Level H Lesson 22
Reflections and Translations
In lesson 22 the objective is, the student will reflect and translate geometric figures
in a four-quadrant graph.
We will have three essential questions that will be guiding our lesson. Number 1,
how do the coordinates of a figure change when you reflect a figure over the x-axis?
Number 2, how do the coordinates of a figure change when you reflect a figure over
the y-axis? Number 3, when translating a figure, how do the coordinates change if
both numbers in the translation are negative?
The SOLVE problem for the lesson is, Davis is using a coordinate plane to make
designs. He began with a trapezoid at the following points: (1, 6), (5, 4), (5,1) and 1, 1). He is gong to translate his figure to make the design. If the first translation is,
negative 5, negative 2, what are the coordinates of the design?
We’re going to begin by completing the S step to study the problem. First we’re going
to underline the question. What are the coordinates of the design? We’re then going
to complete the statement this problem is asking me to find, the coordinates after
the shape is translated.
Right now you’re going to pair up and face your partner. Student A or the first
student, is going to pretend they are looking in a mirror. Student B will be the
reflection of the student A. Student A should kick their right leg out. What leg should
student B kick out to be a reflection? That would be your left leg. So if you were
looking in a mirror which is normally where you would see your reflection and you
kicked out your right leg. It would look like your left leg was moving.
If you touch your head with your left arm, what arm does your mirror image appear to
be touching your head with? That would be your right arm. The opposite side of the
body, is always shown in the reflection. Just as student A looked at student B and it
looked as the opposite leg was kicking and it looked as if the opposite arm was being
lifted. We can also call a reflection a flip. If you have trouble remembering that
you’re going to flip when you do a reflection, you’ll notice that flip and reflection both
have the letters fl.
We’ve cut out a quadrilateral A, B, C, D. I’m going to place my quadrilateral so that
point A is at 4, 7. I’m then going to copy the quadrilateral and trace on to my
coordinate grid. Let’s think again about the mirror. If you were to move closer to the
mirror, what would your reflection do? Your reflection would move closer.
In this example we’re going to have our y-axis be the mirror. Our point of reflection
for A should be the same distance from the mirror or the y-axis as we reflect it across
the y-axis. When you move closer to the mirror do you or the reflection move up or
down? No, when we reflect across the y-axis point A should be the same distance
from the y-axis as it is from the original point. When you move closer to the mirror do
you or the reflection move up or down? No. So when we reflect across the y-axis the
y coordinate will stay the same. So the ordered pair for our reflection for A will be
(negative 4, 7). When we have a reflection, we’re going to label that point as A prime.
We know that our point A was 4 spaces away from the y-axis. So A prime also must
be 4 spaces away from the y-axis.
Our original coordinates for B, C and D are (7, 7), point C is at (7, 2) and point D is at
(2, 4). I’m now going to take my original quadrilateral and I’m going to flip it,
remember our reflection is a flip, and I’m going to copy that in a different color from
the original and then I’m going to list the coordinates. I’ve now copied my reflection
or my flip of my quadrilateral and each point and its prime are equal distance from
the y-axis. Point B prime is now at (negative 7, 7). C prime is now at (negative 7, 2)
and D prime is at (negative 2, 4).
Now we’re going to pretend that our x-axis is the mirror. Instead of there being a
vertical line the mirror is now a horizontal line. If you flip the quadrilateral over the xaxis will the x-coordinates change? And the answer is No. We’re now going to
determine the coordinates of the A double prime. We need to do that by determining
how far A is away from the x-axis. Point A is 7 away from the x-axis, A double prime
should be 7 spaces away from the x-axis also. Our question was, will the x-coordinate
change when the trapezoid is flipped over the x-axis? And the answer is no. The xcoordinate for both A and A double prime will still be 4. How many spaces away from
the x-axis is point A? It is 7. How many spaces away from the mirror should point A
double prime be? 7 Since A double prime is the opposite direction from the mirror,
what should be y-coordinate of A double prime be? It will be negative 7. The new
ordered pair for A double prime will be (4, negative 7).
