Notes of Lessons: Arithmetic, Class II

[We have thought that it might be of use to our readers (in their own families) to publish from month to month during the current year, Notes of Lessons prepared by students of the House of Education for the pupils of the Practising School. We should like to say, however, that such a Lesson is never given as a tour de force, but is always an illustration or an expansion of some part of the children’s regular studies (in the Parents’ Review School), some passage in one or other of their school books.—Ed.]

Group: Mathematics • Class II • Average Age: 9 1/2 yrs • Time:30 min

By Hilda M. Fountain
The Parents’ Review, 1904, pp. 69-70

Addition and Subtraction of Fractions

Objects

I. To assist the cultivation of the pupils’ mental habits of attention, promptness, and accuracy.

II. To exercise their reasoning powers.

III. To make use of what they have previously learnt of reduction of fractions to the lowest common denominator and to lowest terms in teaching the new rules for the addition and subtraction of fractions.

IV. To get the pupils to arrive at the rules by the investigation of examples.

V. To help the pupils to realize the fractions they deal with by means of concrete examples.

Lesson

Step I.—Give each child two strips of paper of the same size, each divided into a number of equal parts, 6 and 4. Draw from the pupils that in one case the whole is divided into 6 parts, and that each part is the fraction ¹⁄₆ of the whole, in the other the whole is divided into four equal parts and each part is ¹⁄₄ of the whole.

Step II.—Ask the children to add fractions:

\(\frac{1}{6}+\frac{2}{6},\ \frac{3}{6}+\frac{5}{6},\ \frac{3}{6}+\frac{1}{6};\ \ \ \ \frac{1}{4}+\frac{2}{4},\ \frac{3}{4}+\frac{1}{4}-\frac{2}{4},\)

illustrating with the bits of paper. Then give other examples with various denominators to be worked mentally, these also giving exercise in addition and subtraction.

\(\frac{1}{7}+\frac{4}{7}-\frac{3}{7},\ \frac{8}{11}-\frac{3}{11}+\frac{2}{11},\ \frac{5}{17}-\frac{3}{17}+\frac{13}{17},\ \frac{18}{25}+\frac{9}{25}-\frac{1}{25}.\)

Step III.—Draw from the children that these fractions are all sevenths, or all elevenths, or all seventeenths, i.e., they have all the same name because the denominators are the same. Ask for the Latin word for name, nomen, nominis, to shew the connection. Hence elicit the rule that things ofthe same name can be added together. Give examples of quantities which cannot be added together. Five shillings and six cows cannot be added together; get the reason from the children. The things have different names and the six and five added together do not make eleven of any kind of thing.

Step IV.—Tell the children to add a halfpenny and a farthing together. Ans.: Three farthings. Draw from them that they say three farthings, because they know that a halfpennyis equal to two farthings. Write on the blackboard:

\(\frac{1}{2}+\frac{1}{4}=\frac{2}{4}+\frac{1}{4}=\frac{3}{4}\)

Ask what change we have made in the halfpenny by writing it down as ²⁄₄. We have called it two farthings and given it the same name and denominator as the one farthing. And then we have added the numerators (2 + 1) for the numerator of the answer. Get the children to give the rule: reduce fractions to their lowest common denominator before adding or subtracting them.

Step V.—Give exercises on addition and subtraction of fractions to be worked mentally.

\(\frac{1}{2}+\frac{1}{3},\ \frac{1}{2}-\frac{1}{5},\ \frac{4}{5}+\frac{1}{2},\ \frac{6}{15}-\frac{2}{5}.\)

Give examples of concrete quantities: What is the difference between ²⁄₃ of a foot and ¹⁄₄ of a foot? How much is ²⁄₃ and ³⁄₈ of a day? What is the difference between ⁵⁄₆ of a year and ³⁄₄ of a year? Let the children notice that in these cases the denominator is the number of inches in a foot, hours in a day, or months in a year.

Step VI.—Give examples of abstract fractions involving addition and subtraction to be worked on the board, the teacher working the first.

\(3\frac{3}{7}+\frac{3}{4}-\frac{5}{14}=3\frac{12+21-10}{28}=3\frac{33-10}{28}=3\frac{23}{28}\ answer.\)

\(\frac{5}{6}+\frac{8}{9}-\frac{3}{7}\ \ \ \ \frac{5}{9}-\frac{1}{4}+\frac{2}{3}\)

Let the children find the lowest common denominator and see that they express the answer in the lowest terms.

Editor’s Note: For a modern adaptation of this lesson, see Addition and Subtraction of Fractions by Richele Baburina.