Matemática Elementar

Problema 1

The number line below goes from $0$ to $1$ and is divided into 4 equal parts.

$0 \quad \underbrace{\rule{8cm}{0.4pt}}_{} \quad 1$ $\quad \frac{1}{4} \quad \frac{2}{4} \quad \frac{3}{4}$

What fraction is at point A, which is the second tick mark from $0$ ?

Point A $= \frac{\underline{\quad}}{\underline{\quad}}$

What fraction is at point B, which is the third tick mark from $0$ ?

Point B $= \frac{\underline{\quad}}{\underline{\quad}}$

Problemas

1. The number line below goes from $0$ to $1$ and is divided into 4 equal parts.

$0 \quad \underbrace{\rule{8cm}{0.4pt}}_{} \quad 1$ $\quad \frac{1}{4} \quad \frac{2}{4} \quad \frac{3}{4}$

What fraction is at point A, which is the second tick mark from $0$ ?

Point A $= \frac{\underline{\quad}}{\underline{\quad}}$

What fraction is at point B, which is the third tick mark from $0$ ?

Point B $= \frac{\underline{\quad}}{\underline{\quad}}$

2. Place each fraction on the number line by writing its letter at the correct tick.

Number line from $0$ to $1$ , divided into thirds:

$0 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 1$

A: $\frac{1}{3}$
B: $\frac{3}{3}$
C: $\frac{2}{3}$

Which fraction equals $1$ (the whole)? $\frac{\underline{\quad}}{\underline{\quad}}$

3. Are these fractions equivalent? Write Yes or No, then explain.

a) $\frac{1}{2}$ and $\frac{2}{4}$

Are they equivalent? $\underline{\quad}$

Both land on the same point on a number line? $\underline{\quad}$

b) $\frac{2}{3}$ and $\frac{1}{3}$

Are they equivalent? $\underline{\quad}$

4. Compare each pair of fractions. Write >, <, or =.

a) Same denominator — compare numerators:

$\frac{3}{8} \quad \square \quad \frac{5}{8}$

$\frac{7}{8} \quad \square \quad \frac{2}{8}$

b) Same numerator — compare denominators:

$\frac{1}{3} \quad \square \quad \frac{1}{6}$

$\frac{1}{4} \quad \square \quad \frac{1}{2}$

(Hint: when numerators are the same, the fraction with the smaller denominator is larger — it has bigger pieces.)

5. A ribbon is $1$ meter long. It is cut into 6 equal pieces.

What fraction of the ribbon is each piece? $\frac{\underline{\quad}}{\underline{\quad}}$

Place $\frac{2}{6}$ , $\frac{4}{6}$ , and $\frac{6}{6}$ on the number line:

$0 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 1$

Which fraction equals the whole ribbon? $\frac{\underline{\quad}}{\underline{\quad}}$

6. Order these fractions from least to greatest:

$\frac{3}{4}, \quad \frac{1}{4}, \quad \frac{4}{4}, \quad \frac{2}{4}$

$\underline{\quad}, \quad \underline{\quad}, \quad \underline{\quad}, \quad \underline{\quad}$

Explain how you know which is smallest: $\underline{\qquad\qquad\qquad\qquad\qquad}$

Gabarito — Fractions on the Number Line

1. The number line is divided into 4 equal parts. Each part is $\frac{1}{4}$ .

Second tick mark from $0$ : Point A $= \frac{2}{4}$
Third tick mark from $0$ : Point B $= \frac{3}{4}$

2. Thirds divide the number line into 3 equal parts:

A ( $\frac{1}{3}$ ) is at the first tick.
C ( $\frac{2}{3}$ ) is at the second tick.
B ( $\frac{3}{3}$ ) is at the third tick — the point labeled $1$ .

$\frac{3}{3} = 1$ (the whole). So $\frac{\mathbf{3}}{\mathbf{3}}$ equals $1$ .

3. a) $\frac{1}{2}$ and $\frac{2}{4}$ : Yes, they are equivalent. On a number line, both land exactly at the halfway point between $0$ and $1$ . ( $1 \times 2 = 2$ and $2 \times 2 = 4$ , so $\frac{1}{2} = \frac{2}{4}$ .) Yes, they land on the same point.

b) $\frac{2}{3}$ and $\frac{1}{3}$ : No. The numerators differ ( $2 \neq 1$ ), and they land at different points on the number line.

4. a) Same denominator — the larger numerator means the larger fraction:

$\frac{3}{8} \; \boxed{<} \; \frac{5}{8} \quad (3 < 5)$ $\frac{7}{8} \; \boxed{>} \; \frac{2}{8} \quad (7 > 2)$

b) Same numerator — smaller denominator means larger pieces, so larger fraction:

$\frac{1}{3} \; \boxed{>} \; \frac{1}{6} \quad \text{(thirds are bigger pieces than sixths)}$ $\frac{1}{4} \; \boxed{<} \; \frac{1}{2} \quad \text{(fourths are smaller pieces than halves)}$

5. Each of the $6$ equal pieces is $\frac{1}{6}$ of the ribbon.

Positions on the number line (divided into sixths):

$\frac{2}{6}$ is at the second tick
$\frac{4}{6}$ is at the fourth tick
$\frac{6}{6}$ is at the last tick (equals $1$ )

The whole ribbon = $\frac{6}{6}$ .

6. All fractions have denominator $4$ , so compare numerators: $1 < 2 < 3 < 4$ .

$\frac{1}{4}, \quad \frac{2}{4}, \quad \frac{3}{4}, \quad \frac{4}{4}$

The smallest is $\frac{1}{4}$ because it has only $1$ out of $4$ equal parts — the fewest pieces of all four fractions.