b 6 3 2 m b Table of Contents - Teacher Created Materials · 2 3 16 m b b 6 3 2 3a 2 a2 b 162 c 2 b...

y = mx + by = mx + b a2 + b2 = c2 y = mx + by = 4xa = 2x + 4x + 2 = 3 b = 6x + 34 × 4 = 16 x + 2 = 3 b = 6x + 34 × 4 = 16

© Teacher Created Materials Publishing #11132 (i2625)—Targeted Mathematics Intervention, Level 6 �

Table of ContentsScope and SequenceScope and Sequence for Targeted Mathematics Intervention . . . . . . . . . . . ii

Teacher ResourcesResearch on the Effectiveness of Intervention . . . . . .2

The Need for Intervention . . . . . . . . . . . . . . . . . . . . .2 What Mathematics Interventions Look Like . . . .3

Developing Students’ Mathematics Vocabulary . . . .5 Vocabulary Warm-up Activities . . . . . . . . . . . . . . .6

Using Concrete Models to Introduce Mathematical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 10 Types of Manipulatives and How

They Are Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Differentiating Student Guided Practice . . . . . . . . . 14 Response to Intervention in the

Mathematics Classroom . . . . . . . . . . . . . . . . . . . . . 14 Differentiation by Specific Needs . . . . . . . . . . . . 14

Solving Real-Life Mathematical Problems . . . . . . . . 16 Problem-Solving Steps . . . . . . . . . . . . . . . . . . . . . . 16 Problem-Solving Strategies . . . . . . . . . . . . . . . . . . 17

Playing Games in Mathematics . . . . . . . . . . . . . . . . . . 19 Games Used in This Program . . . . . . . . . . . . . . . . 20

Program Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Lesson Organization . . . . . . . . . . . . . . . . . . . . . . . . 22 Planning for Intervention . . . . . . . . . . . . . . . . . . . . 28 Pacing Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Correlation to Mathematics Standards . . . . . . . . . . 33 How to Find Your State Correlations . . . . . . . . . 33 NCTM Standards Correlation Chart . . . . . . . . . 33

Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Diagnostic Test Item Analysis . . . . . . . . . . . . . . . . 37

Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Lesson Plans Finding Factors and Multiples . . . . . . . . . . . . . . . . . . . 45

Fractions and Decimals . . . . . . . . . . . . . . . . . . . . . . . . . 51

Fractions, Decimals, and Percents . . . . . . . . . . . . . . . . 58

Adding and Subtracting Decimals . . . . . . . . . . . . . . . 63

Multiplying Fractions and Decimals . . . . . . . . . . . . . 70

Dividing Fractions and Decimals . . . . . . . . . . . . . . . . . 75

Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Evaluating Expressions with Substitution . . . . . . . . 88

Simplifying Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 93

Writing Algebraic Expressions . . . . . . . . . . . . . . . . . . 100

One-Step Linear Equations . . . . . . . . . . . . . . . . . . . . . 105

Two-Step Linear Equations . . . . . . . . . . . . . . . . . . . . . 111

Real-World Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Graphing Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Ratios and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . 130

Types of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Parts of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Grid Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Comparing Perimeter and Area . . . . . . . . . . . . . . . . . 153

Calculating Perimeter and Area of Triangles . . . . . 160

Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Converting Amounts . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Measuring Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Using Probability Diagrams . . . . . . . . . . . . . . . . . . . . . 183

Simple and Compound Probability . . . . . . . . . . . . . 190

Mode, Median, Mean, and Range . . . . . . . . . . . . . . . 195

Bar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Circle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Stem-and-Leaf Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

