Project Impact CURR 231 Curriculum and Instruction in Math Session 5 Chapters 6 & 7

Project Impact CURR 231Curriculum and Instruction in Math

Session 5

Chapters 6 & 7

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 Talk

Number of the Day

24Tell me everything you know aboutthis number. Find 4 different ways to represent thisnumber.

Framework – Ch. 9Responsibilities 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/WeaverTeaching 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




TEACHING COMPUTATION

Definition

Basic Facts

Algorithm(s)


Copyright ©2006 by Pearson Education, Inc.

Upper Saddle River, New Jersey 07458All 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




BASIC FACTS

(The Facts That Are Memorized and Then Used to Figure Out the Facts That Are Not Memorized)




Basic Facts Are Dealt with in Two Major Groups:

Easy Facts Hard Facts




Easy Basic Facts Figure Them Out

(Using the Definition)

Discover Relationships

Memorize Them




Develop Thinking Strategies for Figuring Out the Hard Basic Facts

Those strategies shouldBe Mental Strategies, Not

Mechanical nor Pencil-Paper Strategies

Use Already Memorized FactsTucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



Hard Basic Facts Figure Them Out

(Using Thinking Strategies) Discover That the Same

Relationships Hold When Larger Numbers Are Being Used

Memorize ThemTucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



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

Memorize them




ALGORITHMSTeach MeaningfullyCarefully Model the Operation

Step-by-Step (Let the children see what it looks like)

De-emphasize Rote RulesEmphasize Big Ideas




TEACHING ADDITIONDefinition (Establishing Meaning) Associate Addition with Combining Count to Find the Answer




Developing the Basic FactsEasy 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




Thinking Strategies for Hard Basic Addition Facts One More Doubles Make Ten




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




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




TEACHING SUBTRACTIONDefinition (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




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




Thinking Strategies for Hard Basic Subtraction Facts

Relate to Addition

Subtract from Ten




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




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!








Presentation 6b

Relationships Among Basic Facts for Addition and Subtraction




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.




A CURRENT ISSUE in elementary mathematics education is, When teaching basic facts, what should we be emphasizing? Strategies? Or relationships?




Relationships are the connections that provide a meaningful context for mathematical concepts and skills.

Relationships are the basis for number sense.




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.




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.




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.




+ 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 18Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



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.




+ 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




Children should noticethat certain facts havesomething in common.

5 + 0 = 5

0 + 8 = 80 + 1 = 1

3 + 0 = 3

1 + 0 = 16 + 0 = 6

0 + 4 = 4

2 + 0 = 2

7 + 0 = 7

First RelationshipFirst Relationship

How are these facts related?How are these facts related?Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



When zero is added to any otherWhen zero is added to any other

number, that other number is the sum.number, that other number is the sum.

There are 19 addition facts like this.

We will highlight these 19 facts in red on the addition table.

But, if children understand how these facts are related, it is really just one thing to learn.




+ 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




Children should noticefacts like these havesomething in common.

5 + 1 = 6

1 + 8 = 91 + 1 = 2

3 + 1 = 4

9 + 1 = 106 + 1 = 7

1 + 4 = 52 + 1 = 3

7 + 1 = 8

Second Relationship

How are these facts related?Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



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.




+ 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




Children should noticethat certain pairs of factshave the same answer.

5 + 4 = 91 + 8 = 9

3 + 2 = 5

2 + 4 = 67 + 3 = 10

4 + 5 = 9

2 + 3 = 5

8 + 1 = 94 + 2 = 6

3 + 7 = 10

Third Relationship

How are these pairs of facts related?Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



The sum is not changed when you The sum is not changed when you rearrange the numbers.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.




+ 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




There are four easy facts remaining.

2 + 2 = 4 3 + 3 = 6

5 + 5 = 10

4 + 4 = 8

They are the doubles.

The doubles are usually easy to learn.

We will highlight these 4 facts in brown on the addition table.




