Contents
xMath :: 6-9 :: AdditionAddition
xTable of Contents:
  • Introducton
  • Numeration
  • Addition
    • Stamp Game Addition
      • Dynamic Addition
    • Memorization
      • Strip Board (including Doubles)
        • Complete List of Strip Board Materials
        • First Presentation of the Addition Strip Board
        • Addition Booklets
        • Combination Cards
        • The Combinations of One Number
        • The Combinations of One Number with Zero as an Addend
        • Doubles of Numbers
      • Games and Explorations using Bead Bars
        • Complete List of Materials
        • The Snake Game
        • Sums Less Than Ten
        • Sums Greater Than Ten
        • Changing the Order of the Addends
      • Addition Charts and Combination Card Exercises
        • Passage from Chart I to Chart II
        • Passage from Chart II to Chart III (the Whole Chart)
        • Passage from Chart III to Chart IV (the Half Chart)
        • Passage from Chart IV to Chart V (the Simplified Tables)
        • The Bingo Game for Addition (using Chart VI)
          • Exercise One:
          • Exercise Two:
          • Exercise Three:
          • Group Game One
          • Group Game Two
          • Notes:
    • Bead Frame Addition
      • Static Addition
      • Dynamic Addition
      • Finding the Sum More Abstractly
      • Presentation:
      • Game:
    • More Memorization Exercises
      • Further Explorations using Bead Bars
        • Sums with Parentheses
        • Breaking Down the Addends
        • Addends Greater than Ten
      • Special Cases
        • 0 - Calculating the Sum
        • 1 - Calculating the Second Addend
        • 2 - Calculating the First Addend
        • 3 - Inverse of Case Zero
        • 4 - Inverse of 1st Case, Calculating the Second Addend
        • 5 - Inverse of 2d Case, Calculating the First Addend
        • 6 - Calculating the First and Second Addends
        • Collective Activity:
        • Individual Work:
    • Word Problems
  • Multiplication
  • Subtraction
  • Division
  • Fractions - COMING SOON
  • Decimals - COMING SOON
  • Pre-Algebra - COMING SOON
xStamp Game Addition


DYNAMIC ADDITION

Materials:
...wooden stamps of four types:
......green unit stamps printed with the numeral 1,
......blue tens stamps printed with the numeral 10,
......red hundred stamps printed with the numeral 100, and
......green thousand stamps printed with the numeral 1000
...box with three compartments each containing 9 skittles and one counter in the hierarchic colors;
...four small plates

Presentation:
Using dynamic addition work cards, the teacher presents addition using stamps. The child forms the first addend and then forms the second addend, starting his columns well below the first addend. Now you do the addition. The child slides the rows together and begins to count, starting with units. At ten units the child must stop and change these to a ten. The units are put back and a ten is taken out. The child continues counting and changing. Now let's see what the result is. The number is read and the problem is recorded in his notebook.
top

xMemorization


STRIP BOARD (including doubles)


a. Complete List of Strip Board Materials

Materials:
...addition strip board
...pink box containing pink and blue strips
...mimeographed booklets
...box containing 81 combinations on small cards
...box containing pink tiles with 81 sums
...box of 36 pink rectangles and 36 squares
...Control Charts I-VI

top


STRIP BOARD (including doubles)

b. First Presentation of the Addition Strip Board

Materials:
...addition strip board and strips

Presentation:
In order to show the child how to use the materials, the teacher presents a few short exercises. A blue strip is chosen at random. The child identifies the strip. It is placed along the top row of squares. The teacher takes a pink strip. I am going to add 5 (pink) to this 7 (blue). The pink strip is lain on the board. We can see that 7 plus 5 is 12. I read 12 here at the top, pointing to the top row of numerals. The exercises continues like this.
Then the child adds keeping the first addend the same. The second addends are chosen at random order. In these exercises the blue strip remains on the board throughout.

top


STRIP BOARD (including doubles)

c. Addition Booklets

Materials:
...mimeographed booklets of nine pages each; each page has
...nine combinations with a common first addend
...addition strip board and strips
...control chart I (81 combinations in 9 columns)

Exercises:
The child chooses one page in the booklet. He reads the first combination 3+1=__. The first addend is 3, so the blue strip for 3 is placed on the board. The second addend is 1, so the pink strip for 1 is added. The sum is read at the top, and is written in the booklet near the equal sign. The first addend remains the same, therefore the blue strip remains on the board. The pink strip may be turned face down in its place, to remind us that we've finished with 1. The child follows the order on the form.

