Question

1. a. plot the following points on the number line without measuring.

i. $\frac{1}{3}$
ii. $\frac{5}{6}$
iii. $\frac{7}{12}$

Explanation:

Step1: Find a common denominator

To plot the fractions \(\frac{1}{3}\), \(\frac{5}{6}\), and \(\frac{7}{12}\) on the number line, we first find a common denominator. The least common denominator of 3, 6, and 12 is 12.

Step2: Convert each fraction

• For \(\frac{1}{3}\): Multiply numerator and denominator by 4. \(\frac{1\times4}{3\times4}=\frac{4}{12}\)
• For \(\frac{5}{6}\): Multiply numerator and denominator by 2. \(\frac{5\times2}{6\times2}=\frac{10}{12}\)
• For \(\frac{7}{12}\): It is already with denominator 12.

Step3: Analyze the number line intervals

The number line has marks at 0, \(\frac{1}{2}\) (which is \(\frac{6}{12}\)), and 1 (which is \(\frac{12}{12}\)). We divide the interval between 0 and \(\frac{1}{2}\) (0 to \(\frac{6}{12}\)) and between \(\frac{1}{2}\) and 1 (\(\frac{6}{12}\) to \(\frac{12}{12}\)) into 12 - 6 = 6 equal parts? Wait, actually, since the common denominator is 12, we can think of the number line from 0 to 1 as divided into 12 equal parts (since 12 is the LCD). So each part is \(\frac{1}{12}\).

Step4: Plot each fraction

• \(\frac{4}{12}\) (which is \(\frac{1}{3}\)): Starting from 0, move 4 units of \(\frac{1}{12}\) each.
• \(\frac{7}{12}\): Starting from 0, move 7 units of \(\frac{1}{12}\) each.
• \(\frac{10}{12}\) (which is \(\frac{5}{6}\)): Starting from 0, move 10 units of \(\frac{1}{12}\) each.

We can also compare with the existing marks:

• \(\frac{1}{3}=\frac{4}{12}\) is less than \(\frac{1}{2}=\frac{6}{12}\), so it's between 0 and \(\frac{1}{2}\).
• \(\frac{7}{12}\) is greater than \(\frac{1}{2}=\frac{6}{12}\) (since 7 > 6) and less than 1, so it's between \(\frac{1}{2}\) and 1.
• \(\frac{5}{6}=\frac{10}{12}\) is also between \(\frac{1}{2}\) and 1, and closer to 1 than \(\frac{7}{12}\) is (since 10 is closer to 12 than 7 is).

Answer:

To plot the points:

• \(\boldsymbol{\frac{1}{3}}\) (or \(\frac{4}{12}\)): 4 twelfths from 0, between 0 and \(\frac{1}{2}\).
• \(\boldsymbol{\frac{7}{12}}\): 7 twelfths from 0, between \(\frac{1}{2}\) (6 twelfths) and 1.
• \(\boldsymbol{\frac{5}{6}}\) (or \(\frac{10}{12}\)): 10 twelfths from 0, between \(\frac{1}{2}\) and 1, closer to 1.

(Visually, on the number line, mark \(\frac{1}{3}\) at 4/12, \(\frac{7}{12}\) at 7/12, and \(\frac{5}{6}\) at 10/12.)