Proportional Relationships Unit

Transcription

1 Proportional Relationships Unit Reference Packet Need more help? Try any of the IXL 7 th grade standards for practice throughout the unit.

2 Videos to view for help throughout the unit: Introduction to Ratio Intro to proportional relationships Comparing fractions: tape diagram Solving proportions example Proportional relationships: graphs Equations for proportional relationships Writing proportional equations Intro to rates Solving unit rate problem Interpreting a scale drawing Making a scale drawing

3 Review of Ratios and Rates Ratio: The comparison of two or more quantities. The ratio relationship of the number of pink t-shirts to the number of orange t-shirts, the ratio is 5 : 3. Equivalent Ratios: Find a missing value, using a given ratio. Two ways to find the missing value (Using a Table of values and Using a proportion) A pancake recipe requires 2 cups of flour for every 10 pancakes. If someone has to make 25 pancakes, how many cups of flour will they need? Using a Table of Values Flour Pancakes First fill in the given ratio from the example Then simplify the ratio to get the unit rate to help fill in the table. Continue the pattern in the table, until you reach 25 pancakes. The pattern in the table follows the equation F 5 = P Answer: 5 cups of flour

4 A pancake recipe requires 2 cups of flour for every 10 pancakes. If someone has to make 25 pancakes, how many cups of flour will they need? Using a Proportion Depending on your given values, you can scale up or scale down to find a missing value. Scaling up means to multiply the numerator and denominator by the same factor to find the missing amount. Scaling down means to multiply the numerator and the denominator by the same factor that is less than one to find the missing amount. Given ratio 2 10 =? 25 Think about the comparison: cups of flour pancakes We are trying to find the number of cups Scaling up to find the missing value. To find the factor used to write an equivalent ratio, you can divide the pancake values = 2.5 Then multiply your original ratio by the common factor (this is scaling up) =? =? 25 Multiplying by the common factor will help you find the missing value = 5 25 Answer: 5 cups of flour

5 Rates - a special kind of ratio. Rates: a ratio that compares two quantities with different units. Jenny can type 200 words for every 20 minutes Unit Rates: a rate per unit. Jenny can type at a constant rate of 10 words per minute Unit Price: a price per unit. Chicken costs $3.00 per pound Diet coke is on sale for $10 for 4 packs. Find the cost per 1 pack of diet coke. Three common strategies to determine the unit rate (Finding the Unit Rate, Creating a Rate table, and Using Long Division) Diet coke is on sale for $10 for 4 packs. Find the cost per 1 pack of diet coke. Strategies Notes Finding the Unit Rate 10 4 Set up a ratio with the unit measure in the denominator Divide the numerator and denominator by the value in the denominator Simplify if possible. 2.5 per 1 unit Answer: $2.50 per 1 pack of diet coke Make sure the unit rate is comparing the quantity per 1 unit.

6 Diet coke is on sale for $10 for 4 packs. Find the cost per 1 pack of diet coke. Creating a Rate Table Price? 5 10 Pack Fill in the rate table using the given rate and the rate value if applicable For every $10 there is another 4 packs Value of the rate Given rate Try the step down approach to follow the pattern in the table to find the missing value Answer: $2.50 per 1 pack of diet coke Make sure the unit rate is comparing the quantity per 1 unit. Using Long Division 89 : Write the given rate in fraction form. Then convert the fraction into a decimal. Remember Top number in the house! Answer: $2.50 per 1 pack of diet coke

7 Representing a Direct Proportion: Two quantities are in a direct proportion (or directly proportional) if they have a constant ratio or the same unit rate. The constant ratio is called the constant of proportionality. It is represented by the variable k. Three ways to determine whether quantities are directly proportional (Using a table of values, a graph, or an equation) Table of Values Equation Graph The number of dogs is directly proportional to the number of puppies Dog Puppies Since there is a constant of proportionality (represented by the letter k), we can write the equation of the direct proportion using: y = kx 4 20 = = = 1 5 Think of the equation like a formula, the only number we can substitute in for is k. The unit rate and the constant of proportionality is 0.2. y = 0.2x or y = 1 5 x Notice the graph goes through the origin. The coordinates of the origin are (0,0). The graph also shows a straight line. This is called a linear relationship. The coordinates on the graph simplify to the constant of proportionality: (0.2 is equivalent to 8 < ). The last equation is most commonly used for the graphing of a direct proportion. (5,1) = 8 < (10,2) = A 89 = 8 <

8 Determining whether a relationship is in a direct proportion: The three ways to determine whether there is a direct proportion. Using a table of values. Determine whether each pair of values is equivalent to the ratio value or the unit rate by simplifying the ratio. The unit rate is also called the constant of proportionality. Determine whether the cost of coffee is directly proportional to the weight in pounds (lbs). Cost ($) Coffee (lbs) Non-example: Cost ($) Coffee (lbs) = = = = = = 10 3 Yes, the ratios are directly proportional because the ratios simplify to the same constant of proportionality. No, the two ratios are not in direct proportion because they do not have the same constant of proportionality.

