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xMath :: 6-9 :: Subtraction
xTable of Contents:
  • Introducton
  • Numeration
  • Addition
  • Multiplication
  • Subtraction
    • Stamp Game Subtraction
      • Dynamic Subtraction
    • Memorization Exercises
      • Strip Board
        • Introduction and List of Materials
        • Initial Presentation
        • Subtraction Booklets
        • Combination Cards
        • Decomposition of a Number
        • Decomposition of a Number with Zero as the Subtrahend
      • The Snake Game
      • Subtraction Charts and Combination Card Exercises
        • Passage From Chart I to Chart II
        • The Bingo Game of Subtraction (using Chart III)
          • Exercise:
          • Exercise:
          • Exercise:
          • Group Games
    • Bead Frame Subtraction
      • Static Subtraction
      • Dynamic Subtraction
      • Calculating the Difference a little more Abstractly
        • Presentation:
        • Games:
    • More Memorization Exercises (Special Cases)
      • 0- Calculating the Difference
      • 1- Calculating the Subtrahend
      • 2- Calculating the Minuend
      • 3- Inverse of Case Zero - Calculating the Difference
      • 4- Inverse of the 1st Case- Calculating the Subtrahend
      • 5- Inverse of the 2nd Case- Calculating the Minuend
      • 6- Calculating the Minuend and the Subtrahend
      • Collective Activity:
      • Individual Activity:
    • Word Problems
  • Division
  • Fractions - COMING SOON
  • Decimals - COMING SOON
  • Pre-Algebra - COMING SOON


xStamp Game Subtraction

Dynamic Subtraction

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

Again using the work cards, the child does a subtraction. The first number is formed, and from this number a second quantity is taken away and assembled in columns well below the first group. When the child sees that a change is necessary, the stamp of the higher order is placed between the rows as ten of the lower order are counted out. The ten are then placed in the row, the other stamp is put back in the box. When the subtraction is complete the child reads the result-what was left behind. The problem is recorded in his notebook
Would you like to see how we can check to see if you did the subtraction correctly? We can put these two quantities back together. The stamps are counted and changed. 'Did you get your original number?' That means that the subtraction was done correctly.
The child should also have practice subtracting with zeros in the subtrahend.

ex. 4000 - 785 =


xMemorization Exercises


a. Introduction and List of Materials

The child first dealt with the concept of subtraction with number rods. Later he learned the concept with the decimal system material, and the stamp game. Through memorization the child will master all of the combinations necessary for his work.

...Subtraction Strip Board, which differs from the addition strip board in that the numerals 1-9 are in blue,
...followed by a blue line, and 10-18 in red.
...Box of 17 neutral strips (to limit the minuend) 9 blue strips (to function as the subtrahend) and
...9 sectional pink strips (to serve as the difference)
...Booklet of Combinations (page one deals with 18)
...Box of Subtraction Combinations (same combinations as are found in the booklet)
...Box of blue tiles for bingo game
...Subtraction Charts I, II, III (for control)



b. Initial Presentation

To familiarize the child with the subtraction strip board, the teacher demonstrates. The neutral strips and the blue strips are lain out in the pipe organ arrangement. The teacher chooses a neutral strip. This is used to cover the numerals we don't need
The child then chooses a number to subtract, i.e. 5 The blue 5 strip is placed end to end with the neutral strip. The answer for 13-5 is the first number that shows....8. If by chance the child chooses a subtrahend that would give a difference greater than nine, the teacher explains that the maximum difference that can exist is nine, 13 - 2 for example is not necessary.



c. Subtraction Booklets

...booklets of 18 pages; each page has combinations with a common minuend- the first page deals with 18
...Subtraction Strip Board, strips
...Chart I

The child begins with the first page in his booklet. With 18, a neutral strip is not needed, it is already the last number in the row. The combination is 18 - 9; therefore, the blue strip for nine is placed over the numbers. The first number to show is 9. That's the difference, and it is written in the booklet. 18 - 9 is the only combination possible. The child may try others to prove this.
Going on to the second page, the minuend is now 17, therefore the smallest neutral strip is used to cover 18 (the number greater than 17) The child reads the first combination, takes the blue strip corresponding to the subtrahend and places it over the numbers. The answer is read and is written on the form. For the second combination, we know that the minuend will be the same, therefore the neutral strip doesn't need to be changed.
After completing the page, the child should notice 17 - 9 = 8 that while the minuend is fixed, there is 17 - 8 = 9 a decrease of one unit in the subtrahend, and, therefore, an increase of one unit in the difference.

Notes: The blue strip is used as the subtrahend because the child must realize that he is subtracting a group. In subtraction, the aim is always to break down the ten. Since the neutral strips occupy much space, after this exercise the child may use only the longest, sliding it off the edge to its correct position.