+ 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




How will the emphasis on relationships How will the emphasis on relationships help children to master the easy addition help children to master the easy addition facts?facts?

Consider the following summary of the Consider the following summary of the potential effect of the emphasis on potential effect of the emphasis on relationships.relationships.




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




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.








Presentation 6c

Strategies for Hard Addition and Subtraction Facts




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.




Recall also that A CURRENT ISSUE in elementary mathematics education is:

When teaching basic facts, what should we be emphasizing?

Strategies? Or relationships?









Strategies, on the other hand, are step-by-step procedures based on useful relationships that will efficiently produce a desired result.





Recall that the issue is, Should we be emphasizing strategies or relationships?

Obviously, we should emphasize both.





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.




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).




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.




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/WeaverTeaching Mathematics to ALL Children, Second Edition



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.




5 + 5

This fact is one more than a fact that you already know:

5 + 6 is

and one more




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.




Is this fact one more than a fact that you already know?

6 + 8Do 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.




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.




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.




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.




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.




For each target fact, the child thinks of a double that is close to the target fact.




This fact is close to a double that you already know:

8 + 7

What double that you already know is this fact close to?




This fact is close to 7 + 7.

8 + 7 7 + 7




This fact is close to 7 + 7.

8 + 7 is 7 + 7

and 1 more.Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



Is this fact close to a double that you already know?

6 + 7What double is this fact close to?




6 + 7 is close to 6 + 6.

6 + 7 is 6 + 6

and 1 more.Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



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).

The Doubles strategy has been a relatively successful strategy for the hard basic addition facts.

The Doubles strategy has been a relatively successful strategy for the hard basic addition facts.




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.




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.




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.




We can find the answer to this fact by making ten.

5 + 6




Place the larger number in a ten-frame.

5 + 6




Use part of the other number to fill the ten-frame.

5 + 6




Then you can look at what is left outside the ten-frame and tell what the answer is.

5 + 6 = 10 + 1

= 11




We can find the answer to this fact by making ten.

8 + 4




Place the larger number in the ten-frame.

8 + 4




Fill the ten-frame with part of the other number.

8 + 4




What is left outside tells you the answer. You have 10 and 2 more.

8 + 4 = 12




6 + 9

What number goes in the ten-frame?

How many more do you have to move?

How many are left outside?

What’s the answer?




8 + 5

What number goes in the ten-frame?

How many more do you have to move?

How many are left outside?

What’s the answer?




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.




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.




We will now examine three strategies that have proven to be effective for figuring out answers to hard subtraction facts:

Think of Related AdditionOne MoreSubtract from Ten




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.




Think of a related addition fact.

12 - 7 = ?? + 7 = 12





15 - 8 = ?? + 8 = 15





12 - 5 = ?? + 5 = 12




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.




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.




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.




Do you already know 12 - 3 ?

12

12 - 3 = 9




Then, to find 12 - 4, take away one more.

1212 - 4 = 12 - 3 minus 1 more

= 8




15 - 9 is easy if you already know 15 – 8.

15

15 - 8 = 7




15 - 9 is 15 - 8 minus one more.

15

15 - 8 = 7 minus 1 more = 6

So, 15 - 9 = 6.




13 - 6

If I know 13 - 5 = 8

Then, to find 13 - 6, take away one more.

So, 13 - 6 = 7.




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.




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.




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.).




To find 15 - 8, start with 15.

15 is 10 & 5




To find 15 - 8, take 8 from the ten. You can see that 7 is left.

15 - 8 = 7




To find 12 - 7, start with 12.

12 is 10 & 2




To find 12 - 7, take 7 from the ten. You can see that 5 is left.

12 - 7 = 5




13- 8

To find the answer to 13 - 8




13

Remember that 13 is 10 and 3.

- 810 + 3




13


Take 8 from the 10.

- 810 + 3- 8 2




13


Take 8 from the 10.

