7.3.2B Translations & Reflections on a Coordinate Grid
Graph and describe translations and reflections of figures on a coordinate grid and determine the coordinates of the vertices of the figure after the transformation.
For example: The point (1, 2) moves to (-1, 2) after reflection about the y-axis.
Overview
Standard 7.3.2 Essential Understandings
Students connect their work on proportionality with their work on measurement of two - and three-dimensional shapes by investigating similar objects. They understand that if a scale factor describes how corresponding lengths in two similar objects are related, then the square of the scale factor describes how corresponding areas are related, and the cube of the scale factor describes how corresponding volumes are related. Students apply their work on proportionality to measurement in different contexts, including converting among different units of measurement to solve problems involving rates, such as motion at a constant speed. The big idea of this standard is 'scale factor.' Can students identify similar objects and the scale factor that defines this relationship? Can students use this identified scale factor to calculate measurements?
In addition to similarity, students will be expected to recognize and perform transformations on a coordinate grid. In 4th grade, students are introduced to transformation vocabulary and asked to identify types of transformations on objects. In 7th grade, students extend this work with reflections and translations to the coordinate plane and are introduced to and expected to use appropriate notation. In high school, students will extend the work further to rotations and dilations.
All Standard Benchmarks
7.3.2.1
Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors
For example: Corresponding angles in similar geometric figures have the same measure.
7.3.2.2
Apply scale factors, length ratios and area ratios to determine side lengths and areas of similar geometric figures.
For example: If two similar rectangles have heights of 3 and 5, and the first rectangle has a base of length 7, the base of the second rectangle has length $\frac{35}{3}$.
7.3.2.3
Use proportions and ratios to solve problems involving scale drawings and conversions of measurement units.
For example: 1 square foot equals 144 square inches.
For example: In a map where 1 inch represents 50 miles, .5 inch represents 25 miles.
7.3.2.4
Graph and describe translations and reflections of figures on a coordinate grid and determine the coordinates of the vertices of the figure after the transformation.
For example: The point (1, 2) moves to (-1, 2) after reflection about the y-axis.
7.3.2 Benchmark Group B - Translations & Reflections on a Coordinate Grid
7.3.2.4
Graph and describe translations and reflections of figures on a coordinate grid and determine the coordinates of the vertices of the figure after the transformation.
For example: The point (1, 2) moves to (-1, 2) after reflection about the y-axis.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Be able to, given a rule or word description, reflect a point or translate a point to its new location;
- Given a point and its new image (transformation), be able to identify the rule, either in rule format (x - 6, y - 2) or in description format (6 left and 2 down);
- Be able to reflect over y-axis, x-axis, or the lines y = x or y = -x;
- Be able to identify the new coordinates of the transformed image.
Work from previous grades that supports this new learning includes:
- Graph in the first quadrant;
- Identify coordinates;
- Draw simple geometric figures;
- Understand vertex/vertices.
NCTM Standards
- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships:
Understand relationships among the angles, side lengths, perimeters, areas and volumes of similar objects;
- Specify locations and describe spatial relationships using coordinate geometry and other representational systems:
Use coordinate geometry to represent and examine and properties of geometric shapes.
- Apply transformations and use symmetry to analyze mathematical situations:
Describe sizes, positions and orientations of shapes under informal transformations such as flips, turns, slides and scaling;
- Apply appropriate techniques, tools and formulas to determine measurements:
Solve problems involving scale factors, using ratio and proportion.
Common Core State Standards (CCSS)
7.G (Geometry) Draw, construct and describe geometrical figures and describe the relationships between them.
- 7.G.1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
8.G (Geometry) Understand congruence and similarity using physical models, transparencies, or geometry software.
- 8.G.3. Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.
- 8.G.4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
9.GSRT (Similarity, Right Triangles, and Trigonometry) Understand similarity in terms of similarity transformations.
- 9.GSRT.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Misconceptions
Student Misconceptions and Common Errors
- Students sometimes believe that a reflection over y = x or y = -x is like a rotation of it. The correct reflection will actually look wrong compared to the original drawing. One way to help with this is to have the students physically fold their paper to make sure it does reflect over the correct line
- Students may confuse specific math vocabulary (reflection, rotation, translation) with the more casual math vocabulary for these terms (flip, turn, slide) that they learned in elementary grades.
- Students may identify the incorrect line of symmetry (x-axis vs. y-axis) when reflecting.
- When labeling transformed images, students sometimes mistakenly apply the wrong label. Either they use the wrong letter, forget the prime notation (e.g., $A\rightarrow {A}$ instead of $A\rightarrow {A}'$) , or do the wrong number of primes if there is more than one transformation on the graph (ex. $A\rightarrow {A}'$ instead of $A\rightarrow {A}''$) .
- In translations, students can sometimes count to their points incorrectly or switch the order of the coordinates when writing the point down.
Vignette
In the Classroom
This activity explores transformations - specifically, reflections.
Teacher: Today we are going to work on transformations. Can someone tell me what a transformation is?
Student: A change.
