Prenumeration, Number Operations,

Addition and Subtraction

• Subtizing: Kaufman et al. (1949) coined the term subtizing for the rapid, accurate, and confident judgments of number performed for small numbers of items

How many of each color did you see?

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• What you have experienced is subtizing – a nearly automatic recognition of numbers of items of three or less. This innate ability can be expanded in the classroom through activities until it can serve as a tool for student estimation and sense making which will last long past the prenumeration stage of development.

Sequencing

• Sequencing at the prenumeration level need not be overly elaborate or intricate.

• Events in their lives follow a natural time order with some naturally occurring before and after events.

• Ordering pictures of events along a timeline, using forward and backward sequence.

Discuss a forward sequence and a backward sequence for the flow of action.

Have students make up their own sequence. What happened “firstfirst”? What happened “nextnext”? What happened “lastlast”?

• Other types of sequences – both naturally and artificial should be experienced and be coupled with group activities developing listening skills together with motor activity.

Another type of sequence involves size relationships.

Matching

• Students should have the opportunity to match existing sets of objects and to create patterns of their own using a wide variety of manipulates. In doing so there is another type of matching called substitution match where students use a different manipulative to construct their match of what was used in the original presentation.

The student’s task would be to match this original set, but use a different manipulative, like a soccer ball.

It is not the manipulative it self that is being matched but the number of items. This emphasis upon the number, not the form, is essential for later number development.

Matching is not patterning. Patterning is a more sophisticated task. Patterning would be taking an existing group of units,

and then use it as a building block to build longer units by repeating the base unit.

One-to-One Correspondence

• Although children may have also experiences one-to-many and many-to-one correspondences, such as one body-two arms or four legs-one puppy, it is the one-to-one correspondences such as one hand-one glove that we are looking for here.

Name to Number and Number to Name

to “ONE”

“ONE” to

As this become routine for the students try to extend this activity to assigning the numeral, i.e., 1, 2, 3,…, to both the object and the oral name for the object.

Number

• With the foundations established the transition from prenumeration to full blown numeration and counting is so natural as to almost be missed by the teacher. In this transition, however, there are a few possible places where student difficulties may still take place.

• Some children may not be using the correct name for the numbers.

• Some students who, despite an ability to recognize and use one-to-one correspondences, in counting fail to make use of a single numeral per object match.

With 1-1 Correspondence

1 2 3 4 5 6

Without 1-1 Correspondence

1 2 3 4 5 6

Expanding Subtizing

Basic Number System, Base 10

10x10x

10x10x

10

105

10x10x

10x10

104

10x10x

10

103

10x10

102

10x1

101 100,

Basic Number System, Base 8

8x8x8x

8x8

85

8x8x8x8

84

8x8x8

83

8x8

82

8x1

8180

,

Basic Number Operation

• 1st Step: Representation

7

“seven”

• 2nd Step: Comparison

“Which is bigger?”

or

7 > 33rd Step: “By how much?”

7

- 3

4Subtraction is introducedSubtraction is introduced

• Step 4, How many in all?

7

+ 3

10

Addition is introducedAddition is introduced..

Basic OperationsOn Objects

Fact Families

• Four basic structures for addition and subtraction story problems. Each structure has three numbers. Any one of the three numbers can be the unknown in a story problem. Fact families are generated from this relationship.

Change

ResultInitial

Change

ResultInitial

Join Separate

Part Part

Whole

Part-Part-Whole

Larger Small

Set Set

Difference

Compare

• Join: Result Unknown Sandra had 8 pennies. George gave her

4 more. How many pennies does Sandra have altogether.

• Join: Change Unknown Sandra had 8 pennies. George gave

her some more. Now Sandra has 12 pennies. How many did George give her?

• Join: Initial Unknown Sandra had some pennies. George

gave her 4 more. Now Sandra has 12 pennies. How many pennies did Sandra have to begin with?

Change

ResultInitial

8 + 4 = 12

8 + 4 = 12

8 + 4 =12

Change

ResultInitial

Separate

Separate: Result UnknownSeparate: Result Unknown Sandra had 12 pennies. She gave 4 Sandra had 12 pennies. She gave 4 pennies to George. How many pennies pennies to George. How many pennies does Sandra have now?does Sandra have now?

Separate: Change UnknownSeparate: Change Unknown Sandra had 12 pennies. She gave Sandra had 12 pennies. She gave some to George. Now she has 8 some to George. Now she has 8 pennies. How many did she give to pennies. How many did she give to George?George?

