# RATES, PATTERNS AND PROBLEM SOLVING

Welcome to Algebra I! Algebra at its core is all about using the properties of numbers (how they behave) to manipulate unknowns, called variables. But, in practicality, Algebra is used to recognize patterns, turn them into mathematical relationships, and then use these relationships for useful purposes. Today’s lesson, being the first of the course, is exploratory in nature and will utilize a basic understanding of rates or ratios. Exercise #1: Answer the following rate/ratio questions using multiplication and division. Show your calculation (and keep track of your units!). Rates show up everywhere in the real world, whether it is your pay per hour of work or the texts you can send per month. Rates are all about multiplication and division because they ultimately are a ratio of two quantities, both of which are changing or varying. Exercise #2: A runner is traveling at a constant rate of 8 meters per second. How long does it take for the runner to travel 100 meters? (a) If there are 12 eggs per carton, then how many eggs do we have in 5 cartons? (b) If a car is traveling at 65 miles per hour, then how far does it travel in 2 hours? (c) If a pizza contains 8 slices and there are 4 people eating, how many slices are there per person? (d) If a biker travels 20 miles in one hour, how many minutes does it take per mile traveled? (a) Experiment solving this problem by setting up a table to track how far the runner has moved after each second. time, t (seconds) Distance, D (meters) 1 2 5 10 (b) Create an equation that gives the distance, D, that the person has run if you know the amount of time, t, they have been running. (c) Now, set up and solve a simple algebraic equation based on (b), that gives the exact amount of time it takes for the runner to travel 100 meters.

COMMON CORE ALGEBRA I,

UNIT #1 – THE BUILDING BLOCKS OF ALGEBRA – LESSON #1 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2013 The previous exercise showed how we can take a pattern and extend it into the world of algebra, a world that contains symbols and conventions that may seem strange, but hopefully somewhat familiar from previous work. In the final exercise, we will tackle a larger problems to see how rates, patterns, and algebra can combine to solve a more challenging problem. Exercise #3: A man is walking across a 300 foot long field at the same time his daughter is walking towards him from the opposite end. The man is walking at 9 feet per second and the daughter is moving at 6 feet per second. How many seconds will it take them to meet somewhere in the middle? (a) Draw a diagram to help keep track of where the man and his daughter are after 1 second, 2 seconds, 3 seconds, etcetera. Create a table as well that helps keep track of how far each one of them has traveled as time goes on. Time (seconds) Father’s Distance (feet) Daughter’s Distance (feet) Total Distance (feet) 1 2 5 10 (b) What must be true about the distances the two have traveled when they meet somewhere in the middle? (c) Create equations similar to Exercise #3 to predict the distance the father has traveled and the distance the daughter has traveled. (d) Create and solve an equation to predict the exact amount of time it takes for the father and daughter to meet in the middle. Name: _____________________________________ Date: ____________________ COMMON CORE ALGEBRA I, UNIT #1 – THE BUILDING BLOCKS OF ALGEBRA – LESSON #1 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2013 RATES, PATTERNS AND PROBLEM SOLVING COMMON CORE ALGEBRA I HOMEWORK FLUENCY 1. Answer the following rate questions based on either multiplication or division. Think carefully about which is required (they will be mixed up). Show the calculation and units that you use. 2. If there are 4 quarts in a gallon, and 2 pints in a quart, and 2 cups in a pint, then how many cups are in a gallon? Show your calculation or explain how you arrive at your answer. 3. A person driving along the road moves at a rate of 56 miles per hour driven. How far does the person drive in 1.5 hours? Show the calculation you use in your answer and give your answer proper units. 4. Mr. Weiler has 32 students in his class. He wishes to place them into 8 groups of equal size. Which of the following represents the number of students per group? (1) 256 (3) 6 (2) 2 (4) 4 (a) A child bought 4 bags of rubber bands to make into bracelets. If there are 80 rubber bands per bag, how many total rubber bands did he buy? (b) Kirk has 42 pieces of candy to divide evenly between his three children. If he puts the pieces into three boxes, how many pieces of candy are there per box? (c) A car traveling on the Taconic parkway travels 84 miles in two hours. What is the cars speed (a special type of rate) in miles per hour? (d) A car salesperson earns a $500 fee per car she sells. If she sells 4 cars in one day, how much money does she earn in fees? APP 5. S r ( ( ( ( ( PLICATION Seating in th rows farther (a) Assumin following (b) Jonathan equation Does this your yes/ (c) The corre (d) Accordin (e) Finally, l a simple COMMO NS heaters or au away. An e ng this pat g table: n tries to mat for the num s equation w /no answers. ect equation ng to the form let’s say we equation tha Row, r 1 2 3 4 5 6 7 N CORE ALGEBR eMAT uditoriums is example of a ttern contin thematically mber of seats S r 7 2 work for r . is: S r 2 mula from p know that a at gives you Number o Seats, S 9 11 RA I, UNIT #1 – T THINSTRUCTION s often arran a seating cha nues, fill o model the n and determi 2 , where S is 1? What a 7 . Verify art (c), how certain row this answer. of THE BUILDING B N, RED HOOK, NY nged such th art for a thea out the number of se ines: s the number about for r this equation many seats w has 91 seats . BLOCKS OF ALG Y 12571, © 2013 hat rows clo ater is shown eats in a give r of seats in 2 and r n matches yo are in the 15 s in it. Whic GEBRA – LESSON 3 oser to the st n below. en row. He row, r 3? Show c our table for 5th row? Sho ch row is it? STAG N #1 tage have le tries to com calculations r r 1, r 2 ow your calc ? Try to set GE ess seats than me up with an that suppor 2 , and r 3 culation. up and solve

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