7.3.2.A Similarity & Scaling in 2 Dimensions
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.
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 $\frac{35}{3}$.
Use proportions and ratios to solve problems involving scale drawings and conversions of measurement units.
For example: 1 square foot equals 144 square inches.
Another example: In a map where 1 inch represents 50 miles, $\frac{1}{2}$ inch represents 25 miles.
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 A - Similarity & Scaling in 2 Dimensions
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.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Determine and list corresponding sides and angles of similar figures;
- Find a scale factor relationship between similar figures (divide corresponding sides);
- Create and use ratios to find corresponding side lengths;
- Determine similarity between two figures by setting up ratios of corresponding side lengths, and if they are equivalent (decimal or fractional), understanding that the two figures are similar;
- Find missing values on similar figures by scaling values up (or down);
- List properties of images that are similar, and also properties of images that are not similar;
- Given a scale factor, tell how many times the original figure can fit in the new image;
- Find dimensions of the real image if given the scale drawing of it and the scale factor;
- Use proportions and ratios to complete conversions of measurement units.
Work from previous grades that supports this new learning includes:
- Make comparisons using ratios;
- Simplify and make equivalent ratios;
- Understand the equivalence of fractions and decimals;
- Compare numbers;
- Simplify fractions and ratios;
- Perform data analysis and calculate percents;
- Explore and apply rational number concepts;
- Understand percent defined as a ratio to 100 and connected to equivalent fractions and decimals;
- Compare and subdivide similar figures;
- Connect and compare rates using ratios, decimals and percents;
- Analyze geometric figures;
- Find angle measures;
- Find areas;
- Understand concept of angle, vertex, side length, side, angle measure, degrees;
- Explore similarities of a figure;
- Use factors and multiples;
- Use ratios in fraction form;
- Use maps;
- Find areas and perimeters (rectangles, triangles).
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 confuse congruence with similarity.
- Students sometimes confuse the different kinds of notation used with similarity, such as $\cong$ and $\sim$.
- A common misconception is that when the dimensions of an object are doubled, the area is doubled, too.
- Students may confuse additive thinking vs. multiplicative thinking. For example, suppose there were two similar rectangles with a pair of corresponding side lengths of 4 and 7 cm. The smaller similar rectangle has a width of 3 cm. A student may think the other corresponding side should have a length of 6 cm, because the first set of corresponding side lengths changed by 3 cm, so they might think that the other should also change by adding 3 cm.
Vignette
In the Classroom
In this vignette, students create a rubber band stretcher to use when copying geometric figures.
Teacher: On your table you will find clear transparency film, rulers, angle rulers, protractors, markers, colors, plain paper and rubber bands. We will be making a 'Super Stretcher' that will help you draw a copy of your figure on your labsheet. I will now hand out your lab sheet. Those of you that are left-handed need to raise your hand so I can give you a left-handed paper. OK, you will see a picture of a tree on your paper. Your job is to use your rubber bands to make a rubber band stretcher and trace your tree to make one on your plain paper. Every set of partners needs two rubber bands. The first thing you need to do is make your rubber band stretcher by connecting the two rubber bands using a slip knot. Does everyone have theirs made?
Student: I am having trouble.
Teacher: Put the two rubber bands together, and pull one of them through the other one. There, now you have it.
Teacher: OK, now you will work with your partner. One of you will be using the rubber band stretcher and a crayon. You will put your crayon in the end of your rubber band stretcher, and your finger in the other end, putting it on the anchor point on your picture. Now, you need to have the knot on your rubber band stretcher follow the outline of your tree. Do NOT watch your crayon, only watch your knot.
Student: Wow, mine doesn't look like the tree at all!
Student: Oh, my! That tree looks kind of like the original one.
Teacher: Now, if you are done, switch and the other person needs to do the same thing. Make sure you hold the paper for your partner so that the paper doesn't move.
Student: Mine looks OK.
Teacher: Don't look at the drawing that you are making with the crayon. Only watch your knot. I don't expect these to be perfect.
When everyone has had a chance to do this, the teacher continues.
Teacher: OK. You have many different supplies on your desk. You need to compare your two figures. I'm not going to tell you what I want you to compare, just make sure to compare enough different parts of each so that you have an accurate comparison. You might try comparing your tree with your partner's, too.
Student: Let's trace our trees. Let's use different paper so we can compare them better. (to partner) OK, when I traced mine and put it by your tracing, they look almost the same size. I wonder why that would be.
Student: I think they're the same because we used the same original picture. We also used the same rubber band stretcher.
Teacher: OK, what did we find out?
Student: The trees that we made were almost the same size.
Student: So were ours.
Teacher: OK. How about if we compare the original figure to our new image? What do you see? Let's put the clear transparency that you traced your new image on over the original figure. Explain what you see. Compare them.
Student: Well, my original tree is about half the height of the new image, and it's about half as wide as well.
Student: I think a little more than three of the original will fill in the new image.
Teacher: Why do you say a little more than three?
Student: Well, mine isn't perfect so that's why I said that. Maybe if I would have done the rubber band stretcher perfectly it would fit in there exactly 4 times?
