Project Impact CURR 231 Curriculum and Instruction in Math Session 5 Chapters 6 & 7 Outcomes for today: Participants will understand the importance of explicit vocabulary instruction as it relates to math and strategies for instruction. Participants will learn about ways to support and encourage mathematical conversations in their classrooms.

Outcomes Number Talk –Middle School Example Discuss Framework – Ch. 9 Technology Text – Teaching Math Ch. 6 & 7 Adding, Subtracting, Multiplying and Dividing Lab Time educatorlearningcenter.com Activity Backwards map CASHEE skills/standard through framework (if time) Game Time

Number of the Day 24 Number Talk Tell me everything you know about this number. Find 4 different ways to represent this number. Post on chart paper Observe students and choose students to share out their representation (especially those that represent a concept you have been working on)

Framework – Ch. 9 Responsibilities of Teachers, Students, Parents Skim and review Ch. 9 from the Framework on Technology. Think – Pair – Share your key learnings from this chapter with an elbow partner.

Chapter 6: Adding and Subtracting Whole Numbers: Combining and Separating Quantities Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Presentation 6a Computation Overview Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Definition Basic Facts Algorithm(s) TEACHING COMPUTATION Definition Basic Facts Algorithm(s) Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

DEFINITION (Giving Meaning to the Operation) Relate the Arithmetic Operation to a Physical Operation Learn to Use an Already Available Skill to Figure Out Answers Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

BASIC FACTS (The Facts That Are Memorized and Then Used to Figure Out the Facts That Are Not Memorized) Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Basic Facts Are Dealt with in Two Major Groups: Easy Facts Hard Facts Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

(Using the Definition) Discover Relationships Memorize Them Easy Basic Facts Figure Them Out (Using the Definition) Discover Relationships Memorize Them Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Develop Thinking Strategies for Figuring Out the Hard Basic Facts Those strategies should Be Mental Strategies, Not Mechanical nor Pencil-Paper Strategies Use Already Memorized Facts Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

(Using Thinking Strategies) Hard Basic Facts Figure Them Out (Using Thinking Strategies) Discover That the Same Relationships Hold When Larger Numbers Are Being Used Memorize Them Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

BASIC FACTS Easy Facts Thinking Strategies That Figure them out Are mental strategies Discover relationships Use memorized facts Memorize them Hard Facts Figure them out Recognize the same relationships Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

ALGORITHMS Teach Meaningfully Carefully Model the Operation Step-by-Step (Let the children see what it looks like) De-emphasize Rote Rules Emphasize Big Ideas Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

TEACHING ADDITION Definition (Establishing Meaning) Associate Addition with Combining Count to Find the Answer Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Developing the Basic Facts Easy Basic Addition Facts Figure Them Out by Combining and Counting or by Counting On Discover Relationships Among the Facts Adding Zero Adding One Rearranging Memorize the Easy Addition Facts Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Thinking Strategies for Hard Basic Addition Facts One More Doubles Make Ten Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Hard Basic Addition Facts Figure Them Out Using a Thinking Strategy Recognize That the Same Relationships Hold Adding Zero Adding One Rearranging Memorize the Hard Addition Facts Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Teaching the Addition Algorithm Use a Model That Emphasizes the Basic Units De-emphasize Rote Rules Emphasize Big Ideas Always Add Like Units Too Many to Write? Make a Trade Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

TEACHING SUBTRACTION Definition (Establishing Meaning) Associate Subtraction with Comparison (to find the difference) or Take Away (to find the remainder) Find the Answer by Counting Using Mental Imagery Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Easy Basic Subtraction Facts Figure Them Out by Taking Away and Counting or by Using Mental Imagery Discover Relationships Subtracting Zero Subtracting One Subtracting a Number from Itself Recognizing Fact Families Memorize the Easy Subtraction Facts Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Thinking Strategies for Hard Basic Subtraction Facts Relate to Addition Subtract from Ten Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Hard Basic Subtraction Facts Figure Them Out Using a Thinking Strategy Recognize That the Same Relationships Hold Subtracting Zero Subtracting One Subtracting a Number from Itself Recognizing Fact Families Memorize the Hard Subtraction Facts Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Teaching the Subtraction Algorithm Use a Model That Emphasizes the Basic Units De-emphasize Rote Rules Emphasize Big Ideas Always Subtract Like Units If There Are Not Enough, Make a Trade! Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Chapter 6: Adding and Subtracting Whole Numbers: Combining and Separating Quantities Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Presentation 6b Relationships Among Basic Facts for Addition and Subtraction Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

