Stretching and Shrinking

Stretching and ShrinkingStretching and Shrinking

5.1 Similar Triangles5.1 Similar Triangles

Using Shadows to Find HeightsUsing Shadows to Find Heights

ACOS 7ACOS 7thth Grade Math Grade Math

1 1 Demonstrate computational fluency with addition, subtraction, and multiplication of integers. 5 5 Translate verbal phrases into algebraic expressions and algebraic expressions

into verbal phrases.

6 6 Solve one- and two-step equations 8 8 Recognize geometric relationships among two-dimensional and three-dimensional object

1111Solve problems involving ratios or rates, using proportional reasoning. 12 12 Determine measures of central tendency (mean, median, and mode) and the

range using a given set of data or graphs, including histograms, frequency tables, and stem-and-leaf plots.

Vocabulary TermsVocabulary Terms

CorrespondingCorresponding

RatioRatio

SimilarSimilar

Authentic LiteratureAuthentic LiteratureMy ShadowMy Shadow by Robert Louis Stevenson by Robert Louis Stevenson

POD Shadow MathPOD Shadow Math

Shadowy Science By Jess Brallier

20 Feet

2 Feet

3 Feet

POD AnswerPOD Answer

The flag pole is 30 feet tall.The flag pole is 30 feet tall.

33 = = 22

X 20X 20

(3)20 = 60

X(2) = 60

2 2

X = 30

Real LifeReal Life

Treasure Hunt Video ClipTreasure Hunt Video Clip

Shadow Measurement Explanation Video Shadow Measurement Explanation Video ClipClip

Peter Pan’s ShadowPeter Pan’s Shadow

Learning ExperienceLearning Experience

In your group, start at your assigned station.In your group, start at your assigned station.Measure the shadow of the unknown in Measure the shadow of the unknown in centimeters. (Twice)centimeters. (Twice)Hold the meter stick steady. Hold the meter stick steady. Measure the meter stick shadow in centimeters. Measure the meter stick shadow in centimeters. (Twice)(Twice)Record your measurements in your journal.Record your measurements in your journal.Use a proportion to figure the measurement of Use a proportion to figure the measurement of the unknown.the unknown.At the end of 5 minutes, rotate clockwise to the At the end of 5 minutes, rotate clockwise to the next station.next station.

Materials Needed for Each GroupMaterials Needed for Each Group

Meter StickMeter Stick

4 Tape Measures4 Tape Measures

JournalJournal

Pen/PencilPen/Pencil

Flag Pole Flag Pole ShadowShadow

Meterstick Meterstick ShadowShadow

Light Pole Light Pole ShadowShadow

Meterstick Meterstick ShadowShadow

Tree Tree ShadowShadow

Meterstick Meterstick ShadowShadow

ICEICE

IllustrateIllustrate

CalculateCalculate

ExplainExplain

Yellow M & MYellow M & M

Flag PoleFlag Pole

Red M & MRed M & M

TreeTree

Green M & MGreen M & M

Light PoleLight Pole

5.1 Follow-up—LINE PLOT5.1 Follow-up—LINE PLOT

Flag Pole Height (centimeters)

ACEACE

In Journal complete ACEIn Journal complete ACE

Questions 1-4 & 7 on pp. 64-67Questions 1-4 & 7 on pp. 64-67

ACE Question 1 AnswersACE Question 1 Answers

1a Sides CD and XW, Sides DE and XY, 1a Sides CD and XW, Sides DE and XY, Sides CE and XY are corresponding sides.Sides CE and XY are corresponding sides.

1b Angles CDE and XWY, Angles CED 1b Angles CDE and XWY, Angles CED and XYW, Angles ECD and YXW are and XYW, Angles ECD and YXW are corresponding angles (congruent angles).corresponding angles (congruent angles).

ACE Question 2 AnswersACE Question 2 Answers

2a Triangle PQR and Triangle PST are 2a Triangle PQR and Triangle PST are similar triangles.similar triangles.2b Sides PQ and PS, Sides PR and PT, 2b Sides PQ and PS, Sides PR and PT, and Sides QR and ST are corresponding and Sides QR and ST are corresponding sides for the similar triangles.sides for the similar triangles.2c Angles PQR and PST, Angles PRQ 2c Angles PQR and PST, Angles PRQ and PTS, and Angles QPR and SPT are and PTS, and Angles QPR and SPT are corresponding angles (congruent angles) corresponding angles (congruent angles) for the similar triangles.for the similar triangles.

ACE Question 3 AnswerACE Question 3 Answer

From the meterstick’s shadow to the From the meterstick’s shadow to the backboard’s shadow, there is a scale backboard’s shadow, there is a scale factor of 3/2 factor of 3/2 ÷1/2 =3. Therefore, the top of ÷1/2 =3. Therefore, the top of the backboard must be 3 times the height the backboard must be 3 times the height of the meterstick or 3 x 1 = 3mof the meterstick or 3 x 1 = 3m

ACE Question 4 AnswerACE Question 4 Answer

From the meterstick’s shadow to the From the meterstick’s shadow to the flagpole’s shadow, there is a scale factor flagpole’s shadow, there is a scale factor of 38/5 = 7.6 times the height of the of 38/5 = 7.6 times the height of the meterstick, or 7.6 mmeterstick, or 7.6 m

ACE Question 7 AnswerACE Question 7 Answer

The scale factor from the stick’s shadow to The scale factor from the stick’s shadow to the monument’s shadow is the monument’s shadow is

42.25 42.25 ÷ 0.5= 84.5÷ 0.5= 84.5

Therefore, the monument is 84.5 times the Therefore, the monument is 84.5 times the height of the 2-meter stick, or 169 mheight of the 2-meter stick, or 169 m

Math ReflectionMath Reflection

Use at least two vocabulary terms to Use at least two vocabulary terms to explain what properties of similar explain what properties of similar triangles are useful for estimating triangles are useful for estimating heights.heights.

Corresponding Corresponding

RatioRatio

SimilarSimilar

Website LinkWebsite Link

http://http://micro.magnet.fsu.edu/primer/java/sciencemicro.magnet.fsu.edu/primer/java/scienceopticsu/shadowsopticsu/shadows//

CorrespondingCorresponding

Corresponding sides or angles Corresponding sides or angles have the same relative position in have the same relative position in similar figures. similar figures.

RatioRatio

A ratio is a comparison of two A ratio is a comparison of two quantities that tells the scale quantities that tells the scale between them. between them.

SimilarSimilar

Similar figures have the same shape. Two Similar figures have the same shape. Two figures are mathematically similar if and only if figures are mathematically similar if and only if their corresponding angles are equal and the their corresponding angles are equal and the ratios of all pairs of corresponding sides are ratios of all pairs of corresponding sides are equal. This ratio image length:original length equal. This ratio image length:original length compares a side in the image to a side in the compares a side in the image to a side in the original. This means that there is a single scale original. This means that there is a single scale by which all sides of the smaller figure “stretch” by which all sides of the smaller figure “stretch” or “shrink” into the corresponding sides of the or “shrink” into the corresponding sides of the larger figure.larger figure.