Teacher: Did anyone use their protractors or angle rulers?
Student: We did. We found that the angles were almost the same. I thought that was odd since the height got bigger, I thought the angle would get bigger, too. Why is that?
Teacher: Who can answer that?
Student: I think it's because if the angle were to get bigger, it would be a wider base then, and they wouldn't be the same shape?
Teacher: Good! What we have just looked at is the effects of enlarging an image and what happens to the side lengths and the angle measures. So what can we conclude if two figures are similar?
Student: They are the same figure, one is just smaller and one is bigger.
Student: The angles are the same?
Teacher: What would happen if the angles aren't the same size?
Student: Then they aren't the same shape because one would be wider in proportion to the other one.
Teacher: Nice use of that word: proportion. That is correct. They have to be 'in proportion' to one another in order to be similar.
Adapted from: Lappan, G. (2009). Stretching and Shrinking: Understanding Similarity, Teacher's Edition. (p. 19-24). Boston, MA: Pearson.
Resources
Teacher Notes
- Remind the students that a "scale factor" is the number we use to scale up or down by multiplying the scale factor by the side lengths, not by dividing. To scale up, the scale factor will be larger than 1; to scale down, students will multiply by a scale factor that is a smaller than 1. For example, a scale factor of $\frac{1}{3}$ will reduce the scale by a factor of 3. Remind students that dividing by 3 is the same as multiplying by $\frac{1}{3}$.
- Try this resizing activity with students:
- Given two similar triangles, use highlighters to color code corresponding parts; set up ratios and proportions to find unknown measures.
From: 2007 Mississippi Mathematics Framework Revised Strategies, p. 50. - Tell students that two figures are similar if they have the same shape. This means that the corresponding angles are equal. The corresponding sides change by the same scale factor, which means that all the sides of the small figure are multiplied by the same number to obtain the lengths of the corresponding sides of the large figure.
Adapted from: http://www.watertown.k12.ma.us/wms/math/math_help/gradeseven/stretching/scale%20factor.htm
- Scale Factor
A common misconception is that when the dimensions of an object are doubled, the area is doubled, too. This applet investigates how changes in the scale factor influence the ratio of perimeters and the ratio of areas between two figures. - Measuring Up: In Your Shadow
Students extend their knowledge of proportions to solving problems dealing with similarity. They measure the heights and shadows of familiar objects and use indirect measurement to find the heights of things that are much bigger in size, such as a flagpole, a school building or a tree. - Measuring Up: Off the Scale
Students use real-world examples to solve problems involving scale as they examine maps of their home states and calculate distances between cities. - Scaling Away
Students will measure the dimensions of a common object, multiply each dimension by a scale factor, and examine a model using the multiplied dimensions. Students will then compare the surface area and volume of the original object and the enlarged model.
- Shopping Mall Math
Students participate in an activity in which they develop number sense in and around the shopping mall. They solve problems involving percent and scale drawings.
Additional Instructional Resources
- In this session, students will build on their intuitive notions of what makes a "good copy" to build a more formal definition of similarity. They will then look at applications of similar triangles, including triangle trigonometry. Video links are included.
- Making a scale model of a 1963 Buick Riviera
- Middle School Math Links
This website has 16 different professional development activities, each including a guide, lesson introduction, the lesson, resources (teaching tips, lesson plans, etc) and interactive lessons. Lessons include Similar Figures and Geometric Proportions.
similar: having the same shape with the same angles and proportions, though of different sizes.
scale factor: a number used as a multiplier in scaling; in similar figures, the ratio of a distance or length in an image to the corresponding distance or length in a preimage. Also called ratio of similitude.
Example: When comparing the following small right triangle to its similar large right triangle, we see a scale factor of $\frac{1}{2}$ or 0.5.
http://www.themathlab.com/dictionary/swords/swords.htm
scale: the scale is the number used to multiply both parts of a ratio to produce an equal, but possibly more informative, ratio. A ratio can be scaled to produce a number of equivalent ratios.
Example: Multiplying the rate of 4.5 gallons per hour by a scale of 2 yields the rate of 9 gallons per 2 hours.
Example: Scales are used on maps to give the relationship between measurements on the map to the actual physical measurements.
corresponding: similar, especially in position or purpose; a number of corresponding diagonal points.
corresponding sides: any pair of sides in the same relative positions in two similar figures.
Example: The segment stretching from A to B corresponds to the segment stretching from D to E.
scale drawing: a drawing that shows a real object with accurate sizes except they have all been reduced or enlarged by a certain amount (called the scale).
conversion: the expression of a quantity in alternative units, as of length or weight.
conversion factor: a factor by which one unit can be converted to another.
notation: figures or symbols used to represent mathematical functions, objects, or ideas.
Examples:
$\sim$ (similar)
$\cong$ (congruent)
$\overline{FG}$ (segment $FG$)
FG (length of FG)
"A little learning is a dangerous thing." Alexander Pope |
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- Can students identify similar objects and the scale factor that defines this relationship?