BASIC FACTS are the facts that are memorized and then used to figure out the facts that are not memorized. They are basic because they are the basis for all the other facts. For each operation, 100 facts are generally considered to be basic facts. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

A CURRENT ISSUE in elementary mathematics education is, When teaching basic facts, what should we be emphasizing? Strategies? Or relationships? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Relationships are the basis for number sense. Relationships are the connections that provide a meaningful context for mathematical concepts and skills. Relationships are the basis for number sense. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Strategies are step-by-step procedures, based on useful relationships, that will efficiently produce a desired result. Effective strategies usually are very automatic, requiring a minimum of decision making as they are executed. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The answer, of course, is that we should emphasize both. Recall that the issue is: Should we be emphasizing strategies or relationships? The answer, of course, is that we should emphasize both. Relationships and strategies both have an important role. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The100 basic addition facts are displayed in the following table. We will begin by considering some of the relationships related to the easy basic addition facts. The100 basic addition facts are displayed in the following table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

+ 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We have highlighted the easy basic addition facts on the next table. For our purposes, we will consider the easy basic addition facts to be the ones with sums of 10 or less. We have highlighted the easy basic addition facts on the next table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

+ 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

How are these facts related? First Relationship Children should notice that certain facts have something in common. 2 + 0 = 2 3 + 0 = 3 5 + 0 = 5 0 + 1 = 1 0 + 8 = 8 1 + 0 = 1 6 + 0 = 6 7 + 0 = 7 0 + 4 = 4 How are these facts related? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

When zero is added to any other number, that other number is the sum. There are 19 addition facts like this. But, if children understand how these facts are related, it is really just one thing to learn. We will highlight these 19 facts in red on the addition table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

+ 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

How are these facts related? Second Relationship Children should notice facts like these have something in common. 3 + 1 = 4 5 + 1 = 6 1 + 1 = 2 1 + 8 = 9 9 + 1 = 10 7 + 1 = 8 6 + 1 = 7 1 + 4 = 5 2 + 1 = 3 How are these facts related? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

There are 17 addition facts like this. When one is added to any other number, the sum is the next number after that other number (the next number in the counting sequence). There are 17 addition facts like this. But, if children understand how these facts are related, it is really just one thing to learn. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

+ 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

How are these pairs of facts related? Third Relationship Children should notice that certain pairs of facts have the same answer. 3 + 2 = 5 4 + 5 = 9 1 + 8 = 9 5 + 4 = 9 2 + 3 = 5 7 + 3 = 10 2 + 4 = 6 8 + 1 = 9 4 + 2 = 6 3 + 7 = 10 How are these pairs of facts related? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The sum is not changed when you rearrange the numbers. There are 12 rearranged pairs: 24 facts, but if children understand how the pairs of facts are related, there are only 12 things to learn to master these facts. We will highlight these 24 facts in green on the addition table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

+ 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

There are four easy facts remaining. They are the doubles. 2 + 2 = 4 3 + 3 = 6 5 + 5 = 10 4 + 4 = 8 The doubles are usually easy to learn. We will highlight these 4 facts in brown on the addition table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

+ 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

How will the emphasis on relationships help children to master the easy addition facts? Consider the following summary of the potential effect of the emphasis on relationships. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

SUMMARY OF RELATIONSHIPS FOR EASY ADDITION FACTS Relationship Number of Facts Things to Learn Adding zero 19 1 Adding one 17 1 Rearranging 0 1 Pairs 24 12 Doubles 4 4 TOTALS 64 19 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

SO, WE CAN SEE THAT ONE BENEFIT OF AN EMPHASIS ON RELATIONSHIPS IS A SHARP REDUCTION IN THE AMOUNT TO BE MEMORIZED. WHEN MEMORIZED MATERIAL IS RELATED AND INTERCONNECTED TO OTHER MEMORIZED MATERIAL, THERE ARE MANY MORE PATHS TO RECALL AND RECALL IS IMPROVED. THEREFORE, A SECOND BENEFIT OF THE EMPHASIS ON RELATIONSHIPS IS IMPROVED RETENTION OF THE FACTS ONCE THEY HAVE BEEN MEMORIZED. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Chapter 6: Adding and Subtracting Whole Numbers: Combining and Separating Quantities Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Presentation 6c Strategies for Hard Addition and Subtraction Facts Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Recall that BASIC FACTS are the facts that we memorize and then use to figure out the facts that we have not memorized. They are basic because they are the basis for all the other facts. For each operation, 100 facts are generally considered to be basic facts. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