AppendicesAppendix A: References Cited . . . . . . . . . . . . . . . . . . 226

Appendix B: Answer Key . . . . . . . . . . . . . . . . . . . . . . . 227

Appendix C: Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Appendix D: Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Appendix E: Contents of Teacher Resource CD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

y = mx + b 3x – y =4y = mx + by = kx y = mx + bx + 2 = 3 b = 6x + 34 × 4 = 16

#11132 (i2625)—Targeted Mathematics Intervention, Level 6 © Teacher Created Materials Publishing�

teacher resources

Research on the Effectiveness of Intervention

For many people, “Math is right up there with snakes, public speaking, and heights” (Burns 1998). Mention the word mathematics and the room grows silent, people squirm in their seats, and small beads of sweat appear on their temples. In an effort to keep others from the disappointment that will certainly ensue, a disclaimer is spoken, “I am not good at math.” Parents hire tutors because they do not feel comfortable helping their fifth graders with math homework. People hail the mathematics professor as a god-like figure. These behaviors certainly can discourage both learners and teachers alike. Even so, mathematics cannot be ignored because it is relevant every day in real life. In fact, it might be nearly impossible to live a day without math. Counting spare coins in a jar, estimating the cost of an item to include tax, and converting recipes for larger groups are just a few of the ways mathematics enters people’s daily routines. Because of this, it is imperative that this mathematics phobia be put to rest. But how can these misconceptions about mathematics be overcome? The cure for mathematics phobia might be as simple as providing students with the necessary skills and opportunities to be successful in mathematics.

The Need for Intervention The National Council for Teachers of Mathematics (NCTM) has some high expectations or goals for students. They want students to become mathematical problem solvers, learn to communicate and reason mathematically, and make connections (NCTM 2000). The task of teaching mathematics in classrooms today appears more difficult because today’s educators understand that curriculum must be differentiated to meet the needs of all students. Students come into mathematics class with different levels of readiness, learning styles, and interests. To meet the needs of all learners, teachers must provide varied levels of time, structure, support, and complexity with different intensities for these learners (Coleman 2003). Targeted Mathematics Intervention incorporates differentiated activities throughout the lessons. The differentiation is seen in the focus on vocabulary, small-group guided practice, and open-ended activities.

The learning differences among students can be illustrated in several ways. The basic mathematics facts require some memorization. Some students have not been taught memorization skills. English language learners have language and communication issues, and this affects every content area, including math. Students who have processing difficulties find the mathematics symbols and numerals confusing. Other students perform poorly in mathematics simply because they have low self-esteem and short attention spans. Some students have even developed the attitude that mathematics is scary or boring based on previous experiences. Personality dictates that some students are passive learners or are simply disorganized. This makes the task of teaching mathematics complicated for teachers. The one-size-fits-all curriculum and instruction cannot possibly accomplish the goal of reaching all students (Tomlinson 2003). For these reasons, mathematics intervention is necessary. Still, the question remains: What kinds of mathematics interventions work?


#11132 (i2625)—Targeted Mathematics Intervention, Level 6 © Teacher Created Materials Publishing14

teacher resources

Differentiating Student Guided Practice

Response to Intervention in the Mathematics ClassroomAll students learn differently and struggle with different mathematical concepts. In one classroom, teachers can have students who are above grade level, on grade level, below grade level, and English language learners. Not all below-grade-level students struggle in the same areas and fit into the same “category.” Because of this, many of the same researchers who created the Reading First initiative developed a system of identification known as Response to Intervention (RTI). The RTI model supports the idea that teachers should look for curricular intervention designed to bring a child back up to speed as soon as he or she begins having difficulties. “RTI has the potential then to allow disabilities to be identified and defined based on the response a child has to the interventions that are tried” (Cruey 2006). Depending on the levels of difficulty they are having with the mathematics curriculum, students are classified as Tier 1, 2, or 3. Specific definitions of these tiers differ from state to state, but the following are general descriptions.

Tier 1 students are generally making good progress toward the standards, but may be experiencing temporary or minor difficulties. These students may struggle only in a few of the overall areas of mathematical concepts. They usually benefit from peer work and parental involvement. They would also benefit from confidence boosters when they are succeeding. Although they are moving ahead, any problems that do arise should be diagnosed and addressed quickly in order to ensure that these students continue to succeed and do not fall behind.

Tier 2 students may be one or two standard deviations below the mean on standardized tests. These students are struggling in various areas and these struggles are affecting their overall success in a mathematics classroom. These students can usually respond to in-class differentiation strategies and do not often need the help of student study teams.

Tier 3 students are seriously at risk of failing to meet the standards as indicated by their extremely and chronically low performance on one or more measures of the standardized test. These students are often the ones who are being analyzed by some type of in-house student assistance team in order to look for overall interventions and solutions. In the classroom, these would be the students who are having difficulties in most of the assignments and failing most of the assessments.