Our original coordinates for B, C and D are (7, 7), (7, 2) and (2, 4). We’re now going
to take our quadrilateral and we’re going to flip it over our x-axis and draw the
reflection. I’ve now flipped and copied my original quadrilateral with the A double
prime point of (4, negative 7). My new coordinates are B double prime (7, negative
7), C double prime (7, negative 2), D double prime (2, negative 4). We now have our
original figure, the reflection over the y-axis and the reflection over the x-axis.
We’re now going to be working with Translations. When you translate words from
English into another language, the meaning stays the same, but the words change.
When you translate a figure on the coordinate plane, the shape stays the same, but
the coordinates change. This can also be called a slide.
We’re going to place the quadrilateral that we cut out back into its original position,
with point A at (4, 7). We’re then going to slide the quadrilateral 3 spaces to the
right, and then 4 spaces up. This translation can be written as 3 , 4 using the carrot
to enclose the two values. These are both positive numbers. So when the
translation asks us to go positive 3 and positive 4 we’re going to go right and up. We
move to the right and up. We’re going to copy the quadrilateral on to the coordinate
grid in a different color. I’ve now completed coping of my new figure that was created
from the translation of 3, 4. My original points were (4, 7) (7, 7) (7, 2) and (2, 4).
My new points are (7, 11) (10, 11) (10, 6) and (5, 8). Each of my new points we’re
calling A triple prime. Our reflection of over the y-axis was our prime. Our reflection
over the x-axis we’re calling our double prime and this new translation we’re calling
our triple prime. Remember that our translation was (3, 4). Notice from A to A triple
prime we added 3. 4 plus 3 is 7. To get our y coordinate we increased by 4. 7 plus
4 is 11. To do our B triple prime our x coordinate was increased by 3. The y
coordinate was increased by 4. From C to C triple prime our x coordinate was
increased by 3 and our y coordinate was increased by 4. And finally for D and D triple
prime x coordinate was increased by 3, and our y coordinate was increased by 4.
We’re now going to put our quadrilateral back in the original position and we’re going
to do a different translation. We’re going to do a translation of negative 10, negative
8. When we did a positive we moved to the right and positive y coordinate meant
moving up. When we have a negative x coordinate we’re going to moved to the left
10. Our negative 8 in the y coordinate position indicates we’re going to go down 8.
Our new position for A quadruple prime is going to be at negative 6 and negative 1.
You’re now going to take a minute and copy your original quadrilateral in a new color.
Our coordinates of our new points are (negative 6, negative 1) (negative 3, negative
1) (negative 3, negative 6) (negative 8, negative 4). In order to find each x-coordinate
we added negative 10, or we subtracted 10. In order to find each new y-coordinate
we added negative 8, or subtracted 8.
We’re now going to be reviewing our reflection and translations from the lesson. Our
first example we had a reflection over the x-axis. We have a quadrilateral A, B, C, D.
And it’s the reflection A prime, B prime, C prime, D prime. When we reflect over the xaxis our x coordinates are going to stay the same. A prime is still at negative 5, but
the y coordinate which was positive 5, is now negative 5. Point B was at (negative 1,
5). Reflected over the x-axis (negative 1, negative 5). Point C was at (negative 6,
positive 2). C prime is now at (negative 6, negative 2). D was at (negative 2, 2). It is
now at (negative 2, negative 2). So when we reflect over the x-axis our x coordinate
stays the same, and the y coordinate is the opposite.
Lets look at the reflection over the y-axis. We have 2 quadrilaterals again. And we
have quadrilateral E, F, G, H, and we’re reflecting it across the y-axis. When we reflect
across the y-axis our y coordinate stays the same. E (negative 6, negative 1) is now E
prime (6, negative 1). So our x coordinate is the opposite. F is (negative 2, negative
2) and it is now at (positive 2, negative 2). G is at (negative 2, negative 4) and G
prime is at (positive 2, negative 4). Point H is at (negative 6, negative 5) and it’s H
prime is reflected across the x-axis is (6, negative 5). So our rule when reflecting
across the y-axis is the y coordinate will stay the same and the x coordinate will be
the opposite.