When teaching basic facts, what should we be emphasizing? Recall also that A CURRENT ISSUE in elementary mathematics education is: When teaching basic facts, what should we be emphasizing? Or relationships? Strategies? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Relationships are the basis for number sense. Relationships are the connections that provide a meaningful context for mathematical concepts and skills. Relationships are the basis for number sense. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Strategies, on the other hand, are step-by-step procedures based on useful relationships that will efficiently produce a desired result. Effective strategies usually are very automatic, requiring a minimum of decision making as they are executed. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Obviously, we should emphasize both. Recall that the issue is, Should we be emphasizing strategies or relationships? Obviously, we should emphasize both. Relationships and strategies both have an important role. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We will now examine some fact strategies for the hard basic addition facts. Although there are many strategies for finding answers to hard basic addition facts, the strategies that are most successful in resulting in memorization of those facts all have two common characteristics. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The two common characteristics of successful strategies for finding answers to hard basic facts are: 1. They are mental strategies (not pencil/paper or mechanical strategies). 2. They always require the child to use facts that have already been memorized (to build on what the child already knows). Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

As we consider selected strategies, look for these two characteristics: 1. They are mental strategies. 2. They always require the child to use facts that have already been memorized. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We will examine each of these. Three strategies that have proven to be successful for figuring out answers to hard addition facts are: One More Building on Doubles Make Ten We will examine each of these. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The easiest strategy for children to use is the One-More strategy. This strategy requires the lowest level of knowledge and skills. It is also very easy for children to visualize. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

This fact is one more than a fact that you already know: 5 + 5 and one more Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Although the One-More strategy is easy, it requires that the child already know a specific fact before finding the answer to the target fact. If the child does not know the required fact, then the child cannot use this strategy to find the target fact. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Is this fact one more than a fact that you already know? 6 + 8 Do I already know 6 + 7 or 5 + 8 ? To use the One-More strategy, the target fact must be one more than a fact that you know. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Consider this fact: 7 + 6 What fact do I already need to know to use the One-More strategy? I need to already know 7 + 5 or 6 + 6. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Although the One-More strategy is easy for children, it is not easy for a teacher to use with a group of more than 2 or 3 children. When using this strategy, it is necessary for the teacher to be aware, precisely, of which facts each child has already mastered. Otherwise, the teacher cannot know which facts the child is ready to figure out. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

With a class full of children, it is virtually impossible for a teacher to keep up with which facts the individual children have mastered. This strategy is not recommended for classroom teachers. However, this strategy is recommended for individuals or small groups, particularly in a remedial setting. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

A strategy that many teachers have found to be effective is the Doubles strategy. Since the doubles are typically among the facts the children master early, this strategy requires children to use the doubles to figure out other hard basic addition facts. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

For each target fact, the child thinks of a double that is close to the target fact. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

8 + 7 This fact is close to a double that you already know: 8 + 7 What double that you already know is this fact close to? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

8 + 7 7 + 7 This fact is close to 7 + 7. 8 + 7 7 + 7 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

and 1 more. 8 + 7 is 7 + 7 This fact is close to 7 + 7. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Is this fact close to a double that you already know? What double is this fact close to? 6 + 7 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

and 1 more. 6 + 7 is 6 + 6 6 + 7 is close to 6 + 6. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The Doubles strategy has been a relatively successful strategy for the hard basic addition facts. However, it is not an easy strategy for facts that are more than two away from a double (for example, 5+8, 4+8, or 9+6). Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Consequently, the Doubles strategy is often used with several other strategies, and children choose the best strategy for each specific fact. This often results in confusion at that point in the process where the child must choose the best strategy. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

A strategy that many teachers feel may be most effective is the Make- Ten strategy. Since the tens are typically among the facts the children master early, this strategy requires children to use the tens to figure out other hard basic addition facts. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

In this strategy, the child will make ten and see how many are left over. Teachers often use a device called a ten-frame to help children visualize the process. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

5 + 6 We can find the answer to this fact by making ten. 5 + 6 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

5 + 6 Place the larger number in a ten-frame. 5 + 6 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

5 + 6 Use part of the other number to fill the ten-frame. 5 + 6 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Then you can look at what is left outside the ten-frame and tell what the answer is. 5 + 6 = 10 + 1 = 11 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

8 + 4 We can find the answer to this fact by making ten. 8 + 4 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

8 + 4 Place the larger number in the ten-frame. 8 + 4 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

8 + 4 Fill the ten-frame with part of the other number. 8 + 4 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

What is left outside tells you the answer. You have 10 and 2 more. 8 + 4 = 12 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

6 + 9 What number goes in the ten-frame? 6 + 9 How many more do you have to move? How many are left outside? What’s the answer? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