Differentiation by Specific NeedsBelow-Grade-Level StudentsBelow-grade-level students will probably need concepts to be made more concrete for them. They may also need extra work with manipulatives and application games. By giving them extra support and understanding, these students will feel more secure and have greater success. • Allow partner work for oral rehearsal of solutions and allocate extra time for guided practice. • Allow for kinesthetic activities where they organize the step-by-step processes on flash cards

before they actually use the information to solve problems. • Have easy-to-follow notes of the most important procedural information already made up for

these students to add to.


#11132 (i2625)—Targeted Mathematics Intervention, Level 6 © Teacher Created Materials Publishing�8

teacher resources

Program Overview (cont.)

Planning for InterventionTargeted Mathematics Intervention can be used in multiple ways according to district requirements, school resources, and student needs. The program has four main goals that make it flexible for various intervention programs. Each goal is summarized below:

Targeted Instruction of Key Content Standards—This 30-lesson program can be used in a variety of settings to help bring students up to grade level or prepare them for standardized tests. The chosen standards are targeted for each grade level to hit on the most important mathematical concepts. The lessons are strong in concrete examples, multiple representations, and the use of various algorithms.

Different from the Regular Classroom Curriculum—Students in mathematics intervention programs do not need their curriculum from the regular classroom to be repeated. They need more engaging curriculum to grab and keep their interest. Furthermore, fun activities such as games and hands-on lessons help all types of learners.

Easy to Use—Teachers in mathematics intervention programs are not always mathematics teachers. Many times, teachers are teaching off grade level or out of their content area. So, this program includes all necessary materials and has straightforward lessons with clear teacher directions. And yet, the lessons are flexible enough that experienced teachers can incorporate their own teaching styles and strategies.

Compact and Portable—Many intervention programs do not take place in well-stocked mathematics classrooms. This program is compact, yet it contains all the materials teachers need to be successful. Teachers can take the box and student books and successfully teach the 30 lessons. However, if a school has extra resources (manipulatives, computers, calculators, etc.), these resources can easily be tied into the program.

Pacing PlansWhen planning the pacing of a curriculum program, educators need to analyze student data to determine standards on which to focus. This program has targeted the 30 most-tested standards for each grade level. These lessons can be taught one a day over six weeks. The Six-Week Pacing Plan (page 30) shows each two-hour lesson. However, many programs are not exactly six weeks long; so 20 key standards have been outlined in the Sample Four-Week Pacing Plan (page 31). This flexible program can also be used over longer periods of time. The Sample After-School Pacing Plan (page 32) shows how the 60 hours can be spread out over six months.

To further adapt the program to instructional time frames that are shorter than six weeks, it is highly recommended that teachers give the Diagnostic Test to determine which standards students have not mastered. Teachers can then use the Diagnostic Test Item Analysis (pages 37–38) to analyze their students’ results and select lessons to target. In addition, teachers can modify the lessons by using the suggested ways to accelerate the curriculum and decelerate the curriculum on the next page.

y = mx + by = mx + b a2 + b2 = c2 y = mx + by = 4xa = 2x + 4

● ●

● ●

● ●

x + 2 = 3 b = 6x + 34 × 4 = 16x + 2 = 3 b = 6x + 34 × 4 = 16

© Teacher Created Materials Publishing #11132 (i2625)—Targeted Mathematics Intervention, Level 6 39

Name ________________________________________

Diagnostic Test

4 Monique’s jump rope is 3.65 feet long. Mario’s jump rope is 4.32 feet long. How much longer is Mario’s jump rope than Monique’s jump rope?

F .67 feet

G 7.97 feet

H 1.67 feet

J 1.33 feet

1 Which number is a factor of 45 and 72?

A 2

B 9

C 4

D 6

2 What is another way of expressing 2128

?

F .7

G 3/28

H .75

J 4/3

5 Lara had .5 of a bag of crackers. She gave .3 of what she had to her friend Nicolas. How much of the crackers did Nicolas get?

