# MACC.912.A-CED.1.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic.

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MACC.912.A-CED.1.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. CCSS Mathematics Practice

At 8:00 AM, the art teacher, Mr. Burns put some scissors out for his students in the scissor rack. By 10:00 AM, Mr. Burns found that one-half of the scissors were gone. By noon, Mr. Burns found that one-third of the scissors that were left from when he checked before were gone. In the afternoon, Mr. Burns found that one-fifth of the remaining scissors were gone. If at the end of the day there were at least 15 scissors left, what is the smallest possible number of scissors he could have started with?

Entry Points Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. – Guess, Check, and Revise – Make a Model or Diagram – Work Backwards – Use a Formula or Equation/Inequality

Guess, Check, and Revise By 10:00 AM, Mr. Burns found that one-half of the scissors were gone. By noon, Mr. Burns found that one-third of the scissors that were left from when he checked before were gone. The problem states that there are at least 15 left in the afternoon…this is already too small. Let’s try a larger number to start with. Let’s try 30.

Guess, Check, and Revise By 10:00 AM, Mr. Burns found that one-half of the scissors were gone. By noon, Mr. Burns found that one-third of the scissors that were left from when he checked before were gone. I cannot find two-thirds of 20. Let’s try a different number. I can find two-thirds of 21, so start with 42. Let’s try 40.

Guess, Check, and Revise By 10:00 AM, Mr. Burns found that one-half of the scissors were gone. By noon, Mr. Burns found that one-third of the scissors that were left from when he checked before were gone. Let’s try a different number. I know I have to divide by 2 and 3, so let’s try multiples of 6. Let’s try 42. The problem states that there are at least 15 left in the afternoon…this is already too small.

Guess, Check, and Revise By 10:00 AM, Mr. Burns found that one-half of the scissors were gone. By noon, Mr. Burns found that one-third of the scissors that were left from when he checked before were gone. Let’s try a different number. I know I have to divide by 2, 3, and 5, so let’s try multiples of 30. Let’s try 54. In the afternoon, Mr. Burns found that one-fifth of the remaining scissors were gone. I cannot find four-fifths of 18.

Guess, Check, and Revise By 10:00 AM, Mr. Burns found that one-half of the scissors were gone. By noon, Mr. Burns found that one-third of the scissors that were left from when he checked before were gone. The answer is 60. Let’s try 60. In the afternoon, Mr. Burns found that one-fifth of the remaining scissors were gone.

Make a Model or Diagram At the end of the day there were at least 15 scissors left. In the afternoon, Mr. Burns found that one-fifth of the remaining scissors were gone. 15 16 4 444 20 pairs of scissors 4

Draw a Diagram By noon, Mr. Burns found that one-third of the scissors that were left from when he checked before were gone. 20 10 30 pairs of scissors

Draw a Diagram By 10:00 AM, Mr. Burns found that one-half of the scissors were gone. 30 60 pairs of scissors

Work Backwards Afternoon: at least 15 scissors One-fifth of the remaining scissors were gone… this means 4/5 is left.

Work Backwards Noon: one-third of the scissors that were left from when he checked before were gone… this means 2/3 is left.

Work Backwards 10:00 AM: one-half of the scissors were gone

Use a Formula or Equation/Inequality By 10:00 AM, Mr. Burns found that one-half of the scissors were gone. By noon, Mr. Burns found that one-third of the scissors that were left from when he checked before were gone. In the afternoon, Mr. Burns found that one-fifth of the remaining scissors were gone.

Mathematically Proficient Students… … monitor and evaluate their progress and change course if necessary.

Solutions Using our multiple methods, we came up with 2 different answers: – 60 – 56 ¼ Can we have a fraction of a scissor? Do we round down or round up?

Solutions 57 By 10:00 AM, Mr. Burns found that one-half of the scissors were gone. 29 58 Find ½ of 57. By noon, Mr. Burns found that one-third of the scissors that were left from when he checked before were gone. 30 60 Since 1/3 is gone, 2/3 must be left. Find 2/3 of 29. 20 In the afternoon, Mr. Burns found that one-fifth of the remaining scissors were gone. Since 1/5 is gone, 4/5 must be left. Find 4/5 of 20. 16 30 58

Guiding Questions How might you represent the number of pairs of scissors that are in the rack at: – 8 a.m.? – 10 a.m.? – 12 noon? – In the afternoon? – End of the day? Would an equation or an inequality be the better choice for representing this scenario? Do you need one or two variables to represent this scenario? What kinds of numbers make sense for the unknown(s) in this scenario? What does the 15 at the end of the day represent?

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