8 + 5 What number goes in the ten-frame? 8 + 5 How many more do you have to move? How many are left outside? What’s the answer? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The Make-Ten strategy is very successful strategy for several reasons The Make-Ten strategy is very successful strategy for several reasons. It is a universal strategy (it will always work) for all hard basic addition facts. The children only need to learn a single strategy. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The Make-Ten strategy is directly related to the base-10 numeration system. Numbers less than one hundred are always named as tens and ones. Finally, the Make-Ten strategy lays the groundwork for regrouping in addition that children will learn to use when adding multi-digit numbers. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We will now examine three strategies that have proven to be effective for figuring out answers to hard subtraction facts: Think of Related Addition One More Subtract from Ten Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The first of these strategies is the Think of Related Addition strategy. This strategy requires the child to think of an addition fact that is related to the target subtraction fact. Of course, this will only be helpful if the child knows that related addition fact. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Think of a related addition fact. 12 - 7 = ? ? + 7 = 12 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Think of a related addition fact. 15 - 8 = ? ? + 8 = 15 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Think of a related addition fact. 12 - 5 = ? ? + 5 = 12 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The Think of Related Addition strategy is popular among many teachers because it emphasizes the very important relationship between addition and subtraction. It is a universal strategy (it will always work) for all hard basic subtraction facts. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The greatest drawback of this strategy is that children who are having difficulty with 15 - 8 = 7 usually do not know 7 + 8 = 15 either. Children who are having difficulty with 12 - 5 = 7 usually do not know 7 + 5 = 12 either. Remember that this strategy will only work if the child already knows the related addition fact. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The next strategy that we will consider is the One-More strategy The next strategy that we will consider is the One-More strategy. This strategy requires the child to think of a known subtraction and then take away one more. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Do you already know 12 - 3 ? 12 12 - 3 = 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Then, to find 12 - 4, take away one more. 12 - 4 = 12 - 3 minus 1 more = 8 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

15 - 9 is easy if you already know 15 – 8. 15 - 8 = 7 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

15 - 9 is 15 - 8 minus one more. 15 - 8 = 7 minus 1 more = 6 So, 15 - 9 = 6. 15 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

13 - 6 If I know 13 - 5 = 8 Then, to find 13 - 6, take away one more. 13 - 6 Then, to find 13 - 6, take away one more. So, 13 - 6 = 7. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The One-More strategy is a very easy strategy for children The One-More strategy is a very easy strategy for children. The process is easy to understand, and it is easy to subtract 1. The strategy’s greatest drawback is that children who are trying to figure out 15 - 8 must already know 15 - 7 = 8, to find the answer to 15 - 7 the child must already know 15 - 6 = 9, etc. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

A child is able to work only on facts that are 1 away from facts the child already knows. Children master facts at different speeds and in different orders; two children will seldom have mastered the same facts. The teacher must be aware of what facts each child knows in order to know what facts they should be working on. This is very difficult for a classroom teacher to manage. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

A strategy that is a general strategy for hard basic subtraction facts is the Subtract-from-Ten strategy. This strategy requires the child already to know the subtraction facts having 10 as the subtrahend (10-7, 10-4, 10-6, etc.). Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find 15 - 8, start with 15. 15 is 10 & 5 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find 15 - 8, take 8 from the ten. You can see that 7 is left. 15 - 8 = 7 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find 12 - 7, start with 12. 12 is 10 & 2 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find 12 - 7, take 7 from the ten. You can see that 5 is left. 12 - 7 = 5 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find the answer to 13 - 8 13 - 8 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Remember that 13 is 10 and 3. 13 10 + 3 - 8 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

- 8 - 8 13 10 + 3 2 Remember that 13 is 10 and 3. Take 8 from the 10. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

- 8 - 8 13 10 + 3 2 + 3 Remember that 13 is 10 and 3. Take 8 from the 10. 13 10 + 3 - 8 - 8 Combine this with the other 3. 2 + 3 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

- 8 - 8 13 10 + 3 5 2 + 3 Remember that 13 is 10 and 3. Take 8 from the 10. 13 10 + 3 - 8 - 8 Combine this with the other 3. 5 2 + 3 13 - 8 = 5 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

- 8 12 Take 8 from the 10. Combine this with the other 2. Altogether, what is the answer? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The Subtract-from-Ten strategy is very easy to teach The Subtract-from-Ten strategy is very easy to teach. It is easy for children to visualize and understand. Some teachers object to the child doing two steps to get the answer instead of just memorizing the fact. Keep in mind, however, that this strategy is not to be used instead of memorizing the facts. This strategy is an effective vehicle for helping the child memorize the facts. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

If the child does not already know the fact, using a strategy provides a quick way for the child to figure out the correct answer. Without an effective strategy, the child’s only options are to guess or to say “I don’t know.” It is far better for the child to have a way to find the answer. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Think – Pair - Share What did you learn from this chapter that will change, improve or enrich the way you have been teaching math? Share your thoughts with a partner.

