A Story of Ratios

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NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosGrade 8 – Module 3


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Ratios

Session Objectives

• Examine the development of mathematical understanding across the module using a focus on concept development within the lessons.

• Identify the big idea within each topic in order to support instructional choices that achieve the lesson objectives while maintaining rigor within the curriculum.



AgendaIntroduction to the ModuleConcept DevelopmentModule Review



Curriculum Overview of A Story of Ratios






L1: What lies behind “same shape”?

Lesson 1, Concept Development

• What does is mean for figures to be considered similar? Is describing them as “same shape” good enough?

• Introduction to dilations• Center• Scale factor

• Dilations do not preserve the lengths of segments. Rather, the length of a dilated segment is equal to the length of the original segment multiplied by the scale factor of dilation.



Exercises 2-6



If the scale factor is r=3, what is the length of segment OP' ?The length of the segment OP' is 9 cm.

Use the definition of dilation to show that your answer to Exercise 2 is correct. |OP’|=r|OP|, therefore, |OP’|=3×3=9 and |OP’|=9.If the scale factor is r=3, what is the length of segment OQ'? The length of the segment OQ' is 12 cm.

Use the definition of dilation to show that your answer to Exercise 4 is correct.|OQ’|=r|OQ|, therefore, |OQ’|=3×4=12 and |OQ’|=12.

If you know that OP=3, OP'=9, how could you use that information to determine the scale factor?



Lesson 1, Student Debrief



L2: Properties of Dilations


• Dilations maps lines to lines, segments to segments, rays to rays, and angles to angles.

• Dilations preserve the measures of angles.• Properties verified experimentally



Problem Set 1



Problem Set 1






L3: Examples of Dilations


• In the past, topics of congruence and similarity only existed in the world of rectilinear figures. Knowledge of the basic rigid motions and dilation in general provide the opportunity to explore these concepts with curvilinear shapes.

• How do we return a dilated figure back to its original size?



Example 2






L4: Fundamental Theorem of Similarity (FTS)


• FTS is explored in terms of dilation.Theorem: Given a dilation with center O and scale factor r, then for any two points P, Q in the plane so the O, P, Q are not collinear, the lines PQ and P’Q’ are parallel, where P’=dilation(P) and Q’=dilation (Q), and furthermore, |P'Q’|=r|PQ|.

• Teacher led activity based lesson using lined paper and a ruler.



Activity• On a lined piece of paper:• Choose a point near the top of the page on a line to use as our center of dilation.

Label the point O.• Draw a ray, from point O, through a whole number of lines, e.g., through 2 lines, or

5 lines, etc. Label this point P.• Draw another ray, from point O, through the same number of lines. Label this

point Q.• Along ray (you may need to extend it), find another point, P’, a whole

number of lines away from O.• Along ray (you may need to extend it), find another point, Q’, the same

whole number of lines away from O.• Connect points P and Q. Connect points P’ and Q’.



Activity• What do you notice about lines PQ and P’Q’?• Lines PQ and P’Q’ are parallel, i.e., they never intersect.

• Write the scale factor that represents the dilation from O to P and O to P’, e.g., if you went 2 lines for P and 5 lines for P’, then scale factor r = 5/2.• Why does it make sense for the scale factor to be r = 5/2 and not r = 2/5?• If triangle OPQ was the original figure and it has been dilated to triangle OP’Q’

(where the lengths of the sides are longer than the original), that means that the triangle has been magnified, therefore the scale factor should be greater than 1.

• Now if triangle OPQ was the original figure and it has been dilated to triangle OP’Q’ (where the lengths of the sides are shorter than the original), that means the triangle has been shrunk, therefore the scale factor should be less than one (but greater than zero).



Activity (cont.)• Measure the distance from O to P, label your diagram. Measure and label

OP’, OQ, and OQ’. • Compare the length of the dilated segment to the original: OP’ to OP, and

OQ’ to OQ. (Use values rounded to nearest tenths place.) What do you notice?

• You should notice that the length of the dilated segment OP’ divided by the original segment OP is equal to OQ’ divided by OQ.



Activity (cont.)

• Measure the lengths P’Q’ and PQ. Compare the lengths as before. What do you notice?

• It is also equal!• Now compare the ratio of the lengths to the scale factor. What do you

notice?• In each case, the ratio of the segment lengths is equal to the scale factor.