A .8 of the crackers

B .15 of the crackers

C .3 of the crackers

D 1.5 of the crackers

3 Kumar used .04 of the gallon of paint to paint his bedroom. What is that number in percent form?

A 4%

B 40%

C 400%

D .04%

6 Solve.

2 57 ÷ 2

3

F 4 23

G 5 114

H 3 821

J 4 114

y = mx + b 3x – y =4y = mx + by = kx

● ●

● ●

● ●

y = mx + bx + 2 = 3 b = 6x + 34 × 4 = 16


Diagnostic Test (cont.)

7 Solve.

64 =

A 1,296

B 24

C 36

D 7,776

10 Which expression shows 10 more than 8 times x?

F 8 x 10

G 8x + 10

H 10 – x

J 8x – 10

8 If e = 2 and f = 8,

6e + 2f =

F 8ef

G 28

H 52

J 90

11 Solve.

7s = 84

A s = 12

B s = 77

C s = 588

D s = 91

9 Simplify.

2t + 3t(3 + t) + t

A 12t

B 15t

C 12t + 3t2

D 9t + 3t2

12 Solve.

3c + 5 = 11

F c = 3

G c = 18

H c = 5 13

J c = 2


x +

2 =

3b

= 6x

+ 3

4 ×

4 =

16x + 2 = 3 b = 6x + 34 × 4 = 16 6 × 8 = 48 6 ÷ 3 = 2 3 + 5 = 8 4 × 4 = 162 + 6 = 8


Finding Factors and Multiples

Learning Objectives • Identify multiples and factors for whole numbers.

• Devise and test divisibility rules for 2, 3, 4, 5, 6, and 10.

Warm-up Activity 10 min.

Skill: Addition and Subtraction

1. Settle the students for the day, and reinforce their mental math skills by solving addition and subtraction problems within a given period of time. Tell the class that you are going to call out addition and subtraction problems.

2. Encourage students to solve the problems as quickly as possible. Give students 10 seconds to write each answer.

3. Provide several two-digit addition and subtraction problems. Here are some examples:

33 – 13 40 + 11 52 + 12 63 – 34 47 – 15

78 – 27 60 – 18 45 + 32 26 + 37 62 + 16

4. After enough time has been given for the problems, discuss the answers with the class. Ask students to explain how they solved each problem. Having them think metacognitively is an important part of improving their mathematical reasoning.

Vocabulary 10 min.

Complete the Sharing Mathematics (page xx) vocabulary activity using the words below. Definitions of these words are included on pages 243–246 in this book and in the Student Guided Practice Book.

• factor

• multiple

• product

• times table

Materials • Student Guided

Practice Book - It’s a Fact! (page 11;

page011.pdf) - We’re Talking Telephone

Numbers (page 12; page012.pdf)

- Using Simpler Numbers (page 13; page013.pdf)

- Standardized Test Preparation 1 (page 14; page014.pdf)

- Match It! Directions (pages 139–140; page139.pdf)

• Punchouts folder - Match It! Cards

(matchit.pdf ) • Transparency folder - Using Simpler Numbers

(trans01.pdf) • PowerPoint folder

on the CD - Finding Factors and

Multiples (lesson01.ppt) (optional)

• Game board - A Visit to the Museum • four spinners

• note cards or small pieces of paper (approximately 80–100)

• paper and pencils

Lesson 1

y = mx + b 3x – y =4y = mx + by = kx y = mx + b6 × 8 = 48 6 ÷ 3 = 2 3 + 5 = 8 4 × 4 = 162 + 6 = 8


Whole-Class Skills Lesson 25 min.

Use the directions below or the PowerPoint presentation to teach this lesson.

1. Tell students, “Today, we are going to examine the times tables and look for factors and multiples.”

2. Review a few times tables to get students thinking about multiplication.

3. Discuss factors of a number. Explain that a factor is a number that divides exactly into another number. For example, 2 divides into 6, so 2 is a factor of 6. Factors can also be described as “numbers multiplied together to get a product.”

4. Start with a specific number such as 12 and ask the class, “What are the factors of 12?” (1, 2, 3, 4, 6, and 12). Emphasize that to find the factors of 12, it is necessary to know the times tables.

5. Ask students to name the factors of 25 (1, 5, and 25). Practice naming the factors of 36, 45, 60, and 100. Then, introduce the term multiple as “the product of a number and another number.”