Chapter 7: Multiplying and Dividing Whole Numbers: Combining Equal-Sized Groups and Separating Quantities into Equal-Sized Groups Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Presentation 7a Teaching Multiplication and Division: Overview of the Developmental Sequence Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Definition Basic Facts Algorithm(s) Recall that when teaching computation with any of the four basic operations, there are three major components: Definition Basic Facts Algorithm(s) Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

TEACHING MULTIPLICATION Definition (Establishing Meaning) Associate Multiplication with Combining Equal-Sized Groups Add to Find the Answer Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Easy Basic Multiplication Facts Figure Them Out by Combining and Counting or by Adding Discover Relationships Multiplying by Zero Multiplying by One Multiplying by Two Rearranging Memorize Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Teaching Thinking Strategies for Hard Basic Multiplication Facts One More Partial Products Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Hard Basic Multiplication Facts Figure Them Out Using a Thinking Strategy Recognize the Same Relationships Multiplying by Zero Multiplying by One Multiplying by Two Rearranging Memorize Them Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Teaching the Multiplication Algorithm Use the Area Model De-emphasize Rote Rules Emphasize Big Ideas Multiply by Ten Use Partial Products Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

TEACHING DIVISION Measurement Division Partition Division Definition (Establishing Meaning) Associate Division with Separating into Equal Parts Measurement Division Partition Division Count or Subtract to Find the Answer Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Easy Basic Division Facts Figure Them Out by Separating into Equal Groups and Counting or Subtracting Discover Relationships Dividing by One Dividing a Number by Itself Fact Families Memorize Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Teaching Thinking Strategies for Hard Basic Division Facts Relate to Multiplication Partial Quotients Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Hard Basic Division Facts Figure Them Out Using a Thinking Strategy Recognize the Same Relationships Dividing by One Dividing a Number by Itself Fact Families Memorize Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Teaching the Division Algorithm Use a Model That Emphasizes the Basic Units De-emphasize Rote Rules Emphasize Big Ideas Using Partition Division Divide One Unit at a Time Trade Remainders for Smaller Units Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Chapter 7: Multiplying and Dividing Whole Numbers: Combining Equal-Sized Groups and Separating Quantities into Equal-Sized Groups Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Presentation 7b Relationships Among Multiplication Facts Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Remember that BASIC FACTS are the facts that are memorized and then used to figure out the facts that are not memorized. They are basic because they are the basis for all the other facts. For each operation, 100 facts are generally considered to be basic facts. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Remember also that A CURRENT ISSUE in elementary mathematics education is, When teaching basic facts, what should we be emphasizing? Strategies? Or relationships? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Relationships are the basis for number sense. Recall also that relationships are the connections that provide a meaningful context for mathematical concepts and skills. Relationships are the basis for number sense. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Strategies are step-by-step procedures, based on useful relationships, that will efficiently produce a desired result. Effective strategies usually are very automatic, requiring a minimum of decision making as they are executed. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We should emphasize both relationships and strategies. Relationships and strategies both have an important role. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We will now consider some of the relationships related to the easy basic multiplication facts. The100 basic multiplication facts are displayed in the following table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

X 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

For our purposes, we will consider the easy basic multiplication facts to be the ones with multipliers of 3 or less. Those easy basic multiplication facts are highlighted in the following table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

X 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

How are these facts related? First Relationship Children should notice that certain facts have something in common. 1 X 0 = 0 5 X 0 = 0 0 X 9 = 0 0 X 7 = 0 3 X 0 = 0 2 X 0 = 0 0 X 8 = 0 How are these facts related? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We will highlight these 19 facts in red on the multiplication table. When zero is multiplied by any other number, the product is always zero. There are 19 multiplication facts like this. But, if children understand how these facts are related, there is only one thing to learn. We will highlight these 19 facts in red on the multiplication table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

X 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

How are these facts related? Second Relationship Children should notice that another group of facts have something different in common. 8 X 1 = 8 4 X 1 = 4 1 X 2 = 2 6 X 1 = 6 1 X 3 = 3 1 X 9 = 9 5 X 1 = 5 How are these facts related? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We will highlight these 17 facts in red on the multiplication table. When one is multiplied by any other number, the product is always that other number. There are 17 multiplication facts like this. But, if children understand how these facts are related they have only one thing to learn. We will highlight these 17 facts in red on the multiplication table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