Teacher: That's a very basic definition of it, but yes, it is a change. Transformations are geometric figures that have been changed by reflection, rotation and translation. We won't be looking at rotations in 7th grade at all, and we are just going to be looking at reflections today. We'll save translations for another day. Does anyone know what a reflection is?
Student: It's what you see when you look in the mirror.
Teacher: That's right, but what about math? How can we use reflections in math? I didn't bring my mirror, so how are we going to do reflections in math class?
Student: It's the same principle as with a mirror. It's taking an image and kind of like seeing the reverse of it, like a reflection.
Teacher: That's right. It is like a reverse of the figure.
Teacher: A reflection of a geometric figure is a mirror image of the object. If we were to place a mirror on a line of reflection, it will give the position of the reflected image.
Teacher: So how could we do that with a piece of paper if we don't have a mirror?
Student: We could just fold the paper and line up the figure and draw it that way?
Teacher: You're right. That is a great way to get started. Everyone take your piece of paper with the image on it and fold it on the x-axis. This is our line of reflection. Which axis is the x-axis?
Student: The one that goes across or horizontally because the y goes high, or up and down.
Teacher: Right. So everyone fold your paper on the x-axis. Do you see where your new figure would be if we reflected it over the x-axis? Go ahead and draw in the corresponding vertices of the new figure.
Student: So we just put a dot where the other dots line up?
Teacher: Yes. That is right. This isn't perhaps the fastest way, but it works, and it is accurate. Anyone have any ideas how we could do it another way?
Student: Well, point H is 4 units above the x-axis, so its reflection would be 4 units below the x-axis. G is also 4 units above the x-axis, so its reflection would also be 4 units below the x-axis. Point F is 7 units above the x-axis, so F' would be 7 units below the x-axis. Then once you have them all plotted, just connect the dots.
Teacher: Wow. Nice job! That is a great explanation. What happens if one of the points is on the x-axis, where is its reflection?
Student: It is itself. Since it is 0 units from the x-axis, then its reflection is also.
Teacher: So when we are done making our new image, F'G'H', how could we double check that it is correct, that we reflected correctly?
Student: Just fold the paper and it should line up with the FGH.
Teacher: Nice job. That's all there is to reflecting across the x-axis. You would do the same across the y-axis, but instead of going above or below the line of reflection, you would go left or right of the y-axis.
Resources
Teacher Notes
- Rotation is not a part of the transformations that students will be studying at this point. For now, students will make reflections over vertical lines, horizontal lines, and the lines y = x and y = -x.
- Sometimes students struggle with this topic because they need review on the coordinate system, plotting points and naming points, rather than the struggle with the concepts of transformations. A quick review of those topics might be beneficial.
- Teachers need to stress the point that students should physically fold their papers to ensure their new reflection is correct.
- Students may find that reflecting over the line y = x or the line y = -x is not as difficult as first thought. A simple way to do this is to count the number of units each point is from the line to be reflected over. If it is 3 units to the left of the line, then they count down 3 units from the line to get the reflection of point. By making a 90-degree angle from the original point over to the line and down (or up), they have reflected it. For example, reflect A over the line y = x. A is 4 units down from the reflection line so A' is 4 units to the left of the line. Alternatively, students might count 4 left and then 4 up; the order doesn't matter.
- Plot the following points on grid paper and translate (slide) the point and name the new coordinates. Use colored pencils to mark new points.
For example, point A: (-2, 4).
Slide 4 units right to point A1: (2, 4).
point B:(4, 0).
Slide 3 units down to point B1:(4, -3).
Source: 2007 Mississippi Mathematics Framework Revised Strategies, p. 41. - Make sure students know how to graph the equations y = x and y = -x.
- This standard does not include dilation, which is addressed in 7.3.1 with scaling and scalar drawings (scale factor, similar figures, etc.).
- Make sure students are aware that the original object (preimage) and its image are congruent - identical in every respect except for their position.
- To translate a figure is to simply slide it somewhere else. But in the move, you may not change the figure in any other way. You cannot rotate it, resize it, or flip it over. You may only slide it side to side, up and down. http://www.mathopenref.com/translate.html
Schielack, J. (2010). Focus in Grade 8, Teaching with Curriculum Focal Points. (p. 80-83). Reston, VA: National Council of Teachers of Mathematics.
Transformation applets
This site leads to applets related to visualizing transformations.
Additional Instructional Resources
- Geogebra Reflections
- Geogebra Translations
- Interactive geometric transformations lessons
- Web-based student activities
This applet allows the user to translate triangles, squares and parallelograms on both the x and y-axes. The user can also reflect the figure around x values, y values and the line x = y.
Note from writer: The terms below are first introduced in the Minnesota State Standards in 4th grade. Students will need to be retaught these terms in the context of doing these transformations on a coordinate grid.
reflection: a transformation that "flips" a figure over a mirror or reflection line.
Example: Reflection of the object ABDC on the left over the line on the right. This line is called the reflection line.
translation: moving without resizing; a 'slide'; every point of the shape must move the same distance and in the same direction.
