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Introduction Intervention IIN v

IntroductionCalifornia Fast Forward Math is a program for students who are at serious risk of not meeting the Standards. Generally, these students are performing two or more years below grade level. These students require an intensive intervention program that focuses on foundational skills and concepts essential for success in a basic grade-level mathematics program. California Fast Forward Math serves as a vehicle for students to accelerate their progress in mathematics in the shortest possible time so that they can begin to make progress using the basic grade-level programs.

The goal of California Fast Forward Math is to provide teachers with materials for students in grades 4-7 that go beyond a basal program to reach back further to help students who are signifi cantly behind and accelerate them back to on-grade level. More importantly, these materials are designed for fl exible use inside or outside of the classroom and to provide intensive intervention related to the topics being taught in the classroom.

California Fast Forward Math can be used in a variety of instructional settings, such as intersession, summer school, before or after school hours, in a tutorial session, or during extended math instruction in the classroom. The diagnostic tests related to classroom topics can be used to provide focused intervention for use in different settings.

Overcoming learning problems in mathematics requires giving attention to the background of the individual students and to the nature of their previous instruction. What is needed are instructional programs that create steady and measurable progress for students, showing them that whatever diffi culties they might have had in the past, they are learning mathematics now. There is research to indicate that teachers can intervene to help these students. As stated in a study by Kroesbergen, Evelyn H. & Van Luit, Johannes E. H. (2003), “the computer cannot remediate the basic diffi culties that the children encounter. The results of the present study show that in general, traditional interventions with humans as teachers, and not computers, are most effective.”

California Fast Forward Math

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vi IIN Intervention Introduction

Structure of the ProgramCalifornia Fast Forward Math was developed for students in Grades 4 to 7 who have signifi cant gaps in their knowledge of mathematics. It includes materials that teachers can use to support instruction of the six topical volumes and the subset of Standards identifi ed in Appendix E of the Mathematics Framework for California Public Schools.

Components • 6 Volumes of Teacher Guides • 17 Student Editions to cover grade level standards by modules • Assessment Guide • OnLine Assessment with prescriptions • Individual Record Forms CD-ROM • iTools Digital Manipulatives CD-ROM • Student Manipulative Kits • Basic Fact Flash Cards • Administrator’s Guide

Teacher Guides The 6 volumes of Teacher Guides are as specifi ed in Appendix E of the Mathematics Framework for California Public Schools:

• Volume 1: Place Value and Basic Number Skills • Volume 2: Fractions and Decimals • Volume 3: Ratios, Rates, and Percents • Volume 4: The Core Processes of Mathematics • Volume 5: Functions and Equations • Volume 6: Measurement

Student Editions The 17 Student Editions are split into modules across the volumes that represent the standards at grade level spans: Module A, Grades K-3 Standards, Module B, Grades 4-5 Standards, and Module C, Grades 6-7 Standards. Volume 1 includes Standards for Modules A and B and Volumes 2-6 each cover Standards for Modules A, B, and C.

Assessment Guide The Assessment Guide covers all types of assessments identifi ed in Chapter 10: Initial, Diagnostic, Formative, and Summative. See Assessment Overview in the following pages for more information.

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Introduction Intervention IIN vii

CD-ROMs Two types of CD-ROMs are available. The Individual Record Forms CD-ROM aligns with the assessment items in the Assessment Guide.

The iTools CD-ROM incorporates interactive on-line manipulatives to help students solve problems and explore mathematical concepts. The iTools CD-ROM can be used when presenting California Fast Forward Math skills as a whole-class or a small group demonstration.

Student Manipulative Kits The Student Manipulative Kits are to be used as part of the 6-page teaching plan, or for direct instruction by the teacher for each skill. Use of these manipulatives with students is a key part of the instruction to enhance learning, to help students at risk, and to appeal to all types of learners. Specifi c references to the manipulatives are cited with each skill in the Teacher Guide. Further use of manipulatives to reteach the skills are in the Alternative Teaching Strategy for each skill in the Teacher Guide.

Basic Fact Flash Cards The Basic Fact Flash Cards are included to help struggling students with basic facts. The goal for mastery of the basic facts is automaticity. A student is considered to have achieved automaticity when he or she can give an answer to a basic fact in less than 3 seconds without using fi nger counting. Benjamin Bloom (1986) described automaticity as the ability to perform a skill unconsciously with speed and accuracy while consciously carrying on other brain functions. Klein (2005) affi rms the importance of automaticity by stating, “Memorizing the basic number facts…frees up working memory to master the arithmetic algorithms and tackle math applications. Students who do not memorize the basic number facts will founder as more complex operations are required. There is no real mathematical fl uency without memorization of the most basic facts.”

