine the probabilities for different events. In Distributed Practice, students practice decimal number, fraction, and percent conversions, as well as operations involving rational numbers.

Lesson Planner

Powers and Place ValueProblem Solving: Calculating Chances

Lesson 2

Unit 7 • Lesson 2 805


What is special about the tens and ones places?We learned that powers are just another way of writing repeated multiplicationproblems.Forinstance,wecanwrite2· 2 · 2 · 2 = 24.

Let’s look at some patterns with place value and powers. This time, let’s use powers with base numbers of 10. In Example 1, we begin with 106, or 1,000,000, and work down to 102, or 100. Notice what is happening to the exponent each time. We can predict what the exponent will be for 10 and 1.

Example 1

Identify the patterns in the exponents.

MillionsHundred

ThousandsTen

Thousands Thousands Hundreds Tens Ones1,000,000 100,000 10,000 1,000 100 10 1

106 105 104 103 102 ? ?

When we get to the tens and ones place, the exponents are 1 and 0. This may seem strange, but it makes sense when we look at the pattern.

101 = 10

100 = 1

There are formal proofs for showing how these statements are true, but for now we’ll just remember them as important math rules. Let’s look at two important rules about exponents.

Powers and Place Value

Powers and Place ValueProblem Solving: Calculating Chances

Lesson 2

Listen for:

• Each place to the left is 10 times bigger than the previous place.

• Each place to the right is 110 of the previous

place.

• We see more zeros as we move to the left.

• Then use the place-value table to continue modeling in Example 1 . Show the powers that go along with the place values in the place-value table beginning at the left-hand side with millions (106), hundred thousands (105), and so on.

• Then ask students to predict the tens and ones place. The logical exponents are 1 and 0. This might seem strange to students, but explain that 101 does indeed equal 10 and 100 = 1.

Building Number Concepts: Powers and Place Value

What is special about the tens and ones places?(Student Text, pages 486–487)

Connect to Prior KnowledgeWe can use place-value tables to work with large numbers, or numbers with many place values. Have volunteers come to the board and write a large number in a place-value chart.

Link to Today’s ConceptIn today’s lesson, we use the place-value table to work with exponents and see patterns with zeros.

DemonstrateEngagement Strategy: Teacher ModelingDraw a place-value table that shows place values from ones to millions on the board. Do this in one of the following ways:

: Use the mBook Teacher Edition for page 486 of the Student Text.

Overhead Projector: Display Transparency 13, and modify as discussed.

Board: Copy the place-value table on the board, and modify as discussed.

• Show students the place-value table. Elicit from students the name and numeric representation of each place-value position across the top row of the table. Start at the right with the ones place, then the tens place, etc., and complete the table all the way to the millions place.

• When the table is completely labeled, ask students to comment on patterns they see in the place-value table.

486


Lesson 2


Lesson 2

Rule for Exponents of 1

Any base with an exponent of 1 is equal to itself.

101 = 10

11 = 1

9,8231=9,823

27.31 = 27.3

Rule for Exponents of 0

Any base with an exponent of 0 is equal to 1.

100 = 1

10 = 1

9,8230 = 1

27.30 = 1

Let’s see how we use these rules.

Example 2

Write each power as a number.131

Since the exponent is 1, we know the answer is 13.

131 = 13

270

Since the exponent is 0, we use the rule for zero exponents. Any number with a zero exponent is equal to 1.

270 = 1

These rules will help us when we solve computation problems with exponents.

Any number with an exponent of 1 is equal to itself. Any number with an exponent of zero is equal to 1.

Reinforce UnderstandingUse the mBook Study Guide to review lesson concepts.

Apply SkillsTurn to Interactive Text, page 251.

container. Draw one paper from the container, and share the answers with the class. Invite that student to explain the answers.

Reinforce UnderstandingIf you feel students need more practice with this

concept, use the following whole powers:

130 (1)

11 (1)

220 (1)

What is special about the tens and ones places? (continued)

Demonstrate• Read the top of page 487 in the Student

Text. Go through the examples for each rule to illustrate it.