Separate: Initial UnknownSeparate: Initial Unknown Sandra had some pennies. She gave Sandra had some pennies. She gave 4 to George. Now Sandra has 8 pennies 4 to George. Now Sandra has 8 pennies left. How many pennies did Sandra have left. How many pennies did Sandra have to begin with?to begin with?

12 – 4 = 8

12 – 4 = 8

12 – 4 = 8

Part-Part-Whole: Whole UnknownPart-Part-Whole: Whole Unknown George has 4 pennies and 8 nickels. George has 4 pennies and 8 nickels. How many coins does he have?How many coins does he have?

George has 4 pennies and Sandra has 8 George has 4 pennies and Sandra has 8 pennies. They put their pennies into a pennies. They put their pennies into a piggy bank. How many pennies did they piggy bank. How many pennies did they put into the bank?put into the bank?

Part-Part-Whole: Part UnknownPart-Part-Whole: Part Unknown George has 12 coins. Eight of his coins George has 12 coins. Eight of his coins are pennies, and the rest are nickels. How are pennies, and the rest are nickels. How many nickels does George have?many nickels does George have? George and Sandra put 12 pennies into George and Sandra put 12 pennies into the piggy bank. George put in 4 pennies. the piggy bank. George put in 4 pennies. How many pennies did Sandra put in?How many pennies did Sandra put in?

Part Part

Whole

4 + 8 = 12

4 + 8 = 12

Compare: Difference UnknownCompare: Difference Unknown George has 12 pennies and Sandra has 8 George has 12 pennies and Sandra has 8 pennies. How many more pennies does George pennies. How many more pennies does George have than Sandra?have than Sandra?

George has 12 pennies. Sandra has 8 pennies. George has 12 pennies. Sandra has 8 pennies. How many fewer pennies does Sandra have than How many fewer pennies does Sandra have than George?George?

Compare: Larger UnknownCompare: Larger Unknown George has 4 more pennies than Sandra. Sandra George has 4 more pennies than Sandra. Sandra has 8 pennies. How many pennies does George has 8 pennies. How many pennies does George have?have?

Sandra has 4 fewer pennies than George. Sandra Sandra has 4 fewer pennies than George. Sandra has 8 pennies. How many pennies does George has 8 pennies. How many pennies does George have?have?

Compare: Smaller UnknownCompare: Smaller Unknown George has 4 more pennies than Sandra. George George has 4 more pennies than Sandra. George has 12 pennies. How many pennies does Sandra has 12 pennies. How many pennies does Sandra have?have?

Sandra has 4 fewer pennies than George. George Sandra has 4 fewer pennies than George. George has 12 pennies. How many pennies does Sandra has 12 pennies. How many pennies does Sandra havehave

Difference

Larger Small

Set Set

12 – 8 = 4

12 – 8 = 4

12 – 8 = 4

Used Model Based Problems• Many students will use counters or

number lines (models) to solve story problems. The model is a thinking tool to help them both understand what is happening in the problem and a means of keeping tract of the number and solving the problem. Problems can also be posed using models when there is no context involved.

Part-Part-Whole Model for 5 + 3 = 8 and 8 – 3 = 5

Addition Models

Models for 9 – 4 as a Missing-Part Problem

Models for the difference between 8 and 5.

Word Problems

• Join

• Separate

• Part-Part-Whole

• Compare

Helping Children Master the Basic Facts

Mastery of basic fact means that a child can give a quick response (in about 3 seconds) without resorting to nonefficient means, such as counting. Work towards mastery of addition and subtraction facts typically begins in the first grade. Most books include all addition and subtraction facts for mastery in the second grade, although much additional drill is usually required in grade 3 and even after. Multiplication and division facts are generally a target for mastery in the 5th grade. Unfortunately, many children in grade 8 and above do not have a complete command of the basic facts.

A Three Step Approach to Fact Mastery1. Help children develop a strong

understanding of the operations and of number relationships.

2. Develop efficient strategies for fact retrieval through practice.

3. Then provide drill in the use and selection of strategies once they have been developed

One-More-Than and Two-More-Than Facts• Each of the 36 facts highlighted in the chart has at

least one addend of 1 or 2. These facts are a direct application of the one-more-than and two-more-than relationships.

Join or part-part-whole problems in which one of the addends is a 1 or 2 are easy to make up. For example: When Tommy was at the circus, he saw 8 clowns come out in a little car. Then 2 more clowns came out on bicycles. How many clowns did Tommy see in all?

