Matemática Elementar

Problema 1

The coordinate plane has two number lines that cross at the origin $(0, 0)$ .

The horizontal line is the x-axis.
The vertical line is the y-axis.
An ordered pair is written as $(x, y)$ .

Name the coordinates of each point:

| Point | Coordinates | |-------|-------------| | A — 3 units right, 4 units up | $(\underline{\quad},\; \underline{\quad})$ | | B — 5 units right, 0 units up | $(\underline{\quad},\; \underline{\quad})$ | | C — 0 units right, 2 units up | $(\underline{\quad},\; \underline{\quad})$ | | D — 6 units right, 6 units up | $(\underline{\quad},\; \underline{\quad})$ |

Problemas

1. The coordinate plane has two number lines that cross at the origin $(0, 0)$ .

The horizontal line is the x-axis.
The vertical line is the y-axis.
An ordered pair is written as $(x, y)$ .

Name the coordinates of each point:

2. Plot each point on the coordinate plane (first quadrant, both axes from $0$ to $10$ ).

$P = (2, 5)$ : move $2$ right, $5$ up
$Q = (7, 3)$ : move $7$ right, $3$ up
$R = (4, 4)$ : move $4$ right, $4$ up
$S = (0, 8)$ : move $0$ right, $8$ up

What do you notice about the position of point $S$ ? $\underline{\qquad\qquad\qquad}$

3. A city map uses a coordinate grid. The library is at $(3, 7)$ and the school is at $(3, 2)$ .

a) Which coordinate is the same for both buildings? $\underline{\quad}$

b) What does this tell you about where they are located?

$\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$

c) How far apart are they along the y-axis?

$|7 - 2| = \underline{\quad} \text{ blocks}$

4. A treasure map has four landmarks:

Tree: $(1, 1)$
Rock: $(5, 1)$
Cave: $(5, 6)$
Treasure: $(1, 6)$

Plot these four points. Connect them in order (Tree → Rock → Cave → Treasure → Tree).

What shape do they form? $\underline{\qquad\qquad}$

Find the length of each side:

Bottom (Tree to Rock): $|5 - 1| = \underline{\quad}$ units
Right side (Rock to Cave): $|6 - 1| = \underline{\quad}$ units

5. The table shows how a plant grows over time.

| Weeks ( $x$ ) | Height in cm ( $y$ ) | |------------|-------------------| | $0$ | $2$ | | $1$ | $4$ | | $2$ | $6$ | | $3$ | $8$ | | $4$ | $10$ |

Write the ordered pairs: $\underline{\quad\quad}, \underline{\quad\quad}, \underline{\quad\quad}, \underline{\quad\quad}, \underline{\quad\quad}$

Plot the points on a coordinate plane with $x$ from $0$ to $5$ and $y$ from $0$ to $12$ .

How much does the plant grow each week? $\underline{\quad}$ cm

6. Point $A$ is at $(2, 3)$ . Point $B$ is at $(8, 3)$ .

a) What is the distance from A to B?

$|8 - 2| = \underline{\quad} \text{ units}$

b) Point $C$ is at $(8, 9)$ . What is the distance from B to C?

$|9 - 3| = \underline{\quad} \text{ units}$

c) What is the perimeter of triangle $ABC$ if $AC = 9$ units (given)?

$P = AB + BC + AC = \underline{\quad} + \underline{\quad} + \underline{\quad} = \underline{\quad} \text{ units}$

Gabarito — The Coordinate Plane

1. The first number in an ordered pair is the x-coordinate (how far right); the second is the y-coordinate (how far up).

| Point | Coordinates | |-------|-------------| | A | $(3,\; 4)$ | | B | $(5,\; 0)$ | | C | $(0,\; 2)$ | | D | $(6,\; 6)$ |

$P = (2, 5)$ : 2 right, 5 up
$Q = (7, 3)$ : 7 right, 3 up
$R = (4, 4)$ : 4 right, 4 up
$S = (0, 8)$ : 0 right, 8 up

Point $S$ lies on the y-axis because its x-coordinate is $0$ — it is not to the right of the origin at all.

3. a) The x-coordinate is the same: both have $x = 3$ .

b) The library and school are on the same vertical line (both are $3$ blocks to the right on the grid, directly above/below each other).

c) $|7 - 2| = \mathbf{5}$ blocks apart along the y-axis.

4. The four points form a rectangle.

Bottom (Tree to Rock): $|5 - 1| = \mathbf{4}$ units
Right side (Rock to Cave): $|6 - 1| = \mathbf{5}$ units

(The shape has two sides of length $4$ and two sides of length $5$ , confirming it is a rectangle.)

5. Ordered pairs: $(0, 2),\; (1, 4),\; (2, 6),\; (3, 8),\; (4, 10)$

The plant grows $\mathbf{2}$ cm each week (the y-value increases by $2$ every time $x$ increases by $1$ ).

6. a) $A = (2, 3)$ , $B = (8, 3)$ — same y-coordinate, so the distance is horizontal:

$|8 - 2| = \mathbf{6} \text{ units}$

b) $B = (8, 3)$ , $C = (8, 9)$ — same x-coordinate, so the distance is vertical:

$|9 - 3| = \mathbf{6} \text{ units}$

c) $AC = 9$ units (given).

$P = 6 + 6 + 9 = \mathbf{21} \text{ units}$