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Name

IIN 43

Add and SubtractLike Fractions

Skill 54

numerator

denominator

Vocabulary

shaded parts

total parts

numerator

denominator

shaded parts

total parts

numerator

denominator

Read the problem. Use the picture to help you fi nd the answer.Michael divides his paper into 8 equal parts. He shades 6 of the parts. What fraction tells how much of the paper is shaded?

Sarah’s paper shows these equal parts:

REASONING Sarah says that she and Michael shaded the same amount of the paper. Is Sarah correct? Explain your answer.

Fill in the blanks with one of the following words: numerator and denominator. Use each word only once.

1. The is the part of a fraction above the line, which tells how many of the parts are being counted.

2. The is the part of a fraction below the line, which tells how many equal parts there are in the whole or in the group.

Use the fractions below.

3. Circle the numerators.4. Put a box around the denominators.

4__5

4__8

2__4

1__2

1__5

Possible explanation: Yes, Sarah is correct. When the two

sheets of paper are lined up, the same amount of paper is

shaded.

3

6

4

6

4

8

numerator

denominator

NS 3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context. NS 3.2 Add and subtract simple fractions. also MR 2.3

MIECA09ASEK-32A_SKILL05443 43 2/8/07 10:03:00 AM

Each skill begins with Get Ready. This section is designed to be used with individual or small groups of students where the teacher gives direct instruction on prerequisite skills. It serves as an introduction to each skill and addresses reasoning to support the Standard for the skill.

The Vocabulary section addresses the specialized vocabulary of mathematics. Particular attention should be given to the needs of lower level students, including the academic language of instruction. In this section, thorough discussion of mathematics terms and a simple activity provide opportunities to reinforce vocabulary.

The California Mathematics Standard that relates to each skill is provided for the student at the bottom of this page.

viii IIN Intervention Introduction

Administrator’s Guide The Administrator’s Guide is a brief description of how and when to use California Fast Forward Math. It includes most of the information from the front of the Teacher Guide in a convenient form for administrators or curriculum coordinators who are planning curriculum to be covered in the school year.

Student EditionConcepts are developed in a logical order and increase in depth and complexity. The six volumes are split into modules that develop the Standards by grade level spans: Module A, Grades K-3 Standards, Module B, Grades 4-5 Standards, and Module C, Grades 6-7 Standards. Through initial and diagnostic assessments, students are placed into the appropriate volume and module. Concepts are introduced at a reasonable pace in the form of a 6-page lesson plan for the student and shown in the following examples.

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44 IIN Intervention Volume 2 • Fractions and Decimals

Add Fractions One way to add fractions is to use fraction bars.2__6

� 1__6

Step 1Line up two 1_

6 fraction bars under the bar for 1.

2__6

Subtract Fractions One way to subtract fractions is to use fraction bars.5__6

� 3__6

Step 1Line up the fraction bars for 5_

6 and 3_6 under the bar

for 1.

REASONING What’s the Error? Lisa says that 2_5 � 3_5 � 5__

10.Do you agree? Explain your answer.

Skill 54

Step 2Add one more 1_

6 fraction bar. Count the number of 1_

6 bars

2__6

� 1__6

� 3__6

The sum is .

Step 3Find the largest fraction bar that is the same length.

2__6

� 1__6

� 3__6

3_6 is the same as .

Step 2Compare the bars to fi nd the difference.

The difference is .



Possible response: No, I do not agree. You are fi nding the total

number of fi fths. So, the sum is in fi fths, not tenths. 2_5 � 3_5 � 5_

5, or 1.

3

6

1

2

2

6

1

3


Skill 54

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Volume 2 • Fractions and Decimals Intervention IIN 45

Remember• Model both fractions.

• Line them up under the 1 bar.

• Find the largest fraction bar, or fraction bars with the same denominator, that are the same length.

Name

Model 1_8 . Add 3_

8 .

Find the largest bar that is the same length.

Skill 54

Find the sum. Use fraction bars to help you.

1. Find 1_8 � 3_

8.

So, 1_8 � 3_

8 � , or .

Find the sum or difference. Use fractions bars to help you.

2. 3. 4.

5__8

� 3__8

� , or 1__6

� 3__6

� , or 3__4

� 1__4

� , or

5. 2__8

� 4__8

� , or 6. 2__5

� 1__5

� 7. 5__8

� 1__8

� , or

8. 2__8

� 5__8

� 9. 6__8

� 3__8

� 10. 4__5

� 1__5

�

11. 6__6

� 2__6

�B ____

B , or

B ____

B

12. REASONING James used fraction bars to fi nd 6_6 � 4_6.

He says the difference is 2_6, or 1_4. What is his error?