If the child is just writing the numbers in succession on the column, the aim of memorization is not being met. Therefore, the child may complete a page in any order. A booklet with random order problems for one number can be introduced as well.
example:


3+2=__
3+4=__
3+7=__
3+9=__
3+1=__
3+1=__
3+5=__
3+8=__
3=3=__
3+6=__

Control of error: Chart I. The child simply compares his page to a column on the chart.

top


STRIP BOARD (including doubles)

d. Combination Cards

Materials:
...box of combination cards
...addition strip board and strips
...paper
...Control Chart I

Exercise:
The child fishes for a combination. He reads it and writes it down on his paper 7+6=___. The first addend is 7, so the blue strip for 7 is placed on the board. The second addend is 6, so the pink strip is placed directly next to the blue strip on the board. The sum is read at the top and is written on the paper next to the equal sign. The strips are put back in their places. The child fishes for another combination, and the exercise continues.

Control of error: Control Chart I. The child looks at the first addend, finds the column where the combinations have that first addend, then looks for his combination.

top


STRIP BOARD (including doubles)

e. The Combinations of One Number

Materials:
...addition strip board and strips
...paper
...Control Chart I

Exercise:
Let's see all of the different ways to form 10. The blue strip for one is placed on the board. What do we need to make 10? The pink strip for 9 is added, 1+9=10. The child continues in order, making combinations until 1+9=10. The child then writes the combinations and sums on his paper.
Now the child observes that the pink strip decrease in size as the blue strips increase. Also it is observed that 9+1 is the same as saying 1+9. The 9+1 are held up to the 1+9 strips to compare. If I remember that 1+9=10, then I also remember that 9+1= 10. We can eliminate one of these combinations. The strips are put back in their places and 9+1=10 is crossed off the list. We can do the same for 8+2. It is the same as 2+8. This continues until only five combinations remain. It is sufficient to know these five combinations to know the combinations which form 10. The same is done for all numbers 2-18.

Control of error: Control Chart I. The child looks all the combinations he has made, noticing the tens in red on the diagonal.

top


STRIP BOARD (including doubles)

f. The Combinations of One Number with Zero as an Addend

Materials: addition strip board and strips
paper
Control Chart 1 and /or Chart II (45 combinations)

Exercise:
Let's find all of the combinations that make 9. 0+9=___? Our first addend is zero, so we place nothing on the board. Our second addend is 9, so we place the pink strip for 9 on the board. Our sum is 9. This continues until 9+0 is the last combination.

We notice that the first strip is all pink and the last strip is all blue.
Zero doesn't change the number in addition.
As before the child eliminates the unnecessary combinations.

Control of error: Control Chart 1 and/or Chart II

top


STRIP BOARD (including doubles)

g. Doubles of Numbers

Materials: addition strip board and strips
paper
Control Chart 1

Exercise:
We find the doubles of numbers by taking a blue strip: 1 and the same second addend; the pink strip for 1. The strips are placed on the board, and the combination is written on the paper. The sum is read at the top. The one strips are returned to their places, and the twos are added, etc.

N.B. Here the teacher helps the child reflect on his work, thus noticing that not only is 9+9=18, but also that 1/2 of 18 = 9. The possibility for dialogue here is very great and is a way of engaging langugae in the course of understanding mathematics.