• Be sure students see that it does not matter what the base number is. Anything to the first power is itself, and anything to the zero power is 1. There are formal proofs to show these powers, but they require knowledge of operations with exponents, which is beyond the scope of this material. For now we simply present them as rules.

DiscussCall students’ attention to the Power Concept, and point out that it will be helpful as they complete the activities.

Demonstrate• Use Example 2 to illustrate how to write

each power as a number. Explain that because the exponent in 131 is 1, we know the answer is the number itself, 13. For 270, we use the rule for zero exponents. Any number with a zero exponent is 1.

Check for UnderstandingEngagement Strategy: Think, Tank

Distribute small pieces of paper to students and have them write their names on them. Write the exponents 180, 321, 160, and 491 on the board, and have students write each power as a number (1, 32, 1, 49). Have them write all their answers on one piece of paper. When students finish, collect the pieces of paper and put them into a

Any number with an exponent of 1 is equal to itself. Any number with an exponent of zero is equal to 1.

487


Unit7•Lesson2 251

Name Date

Uni

t 7

ApplySkillsPowersandPlaceValue

Activity1

Fillinthemissinginformationinthetables.

1. A base of 2

25 24 23 22 21 20

32 16 8 4 2 1

2. A base of 4

45 44 43 42 41 4º

1,024 256 64 16 4 1

3. A base of 5

55 54 53 52 51 50

3,125 625 125 25 5 1

4. A base of 3

35 34 33 32 31 30

243 81 27 9 3 1

5. A base of 10

105 104 103 102 101 100

100,000 10,000 1,000 100 10 1

Lesson2 ApplySkills

Apply Skills(Interactive Text, page 251)

Have students turn to page 251 in the Interactive Text, which provides students an opportunity to practice powers and place value on their own.

Activity 1

Students fill in the missing powers or numbers in the tables for different bases. Monitor students’ work as they complete the activity.

Watch for:

• Can students determine the power for the number based on the number itself and its position in the place-value table?

• Can students calculate the numeric value of the power? (Students can use mental math, paper and pencil, or a calculator.)

Reinforce Understanding Remind students that they can review lesson concepts by accessing the online mBook Study Guide.


Lesson 2


Lesson 2

What are the chances of winning?Before entering a game or contest, people want to know what the chances are of winning. Let’s say there is a school lottery for special prizes. There are 100 tickets numbered 1 to 100. Each student will get one ticket. Someone will draw the tickets and announce the winners. After each ticket is drawn and announced, it is placed back in the drawing. Different kinds of prizes are given depending upon the number picked. Here are the numbers and the prizes. How do we figure out the probability of winning each one of the prizes?

Number PrizeAny even number Piece of candy

Any number from 1 to 10 Pen

Any number divided evenly by 25 Baseball hat

The winning number (only one number will be chosen from 1 to 100)

Sweatshirt

We might pick any number from 1 to 100. That means there are 100 possible choices. We can say the total number of possibilities, or sample space , is 100. Now all we have to do is figure out the probabilities for each prize. Remember that the total number, or sample space, for each probability is 100.

Example 1

Calculate the chances of winning each of the prizes.

Number PrizeAny even number Piece of candy

Any number from 1 to 10 Pen

Any number divided evenly by 25 Baseball hat

The winning number (only one number will be chosen from 1 to 100)

Sweatshirt

Problem Solving: Calculating ChancesVocabulary

sample space

Problem Solving: Calculating Chances

What are the chances of winning?(Student Text, pages 488–489)

Connect to Prior KnowledgeBegin by reviewing probabilities with a coin or a deck of cards.

Link to Today’s ConceptIn today’s lesson we discuss the chance of winning a certain game.

Demonstrate• Have students look at page 488 of the

Student Text. Read the scenario with students while explaining the school lottery game.

• Be sure to explain to students that the numbers are returned to the mix after each drawing. That means the probabilities for each of the drawings are based on all 100 numbers. That is, the total number of possibilities, or sample space , is always 100.