Possible response: The fraction bars for 2_6 and 1_4 are not

the same length. The correct answer is 2_6, or 1_3.

4

8

1

2

2

8

1

4

4

6

2

3

2

4

1

2

3

4

6

8

3

5

3

4

6

8

7

8

3

8

3

5

4

6

2

3


The section called Learn the Math provides direct instruction for the teacher. It has clear models and contains guidance for students to help build understanding of a skill that supports a Standard.

Mathematical reasoning is embedded to guide instruction on the Standard and to serve as a check point for understanding of the lesson.

The section called Do the Math/Practice the Skill provides guided practice for the student. It includes scaffolded exercises for the skill followed by exercises with the prompts removed. If the teacher observes that a student has diffi culty with these exercises, the Hands-On Activities on the Alternative Teaching Strategy page found in the Teacher Guide is recommended.

Mathematical reasoning is embedded to guide instruction on the Standard and to serve as a check point for understanding of the lesson.

Introduction Intervention IIN ix

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Skill 54

What are you asked to fi nd?

What information do you need to solve the problem?

How long did Ricky practice on Tuesday?

How long did Ricky practice on Wednesday?

You can make a model to solve the problem.

Model 5_6 and 2_

6 with fraction bars.

Compare the bars to fi nd the difference.


5_6 � 2_

6 � , or

So, Ricky practiced hour more on Wednesday than on Tuesday.

What is the answer to the question?

How can you check your answer?

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Ricky practiced guitar for 2_6 hour on Tuesday. He practiced for

5_6 hour on Wednesday. How much more time did Ricky spend practicing on Wednesday than on Tuesday?

?

how much time Ricky practiced on Tuesday and on Wednesday

2_6 hour

5_6 hour

Ricky practiced 1_2 hour more on

Wednesday.I can line up 2_

6 and 1_2 under the 5_

6 bars and check to see that 2_

6 and 1_2 are the

same length as 5_6.

how much more time Ricky spent practicing on Wednesday

3

6

1

2

1

2


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Show YourWork

Show YourWork

Name Skill 54

Find the sum or difference. Use fraction bars to help you.

1. 2__4

� 1__4

� 2. 3__8

� 3__8

� , or

3. 5__6

� 1__6

� , or 4. 3__5

� 2__5

�

Problem Solving

5. A recipe calls for 3_4 cup of fl our. Sierra has 2_

4 cup of fl our. How many more cups of fl our does Sierra need for the recipe?

6. Christine walked 6_8 mile to school and 1_

8 mile to the library. How far did Christine walk altogether?

Show your answers to your teacher.

QuizFind the sum or difference. Use fraction bars to help you.

1. 4__8

� 3__8

� 2. 1__5

� 3__5

�

3. 6__8

� 2__8

� , or 4. 4__6

� 1__6

� , or

Problem Solving

5. Haley spent 3_8 of her allowance on snacks. She spent 5_

8 of her allowance on music. How much more of her allowance did Haley spend on music than snacks?

6. Tony practiced basketball for 4_6 hour on Saturday and 2_

6 hour on Sunday. How long did Tony practice basketball on the weekend?

6. 6_8 � 1_

8 � 7_8

5. 5_8 � 3_

8 � 2_8, or 1_

4

6. 4_6 � 2_

6 � 6_6,

or 1

5. 3_4 � 2_

4 � 1_41_

4 c

3

4

6

8

3

4

1

5

2

3

4

6

7_8 mi

7

8

4

8

1

2

4

5

3

6

1

2

2_8, or 1_4 more

6_6, or 1 hr


The section called Use the Math/Practice Problem Solving gives students practice in solving problems requiring the mathematics of the skill. The heuristic (What’s the Problem, Break it Down, Solve It, and Check It) allows students to step through the problem in a systematic way to promote generalization and transfer of skills and knowledge to a problem situation.

The section called Independent Practice provides problems that check the student’s profi ciency in the skill. The student is encouraged to check with the teacher before proceeding to the Quiz.

The section called Quiz parallels the problem set for Independent Practice that checks profi ciency in the skill. If a student does not achieve at the mastery level indicated in the Teacher Guide, the Hands-On Activity in the Teacher Guide is one suggestion for reteaching the skill.

x IIN Intervention Introduction

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AnotherAnother

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Find the sum or difference for each problem. Match it with a value below. Then write the circled letter on the line in front of the problem.

