Addition and Subtraction of Fractions

For personal home schoolroom use only. Permission must be granted by the author for any other usage.

Notes of Lessons
Lesson: Elementary Arithmetic.
Form II (Approximately 4th grade)
Time: 30 Minutes

Adapted and expanded by Richele Baburina from “Notes of Lessons” by Hilda M. Fountain, The Parents’ Review, Vol. 15, 1904, pp. 69-70.

Objects

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

II. To exercise their reasoning powers.

III. To make use of what they have previously learned of reducing 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 students to arrive at the rules by the investigation of samples.

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

Lesson

Step I.—Give each child four strips of paper of the same size. Show how to fold the first two strips into 6 equal parts by folding one first in half and then into thirds. Have the children fold two strips of paper into six equal parts. See if the children can tell you how to fold the remaining two strips of paper to be divided into four equal parts: by folding the strips in half and then in half again. Instruct the children to fold the remaining strips of paper into four equal parts.

Draw from the students 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 part the whole is divided into four equal parts and each part is ¹⁄₄ of the whole.

Step II.—Using the strips of paper divided into 6 equal parts, ask the children to count the parts, i.e., ¹⁄₆, ²⁄₆, ³⁄₆, ⁴⁄₆, etc. Ask them to show you ¹⁄₂ of the strip of paper. Using the strips of paper divided into four equal parts, ask the children to show you ¹⁄₂ of it and ³⁄₄ of it.

Step III.—Ask the children to add and subtract fractions while using their strips of paper, reducing to lowest terms if possible:

\(\frac{1}{6}+\frac{2}{6}=\frac{3}{6}\ or\ \frac{1}{2}\)
\(\frac{3}{6}+\frac{5}{6}=\frac{8}{6}\ or\ 1\frac{1}{3}\)
\(\frac{3}{6}+\frac{1}{6}-\frac{2}{6}=\frac{2}{6}\ or\ \frac{1}{3}\)
\(\frac{1}{4}+\frac{2}{4}=\frac{3}{4}\)
\(\frac{3}{4}+\frac{1}{4}-\frac{2}{4}=\frac{2}{4}\ or\ \frac{1}{2}\)

Step IV.—Give other examples with various denominators to be worked mentally.

\(\frac{1}{7}+\frac{4}{7}-\frac{3}{7}=\frac{2}{7}\)

Draw from the children that these fractions are all 7ths, i.e., they have all the same name.

\(\frac{8}{11}-\frac{3}{11}+\frac{2}{11}=\frac{7}{11}\)

Draw from the children that these fractions are all 11ths, i.e., they have all the same name.

\(\frac{5}{17}-\frac{3}{17}+\frac{13}{17}=\frac{15}{17}\)

Draw from the children that these fractions are all 17ths, i.e., they have all the same name.

Step V.—Ask: Can we add cats and dogs together? Elicit, “no.” How could we? We could if we gave them the same name, such as “animals.”

Elicit the rule that things of the same name can be added together. Five pennies and six cows cannot be added together; why? Get the reason from the children. These have different names and the six and five added together do not make eleven of any kind of thing.

This “same name” is the denominator, which comes from the Latin nomen which means name. The denominator names the kind of thing we are dealing with while the numerator, from the Latin numerare, “to number” numbers how many of the kind. So, if we have the fraction ²⁄₃, it means we have two thirds, just the same way we could have two dogs, two cats, two popsicles, two cups of coffee, etc.

Have students name the numerator and denominator of ¹⁄₆, ²⁄₃, ³⁄₄, ⁷⁄₈.

Step VI.—Tell the children to add a half dollar and a quarter together. Answer: Three quarters. Draw from them that they say three quarters because they know that a half-dollar is equal to two quarters. 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 with the half-dollar by writing it down as ²⁄₄. We have called it two quarters and given it the same name and denominator as the one quarter. 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 VII.—Give exercises on addition and subtraction of fractions to be worked mentally.

\(\frac{1}{2}+\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{5}{6}\)
\(\frac{1}{2}-\frac{1}{5}=\frac{5}{10}-\frac{2}{10}=\frac{3}{10}\)
\(\frac{4}{5}+\frac{1}{2}=\frac{8}{10}+\frac{5}{10}=\frac{13}{10}=1\frac{3}{10}\)
\(\frac{6}{15}-\frac{2}{5}=\frac{6}{15}-\frac{6}{15}=0\)

Step VIII.—Give examples of concrete quantities:

What is the difference between ²⁄₃ of a foot and ¹⁄₄ of a foot?
What name can we give it? Inches or 12.
\(\frac{8}{12}-\frac{3}{12}=\frac{5}{12}\)
How much is ²⁄₃ and ³⁄₈ of a day?
What name can we give it? Hours or 24.
\(\frac{16}{24}+\frac{9}{24}=\frac{25}{24}\)
If we have 25 hours, what have we? 1 day and 1 hour.
What is the difference between ⁵⁄₆ of a year and ³⁄₄ of a year?
What name can we give? Months or 12.
\(\frac{10}{12}-\frac{9}{12}=\frac{1}{12}\)
Answer: 1 month.

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 IX.—Give examples of abstract fractions involving addition and subtraction to be worked on paper or a slate, the teacher working the first.

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

\(\frac{5}{6}-\frac{1}{4}+\frac{2}{3}=\frac{10}{12}-\frac{3}{12}+\frac{8}{12}=\frac{7}{12}+\frac{8}{12}=\frac{15}{12}=1\frac{3}{12}=1\frac{1}{4}\)

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