6. Ask students, “How do you know whether a number is a multiple of 10?” (It ends in 0.) “How about 5?” (It ends in 5 or 0.)

7. Ask students, “How can I find out whether a number is a multiple of 3?” Discuss the possible ways of finding multiples of 3.

• Summarize that a number is a multiple of 3 if:

- it is in the 3 times table. - it is divisible by 3. - its digits added together make a multiple of 3.

8. Ask students if they can suggest how to identify multiples of 2, reminding students that all of these numbers are even. Multiples of 6 follow a special rule: they are divisible by both 2 and 3. Use the rules for 2 and 3 for numbers to see if they’re divisible by 6 also.

9. Look together at multiples of 4. Many students will notice that these numbers are all even. Unlike multiples of 2, however, not all even numbers are multiples of 4. For example, 22 is a multiple of 2 but not a multiple of 4. Write the following numbers on the board or overhead: 72, 612, 104, 224, 964. Have students look at the last two digits of each number. If those two digits are divisible by 4, the whole number is.

10. Finish by writing some two-, three-, and four-digit numbers on the board or overhead and together testing whether they are divisible by 2, 3, 4, 5, 6, and 10.

Lesson 1

Finding Factors and Multiples (cont.)


x +

2 =

34

× 4

= 16

6 × 8 = 48 6 ÷ 3 = 2 3 + 5 = 8 4 × 4 = 162 + 6 = 8 6 × 8 = 48 6 ÷ 3 = 2 3 + 5 = 8 4 × 4 = 162 + 6 = 8


Differentiated Guided Practice 25 min.

● Below Level—Teacher Directed • Help students work through It’s a Fact! (SGPB page 11) by completing together one or two

items in each number. Have students complete the rest of the activity sheet on their own.

• If time allows, have students complete We’re Talking Telephone Numbers (SGPB page 12) in pairs.

■ On/Above Level—Student Directed • Have students work through It’s a Fact! (SGPB page 11) with partners.

• As a fun extension, have students complete We’re Talking Telephone Numbers (SGPB page 12) in pairs, including the Challenge.

Problem Solving 20 min.

• Place the Problem-Solving Strategy Transparency: Using Simpler Numbers on the overhead. Use the guiding callouts on page 48 to introduce the strategy to the students. A copy of this transparency is also included in the Student Guided Practice Book on page 13. Students can follow along and make notes as you review the transparency.

Test Preparation 10 min.

• Have students complete Standardized Test Preparation 1 in the Student Guided Practice Book (page 14). Give them about seven minutes. Then, have students trade papers and grade their work.

Learning Game 20 min.

Match It! • While students are completing the test

preparation questions, set up four game stations. Each station needs one A Visit to the Museum game board, a set of Match It! Cards, and a handful of Counters.

• Review with students the Match It! Directions (SGPB pages 139–140). Answer any questions students have about how to play. You may want to model one round of play.

• Allow them time to play the game. Move among the students checking for understanding as they complete the mathematics problems. Make sure you stop the students with about five minutes left so that students can clean up the game stations.

Lesson 1




Lesson 1


Problem-Solving Transparency Callouts

❶

❷

❸

❹&❼

❻

❺

x + 2 = 3 b = 6x + 34 × 4 = 16

© Teacher Created Materials Publishing #11332—Targeted Mathematics, Guided Practice Book 13

Using Simpler Numbers

Problem: Painting HousesThe Problem

Understanding the Problem

Planning and Communicating a Solution

Reflecting and Generalizing

y = mx + by = mx + b a2 + b2 = c2 y = mx + by = 4xa = 2x + 46 × 8 = 48 6 ÷ 3 = 2 3 + 5 = 8 4 × 4 = 162 + 6 = 8 6 × 8 = 48 6 ÷ 3 = 2 3 + 5 = 8 4 × 4 = 162 + 6 = 8


Lesson 1

Student Pages and Punchouts Needed for the Lesson


It’s a Fact! (SGPB pages 11;

page011.pdf)

We’re Talking Telephone Numbers

(SGPB page 12; page012.pdf)


(SGPB page 13; page013.pdf)

x + 2 = 3 b = 6x + 34 × 4 = 16


It’s a Fact!

