Proportional Reasoning

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Chapter 5

• Proportional Reasoning

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5-1 Ratios and Rates

• Ratio comparison of two quantities by division
  (order matteres)
• 17/25, 17 to 25, 1725
• A recipe for homemade ice cream calls for 6 cups
  of cream, 1 cup of sugar, and 2 cups of fruit.
• Write the ratio of fruit to cream.
• Write the ratio of sugar to fruit.

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Rates

• Rate ratio that compares two quantities
  measured in different units
• Suppose Mr. Smith drove 75 miles in 3 hours. His
  rate was 75mi./3hrs.

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Unit Rates

• Unit Rate ratio where the second quantity is in
  a rate of one unit
• To find the unit rate, simply divide the
  numerator by the denominator.
• Mr. Smiths unit rate was 75 3 25mi/1hr. or
  25 miles per hour.

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Practice with Rates

• Belinda biked 36 miles in 4 hours.
• A recipe for a garden weed killer involves mixing
  11 tablespoons of liquid soap with 2 quarts of
  water.

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Simplifying Ratios

• Find the ratio of protein grams to serving grams.
• Find the ratio of fat grams to calories.

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Think and Discuss

• 1. Tell how to identify whether a ratio is true.
• 2. Give an example of a ratio that can be
  written as a unit rate.

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5-2 Identifying and Writing Proportions

• Equivalent ratios ratios that name the same
  comparison
• Ex. 10/6 and 25/15 (both reduce to 5/3)
• Proportion equation stating that two ratios are
  equivalent
• 10/6 25/15

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Comparing Ratios by Reducing

• Method 1
• 2/7, 6/21
• 8/24, 6/20

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Comparing Ratios with Cross-Multiplication

• Method 2
• 24/51, 72/128
• Multiply 24 x 128 and multiply 51 x 72
• If products are the same, the ratios are
  proportional

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Cross-multiplication Practice

• Determine if the ratios are proportional.
• 150/105, 90/63
• 20/25, 12/18

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Finding Equivalent Ratios

• Find a ratio equivalent to each ratio by
  multiplying the numerator and denominator by the
  same number.
• Find an equivalent ratio by dividing the
  numerator and denominator by a common factor.
• 8/14
• 4/18

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Equivalent Ratios Practice

• 3/5
• 28/16
• 6/18
• 21/36

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Think and Discuss

• 1. Describe what it means for ratios to be
  proportional.
• 2. Explain how to determine if ratios are
  proportional.
• 3. Give an example of a proportion.

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5-3 Solving Proportions

• Solve proportions using cross-multiplication.
• p/6 10/3
• P x 3 3p
• 6 x 10 60
• 3p 60
• P 20

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Practice

• 16/4 8/x
• 9/15 m/5
• 18/25.8 11/h

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Think and Discuss

• 1. Describe the error in these steps 2/3
  x/12. 2x 36. x 18.
• 2. Show how to use cross products to decide
  whether the ratios 645 and 215 are proportional.

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5-4 Dimensional Analysis

• You must MEMORIZE the basic units of measurement
  for the metric and customary systems.
• Length, Capacity, Mass/Weight, Time

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Length

• Metric

• Customary

• 10 mm 1 cm
• 100 cm 1 meter
• 1,000 m 1 km

• 12 in. 1 foot
• 3 ft 1 yard
• 5,280 ft 1 mile

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Capacity

• Metric

• Customary

• 1,000 ml 1 liter (L)

• 8 oz. 1 cup (c)
• 2 c 1 pint (pt)
• 2 pt 1 quart (qt)
• 4 qt 1 gallon (gal)

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Mass/ Weight

• Metric

• Customary

• 1,000 mg 1 gram
• 1,000 g 1 kg

• 16 oz. 1 lb.
• 2,000 lbs. 1 ton

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Time

• 60 seconds 1 min.
• 60 min. 1 hour
• 24 hours 1 day
• 7 days 1 week
• 52 weeks 1 year
• 365 days 1 year

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Unit Conversion

• When converting one unit of measurement to
  another, you must know which unit is bigger.
• 5 lbs. to oz.
• Pounds are bigger than ounces
• 34 cm to
• Centimeters are smaller than meters

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Bigger to Smaller Unit

• When converting a bigger unit to a smaller unit,
  you multiply
• 5 lbs. to oz.
• Step 1 1 lb. 16 oz.
• Step 2 5 x 16 80 oz.