9 Using a graph A graph represents a direct proportion when it is linear (straight line) and when it passes through the origin (0, 0). Determine whether the graph shows a direct proportion relationship Non-example: (The graph above is linear and passes through the origin). (The graph above is linear but does NOT pass through the origin). (The graph above is linear and passes through the origin). (The graph passes through the origin but IS NOT linear). Yes, each graph is a representation of a direct proportion. No, both graphs are not representing a direct proportion.

10 Using an equation. An equation that represents a direct proportion can be written in the form: y = kx Determine whether an equation shows a direct proportion relationship. To determine whether an equation shows a direct proportion relationship, one strategy we can apply is the Zero Zero Test: Since a direct proportion must pass through the origin, we can test any equation to determine if it represents this relationship by substituting the value zero in for x and y. Non-example y = 4x 0 = 4(0) 0 = 0 y + 2 = 5x = 5(0) 2 0 Yes, the equation does represent a direct proportion. No, the equation does not represent a direct proportion. 3y = 1 2 x 3(0) = 1 2 (0) 0 = 0 y = 2x 6 0 = 2(0) 6 0 = Yes, the equation does represent a direct proportion. Since the coordinate (0,0) makes the equation true, the line must pass through the origin, and the equation is directly proportional. No, the equation does not represent a direct proportion. Since the coordinate (0,0) does not make the equation true, the line does not pass through the origin and therefore is not directly proportional.

11 Scale, Scale Factor, and Scale Drawings Scale a comparison of length in a scale drawing to the corresponding length in the actual object Ratio Value Scale: 1 : 25 Every 1 unit on the map represents an actual 25 units on the ground. Rate Value Scale: 2 cm: 5 ft Every 2 cm on the drawing represents an actual 5 feet. Scale factor: the ratio of a length in a scale drawing/scale model to the corresponding length in the actual figure. It can be expressed as a fraction, decimal, or percent. Scale Factor is also known as the constant of proportionality scale factor = scaled length original length original length scale factor = scaled length Enlargement / Scaled Up (Increase in size) Reduction / Scaled Down (Decreases in size) Scale factor = a8.< 89.< = a 8 = 3 Scale factor = b.c A.d = a 8 = 3 All pairs of corresponding sides have a ratio value equivalent to the scale factor. Scale factor = d a9 = 8 < Scale factor = : A9 = 8 < Scale factor is greater than 1. Scale factor is less than 1, but greater than 0. What to look for to determine whether figures are similar: Have the same shape but are not necessarily the same size. The corresponding angels are congruent, or have the same measure The corresponding sides are proportional. The original figure and a scale drawing are similar.

12 Quick Refresher on units of measurement: Your knowledge of proportions can help you convert between systems of measure. Metric System: Length 1 centimeter = 10 millimeters 1 meter = 100 cm = 1,000 mm 1 kilometer = 1,000 m Mass 1 gram = 1,000 milligrams 1 kilogram = 1,000 g Capacity 1 liter = 1,000 milliliters 1 kiloliter = 1,000 liters Customary System: Length 1 foot = 12 inches 1 yard = 3 feet 1 mile = 5,280 ft 1 mile = 1,760 yd Weight 1 pound (lb) = 16 ounces (oz) 1 ton = 2,000 lbs Capacity 1 cup = 8 fluid ounces 1 pint = 2 cups 1 quart = 2 pints 1 gallon = 4 quarts Converting Between Systems Length 1 inch = 2.54 cm 1 cm = 0.39 in 1 meter = 3.28 ft 1 m = 1.09 yd 1 mile = km Weight 1 oz = g Capacity 1 liters = qt 1 gal = liters 1 c = 236 ml Proportional relationships can help in converting between the units of measurement. Example using proportions: How many milliliters of water are in 3.4 liters? 3.4 L = ml 1 liter 1000mL = 3.4L x x = 3.4 (1000) x = 3400 ml Example using proportions: How many yards are there in 90 inches? 90 inches = yards 1yard 36 inches = x 90 inches 36x 36 = x = 2.5 yards Example using common factors: How many milliliters of water are in 3.4 liters? 3.4 L = ml Example using common factors: How many yards are there in 90 inches? 90 inches = yards Since 1 L = 1,000 ml then you can just multiply 3.4 x 1,000 = 3,400 ml Since 1 ft = 12 inches and 1 yd = 3 feet then 1 yd = 36 inches. Now you can divide 90 36= 2.5 yd