Example: If the shape gets moved 30 units in the "x" direction, and 40 units in the "y" direction, we can write:
● (x,y) → (x + 30, y + 40)
This expression says "all the x and y coordinates will become x + 30 and y + 40"
Source: http://www.themathlab.com/dictionary/rwords/rwords.htm
transformation: the movement of a figure in a plane from its original position, the preimage, to a new position, the image. Also called a map. Transformations can occur when there is a reflection, a rotation, a translation, or a glide reflection of the original image.
notation: figures or symbols used to represent mathematical functions, objects, or ideas.
Examples:
J and J' (labels for points before and after transformation)
translation notation (x, y) → (x + 3, y - 2)
ABCD and A'B'C'D' (labels for a quadrilateral before and after transformation)
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- Can students distinguish between the original figure and the one after the transformation?
- Do students understand the difference between a reflection and a translation?
- If students are given the transformed image's coordinates and the rule used to transform it, can they come up with the original figure's coordinates?
- Mississippi Department of Education. (2007). 2007 Mississippi Mathematics Framework Revised Strategies, (p. 39). Jackson, MS: Mississippi Department of Education.
- Schielack, J. (2010). Focus in Grade 8, Teaching with Curriculum Focal Points. Reston, VA: National Council of Teachers of Mathematics.
- Schifter, D. (February, 1999). Learning Geometry: Some Insights Drawn from Teacher Writing. In Teaching Children Mathematics, 5 (5), 360-366.
- Pintozzi, C. (2001). Chapter 20: Transformations and Symmetry. In Mathematics Review. (p. 272-80). Woodstock, GA: American Book.
- Translation of a polygon
http://www.mathopenref.com/translate.html
Assessment
1.
Answer: a
Source: Minnesota Grade 7 Mathematics MCA-III Item Sampler Item, 2011, Benchmark 7.3.2.4
2.
The point (3, - 4) is translated 2 units to the right and 3 units up. What are the coordinates of the image of (3, - 4)?
Answer: (5, -1)
Source: New England Common Assessment Program, Released items, Grade 8, http://www.ride.ri.gov/Assessment/DOCS/NECAP/2010_Items/Gr8/NECAP_2010_Gr8_Math_Released_Items.pdf
3.
Answer: d
Source: California Geometry released exam questions
http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqgeomapr15.pdf
4.
Answer: b
Source: Texas 2008-09 Released TAKS exam 8th grade math
http://www.tea.state.tx.us/student.assessment/taks/released-tests/
Differentiation
- Review the coordinate grid system and plotting points with students before starting to perform transformations.
- Advise students to fold the paper to get the correct reflection; use different colors for each new figure. Use fairly large grid paper (1/2 inch, or not less than 1 cm grid paper) to plot and to be able to see the new coordinates. Give students multiple choice options rather than open-ended problems to help them see choices, which they can eliminate as they work.
- Print pages from this website out and have the students compile them in a booklet.
http://owensville.k12.mo.us/~smart_lessons/5/J.%20Mahurin/Math/Geometric%20Shapes/transformationandgraphing-sb.pdf - Have students trace the original figure to be reflected or translated. Have them cut it out. They can then take the cut-out figure and reflect it or translate it. By using a physical figure, the students will be less likely to make errors and be able to get the right shape in their image because they can reflect it or translate it and then trace the figure. (Make sure they have labeled the cut out shape with the correct letters, which they can then transfer to the new image.)
- Review reflection in the line y = x.
A reflection in the line y = x can be seen in the picture below in which A is reflected to its image A'. The general rule for a reflection in the y-axis: (A, B) → (B, A)
http://www.mathwarehouse.com/transformations/reflections-in-math.php
- A way to remember what translation means is "tranSLate means SLide"
Gliding through the Galaxy using Transformations
This website offers an elementary geared lesson on transformations in the Solar System, along with activities and ideas appropriate for struggling learners.
- Have students highlight the line that the figure is to be reflected over.
- Review which axis is the x axis and which is the y-axis ('y' goes 'high).
- Print these pages out and have the students compile them in a booklet:
http://owensville.k12.mo.us/~smart_lessons/5/J.%20Mahurin/Math/Geometric%20Shapes/transformationandgraphing-sb.pdf - A way to remember what translation means is "tranSLate means SLide"
- Have students do multi-step transformations for one figure. For example, translate the image using the rule (x+2, y-4), and then reflect that new image about the line, y = x.
- Have students reflect over lines other than y = x or y = -x
- Extend the lesson for students to learn about rotation.
Parents/Admin
Administrative/Peer Classroom Observation
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Students are: (descriptive list) |
Teachers are: (descriptive list) |
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identifying the line that is being reflected over; it is not necessarily to be able to name the equation for it. |
reminding students to fold their papers to double check that the reflection was done correctly. |
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using a coordinate grid to reflect and translate figures. |
requiring students to write the translations using proper notation (A'B'C'). |
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identifying the new points of reflection or translation. |
having students perform two translations on the same figure. |
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folding their papers to double check that their reflections are done correctly. |
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