Combine this with the other 3.

- 810 + 3- 8 2 + 3




13


Take 8 from the 10.

Combine this with the other 3.

13 - 8 = 5

- 810 + 3- 8 2 + 35




Take 8 from the 10.

Combine this with the other 2.12

- 8Altogether, what is the answer?




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 EditionCopyright ©2006 by Pearson Education, Inc.


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.




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




Presentation 7a Teaching Multiplication and Division:Overview of the Developmental Sequence




Recall that when teaching computation with any of the four basic operations, there are three major components:

Definition

Basic Facts

Algorithm(s)




TEACHING MULTIPLICATIONDefinition (Establishing Meaning)Associate Multiplication with

Combining Equal-Sized GroupsAdd to Find the Answer




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




Teaching Thinking Strategies for Hard Basic Multiplication Facts

One More

Partial Products




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




Teaching the Multiplication Algorithm Use the Area Model De-emphasize Rote Rules Emphasize Big Ideas

Multiply by Ten

Use Partial Products




TEACHING DIVISION

Definition (Establishing Meaning) Associate Division with Separating into

Equal PartsMeasurement DivisionPartition Division

Count or Subtract to Find the AnswerTucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



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




Teaching Thinking Strategies for Hard Basic Division Facts

Relate to Multiplication

Partial Quotients




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




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








Presentation 7b Relationships Among Multiplication Facts




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.




Remember also that A CURRENT ISSUE in elementary mathematics education is, When teaching basic facts, what should we be emphasizing? Strategies? Or relationships?




Recall also that relationships are the connections that provide a meaningful context for mathematical concepts and skills.










We should emphasize both relationships and strategies.





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.




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




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.




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




Children should notice that certain facts have something in common.

5 X 0 = 0 1 X 0 = 0

3 X 0 = 00 X 7 = 0

2 X 0 = 0 0 X 8 = 0

0 X 9 = 0

First Relationship




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.




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




Children should notice that another group of facts have something different in common.

4 X 1 = 48 X 1 = 8

6 X 1 = 61 X 3 = 3

5 X 1 = 5 1 X 9 = 9

1 X 2 = 2

Second Relationship




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.




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




Children should realize that 2 times a number is the same as that number added to itself.

4 X 2 = 4 + 42 X 8 = 8 + 8

6 X 2 = 6 + 62 X 3 = 3 + 3

5 X 2 = 5 + 5

2 X 9 = 9 + 9

Third Relationship




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.




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




Children should notice that certain pairs of facts have the same answer.

5 X 3 = 153 X 5 = 15

4 X 3 = 123 X 4 = 12

6 X 3 = 183 X 6 = 18

7 X 3 = 213 X 7 = 21

8 X 3 = 243 X 8 = 24

9 X 3 = 273 X 9 = 27

Fourth Relationship

How are these pairs of facts related?




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.




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




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.







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

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.




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 9Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



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.








Presentation 7c Strategies for Finding Answers to Hard Basic Multiplication Facts




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.




Recall that A CURRENT ISSUE in elementary mathematics education is, When teaching basic facts, what should we be emphasizing? Strategies? Or relationships?














So, should we be emphasizing strategies or relationships?

We should emphasize both.





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 ProductsTucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



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.




To find 6 X 8, visualize 6 rows of eight.




That’s 5 eights and one more eight.




If I know 5 X 8 = 40, then I just add one more eight.

6 X 8 = 48

40

8




To find 9 X 5, visualize 9 rows of five.




That’s 8 fives and one more five.




If I know 8 X 5 = 40, then I just add one more five.

9 X 5 = 45

40

5




7 X 8 = ?

6 X 8 = 48

7 X 8 is 6 X 8 and one more 8.

7 X 8 = 48 + 8 or 56

Do I know 6 X 8 ?




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.




To find 8 X 7, visualize 8 rows of seven.




To find 8 X 7, visualize 8 rows of seven.