• Go through each of the numbers that would win each of the prizes.

• Go through Example 1 , and tell students we calculate each of the probabilities for each of the prizes listed.

488



Lesson 2

Any Even Number

• Thereare50evennumbers(2,4,6,8,upto100)sothechancesare:

50100 = 0.50 = 50%

Any Number From 1 to 10

• Thereare10ofthesenumberssothechancesare:

10100 = 0.10 = 10%

Any Number Divided Evenly by 25

• Thereare4ofthesenumbers(25,50,75,100)sothechancesare:

4100 = 0.04 = 4%

The Winning Number

• Thereisonly1ofthesenumberssothechancesare:

1100 = 0.01 = 1%

Improve Your Skills

We have to think carefully about setting up the probability before we write it down. In the example above, there are only 4 numbers between 1 and 100 that we can divide evenly by 25. There are not 25 numbers.

When we work with probabilities, we must ask ourselves, “Did I set up the problem correctly? Am I thinking about the right numbers?” It’s important that we know what rational numbers look like, and that we can convert easily from one form of rational number to another.

Problem-Solving ActivityTurn to Interactive Text, page 252.

Reinforce UnderstandingUse the mBook Study Guide to review lesson concepts.

Improve Your Skills

• Read the box at the bottom of the page. Tell students that they need to use good problem-solving skills to talk about probabilities.

• Tell students that they need to think about the following questions: What is the problem asking for? What is the part? What is the whole?

Demonstrate• Have students turn to page 489 of the

Student Text to continue looking at Example 1.

• Show students how to calculate the probabilities of winning the prizes in the school lottery. Start with the first prize. Any even number wins a piece of candy.

• Explain that there are 50 even numbers and 50 odd numbers between 1 and 100. So the chances of winning a piece of candy are 50

out of 100, or 501 00 , which is equal to 0.50,

or 50 percent.

• Walk through each of the remaining prizes in the same manner.

• Make sure students understand each of the forms of rational numbers used. The fraction helps us set up the comparison of the part and the whole. Students need to be very careful about how they set up these fractions.

• Explain that in most cases, we convert to decimal numbers. Decimal numbers are often the stepping stone between fractions and percents. We can convert fractions to decimal numbers easily with a calculator (numerator divided by denominator) and round them to the hundredths place.

• Guide students in converting the decimal numbers to percents. Percents are better for actually discussing probability. Most people do not talk about fractional or decimal number probabilities. For example,

saying that he has a 34 chance of winning or

0.75 chance of winning is not as common as a 75-percent chance of winning.

489


Lesson 2

Problem-Solving Activity(Interactive Text, pages 252−254)

Have students turn to pages 252–254 in the Interactive Text, which provide students an opportunity to practice calculating changes.

Go over the sample game board, and explain how to fill out the score card. Divide students into pairs for playing the game.

Monitor students’ work as they complete the activity.

Watch for:

• Can students determine the fraction that represents the wins for each player in each game?

• Can students convert that fraction to a decimal number and round it to the nearest hundredths place?

• Can students compute the average at the end of the tenth game and determine the percentage of wins?

Be sure to go over students’ answers when they complete the activity. The main point to emphasize is that there is about a 50-percent chance for each player to win.

Once you have recorded all the students’ averages, report the findings to the class. Discuss reasons why the number might be off from 50 percent (e.g., rounding the decimal numbers, not completely shuffling the cards, sampling 10 games was not enough to study the data).

Reinforce Understanding Remind students that they can review lesson concepts by accessing the online mBook Study Guide.