Control of error: Control Chart II on which the double of each number is found at the top of each column.

top


GAMES AND EXPLORATION USING BEAD BARS

a. Complete List of Materials

Materials:
...Snake Game reminder beads, without the bridge
...box of ten bars
...box of colored bead bars (9 of each)
...box containing signs for the operations +, -. x, /, =, ()
...Addition Chart I

top


GAMES AND EXPLORATION USING BEAD BARS

b. The Snake Game

Materials:
...reminder beads usually black and white
...ten bars
...colored bead bars
...Addition Chart I

Presentation:
The reminder beads are lain out in a triangle arrangement to facilitate movement. The child is asked to make a snake with the colored bead bars. (The box is then covered again) The child no longer counts bead by bead to arrive at the ten. The first two beads are isolated. The child mentally computes the sum, i.e. 8+9=17. We can replace these with a ten (the ten bar is lain out) and seven (the reminder bead bar connects the ten to the rest of the snake). The 8 and 9 are put into the empty box. Go on adding the seven reminder bar to the next bar of the snake. These are isolated, added, the ten bar and corresponding reminder bar replace them in the snake, and the colored bead bar is returned to the empty box while the first reminder bead bar is replaced in its place in the triangle.
The child may make combinations of more than two bead bars, keeping the sum less than 19. If the child does not remember a combination, he may use the Addition Chart I. Command cards may be made.

Control of Error: As before, the child controls his work by matching. The ten bars and reminder bead bar (if any) are grouped together to one side. The colored bead bars from this snake are arranged in hierarchic order. the large box of colored bead bars is reopened, in case exchanges are necessary. The child takes a ten bar and a colored bead bar, i.e. 8. What must be added to eight to make ten? A colored bead bar of two is united with the 8 and placed next to the ten. If the child doesn't have a bead bar of two, an exchange must be made. Combinations of two numbers to make ten should always be used when controlling. The child sees that his addition was exact when the colored beads are all matched to ten bars (and the black reminder bar).

top


GAMES AND EXPLORATION USING BEAD BARS

c. Sums Less Than Ten

Materials:
...ten bars
...colored bead bars
...box of signs for operations

Presentation:
The teacher sets up an addition of two numbers, using colored bead bars and the plus and equals signs. the child reads the problem, computes the addition problem in his head and puts out the appropriate bead bar for the sum.
The child also adds combinations of these numbers. Sums are always less than ten. Command cards may be made for the work. The Addition Chart I may be used for control, or to help the child remember the combinations.

top


GAMES AND EXPLORATION USING BEAD BARS

d. Sums Greater Than Ten

Materials: same as above

Presentation:
As before, the child combines two or more numbers, whose sum will be greater than ten. The colored bead bar is placed below the ten bar to facilitate counting.
For example:

A child chooses 14 and 12 in beads to add. Tell the child to add the units first and then the tens to achieve the sum. 4 + 2 = 6 and 2 tens are twenty. The sum is 26.

The child chooses 18 + 25 in beads. Again units are added first. Since that sum is 13. Place a three bar below and carry the one ten mentally. Carrying the one ten add the other tens. The sum is 4 tens and the total sum is 43. This exercise gives a child experience with horizontal as well as vertical addition, a fact often overlooked in preparation for standardized tests.

Command cards may be made. The chart may be used as control.

top


GAMES AND EXPLORATION USING BEAD BARS

e. Changing the Order of the Addends

Materials: same as above

Presentation:
The teacher sets up an addition of two numbers, and the child completes the sum, placing the corresponding bead bars in their places. This equation is correct. The teacher switches the places of the two colored bead bars. Is it still correct? Perhaps it was a coincidence; let's try another.

Aim: to give the concept of the commutative property of addition (although it is not named as such at his age)

top


ADDITION CHARTS AND COMBINATION CARD EXERCISES

a. Passage from Chart I to Chart II

A group of children or individual children may copy Control Chart I or the teacher may make up 36 pink rectangles which are the dimensions of the space for the combinations.

Let's see how many combinations we can eliminate. We start from the first row 1+1=2. We must leave that...1+2=3. We can read along the diagonal the combinations that make 3. 2+1=3. We can cancel this combination. A card is placed over it, or it is crossed out. We'll go on to the combinations that make 4. They are 1+ 3 = 4, 2+ 2 = 4 and 3 +1 = 4. We must leave the first two. But what of 3 + 1 = 4 ? This can be eliminated.