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Smaller to Bigger Unit

• When converting a smaller unit to a bigger unit,
  you divide.
• 34 cm to m
• Step 1 1 m 100 cm
• Step 2 34 100 .34 m

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Review the steps

• Step 1 Decide which unit is bigger
• Step 2 Bigger to smaller multiply
• Smaller to bigger divide
• Step 3 Write down your conversion factor that
  you must memorize
• Step 4 Multiply or divide

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Conversion Practice

• A bucket holds 16 quarts. How many gallons of
  water will fill the bucket? Use a conversion
  factor to convert the units.
• Convert 80 miles to feet.

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More Practice

• A phone service in the United States charges
  1.99 per minute for a call to Australia. How
  many dollars per hour is the phone service
  charging?
• An oil drum holds 55 gallons. How many quarts of
  oil will it take to fill the drum?

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Even More Practice

• If orange juice sells for 1.28 per gallon, what
  is the cost per ounce?
• Convert 48 inches of ribbon to feet.

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Think and Discuss

• Explain the 4 steps needed to convert a unit of
  measurement.
• Compare the process of converting feet to inches
  with the process of converting feet per minute to
  inches per minute.

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5-6 Similar Figures and Proportions

• Similar shapes with the same shape but not the
  same size
• Corresponding sides matching sides of two or
  more polygons
• Ratios of corresponding sides are proportional
• Corresponding angles matching angles of two or
  more polygons
• Corresponding angles are equal of similar shapes

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Are they similar?

• Identify the corresponding sides in the pair of
  shapes.
• Determine whether the ratios of the lengths of
  the corresponding sides are proportional.
• Reduce each ratio.
• If the sides are proportional, then the
  corresponding angles must be equal.

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Example 1
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Example 2
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Practice
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Practice
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Think and Discuss

• Explain what corresponding means in reference to
  sides and angles.
• Explain whether all rectangles are similar. Give
  specific examples to justify your answer.
• Are all squares similar?

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5-6 Using Similar Figures

• Indirect measurement method of using
  proportions to find an unknown length or distance
  in similar figures
• 3 different scenarios
• Missing side
• Lake
• Shadow

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Missing Sides

• Write a proportion using corresponding sides.
• Put a variable in for the unknown side.
• Cross-multiply
• Solve for the variable.

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Lake Method

• Find the distance across a body of water by using
  similar triangles.

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Shadow Method

• Use shadows to figure out the height of tall
  objects.

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Think and Discuss

• Name two objects that would make sense to measure
  using indirect measurement (shadow method).

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5-7 Scale Drawings and Models

• Scale model proportional model of a 3-D object
• Model trains, cars, airplanes, etc.
• Scale factor ratio at which the scale model is
  related to the actual object
• Scale drawing proportional drawing of an object

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Identifying Scale Factor
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Using Scale Factors

• A photograph of Vincent van Goghs painting
  Still Life with Irises Against a Yellow
  Background has dimensions 6.13 cm and 4.90 cm.
  The scale factor is 1/15. Find the size of the
  actual painting.
• Photo 1
• Painting 15

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Measurement Application

• On a map of Florida, the distance between Hialeah
  and Tampa is 10.5 cm. What is the actual
  distance between the cities if the map scale is 3
  cm 80 miles?
• Map
• Actual

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Practice

• On a road map with a scale of 1.5 in. 60 miles,
  the distance between Pittsburgh and Philadelphia
  measures 7.5 inches. What is the actual distance
  between the two cities?

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Practice

• A photograph was enlarged and made into a poster.
  The poster is 20.5 inches by 36 inches. The
  scale factor is 5/1. Find the size of the
  original photograph.

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More Practice

• Identify the scale factor.

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Think and Discuss

• 1. Given a scale factor of 5/3, explain how you
  can tell whether the model is bigger or smaller
  than the original object.
• 2. Describe how to find the scale factor if an
  antenna is 60 ft. long and a scale drawing shows
  the length is 1 foot long.