Our last example is a Translation of (8, negative 7). Because our x coordinate is
positive, it means we’ll be moving towards the right. Our y coordinate is negative so
we’ll be moving down. Our original coordinate point J was (negative 5, 6) When we
add 8 to negative 5 we have 3. 6 minus 7 is negative 1. Point K was at negative 3.
Negative 3 plus 8, is positive 5. 6 minus 7 again is negative 1. Point L had an x
coordinate of negative 5, by adding 8 we have a new x coordinate of 3. The y
coordinate was 1 minus 7 is a negative 6. Our last point M has an x coordinate of
negative 3 by adding 8 we have a positive 5. Our y coordinate was 1 minus 7 is
negative 6. In our last example x, y was changed by x adding 8, and y subtracting 7
or adding a negative 7. Whatever the value is to the x coordinate either subtracting
or adding and the same thing goes for the y coordinate you’re either going to add or
subtract the value of the translation.
We’ll now go back to the SOLVE problem from the beginning of the lesson. Davis is
using a coordinate plane to make designs. He began with a trapezoid at the
following points: (1, 6), (5, 4), (5, 1) and (1, negative 1). He is going to translate his
figure to make the design. If the first translation is <negative 5, negative 2>, what
are the coordinates of the design?
At the beginning of the lesson we completed the S step by underlining the question
and completing the statement this problem is asking me to find the coordinates after
the shape is translated.
When we organized the facts the first thing we’re gong to do is identify our facts.
Davis is using a coordinate plane to make designs. That’s a fact. He began with a
trapezoid at the following points: (1, 6), (5, 4), (5, 1) and (1, negative 1). That is a
fact. He’s going to translate his figure to make the design. That is a fact. If the
translation is <negative 5, negative 2>, is our final fact. We’re then going to go back
and eliminate any unnecessary facts. We don’t need to know that he is using a
coordinate plane in order to find the coordinates of the translation. We’re also going
to cross out the fact that he’s translating his figures. Because we have the
information we need in the translation of < negative 5, negative 2>.We complete the
O step by listing the coordinates before the translation and what the actual
translation will be.
We’re going to move to the L step where we line of a plan by choosing a operation.
We know that we’re going to add the values of the translation to each of our points.
Add the first value in the translation to all the x-values, and add the second value in
the translation to all the y-values.
When we verify our plan with action, we want to start by estimating our answer. Our
estimate here is not an actual value, but the idea that all of the coordinates will be
less than the original. The reason we say that is because we’re adding but we’re
adding a negative value. Then we carry out our plan. Our first coordinate (1, 6) we
subtract 5, 1 minus 5 is negative 4, and 6 minus 2 is 4. When we add negative 5,
negative 2 to our coordinates (1, 6) our new coordinate is negative (4, 4). When we
add a negative 5, to 5 our x-coordinate will be 0. When we add negative 2 to 4 our y
coordinate is 2. Our original point 5, 1 becomes 0, negative 1. When we add
negative 5, negative 2. Our last point (1, negative 1) is translated to (negative 4,
negative 3).
We’re now going to move to the E step. We’re going to examine your results. Does
your answer make sense? Compare your answer to the questions. Yes it does,
because I translated the figure by adding. Is your answer reasonable? Compare your
answer to the estimate. Yes, because all of my coordinates are smaller than they
were. Is your answer accurate? We go back and check our work. Write your answer
in a complete sentence. The coordinates of the shape after the translation will be
(negative 4, 4), (0, 2), (0, negative 1), and (negative 4, negative 3).
We’re now going to go back and answer the essential questions from the beginning
of the lesson.
How do the coordinates of a figure change when you reflect a figure over the x-axis?
The x-coordinate stays the same, the y-coordinate is the opposite.
How do the coordinates of a figure change when you reflect a figure over the y-axis?
The y-coordinate stays the same, the x-coordinate is the opposite.
When translating a figure how do the coordinates change if both numbers in the
translation are negative? The coordinates will have a lower value for both the x and y
coordinates.