1. 3__8

� 1__8

� , or 2. 2__5

� 1__5

�

3. 3__6

� 1__6

� , or 4. 3__6

� 2__6

�

5. 3__6

� 1__6

� , or 6. 3__8

� 1__8

� , or

7. 1__8

� 5__8

� , or 8. 6__8

� 1__8

�

9. 4__5

� 2__5

�

Use your answers to decode the phrase below. Match the letter of the answer from each problem on the blank.

! 3 5 8 1 4 7 6 9 2

Write a problem about students eating pizza that can be solved with the number sentence 3_

8 � 1_8 �

B ____

B .

R L E A H S M T U R L E A H S M T U1__6

1__4

2__5

1__3

1__2

3__5

2__3

5__8

3__4

Skill 54

Possible response: Adam ate 3_8 of the pizza. Karen ate 1_8 of

the pizza. How much of the pizza was eaten? 3_8 � 1_8 � 4_

8, or 1_2

H

M

A

U

E

S

R

L

T

M A T H R U L E S

4

8

4

6

2

6

6

8

2

5

3

4

1

3

2

3

1

2

3

5

1

6

2

8

1

4

5

8


The section called Another Look provides another opportunity to engage students to apply the skill in a different context.

The section called Write Math is provided so students can write about their learning and use appropriate academic language in mathematics.

Introduction Intervention IIN xi

Teacher Guide The Teacher Guide is an easy-to-use planning tool that includes an overview of the program and suggestions for managing individualized instruction. In the six volumes, there are simple and easy-to-follow guidelines for direct instruction in the form of a 4-page lesson plan as shown in the following examples.

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Skill 54

CA Standards Grade 3, NS 3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context.

NS 3.2 Add and subtract simple fractions. also MR 2.3

ObjectiveTo add and subtract simple fractions and to determine equivalent values

Vocabularynumerator The part of a fraction above

the line, which tells how many parts are being counted

denominator The part of a fraction below the line, which tells how many equal parts there are in the whole or in the group

Manipulativesfraction bars, fraction circles

Materialscrayons or colored pencils, number line

MATH BACKGROUND

• Fraction bars representing like fractions can be placed together and counted to represent the process of adding like fractions.

• You can subtract like fractions by lining up the bars and comparing to find the difference.

• When you add and subtract like fractions, the denominators stay the same and you add the numerators.

• A fraction is in simplest form when it uses the largest fraction bar or bars possible.

Teach page 43.

• Guide students through the sample problem.What number is the numerator? 6What number is the denominator? 8What fraction tells how much paper Sarah shaded? 3_

4

• REASONING COMMON ERROR Some students may not fold the paper to show equal parts. Use fraction bars or paper folding activities to practice making equivalent fractions.

• Assign Exercises 1–4.• Review vocabulary with students.

• VOCABULARY COMMON ERRORStudents might reverse the numerator and denominator when writing fractions. To correct this, have them draw models to represent the numerator and denominator.

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Name

IIN 43

Add and SubtractLike Fractions

Skill 54

numerator

denominator

Vocabulary

shaded parts

total parts

numerator

denominator

shaded parts

total parts

numerator

denominator

Read the problem. Use the picture to help you fi nd the answer.Michael divides his paper into 8 equal parts. He shades 6 of the parts. What fraction tells how much of the paper is shaded?

Sarah’s paper shows these equal parts:

REASONING Sarah says that she and Michael shaded the same amount of the paper. Is Sarah correct? Explain your answer.

Fill in the blanks with one of the following words: numerator and denominator. Use each word only once.

1. The is the part of a fraction above the line, which tells how many of the parts are being counted.

2. The is the part of a fraction below the line, which tells how many equal parts there are in the whole or in the group.

Use the fractions below.

3. Circle the numerators.4. Put a box around the denominators.

4__5

4__8

2__4

1__2

1__5

Possible explanation: Yes, Sarah is correct. When the two

sheets of paper are lined up, the same amount of paper is

shaded.

3

6

4

6

4

8

numerator

denominator

NS 3.1 Compare fractions represented by drawings or concrete materials to show equivalency and to add and subtract simple fractions in context. NS 3.2 Add and subtract simple fractions. also MR 2.3

Add and Subtract Like Fractions

Skill 54

MIECA09ATEK-32A_SKILL05429 29 2/9/07 12:55:30 PM


Teach page 45.

• Guide students through Exercise 1. How many 1_

8 fraction bars did you model? 4 What sum did you model? 4_

8What is the largest fraction bar that is the same as 4_

8? 1_2

• Assign Exercises 2–4. Check students’ answers.What fraction is the same as 2_

8? 1_4

4_6? 2_

32_4? 1_

2

• Assign Exercises 5–12. Review the steps in the Remember section with students. Monitor students’ work.

