Answer to Question #342136 in Algebra for licks

Question

In modern mathematics teaching we want relational understanding, in addition to instrumental understanding. Describe how a teacher would teach for both instrumental and relational understanding of long division 357÷19.

Teaching long division of 357÷19 for instrumental understanding

Teaching long division of 357÷19 for relational understanding

 

 

Answer 1

Instrumental understanding:

Step 1:

Start by setting it up with the divisor 19 on the left side and the dividend 357 on the right side like this:

Step 2:

The divisor (19) goes into the first digit of the dividend (3), 0 time(s). Therefore, put 0 on top:

    

Step 3:

Multiply the divisor by the result in the previous step (19 x 0 = 0) and write that answer below the dividend.

Step 4:

Subtract the result in the previous step from the first digit of the dividend (3 - 0 = 3) and write the answer below.

Step 5:

Move down the 2nd digit of the dividend (5) like this:

Step 6:

The divisor (19) goes into the bottom number (35), 1 time(s). Therefore, put 1 on top:

Step 7:

Multiply the divisor by the result in the previous step (19 x 1 = 19) and write that answer at the bottom:

Step 8:

Subtract the result in the previous step from the number written above it. (35 - 19 = 16) and write the answer at the bottom.

Step 9:

Move down the last digit of the dividend (7) like this:

Step 10:

The divisor (19) goes into the bottom number (167), 8 time(s). Therefore put 8 on top:

Step 11:

Multiply the divisor by the result in the previous step (19 x 8 = 152) and write the answer at the bottom:

Step 12:

Subtract the result in the previous step from the number written above it.(167 - 152 = 15) and write the answer at the bottom.

You are done, because there are no more digits to move down from the dividend.

The answer is the top number and the remainder is the bottom number.

Therefore, the answer to 357 divided by 19 calculated using Long Division is:

18 Remainder(15)

The answer to 357 divided by 19 can also be written as a mixed fraction as follows:

$18 \cfrac {15}{19}$

Note that the numerator in the fraction above is the remainder and the denominator is the divisor.

Answer for relational understunding.

We use base-10 (Dienes) blocks for relational understanding.

each cell in the image is equal to 1 and the long line consists of 10 cells, so it is equal to 10 The line consisting of 7 cells is equal to 7 The line consisting of 9 cells is equal to 9

357 is represented by 19s. To do this, remove 1 cell (given in red) from every second line in 357 and continue it by setting it to the side:

The result is the following:

we count how many 19 taliks there are. that is, as follows:

that is, 18 of the 19s out of 357. and 15 residues remain.

Answer: 18 and residue is 15