We have another chart on which all of these combinations which were crossed out. are eliminated. Something else is different. All of the combinations that make the same sum are arranged on a horizontal row. Now each column begins with a combination in which the addends are the same. But this chart contains all of the same combinations as before, just arranged slightly differently. Here you can find all of the combinations needed to do your work. The teacher gives an example or two to show that even though 8+2 is not listed, we find the sum when we look at 2+8.

top


ADDITION CHARTS AND COMBINATION CARD EXERCISES

b. Passage from Chart II to Chart III (the Whole Chart)

Materials:
...box of combination cards
...paper
...Control Chart I and III (which has only sums)

Exercise:
The child fishes for a combination. He reads it and writes it down on his paper. What is the first addend? Place your finger on the blue row at the first addend. Place your finger of the other hand on the pink column at the second addend. Move along the row and column until your fingers meet. The meeting place is at the sum. Write the sum on your paper. Fish again, etc.

Control of error: Control Chart I

top


ADDITION CHARTS AND COMBINATION CARD EXERCISES

c. Passage from Chart III to Chart IV (the Half Chart)

Materials:
...box of combination cards
...Control Chart I and IV (which has only sums)
...paper

Exercise:
In the same way as the child passed from Chart I to Chart II, a group activity may be conducted to show that some of these sums on Chart III can be eliminated. The combinations are reconstructed, going along the diagonal. This 4 was made by 1+3, this by 2+2, and this by 3+1, so we can eliminate it. When the elimination is complete, we have Chart IV.
The child fishes for a combination, reads it and writes it down on his paper, i.e. 5+9. On the pink column at the left, he places his finger on the first addend and goes all the way to the end of that row. Then he places his other finger on the second addend (in the same pink column) As before the two fingers are moved toward each other, and at the meeting place in the sum. When the child pulls out a combination, which has the first addend greater than the second, he will find that he can't do it. We simply reverse the order of the addends.

Control of error: Control Chart I

top


ADDITION CHARTS AND COMBINATION CARD EXERCISES

d. Passage from Chart IV to Chart V (the Simplified Tables)

Materials:
...box of combination cards
...Control Chart I and V (which has only 17 sums on two diagonals: external is even, internal is odd)
...paper

Exercise:
The child fishes for a combination, reads it and writes it down on his paper, i.e. 4+6. The first finger is placed at 4 on the left-hand column and moved along to the end of the row. The other finger is placed at the second addend, 6 on the left hand column, and is also moved along to the end of the row. The two fingers are then moved toward each other along the diagonal one step each at a time. Where they meet we read the sum. The next combination is 3+9=___ to emphasize that each finger goes the same distance. Then 2+9=___ is used as an example. The two fingers must meet on the internal diagonal. Other examples are given.

To bring a new slant of interest to this activity, the teacher brings into focus that when both addends are even, the sum is even. When both addends are odd, the sum is even. When one addend is even and the other odd, the sum is odd.

Control of error: Control Chart I

top


ADDITION CHARTS AND COMBINATION CARD EXERCISES

e. The Bingo Game for Addition (using Chart VI)

Materials:
...box of combination cards
...Chart VI (which has only the first and second addends; the rest is blank)
...Control Charts I and III
...box of 81 pink tiles for the sums

i. Exercise One:
The tiles are randomly arranged on the table face up. The child fishes for a combination, reads it and writes it down. The child thinks of the sum, looks for a tile with that sum , and looks for the place to put it on Chart VI. The first finger is put on the first addend, the other finger on the second addend. Where they meet is where the tile belongs. The sum is written on the paper. The child fishes again, etc.
ii. Exercise Two:
The tiles are in the box. The child fishes for a tile and reads the numeral. On his paper he writes the numeral and the equal sign. He thinks of a combination and writes it to complete the sentence. Then those two addends are used to find the tile's position.
iii. Exercise Three:
The tiles are placed in piles that have common sums. The child takes one pile, i.e. the pile of 8's. What does 8 equal? The child thinks of a combination, writes the sentence and uses the addends to find the corresponding position for the tile. He continues thinking of combinations until all of the tiles of that pile have been placed on the board. He notices that a diagonal is formed. the child does not need to do all the piles in one sitting; however he must complete whole piles he has chosen.
If the child arranges the piles in order, he may find an ascending and descending stair.

