Point A is translated to point B. Write the coordinates of both points. What do you notice?

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

Coordinates of point A: $(2,3)$
Coordinates of point B: $(6,2)$
To get from A to B, the x-coordinate changes from 2 to 6 (an increase of 4), and the y-coordinate changes from 3 to 2 (a decrease of 1).
What you notice: The point has been translated 4 units to the right and 1 unit down. The x-coordinate increased by 4, and the y-coordinate decreased by 1.

The coordinates are $\boxed{\text{A}=(2,3), \text{B}=(6,2)}$.
What you notice: $\boxed{\text{The x-coordinate increased by 4, and the y-coordinate decreased by 1.}}$.

• Point C is translated to point D.

Point C is at $(3,3)$.
Point D is at $(3,1)$.

Coordinates of point C: $(3,3)$
Coordinates of point D: $(3,1)$
To get from C to D, the x-coordinate remains the same (3), and the y-coordinate changes from 3 to 1 (a decrease of 2).
What you notice: The point has been translated 2 units down. The x-coordinate stayed the same, and the y-coordinate decreased by 2.

The coordinates are $\boxed{\text{C}=(3,3), \text{D}=(3,1)}$.
What you notice: $\boxed{\text{The x-coordinate stayed the same, and the y-coordinate decreased by 2.}}$.

• Teddy plots a point that has the coordinates $(5, 4)$. He translates the point so that it has the same x-coordinate, but a different y-coordinate. Has he translated the point up/down or left/right?

If the x-coordinate remains the same, the point has not moved horizontally (left or right).
If the y-coordinate changes, the point has moved vertically (up or down).

Since the x-coordinate is the same and the y-coordinate is different, Teddy has translated the point up/down.

He has translated the point $\boxed{\text{up/down}}$.

• The point is translated 4 squares to the right and 2 squares down.

The initial point is at $(4,2)$.

Translate 4 squares to the right: add 4 to the x-coordinate.
Translate 2 squares down: subtract 2 from the y-coordinate.
New x-coordinate: $4+4 = 8$
New y-coordinate: $2-2 = 0$
The new coordinates are $(8,0)$.

Initial coordinates: $(4,2)$
New coordinates: $(8,0)$
What you notice: The x-coordinate increased by 4, and the y-coordinate decreased by 2.

The coordinates are $\boxed{\text{Initial}=(4,2), \text{New}=(8,0)}$.
What you notice: $\boxed{\text{The x-coordinate increased by 4, and the y-coordinate decreased by 2.}}$.

• Complete the table.

Original coordinates: $(5,2)$
New coordinates: $(2,5)$
Change in x: $2-5 = -3$ (3 left)
Change in y: $5-2 = 3$ (3 up)
Translation: 3 left and 3 up.

Original coordinates: $(6,7)$
Translation: 1 left and 1 down
New x-coordinate: $6-1 = 5$
New y-coordinate: $7-1 = 6$
New coordinates: $(5,6)$.

The completed table is:

• A triangle is translated so that point A translates to point B.

Point A is at $(1,6)$.
Point B is at $(5,4)$.

Change in x: $5-1 = 4$ (4 units right)
Change in y: $4-6 = -2$ (2 units down)
The translation is 4 units right and 2 units down.

Let's call the bottom-left vertex $V_1$ and the bottom-right vertex $V_2$.
$V_1 = (1,3)$
$V_2 = (2,3)$

For $V_1' = (1+4, 3-2) = (5,1)$
For $V_2' = (2+4, 3-2) = (6,1)$

The coordinates of the other vertices of the translated triangle are $\boxed{(5,1) \text{ and } (6,1)}$.

How did you work this out?
I first found the translation vector by comparing the coordinates of point A $(1,6)$ and its translated position B $(5,4)$. The x-coordinate changed by $5-1=4$ (4 units right), and the y-coordinate changed by $4-6=-2$ (2 units down). Then, I identified the coordinates of the other two vertices of the original triangle, which are $(1,3)$ and $(2,3)$. Finally, I applied the same translation (4 units right, 2 units down) to each of these vertices to find their new coordinates: $(1+4, 3-2) = (5,1)$ and $(2+4, 3-2) = (6,1)$.