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Intensive Mathema cs Interven on Mul plica on, Division, & Frac ons E
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Instruc on Manual by Daniel C. Sinclair PA
Angela C. Sinclair G
Artwork and Design by ES
Daniel C. Sinclair Angela C. Sinclair 800‐745‐8212  www.Teach4Mastery.com LY
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Table of Contents
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Purpose, Goal, Methodology, and Philosophy 
What Mathema cal Concepts does Percep ons Cover? This is designed as a 16week intensive intervention
for students entering 5th
grade or higher. A basic
understanding of
multiplication and division
facts is helpful.
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Mathema cs is the Study of… 
Individual or Collabora ve Learning 
How to Use Percep ons 
Flowchart of Percep ons Program 
Tools Needed for Percep ons 
Using Manipula ves in Math  Built-in Progress
Monitoring for every
lesson
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Introduc on to the Manipula ves Used 
Progress Monitoring & Student Self‐Assessment Rubric 
Mastery ‐ What does it mean?  16 weeks equals 80
days

Suggested Pacing Guide 
Key Academic Vocabulary and Symbols  Each lesson is designed
to take 2 days, although
some may be one or
three days. The 32
Lessons equal 64 of the
80 days, allowing for
days of testing and days
for application, additional
practice, or make-up.
Unit 1 - Understanding Multiplication
and Division Example: Unit 1
Lesson 5: Beginning Division Daily Sequence
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Lesson 2: Factors and Products Lesson 3: Beginning Factoring Lesson 4: Factoring and Prime & Composite Numbers Lesson 6: Solving for Unknowns Unit 2 - Multi-Digit Multiplication
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and Division Day 4 & 5 Lesson 2
Lesson 7: Distribu ve Property and N
Day 6 & 7 Lesson 3
Making Rectangles and Squares ES
Day 2 & 3 Unit Pre-Test &
Lesson 1
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Day 1 Level Pre-Test
Lesson 1: Mul plica on: Repeated Addi on Place‐Value Nota on (Expanded Form) Lesson 8: Two‐Digit Mul plica on Day 10 & 11 Lesson 5
Lesson 9: Two‐Digit Mul plica on with Regrouping Day 12 & 13 Lesson 6
Lesson 10: Long Division Day 14 Unit 1 Post-Test
Lesson 11: Division with Remainders Lesson 12: Mul ‐Digit Mul plica on Lesson 13: Mul ‐Digit Division LY
Day 8 & 9 Lesson 4
Unit 3 - Introduction to Fractions Lesson 14: Interrela onships: Combine & Compare Lesson 15: Ra onal Numbers: Ra os & Frac ons SA
Lesson 16: Parts of a Frac on and Frac on of a Number Lesson 17: Types of Frac ons and Comparing/Combining Frac ons Lesson 18: Add & Subtract Frac ons with Same Denominators & Decomposing Frac ons Lesson 19: Equivalent Frac ons M
Lesson 20: Finding Equivalencies, Simplifying, and GCF E
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Unit 4 – Operations with Fractions Part 1
Lesson 21: Adding and Subtrac ng Frac ons Lesson 22: Cross‐Mul plica on (Concrete) Lesson 23: Cross‐Mul plica on (Abstract) Lesson 24: Changing Improper Frac ons & Mixed Numbers Lesson 25: Addi on with Improper Frac ons & Mixed Numbers PA
Lesson 26: Subtrac on with Improper Frac ons & Mixed Numbers Lesson 27: Adding Mul ple Frac ons Lesson 28: Mul plica on with Frac ons Lesson 30: Division with Frac ons Using the Reciprocal ES
Lesson 29: Division with Frac ons G
Unit 5 – Operations with Fractions Part 2 Lesson 31: Canceling (Pre‐reducing) when Mul plying Frac ons Lesson 32: Solving for Unknowns in Frac ons Lesson 33: Solving for Unknowns (Op onal) LY
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Purpose, Goal, and Philosophy
Perceptions - A 16 Week Intensive Intervention
by Teach 4 Mastery, Inc.