- Can students use the identified scale factor to calculate measurements?
- Can students find the corresponding sides and angles of two similar figures?
- Can students use multiple strategies to find the missing side lengths?
- Do students understand the difference between the everyday use of the word "similar" and the mathematical meaning or use of the word?
- Can students determine what is the same and what is different in two similar figures?
- Do students understand how ratios relate to similarity?
- In what ways can students apply the ideas about similarity to use in the everyday world?
- 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: b
Source: Minnesota Grade 7 Mathematics MCA-III Item Sampler Item, 2011, Benchmark 7.3.2.1
2.
Answer: b
Source: Minnesota Grade 7 Mathematics Modified MCA-III Item Sampler Item, 2011, Benchmark 7.3.2.2
3.
Possible answers: (The numerators and denominators can be flipped around, and left side could be written on right side.)
$\frac{p}{r}=\frac{g}{f}$;$\frac{p}{q}=\frac{g}{h}$;$\frac{r}{q}=\frac{f}{h}$
Source: Minnesota Grade 7 Mathematics MCA-III Item Sampler Item, 2011, Benchmark 7.3.2.2
4.
Answer: b
Source: Minnesota Grade 7 Mathematics Modified MCA-III Item Sampler Item, 2011, Benchmark 7.3.2.3
5.
Answer: b
Source: Minnesota Grade 7 Mathematics MCA-III Item Sampler Item, 2011, Benchmark 7.3.2.3
6.
Answer: d
Source: California Grade 7 Released Exam 2009 http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqmath7.pdf
7.
Find the area of the following shapes after the transformations have been made.
1. |
A square has an area of 19. If the side length is increased by a factor of 5, what is the new area of the square? New area = |
2. |
A square has an area of 22. If the side length is increased by a factor of 2, what is the new area of the square? New area = |
Answers: 1) 475 (19 × 5 × 5)
2) 88 (22 × 2 × 2)
"Things don't change. You change your way of looking, that's all.
-Carlos Castaneda
Differentiation
- Provide calculators.
- Provide labeled diagrams of all images used.
- Keep all diagrams in the same orientation.
- Provide grid paper for measurement instead of or in addition to standard rulers.
- Relations and sizes
- Review and demonstrate the definition of scale.
- Knowledge Cards are great tools for helping students rehearse procedures, remember laws and theorems, and identify important mathematical terms and topics.
Harvey F. Silver, H.F., Brunsting, J.R., and Walsh, T. (2007). Knowledge Cards. In Math Tools 3-12: Math Tools, Grades 3-12: 64 Ways to Differentiate Instruction and Increase Student Engagement. (p. 21). Thousand Oaks, CA: Corwin Press.
- Scaling up your home
In this math activity, the student draws a bedroom to scale and creates a fantasy home.
- Use this activity to work on similarity ratios.
$\triangle{TUV}\sim\triangle{TWX}$
Since we know that the two triangles are similar, all that we have to do in order to find the similarity ratio is to match a pair of corresponding sides, and then to divide.
Step 1) Match a pair of corresponding sides.
TU and TW are corresponding.
Step 2) Divide the corresponding sides.
TW ÷ TU = $\frac{40}{10}=\frac{4}{1}$. Therefore, the similarity ratio is $\frac{4}{1}$ from $\triangle{TWX}$ to $\triangle{TUV}$.
Alternately, you could have found the similarity ratio from the smaller triangle, $\triangle{TUV}$, to the larger $\triangle{TWX}$.
TU ÷ TW = $\frac{10}{40}=\frac{1}{4}$.
http://www.mathwarehouse.com/geometry/similar/triangles/
- Use this baseball bat activity to explore scale factors.
Have one student model a batting stance with a regulation size baseball bat. Have another student model the batting stance with a souvenir miniature baseball bat. Ask the class, "Why does the student with the souvenir bat look so funny?"
Students work in individually or in pairs
● 1 regulation-sized baseball bat
● 1 souvenir miniature baseball bat
● 1 piece of posterboard for each student
● 1 metric measuring tape for each student
1. Measure the length of the regular bat in centimeters.
2. Measure the length of the souvenir bat in centimeters.
3. Measure the height of the student holding the regular bat.
4. Determine how tall a person should be in order to be proportional to the souvenir bat.
5. Using the posterboard, draw a person the correct height.
6. Have students show their work and/or explain in words the process for determining the correct height of the figure on the posterboard.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) |
Teachers are: (descriptive list) |
measuring side lengths, measuring angles and tracing figures to find relationships between side lengths and angles. |
using the term similar in correct context. |
comparing figures. |
using many models to demonstrate how to find missing side lengths. |
comparing angles. |
consistently reviewing the characteristics that similar images have in common. |
matching up corresponding sides. |
pointing out similarities and differences in strategies used to find missing values. |
using various problem-solving strategies to determine if two (or more) figures are similar. |
using ratios to define similarity. |
Parent Resources
- The world of math online
- Resizing activity
- Math at work
This website includes an interview with the maker of Hot Wheels cars and how math is so much a part of his job. - Scale and scale factor video