Division as Unknown Group Size and Unknown Group Length
The solution for 2,051 ÷ 7
Unknown Group size: 293 items per group for each of the 7 groups.
Unknown group length: 293 groups each containing 7 items.
Similarities
In both methods, the groups are presumed to be equal. For example, in the equation 2,051 ÷ 7, it is presumed that the unknown resulting group will contain an equal number of 7. On this basis, the dividend can then be split into an unknown number of groups but ensuring that each group constitutes 7 items.
Both methods involve distributing the items until the final answer is arrived at.
Differences
In the unknown group size, the number of items per group is not known while for the unknown group length the number of groups is unknown.
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For the division as unknown group size, the answer is obtained by equal sharing strategy in which the 2051 items are dealt out to the seven groups until all the items are finished. The group size is then determined by counting the number of items in one group. In the unknown group length, groups of 7 items are drawn from the total 2,051 items until all the items are done. The group length is then determined by counting all the total number of groups containing the 7 items.
The Scaffolding and Standard Long Division Method
Figure 1: Scaffolding Division
Figure 2: Standard Division
Similarities
Both methods use a drop-down division algorithm, in which the solution is worked downwards following a repeated subtraction approach.
The arrangement of figures is the same for both methods. The dividend is below the division bar, with the divisor to the right of the division bar. All figures are arranged in order of their place values and the equation solved starting with the figures in the highest place value positions. This arrangement makes the working easier and minimizes confusions.
Differences
In the standard long division method, the answers are written on the right side of the working solution. This way, there is more room for accumulating many parts of the answer before providing the final answer. This allows room for errors and making corrections in the calculation process. In the scaffolding method, on the other hand, the answer to the division is written at the top, giving limited room for writing smaller parts of the answer. Conversely, there is no room for error and correction as only one correct number is provided for each calculation.
The scaffolding method has a sidebar chart, while the standard long division method does not.
Doubling Sidebar Chart
The double division method using the sidebar chart is an easier alternative for teaching students the division of long numbers with less frustration. The double division eases the need to memorize multiplication facts or the guesses and estimates of how many times a number goes into another, as is the case with other division methods. The method deals with whole numbers rather than individual numbers. This involves sequentially subtracting multiples of the divisor from the dividend, from the highest possible multiples to the lowest. Although it may be a longer method, arriving at the final quotient is easier. This especially is through the use of the sidebar chart.
The sidebar chart shows all the possible multiples used in solving the problem, from the highest value to the lowest. The first step in generating the chart is sequentially doubling the divisor and indicating the resulting values. The highest value that is equal to or less than the dividend is chosen and subtracted from the dividend. Students should be careful in choosing the correct multiplier value, as every value that is chosen will form a digit of the quotient. It should be noted that the unoccupied place values to the right of the multiples are filled with zeros for ease of subtraction. The exact number of zeros is also added to the right of the multiplier value chosen and the resulting value written on the right of the equation. For example, if the first value chosen is a multiple of 4 of the dividend, then the first digit of the quotient will be 4. This is repeated until the value left at the bottom of the solution is zero. The figures indicated to the right or left of the equation constitute the sidebar chart. To obtain the final answer to the problem, the values to the right of the equation are added (Cooper, 2015).
Transition from the Scaffolding Method to Standard Division Algorithm
Both the scaffolding method and the standard division algorithm have a similar arrangement, in which the divisor is to the left of the dividend. The dividend is to written below the division bar. This arrangement may stir confusion among some students. It is therefore important to introduce the students to the standard division method by highlighting this similarity, to pre-empt the possible confusion of the two methods. The advantages of using the standard division algorithm over the scaffolding method are then elaborated. Most students abhor procedures involving multiplication factors and making estimates of complex number multiplications. The use of simpler multiples makes the standard division algorithm easier. According to Reys et al (2014), proper explanation of this advantage creates a positive attitude and acceptance of the new method.
Once the students are receptive to the standard division algorithm, real division problems and their solutions are then gradually introduced starting with simpler problems. Giving the students a platform to solve more problems with limited assistance creates a better understanding and grasping of the content.
References
Auer, M. F. F. (2016). Solving multiplication and division problems: latent variable modeling of students' solution strategies and performance .
Cooper, J. (2015, February). Growth of mathematical knowledge for teaching–The case of long division. In CERME 9-Ninth Congress of the European Society for Research in Mathematics Education (pp. 2081-2088).
Long, C., & Qing-hong, Y. (2014, May). A group division method based on collaborative learning elements. In Control and Decision Conference (2014 CCDC), The 26th Chinese (pp. 1701-1705). IEEE.
Reys, R. E., Lindquist, M., Lambdin, D. V., & Smith, N. L. (2014). Helping children learn mathematics . John Wiley & Sons.