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Purpose (Reason) for this Intervention M
Research1 has shown that a student’s understanding of frac ons and whole‐number division are early predictors of the student’s mathema cal achievement in high school. This was found to be more important than the student’s knowledge of other mathema cal areas along with other social and economic factors. E
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The research further supported that mastery of frac ons and division is important if students are to develop understanding of algebra and other aspects of high school mathema cs. The Goal of this Intervention
1. Provide a supplementary interven on covering mul plica on, division, and frac ons 2. To increase the conceptual understanding of frac ons and division in teachers PA
3. To increase the conceptual understanding of frac ons and division in students Methodology and Philosophy - Construct, Express, & Apply
mathema cal concepts and ideas G
Construct: Demonstrate through both concrete and semi‐concrete representa ons of ES
Whenever the instruc ons indicate to construct, we recommend using the MasterPiecesTM or the MasterFrac onsTM to build or draw the problem given and develop conceptual understanding. Express: Demonstrate through ar s c, wri en, and verbal expression, fluency and accuracy of the concept and computa on of the mathema cal process O
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Whenever the instruc ons indicate to solve, show, draw, etc... we recommend using drawing, wri en, and verbal communica ons to have students express what they have learned. Get crea ve on how the students will express to you or to others the understanding they have of a given concept and process. Ideally, a student who has mastered a concept can hear the problem verbally and express back the computa on, concept, and applica on of the problem in a wri en or verbal manner. Apply: Demonstrate the correct applica on of the mathema cal concept and computa on into real world se ngs Word Problems & Mastery Challenges ‐ Being able to apply what has been learned is the greatest sign of achievement! 1
A 20 year longitudinal study on Early Predictors of High School Mathema cs Achievement by Robert S. Seiglery et al. 5 061514 Lesson 5: Beginning Division
Lesson Objectives:
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1. Understand that division is the inverse opera on of mul plica on 2. Understand that the divisor and quo ent are new names for factors 3. Understand that dividend is the new name for our product 4. Learn the following terms or symbols: Inverse Mul plica on Division Divisor Dividend Quo ent Fig. A Step 1: Both Teacher and Student Watch Video Step 2: Explicitly Instruct Students Using Manipula ves Construct, Express, & Apply Step 3: Student Proceeds to Workbook Pages for Prac ce Step 4: Student Demonstrates Mastery ‐ Teaching Back to You Now take what we have learned about mul plica on and use it in the inverse opera on known as Division. The term inverse means to reverse the posi on, order, or direc on of what you have been doing. With that in mind look at these opera ons: Mul plica on: increasing one factor by another factor to determine the product PA
Example
Lesson Steps:
How many 2s out of 6? Division: given the dividend (product) and the divisor (one factor), now look to find how many mes the divisor can be G
counted out of the dividend ES
In mul plica on we are looking for the product, and in Make Equal Groups division we are looking for the missing factor (quo ent). It is important to realize the connec on between these two opera ons. Make a Rectangle ? © Teach4Mastery, Inc. many groups of 2s we could count out of 6. We could make 3 equal groups of 2s. If we were to put those 3 groups together, Product Dividend LY
Factor Divisor 6 Example (Fig A): We have 6 units, and we want to know how N
2 Missing Factor Quo ent O
2 x ? = 6 is the same as 6 ÷ 2 = ? what shape would they make? A rectangle As a helpful word associa on, the words Quo ent and Ques on both begin with the le er “Q.” In division, what is the ques on? The Quo ent! CCSS covered: 3.OA.2, 3.OA.4, 3.OA.5, 3.OA.6 6.EE.2b 23 061514 Lesson 5: Beginning Division
The rectangle (Fig. B) is what makes up the area represen ng the dividend. Since we are doing the inverse opera on bo om and right to the top and le . Fig. B How Far Over How Far Up Rectangle How Far Up How Far Over E
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(division) of mul plica on, let’s move our factors from the Division Terms Mul plica on Terms Factor Quo ent Factor Product Divisor Dividend PA
The key is to realize that they are both represen ng the same thing. The only difference is whether you know how much you have (Dividend) or only the factors. ES
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© Teach4Mastery, Inc. 24 E
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Intensive Mathema cs Interven on Mul plica on, Division, & Frac ons Student Workbook by PA
Daniel C. Sinclair Angela C. Sinclair G
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Artwork and Design by Daniel C. Sinclair Angela C. Sinclair N
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800‐745‐8212  www.Teach4Mastery.com ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com 06‐15‐14 Student Self‐Assessment Rubric E
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We at Teach4Mastery are so excited to help you understand math be er, and we hope our methods and instruc on will make it easier for you. On the back side of each of your student pages, you will find the following shown below. This will assist you in le ng your teacher know if you are understanding the materials or need addi onal help. A er you complete each page, shade in the symbol that best describes how you feel about the work you just finished. STOP means “I don’t get it. I need more help in understanding the concept or comple ng the problems.” SLOW means “I’m beginning to understand, but I will need addi onal work to be able to complete the problems with confidence.” GO means “I got it! I understand the concept and can do the problems given with confidence.” GO ES
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STOP Here is where you will find your student rubric in your workbook. Ask your teacher if you have any ques ons. STOP SLOW GO LY
