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There are five main topics we will be learning. Unit Rates Proportional Relationships o Table o Graph o Equation Complex Fractions as Unit Rates Real World Applications Scale Drawings Vocabulary: As the unit progresses please have a discussion about these words. The definitions can be found within the unit and online through websites. Commission Constant of Proportionality Enlargement Independent Variable Proportional Reduction Scale Factor Variable Complex Fractions Dependent Variable Equation Markup Rates Scale Unit Price Constant Discount Equivalent Ratios Origin Ratios Scale Drawing Unit Rate Activities: 1. Better Buy- While shopping have your children compare prices on products that are the same but come in different quantities. They can also compare different brands and see which brand is the better buy when you compare unit prices. 2. Trips- Have them calculate how far the car can go on a tank of gas based on the miles per gallon your car gets. Have them calculate the cost of gas you will need to spend to travel to a destination based on miles per gallon and the price of gas. 3. Time management- They can calculate the average time it takes them to do a task like vacuuming, dishes, showers, brushing teeth, etc… Use this time to create a schedule. 4. Room plans- Create a scale drawing with them of their rooms and use paper to cut out scale versions of their furniture. Use these pieces to rearrange their room. 5. Project- There are a lot of different examples for projects, but here is one idea. Tell them to determine the cost and materials they would need to put new flooring in a room in the house. Topics: Unit Rates A ratio is a comparison of two numbers. It can be written in a few forms including: 𝑎 a:b, a to b, and 𝑏 . When you multiply or divide these two quantities by a constant number, equivalent ratios are formed. By dividing the ratios, a unit rate is formed. Examples in the real world are unit prices, miles per gallon (mpg), miles per hour (mph), etc… 𝑦 To get the unit rate or unit price, the formula is 𝑥 = 𝑘.This is also known as the constant of proportionality. Proportional Relationships There are three ways to represent proportional relationships: table, graph and an equation. Constant of Proportionality: Equation 𝑦 = 𝑘 𝑥 y= kx k is the constant of proportionality. Also known as a unit rate or unit price. It is comparing a quantity to one single unit of another quantity. y is the dependent variable (output) x is the independent variable (input) Table Graph A table is in proportion if you get a constant of proportionality when you divide the two numbers. A graph is proportional if it is a straight line and goes through the origin. X (input) Y (output) 0 0 1 3 2 6 3 9 By manipulating the formula to find the constant of proportionality, you get the equation. 𝑦 = 𝑘 𝑥 (𝑥) 𝑦 = 𝑘(𝑥) 𝑥 𝑦 = 𝑘𝑥 A table shows multiple equivalent ratios. To test to see if the table is proportional, you divide y by x and see if they are all equal. If they are all equal, then the number is the constant of proportionality. With the table being arranged as a comparison of x to y, it lends itself to creating a graph. A graph has to have two characteristics of being a straight line and goes through the origin. Both of these characteristics need to be on the graph. With a graph of the proportional relationship you can see the unit rate at (1,r) r is the same as the constant of proportionality. In the above example, you can see how the graph is straight, includes the origin (0,0) and has a constant of proportionality 3 by looking at (1,3). Fractional Unit Rates Many real world questions do not use whole numbers. Fractions and decimals can be used in a ratio. This is known as a complex fraction where either the numerator, denominator, or both are fractions. The process to find the unit rate is still the same, divide the ratio to get a denominator of one. Dividing fractions in a ratio is the same as dividing fractions. 1.) Make sure they are either a fraction or an improper fraction not a mixed number. 2.) Keep the first number the same. 3.) Change division to multiplication. 4.) Take the reciprocal (flip) the second number. 5.) Also known as “invert and multiply” “keep, change, flip” Example: You can read 3½ pages in 10 minutes. How many pages can you read in an hour. *Note there are many ways to answer this! Method 1: Two things to remember: 1 3 2 1 6 10 1.) 10 minutes is 60 , 𝑜𝑟 hour. 1 2.) 3 2 = 7 1 ÷ 2 6 7 × 2 6 1 = 42 2 1 6 of an 7 2 or 21 pages per hour Method 2: (this only works on certain denominators) There are 6 (10 minutes) in an hour. 1 32 1 6 × 6 6 = 21 1 or 21 pages per hour *With Method 2, there are only certain times this will work. It is easier and quicker if you can recognize a number you can multiply to get the denominator to 1. There are 4 (15 minutes, ¼ of an hour) in 1 hour. So, multiply by four. There are 2 (6 inches, ½ of a foot) in 1 foot. So, multiply by two. There are 10 (10¢ (dime), 1⁄10 of a dollar) in 1 dollar. So, multiply by ten. If you see this relationship, you can multiply instead of dividing. Real world Problems There are examples of finding the cost of flooring using proportional relationships and given information. There are questions like using proportions to modify a recipe to create a different number of servings. The next unit is about percentages and will go in depth with percentage problems. In this unit we will “plant the seed” on discounts, markups and changes based on fractions. This will then tie into the percentages for the next unit. Scale Drawings Scale drawings are reductions or enlargements of a two-dimensional picture. The dimensions from the original to the scale drawing are proportional and have a scale factor. The scale factor is a constant of proportionality. Scale Drawings When the units are the same: The scale factor is the same as the constant of proportionality. 𝑛𝑒𝑤 = 𝑠𝑐𝑎𝑙𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 When the scale factor is greater than one, it is an enlargement. When the scale factor is less than one, it is a reduction When the units are different: The scale is a ratio. i.e. 1 in. = 4 ft. You set up proportions to find the missing length. For example: 2 ½ in = _____ft. 1 2 2 𝑖𝑛 𝑥 𝑓𝑡 = 1𝑖𝑛 4𝑓𝑡 Method 1: You multiply 1 x 2 ½ to get 2 1/2 , so multiply 4 x 2 ½ to get 10ft. Method 2: Use cross products (multiplication) to get the answer. 1 2 2 𝑥 = 1 4 1(x) = 2 ½ (4) X = 10