• REASONING COMMON ERRORStudents may incorrectly line up fraction bars when comparing to find equivalent values. Use fraction bars to practice making equivalent fractions by left aligning the bars against a ruler or straightedge.

For students who are not successful on Do the Math, use the hands-on activities on the Alternative Teaching Strategy page.

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Add Fractions One way to add fractions is to use fraction bars.2__6

� 1__6

Step 1Line up two 1_

6 fraction bars under the bar for 1.

2__6

Subtract Fractions One way to subtract fractions is to use fraction bars.5__6

� 3__6

Step 1Line up the fraction bars for 5_

6 and 3_6 under the bar

for 1.

REASONING What’s the Error? Lisa says that 2_5 � 3_5 � 5__

10.Do you agree? Explain your answer.

Skill 54

Step 2Add one more 1_

6 fraction bar. Count the number of 1_

6 bars

2__6

� 1__6

� 3__6

The sum is .


2__6

� 1__6

� 3__6


Step 2Compare the bars to fi nd the difference.

The difference is .



Possible response: No, I do not agree. You are fi nding the total

number of fi fths. So, the sum is in fi fths, not tenths. 2_5 � 3_5 � 5_

5, or 1.

3

6

1

2

2

6

1

3

Skill 54©

Har

cour

t


Remember• Model both fractions.

• Line them up under the 1 bar.

• Find the largest fraction bar, or fraction bars with the same denominator, that are the same length.

Name

Model 1_8 . Add 3_

8 .


Skill 54

Find the sum. Use fraction bars to help you.

1. Find 1_8 � 3_

8.

So, 1_8 � 3_

8 � , or .

Find the sum or difference. Use fractions bars to help you.

2. 3. 4.

5__8

� 3__8

� , or 1__6

� 3__6

� , or 3__4

� 1__4

� , or

5. 2__8

� 4__8

� , or 6. 2__5

� 1__5

� 7. 5__8

� 1__8

� , or

8. 2__8

� 5__8

� 9. 6__8

� 3__8

� 10. 4__5

� 1__5

�

11. 6__6

� 2__6

�B ____

B , or

B ____

B

12. REASONING James used fraction bars to fi nd 6_6 � 4_6.

He says the difference is 2_6, or 1_4. What is his error?

Possible response: The fraction bars for 2_6 and 1_4 are not

the same length. The correct answer is 2_6, or 1_3.

4

8

1

2

2

8

1

4

4

6

2

3

2

4

1

2

3

4

6

8

3

5

3

4

6

8

7

8

3

8

3

5

4

6

2

3

Teach page 44.

• Guide students through Steps 1–3 of the addition problem. In Step 2, ask: What is the sum? 3_

6 . Look at Step 3. How do you know 3_

6 is the same as 1_2? The 1_

2 fraction bar is the same length as the three 1_

6fraction bars.

• Guide students through Steps 1–3 of the subtraction problem. In Step 2, ask: What is the difference? 2_

6 Look at Step 3. Howdo you know 2_

6 is the same as 1_3? The 1_

3fraction bar is the same length as the two 1_

6 fraction bars.

• REASONING COMMON ERRORStudents may add or subtract the denominators, as well as numerators. Use fraction bars to practice adding and subtracting fractional parts of the same size.


The California Math Standard that relates to each skill is provided for the teacher at the top of this page. The Math Reasoning Standard that supports the skill is also included.

The section on Math Background includes critical information for the teacher about understanding and teaching the Standard.

A reduction of the Student Edition is included at point of use for the teacher to view the answers and guided questions for each section in the Student Edition.

The section called Reasoning/Common Error and Vocabulary/Common Error calls attention to errors students commonly make or ideas students often misunderstand and includes suggestions for how to correct this.

The Learn the Math section in the Teacher Guide provides tips and guided questions for direct instruction by the teacher. It highlights questions teachers can ask to encourage students to think critically.

The Do the Math section in the Teacher Guide includes questions to enhance the guided practice for the student. It includes a reference to the Alternative Teaching Strategies for use with students who are struggling with the content.

xii IIN Intervention Introduction

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4_5 � 2_

5

2_8 � 3_

8

3_8

4_8

5_8

6_8

7_8 10 1_

82_8

3_5

4_5 11_

52_50


OPTIONAL

OBJECTIVE To use models to add and subtract simple fractions MANIPULATIVES/MATERIALS fraction circles, number lines

Alternative Teaching Strategy

Have students work with partners and take turns modeling the fraction operations. Use fraction circles to add and subtract fractions.