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SLOW ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com Beginning Division 5‐A Division: How Far Over Quo ent How Far Up Area Divisor SA
Dividend Factor As shown in Mul plica on Terms: M
Factor Product E
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Construct, then use the rectangles to draw your lines to show your answer. (____) 12 25 (____) ES
( 4 ) ( 5 ) G
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( 3 ) (____) ( 2 ) 24 10 N
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(____) ( 4 ) 15 LY
( 3 ) (____) 28 ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com 06‐15‐14 5‐A (____) (____) ( 7 ) ( 6 ) 42 M
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35 ( 4 ) (____) E
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(____) 36 ( 9 ) 54 ES
45 (____) G
( 5 ) 48 PA
(____) ( 8 ) O
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Adrienne made 36 cupcakes for Emmie’s birthday. If there are 12 people at her party, how many cupcakes could each person have? Maggie is going to Grandma and Grandpa’s house for one week. Her mother packed 14 ou its. If she wore the same amount of ou its each day she was there, how many ou its could she wear each day? STOP SLOW GO ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com Beginning Division 5‐B Division: How Far Over SA
How Far Up Quo ent __________ ________________ Dividend As shown in Mul plica on Terms: Factor M
Factor ________________ E
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Construct, then use the rectangles to draw your lines to show your answer. (____) 18 24 ES
( 2 ) ( 6 ) G
(____) PA
( 3 ) (____) ( 7 ) 16 (____) 28 N
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(____) ( 3 ) 15 LY
( 5 ) (____) 21 ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com 06‐15‐14 5‐B (____) ( 6 ) (____) ( 7 ) 49 M
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54 (____) E
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( 8 ) (____) 64 ( 9 ) 72 ES
30 (____) G
( 5 ) 40 PA
(____) ( 4 ) LY
If Bentley has three friends over, how many cookies will each person get? N
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Bentley’s grandmother made 16 cookies for Bentley and her friends. If Bentley has one friend over, how many cookies will each person get? Bobby’s gumball machine only had 36 gumballs le in it. Each me he turned the lever, 4 gumballs came out. How many mes can he turn the lever before the machine will be empty? STOP SLOW GO ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com Beginning Division 5‐C Division: How Far ________ _______________ Divisor SA
How Far _____ ________________ ______________ As shown in Mul plica on Terms: ________________ M
Factor ________________ E
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Construct, then use the rectangles to draw your lines to show your answer. (____) 30 PA
( 5 ) (____) ( 3 ) 27 ES
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(____) ( 2 ) ( 6 ) 18 (____) 36 ( 4 ) LY
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(____) (____) ( 5 ) 32 35 ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com 06‐15‐14 5‐C (____) ( 5 ) (____) ( 8 ) 72 M
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(____) 48 ( 8 ) 56 ES
81 (____) G
( 9 ) 63 PA
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Michelle bought the family size pack of toilet paper that had 48 rolls. If her family goes through 8 rolls a week, how long with that family pack last? Lizzy had 54 old CDs. She decided that was too many. So, she divided them up evenly among 6 friends. How many CDs did each of her friends get? STOP SLOW GO ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com Mastery Challenge 5 E
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Ezekiel’s total budget for his birthday party was $100. He wanted to go to the zoo with his friends. Each cket cost $6.00 each. Since they were going to be there all day, he would need to purchase lunch for each person. Each lunch cket was $4.00. How many friends could he bring if both of his parents were also coming? ES
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Jessica processed 18 shipments in 3 hours. How many shipments did she average per hour? If the company sent out an average of 24 shipments per hour, how many employees did they have processing shipments? (Please assume that everyone ships the same amount as Jessica per hour.) LY
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Nick was training for a marathon. He needed to run 54 miles per week. He ran in the mornings on Monday, Wednesday, and Friday, and then ran in the a ernoon on Tuesday, Thursday, and Saturday. On average, how many miles would he need to run in each day in order to meet his goal? In addi on to his running, his training coach suggested that he ride his bike 48 miles per week. If he rode his bike each day that he ran, how many miles would he need to bike each day to complete this goal? How many miles was Nick running and biking each day? ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com 06‐15‐14 MC‐5 E
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Anne e’s grandmother was a professional caterer. Her grandmother gave her the recipes and the shopping lists for them, however all of her lists were for 16 people. Anne e needed to serve her family of only 4 people. What number must Anne e divide by to convert her grandmother’s recipes for her family? ________ 2 cups = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon Her grandmother’s shopping list called for 1 gallon of milk, how many quarts would Anne e need to purchase? _________________________ If the milk only came in pints, how much would she need? ____________________________________________ If you converts the pints into cups, how many cups of milk would she need?__________________________________ Her list also called for... Two pints of melted bu er, how much bu er would Anne e need?____________________________________ One dozen eggs, how many eggs would Anne e need? _____________________________________ One quart of sour cream, how much sour cream would Anne e need? _______________________________ Two quarts of bu ermilk, how much bu ermilk would Anne e need? ________________________________ ES
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STOP SLOW GO ©Teach4Mastery, Inc. 800‐745‐8212 www.Teach4Mastery.com