X 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

How are these facts related? Third Relationship Children should realize that 2 times a number is the same as that number added to itself. 2 X 8 = 8 + 8 4 X 2 = 4 + 4 6 X 2 = 6 + 6 2 X 3 = 3 + 3 2 X 9 = 9 + 9 5 X 2 = 5 + 5 How are these facts related? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

2 times a number is the same as that number plus itself. There are 15 multiplication facts like this. But, since the doubles have already been memorized as addition facts, there is nothing new to learn. We will highlight these 15 facts in yellow on the multiplication table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

X 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

How are these pairs of facts related? Fourth Relationship Children should notice that certain pairs of facts have the same answer. 4 X 3 = 12 3 X 4 = 12 5 X 3 = 15 3 X 5 = 15 6 X 3 = 18 3 X 6 = 18 7 X 3 = 21 3 X 7 = 21 8 X 3 = 24 3 X 8 = 24 9 X 3 = 27 3 X 9 = 27 How are these pairs of facts related? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The product is not changed when you rearrange the numbers. There are 6 rearranged pairs: 12 facts, but if children understand how the pairs of facts are related, there are only 6 things to learn. We will highlight these 12 facts in blue on the multiplication table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

X 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The fact, 3 X 3, will also have to be memorized. This is one thing to learn. We will highlight this fact in green on the multiplication table. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

X 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

How will the emphasis on relationships help children to master the easy multiplication facts? Consider the following summary of the potential effect of the emphasis on relationships. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

SUMMARY OF RELATIONSHIPS FOR EASY MULTIPLICATION FACTS Relationship Number of Facts Things to Learn X 0 19 1 X 1 17 1 X 2 15 0 Rearranged Pairs 12 6 3 X 3 1 1 TOTALS 64 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

SO, WE CAN SEE THAT ONE BENEFIT OF AN EMPHASIS ON RELATIONSHIPS IS A SHARP REDUCTION IN THE AMOUNT TO BE MEMORIZED. WHEN MEMORIZED MATERIAL IS RELATED AND INTERCONNECTED TO OTHER MEMORIZED MATERIAL, THERE ARE MANY MORE PATHS TO RECALL AND RECALL IS IMPROVED. THEREFORE, A SECOND BENEFIT OF THE EMPHASIS ON RELATIONSHIPS IS IMPROVED RETENTION OF THE FACTS ONCE THEY HAVE BEEN MEMORIZED. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Chapter 7: Multiplying and Dividing Whole Numbers: Combining Equal-Sized Groups and Separating Quantities into Equal-Sized Groups Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Presentation 7c Strategies for Finding Answers to Hard Basic Multiplication Facts Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Remember that BASIC FACTS are the facts that we memorize and then use to figure out facts that are not memorized. They are basic because they are the basis for all the other facts. For each operation, 100 facts are generally considered to be basic facts. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Recall that A CURRENT ISSUE in elementary mathematics education is, When teaching basic facts, what should we be emphasizing? Strategies? Or relationships? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Relationships are the basis for number sense. Relationships are the connections that provide a meaningful context for mathematical concepts and skills. Relationships are the basis for number sense. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Strategies are step-by-step procedures, based on useful relationships, that will efficiently produce a desired result. Effective strategies usually are very automatic, requiring a minimum of decision making as they are executed. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

So, should we be emphasizing strategies or relationships? We should emphasize both. Relationships and strategies both have an important role. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We have already been introduced to several effective strategies for finding answers to hard addition and hard subtraction facts. Now, we examine two effective strategies for finding answers to hard multiplication facts: One More Partial Products Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The first strategy for figuring out hard basic multiplication facts that we will consider is the One-More strategy. This strategy requires the child to build on a known multiplication fact. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find 6 X 8, visualize 6 rows of eight. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

That’s 5 eights and one more eight. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

40 8 If I know 5 X 8 = 40, then I just add one more eight. 6 X 8 = 48 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find 9 X 5, visualize 9 rows of five. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

That’s 8 fives and one more five. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

40 5 If I know 8 X 5 = 40, then I just add one more five. 9 X 5 = 45 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