With the area models, fractions are based on regional parts of a whole. Manipulatives such as fraction circles demonstrate the area model.

Direct students to solve the following problems by using fraction circles.

1. 2_8 � 4_

8

2. 5_6 � 2_

6

Ask: What models did you make?Responses might include:• I modeled 2_8 and 4_

8 by using two 1_8 pieces

and four 1_8 pieces on top of the one whole

and combined them to make 6_8.

• I modeled 5_6 by using five 1_6 pieces. I took

away two 1_6 pieces, leaving three 1_6 pieces. So, the difference is 3_6.

Ask: Are there other larger fraction pieces that cover the same amount of area? Explain.Responses might include:• Yes, I can cover the same area for the 6_8

pieces with three 1_4 pieces to make 3_

4.• Yes, I can cover the same area for the 3_6

pieces with one 1_2 piece to make 1_

2.

Continue this procedure with other simple fractions.

Have students work with partners to take turns drawing and labeling a number line to show addition and subtraction of fractions.

By using the number line as a tool for tracking their calculations, students can use count on or count back strategies to find sums and differences.

Direct students to solve the following problems.

3.

Ask: How can you count on to find the sum?Responses might include:• I can start by labeling a point for 2_8 and

count on to the right by drawing 3 jumps of 1_

8 to end on 5_8.

4.

Ask: How can you count back to find the difference?Responses might include:• I can start by labeling a point for 4_5 and

count back to the left by drawing 2 jumps of 1_

5 to end on 2_5.

Continue this procedure with other simple fractions.



Skill 54

What are you asked to fi nd?

What information do you need to solve the problem?

How long did Ricky practice on Tuesday?

How long did Ricky practice on Wednesday?

You can make a model to solve the problem.

Model 5_6 and 2_

6 with fraction bars.

Compare the bars to fi nd the difference.


5_6 � 2_

6 � , or

So, Ricky practiced hour more on Wednesday than on Tuesday.

What is the answer to the question?

How can you check your answer?

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Ricky practiced guitar for 2_6 hour on Tuesday. He practiced for

5_6 hour on Wednesday. How much more time did Ricky spend practicing on Wednesday than on Tuesday?

?

how much time Ricky practiced on Tuesday and on Wednesday

2_6 hour

5_6 hour

Ricky practiced 1_2 hour more on

Wednesday.I can line up 2_

6 and 1_2 under the 5_

6 bars and check to see that 2_

6 and 1_2 are the

same length as 5_6.

how much more time Ricky spent practicing on Wednesday

3

6

1

2

1

2

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Show YourWork

Show YourWork

Name Skill 54

Find the sum or difference. Use fraction bars to help you.

1. 2__4

� 1__4

� 2. 3__8

� 3__8

� , or

3. 5__6

� 1__6

� , or 4. 3__5

� 2__5

�

Problem Solving

5. A recipe calls for 3_4 cup of fl our. Sierra has 2_

4 cup of fl our. How many more cups of fl our does Sierra need for the recipe?

6. Christine walked 6_8 mile to school and 1_

8 mile to the library. How far did Christine walk altogether?

Show your answers to your teacher.

QuizFind the sum or difference. Use fraction bars to help you.

1. 4__8

� 3__8

� 2. 1__5

� 3__5

�

3. 6__8

� 2__8

� , or 4. 4__6

� 1__6

� , or

Problem Solving

5. Haley spent 3_8 of her allowance on snacks. She spent 5_

8 of her allowance on music. How much more of her allowance did Haley spend on music than snacks?

6. Tony practiced basketball for 4_6 hour on Saturday and 2_

6 hour on Sunday. How long did Tony practice basketball on the weekend?

6. 6_8 � 1_

8 � 7_8

5. 5_8 � 3_

8 � 2_8, or 1_

4

6. 4_6 � 2_

6 � 6_6,

or 1

5. 3_4 � 2_

4 � 1_41_

4 c

3

4

6

8

3

4

1

5

2

3

4

6

7_8 mi

7

8

4

8

1

2

4

5

3

6

1

2

2_8, or 1_4 more

6_6, or 1 hr

AnotherAnother

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Find the sum or difference for each problem. Match it with a value below. Then write the circled letter on the line in front of the problem.

1. 3__8

� 1__8

� , or 2. 2__5

� 1__5

�

3. 3__6

� 1__6

� , or 4. 3__6

� 2__6

�

5. 3__6

� 1__6

� , or 6. 3__8

� 1__8

� , or

7. 1__8

� 5__8

� , or 8. 6__8

� 1__8

�

9. 4__5

� 2__5

�

Use your answers to decode the phrase below. Match the letter of the answer from each problem on the blank.