We will break 8 X 7 into two easy parts.




4 X 7

4 X 7

But 4 X 7 = 28Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



4 X 7 = 28

4 X 7 = 28

So, 8 X 7 = 28 + 28 = 56Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



6 X 9

Can I change this to two easier facts that I know?




3 X 9

3 X 9

But 3 X 9 = 27Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



6 X 9

So 6 X 9 = 27 + 27 = 54




8 X 8

Can I change this to two easier facts that I know?

It is 4 X 8 and 4 more eights.

That is (4 X 8) + (4 X 8)




The Partial-Products strategy is a very good strategy to teach.

It is easy for children to visualize and understand.




Not only does it help the child to figure out hard basic multiplication facts, but it also provides a strongbackground for the use of partial products in multi-digit multiplication.








Presentation 7d Teaching the Big Ideas for the Multiplication Algorithm




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.




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.”




To model 10 X 12,

show 12 ten times.




Group the ones to form tens,

















And group tens to form hundreds.














10 X 12 = 120

We have 1 hundred and 2 tens.




Similarly, 10 X 30




= 300 Similarly, 10 X 30




10 X 24




10 X 24 = 240




10 X 7




10 X 7




10 X 7




10 X 7 = 70




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 = 370

10 X 52 = 520 124X 10

124 0X 108

8 0Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



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 = 260 0




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 = 2040 0 0




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.




Because we can rearrange the numbers being multiplied without changing the answer,

40 X 70 = 4 X 7 X 10 X 10.




40 X 70 = 4 X 7 X 10 X 10.

We have a basic fact

and two multiplications by 10.




40 X 70 =




40 X 70 =




40 X 70 = 28




40 X 70 = 280




40 X 70 = 2800




90 X 30 =




90 X 30 = 27




90 X 30 = 270




90 X 30 = 2700




40 X 800 = 32000

A basic fact and three multiplications by 10




600 X 500 = 30 0000

A basic fact and four multiplications by 10




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.




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.




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.

3

3

7 3 X 7

3 X 7Tucker/Singleton/WeaverTeaching Mathematics to ALL Children, Second Edition



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

10 4





To find the product of 14 X 8, we can break the product into easy partial products.

8

10 48 X 10






8

10 48 X 10 8 X 4




We end up with a basic fact and a multiplication by ten.

8

10 48 X 10 8 X 4





20 8

6





20 8

6

6 X 20(A basic fact and a multiplication by 10)

6 X 8(A basic fact)




We can break the product 43 X 15 into easy partial products.

40 3

5

10

10 X 40

5 X 40 5 X 3

10 X 3




The partial products require only basic facts and multiplication by ten.

40 3

5

10 X 40

10

5 X 40 5 X 3

10 X 3




We can break the product 27 X 34 into easy partial products.

30 4

7

20

20 X 307 X 30 7 X 4




The partial products require only basic facts and multiplication by ten.

30 4

7

20

20 X 307 X 30 7 X 4




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

40

9

6 7X 4 9




First, you multiply 9 times 7.

60 7

40

9

6 7

X 4 9




First, you multiply 9 times 7.

60 7

40

9

6 7

X 4 9




Then, you multiply 9 times 6 tens.

60 7

40

9

6 7

X 4 9




Then, you multiply 9 times 6 tens.

60 7

40

9

6 7

X 4 9




Then, you multiply 4 tens times 7.

60 7

40

9

6 7

X 4 9




Then, you multiply 4 tens times 7.

60 7

40

9

6 7

X 4 9




And finally, you multiply 4 tens times 6 tens.

60 7

40

9

6 7

X 4 9




And finally, you multiply 4 tens times 6 tens.

60 7

40

9

6 7

X 4 9




When we add the partial products together, we have the total product.

60 7

40

9

6 7

X 4 9




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.




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.

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?

Documents

Project Impact CURR 231 Curriculum and Instruction in Math Session 5 Chapters 6 & 7