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

2423222120191817161514131211109876543210

x + 2 = 3 b = 6x + 34 × 4 = 16

#11332—Targeted Mathematics, Guided Practice Book © Teacher Created Materials Publishing12

We’re Talking Telephone Numbers

x + 2 = 3 b = 6x + 34 × 4 = 16



Problem: Painting HousesThe Problem

Understanding the Problem

Planning and Communicating a Solution

Reflecting and Generalizing

Standardized Test Preparation 1 (SGPB page 14; page014.pdf)

x + 2 = 3 b = 6x + 34 × 4 = 16


Standardized Test Preparation 1

Match It! Directions (SGPB pages 139–140;

page139.pdf)

Match It! Cards (matchit.pdf )

#11131 (i2619)—Targeted Mathematics Intervention, Punchouts © Teacher Created Materials

Match It! Question

1

Match It! Question

4

Match It! Question

7

Match It! Question

2

Match It! Question

5

Match It! Question

8

Match It! Question

3

Match It! Question

6

Match It! Question

9

Match It! Cards

What is the next number in this sequence?

25, 36, 47, 58, ___

Which of the decimals listed has the

greatest value?

0.67 0.325

0.38 0.5

Write the number that means eight ten

thousands, seven thousands, four hundreds, nine tens, and three ones?

Compute to find the quotient.

287 ÷ 9 = ___

What is the value of the 6 in 9,654,758?

Solve.

7 467 ) )

Leah bought a green lizard for $12.45 and a small

iguana for $21.79. How much did she spend in all?

What is the product of 0.43 x 10?

Christina rode the Whirly Bird Twister, which cost

$3.55, and the Screamin’ Coaster, which cost $2.35. How much did she spend?

x + 2 = 3 b = 6x + 34 × 4 = 16


Match It! Directions (cont.)

What You Need • A Visit to the Museum game board

• spinner (Divide the spinner into 4 parts and write the numbers 1–4 on it.)

• Match It! cards

• Counters or something else to use as game markers

• pencils and paper

Object of the Game • Match math problems and their answers. Use your memory to do this. Be the first player to move

through the museum.

Setting Up the Game • Place the game board in the middle of all the players.

• Shuffle the 36 cards and place them facedown. Arrange them in 6 rows with 6 cards in each row.

• Each player places a game marker on ENTER.

How to Play the Game • The youngest player goes first. Then, play passes to the left.

• For each turn, flip over two cards. Make sure that everyone can see the cards when you flip them.

• If you flip a mathematics problem, solve it. Then, try to find the correct answer. If you flipan answer, remember where it is and try to find the correct problem. You are trying to matchproblems with their answers.

• If you don’t have a match, your turn is over.

• If you make a pair, spin the spinner and move that many spaces.

• Your turn is over after you move. Keep any pairs you find.

How to Win the Game • The first player to land on EXIT wins!

• Second place goes to whoever has the most matches. (If the winner had the most matches, there isno second place!)



PowerPoint Presentation Slides

Lesson 1


How do you know whether or not

a number is a multiple of 10?

It will end in a 0.

How do you know whether or not

a number is a multiple of 5?

It will end in 5 or 0.

To find the factors for 18, you

need to know what numbers can

be multiplied to get 18.

1 x 18 = 18

2 x 9 = 18

3 x 6 = 18

4 x 7

5 x 3

9 x 2

8 x 9

5 x 11

7 x 8

10 x 6

1 x 3

6 x 4

5 x 9

3 x 6

7 x 12

= 28

= 15

= 18

= 72

= 55

= 56

= 60

= 3

= 24

= 45

= 18

= 84

Today’s Lesson

Finding

Factors and

Multiples

Let’s warm up today

by practicing addition

and subtraction.

Warm-Up Activity

Solve these addition and subtraction

problems as quickly as possible.

12 + 14

24 + 7

38 + 18

11 + 59

22 + 62

68 + 23

23 – 11

46 – 25

74 – 9

59 – 18

61 – 26

36 – 17

Let’s look at the answers.