! 3 5 8 1 4 7 6 9 2

Write a problem about students eating pizza that can be solved with the number sentence 3_

8 � 1_8 �

B ____

B .

R L E A H S M T U R L E A H S M T U1__6

1__4

2__5

1__3

1__2

3__5

2__3

5__8

3__4

Skill 54

Possible response: Adam ate 3_8 of the pizza. Karen ate 1_8 of

the pizza. How much of the pizza was eaten? 3_8 � 1_8 � 4_

8, or 1_2

H

M

A

U

E

S

R

L

T

M A T H R U L E S

4

8

4

6

2

6

6

8

2

5

3

4

1

3

2

3

1

2

3

5

1

6

2

8

1

4

5

8

Teach page 46.

• Point out that Make a Model is the Problem Solving Strategy used.

Teach page 47.

• Assign Exercises 1–6. Remind students that they can use fraction bars to help them.

• Discuss solutions to problems 5 and 6.

Quiz• Determine if students can accurately add

and subtract simple fractions. Success is indicated by 4 out of 6 correct responses.

• Students who were not successful on the Quiz may benefit from the Alternative Teaching Strategy on the previous page.

AnotherAnother Teach page 48.

• Review instructions for the activity with students to decode the phrase.

• WRITE Math COMMON ERRORStudents may confuse the operations and write a problem involving subtraction. Remind students that addition involves combining or joining fractional parts.


The Alternative Teaching Strategy page is to give the teacher an opportunity to reteach the skill by using manipulative-based instruction. The format of One Way and Another Way is to give the teacher more opportunities to introduce the skill to different types of learners.

Use the Math states the problem solving strategy used.

Independent Practice includes tips for the teacher to prepare students for the Quiz. The Quiz includes an indicator to determine profi ciency for the skill.

Write Math/Common Error calls attention to errors students commonly make in writing about mathematics and includes suggestions for how to correct this.

Introduction Intervention IIN xiii

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xiv IIN Intervention Introduction

Assessment Overview The assessments found in California Fast Forward Math are a series of extensive diagnostic tools to identify students’ areas of weakness, place them at an appropriate starting point in the program, monitor their learning, and measure their progress. The entry-level assessments, or initial assessments, identify whether a student is two or more years below grade level and needs to be placed into California Fast Forward Math. The diagnostic assessments are broken down by volumes, analyze students’ comprehension of Standards, and prescribe skills for the Standards found in the California Fast Forward Math Student Editions. Each skill includes a formative assessment, or Quiz, to provide teachers with immediate feedback on students’ performance. The fi nal assessment tool is the summative test. Each summative test is divided into sections by skill so that teachers can assess students only on skills and Standards prescribed by the diagnostic tests.

What do the four types of assessments contain?Initial assessments are used to place students into California Fast Forward Math. They consist of assessment items from one or two grade levels. Each assessment is approximately 50-70 items in length. These items are at least two grade levels below the students’ grade level. The mastery level for these assessments has been set at 75%. If students score below 75%, they should be considered for instruction from California Fast Forward Math.

Diagnostic assessments are taken after students score less than 75% on the Initial assessment. Each skill in the Student Edition has a corresponding section in a diagnostic assessment. This section contains 4 assessment items. Mastery is set at 75% to ensure comprehension of a skill.

Formative assessments provide immediate feedback as students are working on skills. They are found in the Student Edition in the form of a short response quiz. Each quiz assesses content addressed in the skill and contains computation and problem solving items.

Summative assessments show students’ mastery of skills. They are parallel to diagnostic assessment. Each skill in the Student Edition has a corresponding summative assessment. These assessments contain 4 assessment items. Mastery is set at 75% to ensure comprehension of a skill.

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Introduction Intervention IIN xv

How do I decide if students should exit the program?After completing all diagnostic assessments and prescribed skills and summative assessments assigned through the initial assessment, teachers need to determine the next steps for each student. The following tables explain how each diagnostic assessment may be used for students at a given grade level and when teachers may choose to have students exit this program.

How do the California Fast Forward Math assessments work?The following diagram illustrates the recommended testing path that a teacher should follow when assigning instruction and assessments.

Students start with the initial assessment. It should be noted that each grade level has a unique initial assessment. After analyzing the results of the initial assessment, students who are identifi ed as candidates for California Fast Forward Math proceed to take the fi rst of the prescribed diagnostic assessments. (Note: To identify the lowest grade level standards in which students demonstrate weakness, 6th and 7th grade students may need to take an additional initial assessment before beginning the diagnostic assessments.) After completing a diagnostic assessment, a student will be prescribed a series of skills from the Student Edition, followed by sections of the summative assessment to check profi ciency. Students should next work through all of the skills prescribed and then complete the summative assessment before taking the next diagnostic assessment. This will allow students to focus on one area of mathematics before moving on to the next volume of diagnostic assessment. This process of taking diagnostic assessments and completing skills and summative assessments should repeat until a student completes all of the diagnostic assessments prescribed by the initial assessment.