7 X 8 = ? Do I know 6 X 8 ? 6 X 8 = 48 7 X 8 is 6 X 8 and one more 8. 7 X 8 = 48 + 8 or 56 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The second strategy for figuring out hard basic multiplication facts that we will consider is the Partial-Products strategy. This strategy requires the child to break a hard fact that the child doesn’t know into two easier facts that are known. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find 8 X 7, visualize 8 rows of seven. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find 8 X 7, visualize 8 rows of seven. We will break 8 X 7 into two easy parts. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

4 X 7 But 4 X 7 = 28 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

4 X 7 = 28 So, 8 X 7 = 28 + 28 = 56 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

6 X 9 Can I change this to two easier facts that I know? Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

3 X 9 But 3 X 9 = 27 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

So 6 X 9 = 27 + 27 = 54 6 X 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

8 X 8 Can I change this to two easier facts that I know? It is 4 X 8 and 4 more eights. 8 X 8 That is (4 X 8) + (4 X 8) Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The Partial-Products strategy is a very good strategy to teach. It is easy for children to visualize and understand. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

but it also provides a strong Not only does it help the child to figure out hard basic multiplication facts, but it also provides a strong background for the use of partial products in multi-digit multiplication. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Chapter 7: Multiplying and Dividing Whole Numbers: Combining Equal-Sized Groups and Separating Quantities into Equal-Sized Groups Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Presentation 7d Teaching the Big Ideas for the Multiplication Algorithm Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Recall that when teaching the multiplication algorithm, we should emphasize two big ideas: Multiplication by ten Use partial products Let’s examine these two big ideas in more detail. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We want our students to know how easy it is to multiply by 10. However, we do not want to just tell them the rule ( “add” a zero). Rather, we want to develop this idea (this rule) using a clear visual model. We want them to know this rule is correct because they can see that it is correct, not because “the teacher told them.” Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To model 10 X 12, ten times. show 12 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Group the ones to form tens, Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Group the ones to form tens, Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Group the ones to form tens, Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Group the ones to form tens, Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Group the ones to form tens, And group tens to form hundreds. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Group the ones to form tens, And group tens to form hundreds. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Group the ones to form tens, And group tens to form hundreds. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We have 1 hundred and 2 tens. 10 X 12 = 120 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Similarly, 10 X 30 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Similarly, 10 X 30 = 300 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

10 X 24 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

= 240 10 X 24 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

10 X 7 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

10 X 7 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

10 X 7 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

= 70 10 X 7 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

From examples like these, it is easy for children to see that multiplication by 10 has the effect of “adding” a zero to the other factor. 10 X 37 = 37 10 X 52 = 52 124 X 10 8 X 10 124 8 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

When multiplying by 100, add two zeros. Once children understand the effect of multiplication by 10, that concept can be extended as follows. 100 = 10 X 10 So, 100 X 26 is the same as 10 X 10 X 26 When multiplying by 100, add two zeros. 100 X 26 = 26 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Similarly, since 1000 = 10 X 10 X 10, 204 X 1000 = 204 X 10 X 10 X 10. We would add three zeros. 204 X 1000 = 204 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Another extension of the multiply-by-ten concept is also needed. Since 40 = 4 X 10 and 70 = 7 X 10, it follows that 40 X 70 = 4 X 10 X 7 X 10. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Because we can rearrange the numbers being multiplied without changing the answer, 40 X 70 = 4 X 7 X 10 X 10. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

and two multiplications by 10. 40 X 70 = 4 X 7 X 10 X 10. We have a basic fact and two multiplications by 10. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

40 X 70 = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

40 X 70 = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

40 X 70 = 28 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

40 X 70 = 280 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

40 X 70 = 2800 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

90 X 30 = Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

90 X 30 = 27 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

90 X 30 = 270 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

90 X 30 = 2700 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

40 X 800 = 32 000 A basic fact and three multiplications by 10 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

600 X 500 = 30 0000 A basic fact and four multiplications by 10 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Those partial products are added together to get the total product. The second big idea for understanding the multiplication algorithm is partial products. The concept of partial products allows us to break a hard multiplication into easier parts (called partial products). Those partial products are added together to get the total product. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We can use the area model to visualize those partial products. One of the thinking strategies for finding answers to hard basic multiplication facts was the partial-products strategy. To find the product 6 X 7, we can think of three 7’s and three more 7’s. We can use the area model to visualize those partial products. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We can use the area model to visualize those partial products. One of the thinking strategies for finding answers to hard basic multiplication facts was the partial products strategy. To find the product 6 X 7, we can think of three 7’s and three more 7’s. We can use the area model to visualize those partial products. 7 3 X 7 3 3 X 7 3 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

This same partial-products strategy can be applied to multiplication examples other than basic facts. To find the product 14 X 8, we can break the product into easy partial products. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

