1. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 1.3 Estimating and Evaluating

2. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Find estimates

3. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Estimating and Evaluating Along with the reasoning and problem-solving skills covered in the previous sections, critical thinking also involves the ability to estimate and evaluate. These skills are part of a bigger picture of being numerate; that is, having the ability to think about and communicate with numbers. We’ll continue our critical thinking skills by applying reasoning and the steps of problem solving to estimation and evaluation.

4. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Estimating Estimate To estimate is to make an approximate calculation or to roughly judge the value of a quantity.

5. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Estimating Important points about estimation: Estimates are an educated guess, not an exact amount. How precise do you want your estimate to be? When should rounding occur – should you round before any calculations are done or wait and round at the end of all calculations?

6. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation Suppose you are planning a wedding reception for 100 people. This is your first time planning such an event. Estimate the cost of the reception. Solution Let’s use Pólya’s steps to help us with the estimation. 1.Understand the Problem If we are estimating the cost of the reception, then we need to account for all of the possible expenses we might encounter and add the approximate expenses to find an estimate for the total.

7. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) 2.Develop a Plan There are several factors that go into the cost of the reception. Breaking the overall cost into smaller segments by listing as many of these as possible is a great place to start. This is a way to account for all possibilities while solving simpler problems.

8. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) 3.Carry Out the Plan How many cost factors can you name? If you’re not in the reception business, you might seek the help of an online planner or other resource so that you don’t leave anything out. Suppose an online search for “planning a reception checklist” produced the following.

9. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) Venue cost Food Décor Entertainment First of all, you need a venue big enough to comfortably hold 100 people. Since this cost varies based on location, it is best to get a few rough estimates—again, this can be done online. The Wedding Report for 2011 lists the average cost of reception venues as $3228.

10. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) Since all we need is an estimate, we’ll use this average. However, keep in mind that some venues include the cost of food in the total cost of the venue, while other venues do not include this and may still cost tens of thousands of dollars. 1 Food for a reception is usually based on a “per person” amount. As a result, changing the number of guests could drastically change the total cost of the food. 1 The Wedding Report, http://www.theweddingreport.com

11. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) Several wedding sites give total cost averages in the range of $4750. Again, keeping in mind that we’re looking for an estimate and not an exact figure, this will help keep us moving forward on finding the estimate and keep us from being bogged down in finding the least expensive price per head. Decorations can be anything from very simple to spectacular! However, the average reported cost is in the $900 range for reception decorations and flower arrangements. So, we’ll use this for our estimate.

12. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) Last, assuming entertainment is a desired feature of the reception, we’ll need to make an initial decision based on some quick research. Having a DJ or single musician ranges from $300 to $700, while having a live band ranges from $900 to $1600. It might seem like averaging the two of these would give a good general idea, but stop and think for a minute. An average of the lowest and highest amounts would give us an amount of $950. This amount is either far too much for the first category, or not really enough for the higher category.

13. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) So instead of being so far off in either direction, we will simply chose one of the categories and mention it in our estimation. For now, choosing the lower end of having a DJ or single musician, means that entertainment cost is about Now, we’re prepared to combine all of the individual estimates in order to arrive at a total estimate for a wedding reception.

14. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) Since the estimation price is $9378, now might be a good time to consider if rounding should play a part, and if so, how precise it should be. Because this is an estimate, giving prices that end in $78 might appear to be more accurate than we were anticipating.

15. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) Since we are new to the game of planning a reception, friends, relatives or whomever the estimation is for might assume that enough research went in to be confident enough to estimate to the nearest dollar. Although we did do the research, this is unlikely. Rounding to the nearest hundred dollars or the nearest thousand dollars might paint a better picture of the estimate. Rounding to the nearest hundred dollars gives $9400 while rounding to the nearest thousand dollars gives $9000.

16. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) Consider the scale of the project. A $400 difference in price is meaningful in a nine ‑ thousand ‑ dollar project. Therefore rounding to $9400 for the estimation is the better choice. Let’s stop for a moment and consider when a difference of $400 might not make such a large impact. Suppose the estimate was more like $32,378. When considering a budget near thirty-two thousand dollars, $400 is no longer likely to be a game-changer.

17. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) 4.Look Back The last step in the process requires that we consider the answer to determine if it is feasible. If you were just beginning the process of planning a wedding reception, this amount could either cause sticker shock or be quite a nice surprise, depending on your past experience with weddings. Since we based our intermediate figures on national averages, it is safe to assume that this is a middle- of-the-road figure.

18. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Using Pólya’s Problem-Solving Steps in Estimation (cont.) What we have learned in the process is that there are several areas where we might be able to save money, but also places where we might want to splurge, for example, hiring a live band to dance the night away!

19. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Using the estimation prices given in Example 1, what is the estimated cost for the same reception if the number of guests was only 50? Or 200 (assuming that all venues in our previous calculations could accommodate 200 people)? Think carefully about how you might arrive at the estimates.

20. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Answer:The only amount likely to change is the cost of food since it is the only one affected by the number of guests. Therefore, a reasonable estimate would be halving the cost of food for 50 people, or doubling it for 200. Therefore, $7003 or $7000 rounded to the nearest hundred dollars is an appropriate estimate for 50 guests, and $14,128 or $14,000 for 200.

21. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Using Deductive Reasoning in Estimation A public university is expanding its campus by building a new Math and Computer Science Center. Estimate the cost of the new center by using the cost specifications of the most recently constructed building on campus. The previous classroom building had 20,000 square feet (sq ft) of assigned space, with a construction cost of $6,000,000. The new center needs to contain 40,000 sq ft of assigned space.

22. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Using Deductive Reasoning in Estimation (cont.) Solution The problem clearly asks for an estimate of the building cost for the new center based on previous figures. This is both Step 1 and Step 2 in our problem-solving process. We understand the problem and are given a plan of action. We can then use deductive reasoning to carry out that plan. State-funded universities build classroom buildings on roughly the same level of quality.

23. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Using Deductive Reasoning in Estimation (cont.) Therefore, since we are given the construction cost of the first building, we can reasonably conclude that the new center will cost approximately the same amount per square foot. Because we are doubling the square footage of assigned space in the new center, we will need to double the construction cost. Therefore, using the information we are given, a good estimate for the cost of the new Math and Computer Science Center is $12,000,000.

24. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Estimating Using Graphs A 2011 study of student graduation data by Complete College America showed that on average, students end up taking significantly more credits than required to complete their college degree. 2 According to the graph, students graduating with a certificate are required to take 30 credit hours. 2 “Time is the Enemy,” Complete College America

25. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Estimating Using Graphs (cont.) a.Notice that the number of credits that students receiving a certificate take is missing. Estimate the number of credits students receiving a certificate actually take. b.If the average cost of a credit hour is $287 when seeking a certificate, estimate the total cost for courses that students should pay for a certificate. c.If the average cost of a credit hour is $287 when seeking a certificate, estimate the cost for courses that students end up paying if they take the average number of credits for a certificate. Use your answer from part a.

26. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Estimating Using Graphs (cont.) Solution a.We can see that under the Certificate heading, we are told that students should take 30 credits to complete the degree. The legend explains that the green books represent the 30 credits needed for the certificate and the blue books represent the excess credits that the students take while pursuing a certificate. We can compare the two stacks and see that they represent about the same number of credits.

27. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Estimating Using Graphs (cont.) In other words, on average students take twice as many credits as they need to get a certificate. So a good estimate for the number of credits students actually take when receiving a certificate is 30 + 30 = 60. In reality, the actual figure for the average number of credits students take when receiving a certificate is 63.5. b.Students should take 30 hours to receive a certificate based on the graph. For our estimate, we can round the price of a credit hour to make our estimation easier. Since the cost is $287 on average, we can round to $300. Multiplying these two together gives us the following.

28. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Estimating Using Graphs (cont.) So, on average, students receiving a certificate should pay approximately $9000. c.Based on part a., we know that students take twice as many credit hours as needed for a certificate. Therefore, we can double the cost that they should pay in order to find the cost they actually pay.

29. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #2 Answer: On average, students take approximately 17 extra credit hours. Skill Check #2 Use the graph in Example 3 and estimate the number of excess credits students take when earning a bachelor's degree.