12 + 14

24 + 7

38 + 18

11 + 59

22 + 62

68 + 23

23 – 11

46 – 25

74 – 9

59 – 18

61 – 26

36 – 17

= 26

= 31

= 56

= 70

= 84

= 91

= 12

= 21

= 65

= 41

= 35

= 19

A factor is a number that divides

evenly into another number.

Factors can also be described as

“numbers multiplied together to

get a product”

Today we are going to

examine the times tables and

look for factors and multiples.

Let’s rst review some

multiplication facts.

Whole-Class Skills Lesson

How did you get your answers?

12 + 14

24 + 7

38 + 18

11 + 59

22 + 62

68 + 23

23 – 11

46 – 25

74 – 9

59 – 18

61 – 26

36 – 17

= 26

= 31

= 56

= 70

= 84

= 91

= 12

= 21

= 65

= 41

= 35

= 19

So the factors for 18 would be

1 x 18 = 18

2 x 9 = 18

3 x 6 = 18

1 2 3 6 9 18

What are the factors for the

following numbers?

25

36

45

60

1, 5, 25

1, 2, 3, 4, 6, 9, 12, 18, 36

1, 3, 5, 9, 15, 45

1, 2, 3, 4, 5, 6, 10, 12, 15, 20,

30, 60

A multiple is the product of a

number and another number.

How do you know if a number

is a multiple of 3?

• It is in the 3s times table

• It is divisible by 3

• Its digits added together make

a multiple of 3

Can you think of ways to find out

if a number is a multiple of 2?

Hint: Think about even

numbers.

How do you know if a number

is a multiple of 6?

The number has to be divisible by

2 and 3 to be a multiple of 6.

Let’s review the rules for being

divisible by 2 and 3

Divisible by 3:

• It is in the 3s times table

• It is divisible by 3

• Its digits added together make a

multiple of 3

Divisible by 2:

• Number ends in an even number

Can you think of ways to find out

if a number is a multiple of 4?

Here are some examples of multiples

of 4. Do you see any patterns?

16 32 40 84

68 36 92 8

Multiples of 4 are all even.

Remember: Not all even

numbers are multiples of 4.

To figure out if a number is a

multiple of 4, take the last two

digits of the number. If they are

divisible by 4, then the whole

number is a multiple of 4.

Are these numbers multiples of 4?

How do you know?

72

412

104

224

964 YES

YES

YES

YES

12 ÷ 4 = 3

YES 72 ÷ 4 = 18

4 ÷ 4 = 1

24 ÷ 4 = 6

64 ÷ 4 = 16

Is the number 248 divisible by…

2?

3?

4?

5?

6?

10?

YES

NO

YES

NO

NO

NO

Is the number 4,173 divisible by…

2?

3?

4?

5?

6?

10?

NO

YES

NO

NO

NO

NO

Is the number 8,250 divisible by…

2?

3?

4?

5?

6?

10?

YES

YES

NO

YES

YES

YES

x + 2 3x + 2 = 3x +x + b = 6x + 334 ×4 × 4 = 1644


Name ______________________________________

It’s a Fact! 1. Find all the factors of these numbers. a. 16 __________________________ c. 55 __________________________ b. 30 __________________________ d. 100 _________________________

2. Find the factors that these two numbers have in common. a. 20 and 30 ____________________ c. 15 and 45 ____________________ b. 49 and 28 ____________________ d. 10 and 36 ____________________

3. Find the largest factor that these two numbers have in common. a. 10 and 15 ____________________ c. 6 and 21 _____________________ b. 12 and 48 ____________________ d. 35 and 14 ____________________

4. Mark a dot on the graph below where each of the numbers from 1–24 has a factor. The number 5 has been done for you. It is a prime number so the only factors are 1 and 5.

a. Describe the completed graph. Can you see any patterns? ___________________ ___________________________________________________________________ b. Which number has the most factors? _____________________________________ c. Use the graph to find the common factors for 8 and 12. ______________________ d. Find the highest common factor of 12 and 24. ______________________________

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Facto

rs o

f N

um

bers

Numbers

2423222120191817161514131211109876543210

••

x + 2 3x + 2 = 3x +x + b = 6x + 334 ×4 × 4 = 1644


Name ______________________________________

We’re Talking Telephone NumbersDirections: Write your school’s telephone number on the lines below.