Control of error: Control Chart I and III

iv. Group Game One.
The teacher fishes for a combination, shows it to the child and asks, What is 2+3 equal to? If the child responds correctly, he receives the card (flash cards).
v. Group Game Two
The teacher fishes for a tile-say 10. What combinations are equal to 10? Each child gives a different combination until all have been named.

Age: Children's House-7 years

Aim: to give the child the possibility through many different exercises to memorize the combinations necessary for abstract problem solving

vi. Notes:
The pink strips on the addition strip board are segmented so that the child may see how many units are needed to make 10 and how many more are after 10. It is hoped that the child will absorb this aid to memorization. Later when the child is confronted with larger combinations, 24+8=?, he will have memorized 4+8=12 and the rest follows. The point of consciousness to be reached is to look for the combination which makes 10. Therefore, the child will say-I need to add 6 to 4 to make 10,
24 + 8 =
20 + (4 + 6) + 2
20 + (10) + 2

24+6 +(2) brings me up to 30. I have two more units on the right to add...32. Once the combinations are memorized, this type of mental activity naturally follows, thus abstraction. As the child works with various exercises, the teacher must observe and check to see if these points of consciousness are being met.

top

xBead Frame Addition


STATIC ADDITION

Materials:
...small bead frame, form

Presentation:
The teacher initially presents a static addition of two or three 4-digit numbers. The addition problem is written on the form. The first addend is formed on the bead frame. Now we must add the second number. Beginning with the units the teacher moves forward the corresponding number of beads, Invite the child to continue adding the tens, hundreds and thousands. The third addend is added in the same way, units first. The result is read and recorded appropriately on the form.

top


DYNAMIC ADDITION

A second (and all others succeeding) example is dynamic. The problem is written on the form as before and the first addend is formed on the frame. Of the second addend, begin by adding the units, counting the beads as they are moved forward...1,2,3,4 all of the units have been used, so we exchange a ten for ten more units- (move a ten forward, and ten units back) 5, 6, 7, 8, 9 (continue counting and adding unit beads) In fact, 6 + 9 = 15, which is 10 plus 5 units. Continue adding the second addend and then the third addend changing when necessary. Upon completion read and record the sum. The child should have many exercises of this type.

FINDING THE SUM MORE ABSTRACTLY

Presentation:
A dynamic addition problem is written on the form. This time add all of the units first, making the necessary changes. Write the total for the units under the units column and go on to add all of the tens. Where did these two tens come from? Those are the result of changes you made when adding up the units. (The carried over tens are not recorded anywhere.)
After each column the child records the total. The result when completely written corresponds to what is seen on the frame.

Game:
Form 999 on the bead frame. Add one unit. The result, 1000 was obtained by making three changes.

top

xMore Memorization Exercises


FURTHER EXPLORATION USING BEAD BARS


a. Sums with Parentheses

Materials:
...ten bars
...colored bead bars
...box of signs for operations

A. Presentation:
The teacher presents the new symbols: ( ) parentheses. First a combination of three numbers is set up and the sum is completed. These signs are called parentheses; they group things. The same addition is repeated, now with the first two addends in parentheses. Whenever you see these parentheses in arithmetic, it means you must perform the addition inside the parentheses first. The first combination inside the parentheses is added, and the bead bar for the sum is placed below. The signs and bead bar for the next addend (outside the parentheses) are placed below as well. We find that this sum is the same as the original answer.
We haven't changed the addends, and the sum hasn't changed, only the addends have been grouped in a special way.

Aim: to give the concept of the associative property of addition

B. Presentation:
The teacher writes a long addition problem on a slip of paper. The child sets up the problem with the bead bars and signs, and computes the answer. The parentheses are placed around pairs of numbers. Review: We must perform the addition inside the parentheses first. New bead bars are put out for the sums of the pairs. These are added and the sum is found to be the same as the original problem

Note: Command cards may be made up for this work.

top


FURTHER EXPLORATION USING BEAD BARS

b. Breaking Down the Addends

Materials: same as before

Presentation:
On a strip the teacher writes a combination of two or three numbers. The child constructs the problem with bead bars and symbols and computes the answer and puts out the corresponding bead bars for the sum, The child reads the equation.
Now try to do this one; the teacher lays out combinations in parentheses that equal each addend of the first problem, i.e. 1st 9+7+8= 24, 2nd (4+5) + (3+4) + (6+2) = The child computes this as before, adding the first two addends in parentheses, placing the bead bar for the sum below, etc. Those three are added again to obtain the same answer as before. It is noticed that the first and third equations are identical. In the second, each addend was broken down into two smaller addends.