Observations on the Subtraction Chart I:
This chart reproduces all of the combinations in the subtraction booklet. In the first 9 columns the differences are common in horizontal rows. This indirectly shows the invariable property of subtraction; if one adds a number to both the subtrahend and the minuend, the difference is the same, i.e. 1 ­ 1 = 0, 2 ­ 2 = 0, 3 ­ 3 = 09 ­ 9 = 0
In the last nine columns, the subtrahend is consistent in each horizontal row; thus the minuend and the difference increased by one, i.e. 10 ­ 9 = 1, 11 ­ 9 = 218 ­ 9 = 9



d. Combination Cards

...subtraction strip board, strips (neutral and blue)
...combination cards
...Chart I (for control)

The child fishes for a combination, reads it and writes it on his paper, 15 ­ 7 =. The neutral strip is used to cover all of the numbers greater than 15 which are not needed. Next to the neutral strip is placed the blue 7 strip. The difference is the last number showing; this is recorded.
The exercise continues as long as the child would like, and then he controls his work with Chart I.



e. Decomposition of a Number

...subtraction strip board, all of the strips

(In this exercise the pink strips are used for the first time to function as the difference)
Let's see how many ways we can decompose (break down) 9? Nine will be the minuend, therefore a neutral strip is placed over the number greater than 9. The teacher writes down the combination 9 ­ 1 =___. (Note: decomposition always begins at one, removing one unit at a time) The subtrahend is one, so a blue 1 strip is needed. This time it is placed on the first row, under 9. The child guesses the answer and tries to place the pink strip for his answer on the row. If it fits he knows that he is correct. The answer is recorded. The work continues in order until all of the blue strips are used, and a column of combinations has been completed9 ­ 9 = 0
In this exercise the child may recall his work of this fashion in addition, which resulted in elimination of some combinations. In subtraction, all of the combinations are needed and must be learned.
The child tries to decompose other numbers in the same way: i.e. 14. How many ways can 14 be decomposed? The neutral strip identifies 14 as the minuend. Can I do 14 ­ 1? No (the one strip may be tried, but it will not work because 9 is the maximum difference we can have) 14 ­ 2? 14 ­ 4? 14 ­ 4? 14 ­ 5? Yes. The decomposition begins here laying out the blue 5 strip and the pink 9, recording 14 ­ 5 = 9, and so on to 14 ­ 9 = 5.

Control of Error: Chart I



f. Decomposition of a Number with Zero as the Subtrahend

...subtraction strip board, all of the strips

As before, the teacher presents a number to decompose. The neutral strip is lain over the number to limit the minuend. On a piece of paper, the teacher writes, i.e.
7 ­ 0 =___. What must be taken away? nothing. In subtraction also, we see that zero doesn't change anything. On the first row, then, the pink strip for seven is placed and this difference is recorded. 7 ­ 0 = 7 The child continues7 ­ 7 = 0

Control of Error: Chart I



Materials: same materials as for previous snake games: of colored bead bars 1-9 of ten bars of black and white reminder bars (place holders)
...also box, with 9 compartments, for gray bead bars 1-9
...(for bars of 6-9, there is a small space or color change after the fifth bead to facilitate counting)

As before, a snake is made, though this time we add to these colored bead bars, some gray bead bars. (Note: before the first gray bar appears, several colored bead bars should appear to create a large minuend) As before we begin counting, using the black and white reminder bead bars. When we come to a gray bar, we must subtract. The preceding black and white bead bar and the gray bar are isolated. 8 ­ 4 = 4 The 4 black bar is placed in the box cover with the other original colored beads.
Counting continues as usual. The next two bead bars are isolated: black 3 and gray 7. This gray bar means I must subtract. 3 ­ 7 is impossible; therefore I take one ten bar (from the snake's new skin) and place it beside the 3 to make 13. 13 ­ 7 = 6. The black and white 6 is placed in the snake; the black 3 is placed with the other reminder bars; the ten bar is placed back in the box with the other ten bars; the gray 7 is placed in the box cover with the other original colored bead bars.
When the counting is finished, the gold bead bars (and reminder bar) are counted to find the result.

Control of Error: In the box cover are colored bead bars and gray bead bars mixed. First, these are separated into two groups, lain in chronological order. A gray bar is placed with its equivalent colored bar. Two colored bars may be combined to match a gray bar, or vice versa. When all of the gray bars have been matched; the colored bars are paired to be matched with ten bars as usual . When all of the bars have been matched, we know the counting was done correctly.