This same partial-products strategy can be applied to multiplication examples other than basic facts. To find the product 14 X 8, we can break the product into easy partial products. 10 4 8 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

This same partial-products strategy can be applied to multiplication examples other than basic facts. To find the product of 14 X 8, we can break the product into easy partial products. 8 X 10 10 4 8 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

This same partial-products strategy can be applied to multiplication examples other than basic facts. To find the product 14 X 8, we can break the product into easy partial products. 8 X 10 10 4 8 X 4 8 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We end up with a basic fact and a multiplication by ten. 8 X 10 10 4 8 X 4 8 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find the product 28 X 6, we can break the product into easy partial products. 20 8 6 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

To find the product 28 X 6, we can break the product into easy partial products. 20 8 6 6 X 20 6 X 8 (A basic fact and a multiplication by 10) (A basic fact) Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We can break the product 43 X 15 into easy partial products. 40 3 10 5 10 X 40 10 X 3 5 X 40 5 X 3 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The partial products require only basic facts and multiplication by ten. 40 3 10 5 10 X 40 10 X 3 5 X 40 5 X 3 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

We can break the product 27 X 34 into easy partial products. 30 4 20 7 20 X 30 20 X 4 7 X 30 7 X 4 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

The partial products require only basic facts and multiplication by ten. 30 4 20 7 20 X 30 20 X 4 7 X 30 7 X 4 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Consider the multiplication example, 49 X 67. When you use the standard algorithm, every multiplication that you do matches one of the partial products. 60 7 6 7 X 4 9 40 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

First, you multiply 9 times 7. 60 7 6 7 X 4 9 40 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

First, you multiply 9 times 7. 60 7 6 7 X 4 9 40 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Then, you multiply 9 times 6 tens. 60 7 6 7 X 4 9 40 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Then, you multiply 9 times 6 tens. 60 7 6 7 X 4 9 40 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Then, you multiply 4 tens times 7. 60 7 6 7 X 4 9 40 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Then, you multiply 4 tens times 7. 60 7 6 7 X 4 9 40 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

And finally, you multiply 4 tens times 6 tens. 60 7 6 7 X 4 9 40 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

And finally, you multiply 4 tens times 6 tens. 60 7 6 7 X 4 9 40 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

When we add the partial products together, we have the total product. 60 7 40 9 6 7 X 4 9 Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

If the child understands these two big ideas, multiplication by ten and partial products, the standard multiplication algorithm is easier to teach. And, of course, these big ideas are valuable basic concepts for the development of mental-arithmetic skills. Tucker/Singleton/Weaver Teaching Mathematics to ALL Children, Second Edition Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved.

Technology Field Trip to the LAB!!! Take this time to explore several of the internet sites suggested/listed in the text book at the end of most chapters. You will be responsible for reviewing/reflecting on three of those websites to share with the class. educatorlearningcenter.com

Activity - Backward Mapping Refer to resource page, “Backward Mapping CAHSEE Standards.” Write out your assigned CAHSEE standard. Backward map the CAHSEE standard by finding standards from previous grade levels that address critical prerequisite concepts and skills. Instructor’s Notes: Time continued (Slide 14 of 14) Intent: The participants will understand how elementary mathematics standards lay a critical foundation for CAHSEE. Assign CAHSEE standards from the following list: 6th Grade SDAP 3.3, 7th Grade NS 1.2, 7th Grade NS 1.5, 7th Grade MG 2.1, Algebra 4.0. Refer to directions on the slide. Participants may paraphrase or record only specific components of standards if space on the handout is a concern. Explain that participants will think about what prerequisite concepts and skills are required for mastery of the standard, then look through Chapter 2 of the CA Mathematics Framework to list the standards that address those critical prerequisites. Sample answers are provided in Instructor’s Appendix G. Ask the participants to describe any “gaps” you found in your backward mapping, i.e. where a topic seems to skip a grade level. Probability, for example, is addressed in grades 4 and 6 but not in grade 5. Facilitate a discussion around the question at the bottom of the page. Talking points: Students having difficulty at lower grades may be at risk of failing CAHSEE. Some standards that are not key standards contain critical prerequisites for CAHSEE standards. Teachers should take this into account as they make decisions about depth versus breadth. When teachers are aware of “gaps” in the backward mapping, they may reinforce the previous grade level standard so that students are prepared the following year for the next step in complexity. Suggest that this is an activity that can be shared with teachers. 253

Activity GAME TIME!!! Each week, students will take turns leading the class in a math game.

Closing Final thoughts, comments? Making connections – Anything to add to your reflection?