Aim: to give the concept of the dissociative property of addition

top


FURTHER EXPLORATION USING BEAD BARS

c. Addends Greater than Ten

Materials: same as before

Presentation:
The child is asked to set up a given addition with addends greater than ten. At first these should be static operations, i.e. 12 + 14 = When we have an addition like this, we first add the units, then the tens.
After practice such as this, we go on to dynamic operations, ex. 18 + 25 = First we add the units 8 + 5 = 13 We place a three bar here for the units and keep the ten in mind. We then add the tens; 1+2=3 and one ten in my mind makes four tens, put out four ten bars. The sum is 43.

Addition Chart: may be used here if the child doesn't remember a certain combination.

Aim: to practice carrying over
memorization of addition combinations

Notes: All of these games are more popular than the other memorization exercises because they are short, and the result is obtained quickly and easily. The children may invent others

Age: 6-61/2 years

top


SPECIAL CASES

Note: To find out if the child memorized not only the process of calculating in addition, but also the concept, the teacher organizes special combinations, starting with the combination that is familiar to the child.

Materials:
...addition combination booklet
...large sheet of paper or chart,
...red and black pens
...special combination cards

Presentation:
Taking any one page of the booklet, the teacher asks, 'When you work on this page, in this case 1+1, what are you looking for? the sum. In the work you have been doing up till now, you have been calculating the sum. On the chart the teacher writes the title, gives an example, and reads the example to the child.

0 - Calculating the Sum
1 + 2 = ? (One added to two gives you what number?)
The child fills in all the sums for that page.

1 - Calculating the Second Addend
1 + ? = 3 (One added to what number gives you three?)
Let's cover the column of second addends with a strip of paper. Note, this is the first time the child considers a problem of this type. What must we solve here? the second addend.

2 - Calculating the First Addend
? + 2 = 3 (what number added to 2 will give you 3?)
On the model page with totals, the column of first addends is covered. The child sees that in order to complete this combination, the first addend must be found.

3 - Inverse of Case Zero; Calculating the Sum
? = 1 + 2 (what number is obtained by adding one and two?)
The same column of combinations is written on another sheet, without their sums this time, and with the addends to the right of the equal sign. The child sees that the sum must be calculated, as in the first case. The difference is that the problem is set up in reverse order.

4 - Inverse of 1st Case, Calculating the Second Addend
3 = 1 + ? (3 is equal to one plus what number?)
The sums are written in this inverted model, and the column of second addends is covered. The child sees that he must find the second addend.

5 - Inverse of 2d Case, Calculating the First Addend
3 = ? + 2 (three is equal to what number plus 2?)
The column of first addends is covered.

6 - Calculating the First and Second Addends
3 = ? + ? (three was obtained by adding the first number to the second number. What were these numbers?)
In this last case, both columns of addends are covered with strips of paper, leaving only the sum.

Collective Activity:
The teacher passes out the combination cards. In turn the child reads the problem, states what must be found, and finds the case on the chart.

Individual Work:
When the child has understood all of the cases as presented, he may work with these special combination cards. He fishes for one, reads it and writes the equation in his notebook, substituting the red question mark for the right number written in red

Direct Aims: to memorize addition combinations
to make the child realize what must be calculated

Indirect Aim: to prepare first degree equations in algebra, i.e. ( 4 + x = 6).

Notes: On the chart, #0 is not a special case since this is what is familiar to the child. In cases #1, #2, #4, #5, and #6, subtraction is indirectly involved. For this reason fewer activities for memorization of subtraction are necessary.

top

xWord Problems


In these problems the special cases previously examined are recalled. If the child has understood the last activity, he will be able to write a complete equation.

Examples:
1. Yesterday Adam had 3 notebooks. How many did his mother give him, if we know he has 5 notebooks today? 3 + ? = 5

top

 

Creative Commons License