Direct Aim: to memorize subtraction

Indirect Aim: to prepare for algebra: positive and negative numbers



a. Passage From Chart I to Chart II

...Chart II (the numbers in pink function as the minuend; the blue as the subtrahend)
...combination cards
...Chart I

The child fishes for a combination, i.e. 9 ­ 2 =___, reads it and writes it down on his paper. A finger is placed on 9 on the pink strip on the chart.; another finger is placed on 2 on the blue strip. Where the two fingers meet, we find the difference. This is recorded on the paper. The child continues his work in this way, and when finished, he controls with Chart I.



b. The Bingo Game of Subtraction (using Chart III)

...Chart III and box of corresponding tiles
...combination cards
...Chart I (for control of combinations)
...Chart II (for control of placement of the tiles)

A. Exercise:
The tiles are randomly arranged face up on the table. The child fishes for a combination, thinks of the answer, and finds the corresponding tile. After the minuend (pink) and subtrahend (blue) have been established on the chart, the child is able to find the place for the tile. He writes the equation on his paper and continues.

Control of Error: Charts I and II

B. Exercise:
All of the tiles are in the box (or in a sack). The child fishes for a tile, and thinks, what could this be the remainder of? He thinks of a combination and writes it down, i.e.
7 = 14 ­ 7. He puts the tile in its place. He continues in this way, then controls his work.

C. Exercise:
The tiles are arranged on the table in common stacks. The child chooses one stack and thinks of combinations which will yield this difference. He writes down the combination, finds the place on the chart, and so on, continuing until he has finished the stack.
When all of the stacks are arranged in order in a row, what form do they make? a rectangle or parallelopiped.

Group Games
1. The teacher, or a child functioning as the teacher, fishes for a combination and reads it. One of the children guess the difference.

2. The teacher fishes for a tile and the children offer combinations which give that result, until all are given.

Aim: (of all of exercises,) to memorize subtraction combinations


xBead Frame Subtraction


The teacher initially presents a static subtraction. The problem is written on the form and the minuend is formed on the frame. Now we must take away this quantity. Beginning with the units move the beads back to the left. Continue with tens, hundreds and thousands. Read the result and record it.



A second (and all the others following) example is dynamic:
Again the minuend is formed. Beginning with the units we must take away this quantity. Begin counting the units as they are moved back...1,2,3,4,5 we must change one ten to ten units (move one ten back, 10 units forward) and continue...6,7 moving the unit beads back. Subtract tens, hundreds and thousands in the same way. Read and record the result.



With a dynamic subtraction again, the child forms the minuend. As before begin by taking away the units. Record the number of units remaining, in the units column. Subtract the tens and so on. The result on the form will correspond to the difference formed on the frame.

Form 1000 on the frame. Subtract one unit. Form 1000 on the frame and subtract 999. Each time three changes are necessary.


xMore Memorization Exercises (Special Cases)

Note: As in the special cases for addition, these exercises are done after the child has already done much work with the previous exercises.

...subtraction combination booklet
...large sheet of paper or chart, and black pens
...special combination cards

The teacher opens to a page, in the booklet. When you do these combinations, what do you do? calculate the difference.
As before the teacher writes the tile on the chart, gives an example and reads. In all of the other cases, the special case is set up first in the combination booklet or on a sheet of paper, and the child identifies the missing part. Then the special case is added to the chart.

0- Calculating the Difference
16 ­ 9 = ? (If from 16, I take away 9, what will be left?)

1- Calculating the Subtrahend
16 ­ ? = 7 (From 16 I have taken away a certain number, and 7 is left. How much did I take away?

2- Calculating the Minuend
? ­ 9 = 7 (From a certain number I have taken away 9, and 7 is left. What was that number?

3- Inverse of Case Zero - Calculating the Difference
? = 16 ­ 9 (What will be left if from 16 I take 9?)

4- Inverse of the 1st Case- Calculating the Subtrahend
7 = 16 ­ ? (7 is the remainder when from 16 I take away what number?)

5- Inverse of the 2nd Case- Calculating the Minuend
7 = ? ­ 9 (7 is the remainder when from a certain number I take away 9. What is that number?)

6- Calculating the Minuend and the Subtrahend
7 = ? ­ ? (7 is the remainder when from a certain number I take away another number. What is the first number, and what is the second number?)

Collective Activity:
The teacher passes out the special combination cards. Each child reads the combination, states what part is missing and which case it is. (referring to the chart)

Individual Activity:
The child works with the special combination cards as before: drawing one and computing it, then writing the equation in his notebook with the answer in red.

Aims: further memorization of subtraction combinations
reinforcement and understanding of the concept

Notes: This activity develops the ability to work with the same combination under different conditions and from various points of view, thus creating flexibility in the child's mind. Here, also, we see the close relationship between addition and subtraction .


xWord Problems

E. Word Problems

As before word problems are prepared dealing with the seven different cases. These 7 cards are mixed with the 7 addition cards. When the child has done all of these, he may invent his own which will indicate his understanding of these special cases.

1. Rebecca has 6 roasted chestnuts left. Before she had 11. How many did she eat? 6 = 11 ­ 5

Age: between 6 and 7 years



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