Can you think of a word problem using this method?

Facts with Zero• Nineteen facts have zero as one of the addends.

Though such problems are generally easy, some children over generalize the idea that answers to addition are bigger. Word problems, involving zero will be especially helpful. In the discussion, use drawings that show two parts with one part empty.

Doubles• There are only ten doubles facts

from 0 + 0 to 9 + 9, as shown here. These ten facts are relatively easy to learn and become a powerful way to learn the near-doubles (addends one apart). Some children use them as anchors for other facts as well.

Near Double• Near Doubles are also called the “doubles-plus-one” facts and

include all combinations where one addend is one more than the other.

Make-Ten Facts• These facts all have at least one

addend of 8 or 9. One strategy for these facts is to build onto the 8 or 9 up to 10 and then add on the rest. For 6 + 8, start with 8, then 2 more make 10, and that leaves 4 more for 14.

Generic Task• If you did not know the answer to 8 + 5

what are some really good ways you can use to get the answer?

• “Really good” means that you don’t have to count and you can do it in your head.

• See if you can come up with more than one way.

• Share your ideas in your group of three.

Other Strategies,Last Six Facts

• Doubles Plus Two, or Two-Apart Facts Of the six remaining facts, three have addends that differ

by 2: 3 + 5, 4 + 6, and 5 + 7. There are two possible relationships that might be useful here, each depending on knowledge of doubles. Some children find it easy to extend the ideas of doubles. Some children find it easy to extend the idea of the near doubles to double plus 2. For example, 4 + 6 is double 4 and 2 more. A different idea is to take 1 from the larger addend and give it to the smaller. Using this idea, the 5 + 3 fact is transformed into the double 4 facts- double the number in between.

Make-Ten Extended Three of the six facts have 7 as one the addends. The make-ten strategy is frequently extended to the facts as well. For 7 + 4, the idea is 7 and 3 more makes 10 and 1 left is 11. You may decide to suggest this idea at the same time you initially introduce the make-ten strategy. It is interesting to note that Japan, mainland China, Korea, and Taiwan all teach an addition strategy of building through 10 and do so in the first grade.

Counting On Counting on is the most widely promoted strategy. It is generally taught as a strategy for all facts that have 1, 2, or 3 as one of the addends and thus, includes the one and two-more-than facts. For the fact 3 + 8, the child starts with 8 and counts three counts, 9, 10,11. There are several reasons this approach is downplayed in this text. First, it is frequently applied to facts where it is not efficient, such as 8 + 5.It is difficult to explain to young children that they should count for some facts but not others. Second, it is much more procedural than conceptual. Finally, if other strategies are used, it is not necessary.

Ten-Frame Facts The ten-frame model is so valuable in seeing certain number relationships that these ideas cannot be passed by in thinking about facts. The ten-frame helps children learn the combinations that make 10: 5 + 6, 5 + 7, and 5 + 8 are quickly seen as two fives and some more when depicted with

these powerful models.

Strategies for Subtraction Facts

• Subtraction as Think-Addition Students are encouraged to

think, “What goes with this part to make the total?” When done in this think-addition manner, the child uses known addition facts to produce the unknown quantity or part.

Think-addition is most immediately applicable to subtraction facts with sums of 10 or less. These are generally introduce with a goal of master in the first grade. Sixty-four of the 100 subtraction facts fall into this caegory.

• The 36 “Hard” subtraction Facts: Sums Greater Than 10

• Look at the three subtraction facts below, and try to reflect on what thought process you use to get the answers. Even if you “just know them,” think about what a likely process might be.

14 12 15

- 9 - 6 - 6

• Build Up Through 10

This group includes all facts where the part or subtracted number is either 8 or 9. Examples are

13 – 9 and 15 – 8.

• Back Down through 10 Here is one strategy that is

really take-away and not think addition. It is useful for facts where the ones digit of the whole is close to the number being subtracted. For example, with 15-16, you start with the total of 15 and take off 5. That gets you down to 10. Then take off 1 more to get 9. For 14 - 6 just take off 4 and then take off 2 more to get 8. Here we are working backward with 10 as a “bridge”.

• Extend Think-Addition Think-addition remains one of the most powerful ways to think about subtraction facts. When the think-addition concept of subtraction is well developed, many children will use that approach for all subtraction facts.

Closure

1-4 Groups

Pick an addition strategy.

a. List at least three facts for which the strategy can be used.

b. Explain the thinking process or concepts that are involved in using the strategy. Use a specific fact s an illustration.