Why does it work?• The Fundamental Theorem of Similarity:• If D is a dilation with center O and scale factor r, then for any two points P, Q in the

plane so that O, P, Q are not collinear, the lines PQ and P’Q’ are parallel, where P’ = D(P), and Q’ = D(Q), and furthermore,

• Mathematically speaking;

Therefore,



What does it all mean?• Are triangles OPQ and OP’Q’ similar?• Use your transparency to trace angle OPQ. Translate along the vector Does angle OPQ map onto angle OP’Q’? What does that mean? • Dilations preserve the measures of angles. The triangles are similar by AA

criterion. • The lengths of the sides of the triangles are in “proportion” and equal to the scale

factor of dilation.• In general, two figures are said to be similar if you can map one onto

another by a dilation followed by a congruence.






L5: First Consequence of FTS


• Converse of FTS.• Students first experience the effect dilations have on points in the

coordinate plane as an application of FTS.



Exercise 3






L6: Dilations on the Coordinate Plane


• Students generalize what they observed about dilations of points using FTS, that is, students recognize the multiplicative effect that dilations have on the coordinates.

• Given a dilation with scale factor r and center at the origin, a point in the plane (x, y), after the dilation will be located at (rx, ry).



Example 1



Example 1






L7: Informal Proofs of Dilations


• Optional lesson.• Properties observed in Lesson 2 are proved informally.



Discussion






L8: Similarity


• Is dilation enough to prove that two figures are similar?• Similarity is defined as a dilation followed by a congruence.• Show figures are similar by describing the sequence of the dilation and

congruence.



Exercise 1



Exercise 1

Dilate A”B”C” by a scale factor of 2 from the origin. Then translate.






L9: Basic Properties of Similarity


• Similarity is symmetric. If triangle A is similar to triangle B, then triangle B is similar to triangle A.

• Similarity is transitive. If triangle A is similar to triangle B, and triangle B is similar to triangle C, then triangle A is similar to triangle C.



Exploratory Challenge 2






L10: Informal Proof of the AA Criterion for Similarity


• Straightforward proof, then a direct application using the triangle sum theorem from Module 2.

• Students practice presenting informal arguments to show that two triangles are similar.



Exercise 4



Exercise 5






L11: More About Similar Triangles


• Find the length of a segment of a triangle• Verify that triangles are similar• Write ratios of corresponding sides. They are equivalent fractions, why?• Find the number that makes the fractions equal.



Exercise 1



Exercise 1Based on the information given, is ABC~ AB'C'?△ △ Explain. There is not enough information provided to determine if the triangles are similar. We would need information about a pair of corresponding angles or more information about the side lengths of each of the triangles. Assume line BC is parallel to line B'C'. With this information, can you say that ABC~ AB'C'?△ △ Explain.If line BC is parallel to line B'C', then ABC~ AB'C'. Both triangles share A. Another △ △ ∠pair of equal angles is AB'C'and ABC. They are equal because they are ∠ ∠corresponding angles of parallel lines. By the AA criterion, ABC~ AB'C'.△ △Given that ABC~ AB'C', △ △ determine the length of AC'. Let x represent the length of AC'.x/6=2/8We are looking for the value of x that makes the fractions equivalent. Therefore 8x=12, and x=1.5. The length of AC' is 1.5.






L12: Modeling Using Similarity


• Knowledge of similarity leads to ability to take an indirect measurement of• distance across a lake.• length of wood needed for a skate ramp.• height of a building.• height of a tree.



Exercise 1






L13: Proof of the Pythagorean Theorem


• Proof of Pythagorean theorem using similar triangles.• Traditional application of Pythagorean theorem to find lengths of right

triangles.



Proof












L14: The Converse of the Pythagorean Theorem


• Proof of the converse of Pythagorean theorem. Given three lengths that satisfy the Pythagorean theorem, the lengths form a right triangle.

• Students give informal proofs about right triangles using the Pythagorean theorem.



Proof











Biggest TakeawayTurn and Talk:• What questions were answered for you?• What new questions have surfaced?



Key Points• Dilation leads to an understanding of similarity.

• Dilations begin off the coordinate plane, then become fixed to examine the effect dilations and rigid

motions have on coordinates.

Documents

A Story of Ratios