During the last few weeks, Robert was solving problems from 4th grade *Singapore Math*. He had already known most of the algorithms needed to perform mathematical operations. For instance, he knew how to find a fraction of a number.

With some restrictions.

He quickly could write that 3/8 of 40 is 15, but he would hesitate how to find 3/8 of 344 When he could do the division and multiplication in his head, the answer came immediately. If he couldn’t divide in his head, he was not sure what to do. It seemed as if Robert solved the first problem without realizing what mathematical operations he applied and thus he couldn’t extend the method to larger numbers. The only way I could address that was by slowing him down and having him name each math operation as he was performing it.

But Singapore Math introduced the fraction of the number not by presenting rigid algorithm, but as a few sections of a rectangle divided into congruent parts.

To find 3/8 of 40 or 3/8 of 344, the student drew a rectangle and divided it into eight equal parts . Then he shaded 3 of those parts. The whole rectangle representing 40 (or 344) was clearly divided by the number from the denominator and multiplied by numerator.

40 | |||||||
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40:8=5 5×3 =15

When Robert followed this method he didn’t have doubts what to do – he divided and he multiplied appropriately even when 40 was replaced by 344.

Moreover, similar drawing could be used to do the opposite, to find a number knowing the value of its fraction. for instance: Find a number knowing that 3/8 of that number equals 18 (or 345) . I didn’t practice with Robert solving those problems as I was not sure how to do it without confusing him. The Singapore Math offered easy solutions.

18 | |||||||
---|---|---|---|---|---|---|---|

18:3=6

6×8=48

Robert drew a rectangle and divided it into eight sections. He shaded three of them and **above just those three sections**, he wrote 18. Thus he found out easily that one section was 6 and the whole eight sections had to equal 48.

In the past we often used rectangles to represent sums, differences, products, and quotients when Robert had to solve so-called “word problems”. But I have never used them as a way to present the ideas behind the algorithm Robert knew already and the one, he did not learn yet.

*Besides Singapore Math, Robert is still practicing with calendar doing exercises based on 4th grade Saxon Math. He is also practicing other skills with the help from 4th grade Math Sylvan workbook. Although this workbook offers many opportunities to use skills in slightly different contexts, it also has errors, which its publisher is not willing to correct. This is the problem with hastily published workbooks for children and anxious parents, even publishers don’t take them too seriously. Sad.*