InitialAssessment

DiagnosticAssessments

SummativeAssessments

Instruction and Formative

Assessment

Does not require Fast Forward

ExitProgram

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xvi IIN Intervention Introduction

Grade 4 StudentsDiagnostic Assessments by Grade Level of Standards Assessed

Use of Assessment by Student Grade Level

Grades K-2 Grade 4 students will begin the CA Fast Forward Math at this level. Upon completion of all diagnostic assessments, corresponding skills and summative assessments, students may exit program as a Strategic Intervention student.

Grade 3 These assessments are optional for this grade level student. Teachers may wish to assign these assessments and supporting skills to teach students the Standards from one grade level below their current grade level. Upon completion of all diagnostic assessments, corresponding skills and summative assessments, students may exit the program as a Benchmark Intervention student.

Grade 4 These assessments are optional for this grade level student. Teachers may wish to assign these assessments and supporting skills to teach students the Standards from their current grade level.



Grades K-2 Students may need to start with these assessments if the initial assessment indicates weakness at this level. Upon completing, students should continue in CA Fast Forward Math by taking the diagnostic assessments for the grade 3 Standards.

Grade 3 Initial assessments may place grade 5 students in this grade level. Upon completion of all diagnostic assessments, corresponding skills and summative assessments, student may exit program as a Strategic Intervention student.

Grade 4 These assessments are optional for this grade level student. Teachers may wish to assign these assessments and supporting skills to teach students the Standards from one grade level below their current grade level. Upon completion, students may exit the program as a Benchmark Intervention student.

Grade 5 These assessments are optional for students at this grade level. Teachers may wish to assign these assessments and supporting skills to teach students the Standards from their current grade level.

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Introduction Intervention IIN xvii




Grade 3 Students may need to start with these assessments if the initial assessment indicates weakness at this level. Or, students may also have progressed to these assessments after completing the grades K-2 diagnostic assessments and corresponding skills. Upon completing, students should continue in CA Fast Forward Math by taking the diagnostic assessments for the grade 4 Standards.




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xviii IIN Intervention Introduction




Grade 3 Students may need to start with these assessments if the initial assessment indicates weakness at this level. Or, students may also have progressed to these assessments after completing the grade K-2 diagnostic assessments and corresponding skills. Upon completing, students should continue in CA Fast Forward Math by taking the diagnostic assessments for the grade 4 Standards.

Grade 4 Students may need to start with these assessments if the initial assessment indicates weakness at this level. Or, students may also have progressed to these assessments after completing the grade 3 diagnostic assessments and corresponding skills. Upon completing, students should continue in CA Fast Forward Math by taking the diagnostic assessments for the grade 5 Standards.




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Introduction Intervention IIN xix

How do I keep track of students’ progress?The California Fast Forward Math Assessment Guide (AG) includes Individual Record Forms (IRF) to track student progress. IRFs are required for each student for each initial and diagnostic assessment.

Each IRF for an initial assessment contains recommendations for which diagnostic assessments to prescribe next. Following the initial assessment, the teacher transfers students’ results into tables that group assessment items by the grade level Standards and also by the volumes in which they are found. Cumulative test scores are not emphasized, rather the tables identify areas of weakness by volume.

The IRF for a diagnostic assessment requires that the teacher transfer students’ results into tables. These tables correlate assessment items to skills from the Student Edition and to related summative assessment sections. Cumulative scores are not emphasized on diagnostic assessments. Comprehension is evaluated by performance on groups of items that assess an individual skill. Teachers should mark out rows of any skills for which students were not assigned, since the IRF for a diagnostic assessment should be used to record the results of the corresponding summative assessment.

The following table illustrates the use of IRFs to track student progress.

What are in the online assessments?The online assessments are the same as the assessments provided in the Assessment Guide. The online assessment system scores students’ tests and shows the results in a printable report. These reports will display the recommended assessment path for each student. As students take more assessments, the reports will update to show the suggested prescriptions.

Initial assessment IRF shows which diagnostic assessment to prescribe.

Diagnostic assessment IRF shows which skills to prescribe.

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how to combo

Documents

individual students

mathematics framework

grade level standards

basic gradelevel programs

knowledge of mathematics

intensive intervention

california public schools

extended math instruction