____ ____ ____ ____ ____ ____ ____

1. Find the sum of the digits. Is this number a multiple of 2, 4, 5, 10, or 100? Show your work here.

2. Write your home telephone number below. Find the sum of the digits in your home telephone number. Then add it to the sum of the school’s number. Is this new number a multiple of 2, 4, 5, 10, or 100? Show your work here.

3. John wrote his friend’s telephone number on a scrap of paper, but it got torn and smudged. He knows that it was a seven-digit number and that it was a multiple of 2, 4, 5, 10, and 100. Can you suggest what the number was? _______________________________

4. Find two other seven-digit telephone numbers that are multiples of 2, 4, 5, 10, and 100.

______________________________________________________________________

______________________________________________________________________

Challenge

Directions: Find a seven-digit telephone number the sum of whose digits is divisible by 2, 4, 5, and 10.

x + 2 3x + 2 = 3x +x + b = 6x + 334 ×4 × 4 = 1644


Name ______________________________________


Sometimes a problem can seem too difficult to solve. If it has large numbers or complicated number concepts, you can use simpler numbers to help you understand what you need to do. Then, you’ll be ready to tackle the harder problem!

Using simpler numbers can help in several ways. It can help you understand what operations you need to use to solve the problem. Try replacing the large numbers in the problem with smaller numbers. Then, solve the problem. If the answer makes sense for the smaller numbers, then you can use the same operations with the larger numbers.

Another way to use simpler numbers is to break down the problem into smaller parts. As you solve each part, keep track of your answers by drawing pictures or a table. Soon, you may see a pattern that will help you solve the big problem.

Problem: Painting HousesTh e ProblemOne day, 8 painters worked for 12 hours to finish painting half of the outside of a house. Only 4 of the painters plan to stay to finish the other half of the house. How long will it take the 4 painters to complete the other half?

Understanding the Problem • What do we know?

Eight painters worked 12 hours to paint half a house. And, 4 painters will fi nish painting the other half.

• What do we need to find out?

How long will it take for the 4 painters to complete the other half?

Planning and Communicating a Solution • Start with a Simpler Example

If it takes 4 painters 4 hours to paint half of the outside of a house, how long will it take 2 painters to paint the other half?

First, fi nd out how long it would take 1 painter to paint the outside of half the house alone. He would have to work 4 times longer to do the job of the original

4 painters, so he would take 16 hours.

4 x 4 = 16 hours

If 2 painters work on the other half, each will only have to work half as much time as 1 painter. So, it would take them 8 hours to paint the other half of the house.

16 ÷ 2 = 8 hours

• Solve the Original Problem

Start by working out how long it will take 1 painter to complete the work. We know that 8 painters take 12 hours, so 1 painter would have to work 8 times longer to do the job of 8 painters. So, 1 painter would take 96 hours.

8 x 12 = 96 hours

If 4 painters work on the other half, they would each have to work only one-fourth as much time as 1 painter.

• Do you see the answer?

96 ÷ 4 = 24 hours

They would each work 24 hours.

Refl ecting and GeneralizingBy starting with a simpler example, you are able to solve the problem with more ease.

x + 2 3x + 2 = 3x +x + b = 6x + 334 ×4 × 4 = 1644

● ●

● ●

● ●


6 Pick one question from this test. Explain how and why you choseyour answer.

________________________

________________________

________________________

________________________

________________________

________________________

Name ______________________________________

Standardized Test Preparation 1

1 Ming fed 60% of the peanuts to her pet mouse. What is the fractional equivalent?

A 38

B 612

C 35

D 6010

4 Which of these numbers is a multiple of 3, 6, and 9?

F 9

G 21

H 12

J 36

2 Rayna spent $15.99 on a haircut and $19.75 for a wash. How much did it cost altogether?

F $35.50

G $35.74

H $34.74

J $35.76

5 Find the difference between these two decimal numbers.

167.92 — 47.86 = ________

A 89.32

B 120.06

C 145.64

D 310.68

3 Which answer lists all of the factors of 16?

A 1, 2, 8, 16

B 1, 2, 4, 8

C 1, 2, 4, 8, 10

D 1, 2, 4, 8, 16

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