252 Unit7•Lesson2

Name Date

Problem-SolvingActivityCalculatingChances

Chooseapartnerandfollowthedirectionstoplaythegame.Youneed onegameboardandonescorecardforthetwoofyou.Eachofyouwill alsoneedoneitemtomarkyourplaceonthegameboard,suchasapenny orpaperclip.Finally,youandyourpartnerneedtochooseoneitem:

1. acoin(Player1=Heads,Player2=Tails)

2. adeckofcards(Player1=Red,Player2=Black)

3. adie(Player1=numbers1,2,and3,Player2=numbers4,5,and6)

You are going to play a game of equal

chance. Since only two people are

playing, you have a 50% chance of

winning. Therefore, we can predict

that you will win 5 out of 10 games.

Put one marker on each start. Choose

a player to go first. Depending on which

item you and your partner chose, on

each turn, flip a coin, select a card, or

roll a die. If your assigned side, color,

or number turns up, you win the round.

Move your marker up one place on the

game board.

The game ends when one of the players lands on 5. Record your scores by figuring out how

many total rounds you played, and how many wins each player had. Write this number as

a fraction.

For example, if in the first game, player 1 won 5 rounds and player 2 won 3 rounds. 5 + 3 = 8

rounds. So, in the first game, player 1 won 58 of the time and player 2 won 38 of the time.

Play a total of 10 games, recording the scores on the scorecard after each game. Convert the

fractions to decimal numbers. Round to the nearest hundredths. After all 10 games, for each

player add up the 10 decimal numbers and divide by 10 to find the average.

Player 1

Player 2

5 5

4 4

3 3

2 2

1 1

Start Start

SampleGameBoard

Lesson2 Problem-SolvingActivity

Unit7•Lesson2 253

Uni

t 7

Start Start

1

2

3

4

5

1

2

3

4

5

Lesson2 Problem-SolvingActivity


Homework

Go over the instructions on pages 490–491 of the Student Text for each part of the homework.

Activity 1

Students rewrite each of the powers as numbers. Remind students about powers of 0 and 1 and what they mean. Tell them they can use paper and pencil, mental math, or calculators.

Activity 2

Students rewrite repeated multiplication problems as powers.

Activity 3

Students determine the probabilities for different events and choose the correct answer. Instruct students to write the letter of the correct answer on their paper.

Activity 4 • Distributed Practice

Students practice decimal number, fraction, and percent conversions as well as operations involving rational numbers. Remind students that they practice these skills on a regular basis so they continue to improve.


Lesson 2

Activity 1

Rewrite each of the powers as a number.

1. 23 8 2. 32 93. 40 1 4. 81 85. 103 1,000 6. 24 16

Activity 2

Rewrite each of the products using exponents.

1. 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 211 2. 4 · 4 · 4 · 4 44

3. 5 51 4. 10 · 10 102

5. 1 11 6. 5 · 5 · 5 53

Activity 3

Use the information below to answer the questions. Write the letter of the answer on your paper.

Mrs. Green’s class couldn’t decide what kind of party to have at the end of the year. So she put tickets numbered 1 to 10 in a jar for a drawing. Here is the strategy she used for determining the party theme:

Number PrizeEven number Sports/Game party

Odd number less than 5 Pizza party

Odd number 5 or greater Music party

Once all the tickets were in the jar, Mrs. Green shook the jar so that they were all mixed up and then she drew a ticket.1. What are the odds that the class will have a sports/game party? b

(a) 110 (b) 1

2 (c) 15

2. What are the odds that the class will have a pizza party? c(a) 1

10 (b) 12 (c) 1

5

Model 34 Answer: 34 = 81

Homework


Lesson 2

3. What are the odds that the class will have a music party? c(a) 1

10 (b) 210 (c) 3

10

4. Which party has the best chance of being selected? b (a) Pizza (b) Sports/Game (c) Music

Activity 4 • Distributed Practice

Solve.

1. 2.3+4.7+19.46+21.89 48.35

2. 43 ÷ 1

3 4

3. 2.7 ·9 24.3

4. 15 −

16

130

5. 127.30−99.68 27.62

6. Write 45 as a decimal. 0.8

7. Write 0.05 as a percent. 5%

8. Write90%asafraction.9

10

Homework

490

491

lesson 2 skills maintenance lesson planner · skills maintenance working with exponents activity 1...

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