class: center, middle, inverse, title-slide .title[ # Lesson 12: Problem-Solving Guidelines and Hints ] .author[ ### Fei Ye ] .date[ ### May, 2024 ] --- class: center middle ## Unit 2C: Problem-Solving Guidelines and Hints *Primary Source:* PPT for the book "Using & Understanding Mathematics". --- ## A Four-Step Problem-Solving Process - **Step 1:** *Understand the problem.* - Identify and express known information using mathematical expressions. - Identify the goal and unknown information. - Look for clue words, such as operations and relations. - **Step 2:** *Devise a strategy for solving the problem.* - Identify patterns. - Apply known formulas, and/or write down equations. - Use guessing and checking to fill in the gap between know and unknown. - Work backward and identify sub-goals. - **Step 3:** *Carry out your strategy, and revise it if necessary.* - Use analytical and computational tools to carry out the plan. - **Step 4:** *Look back to check, interpret, and explain your result.* .footmark[ These four steps are suggested by George Polya in his book *How to Solve It*. ] --- ## Problem Solving Guidelines and Hints - **Hint 1:** *There may be more than one answer.* - **Hint 2:** There may be more than one method. - **Hint 3:** *Use appropriate tools.* - **Hint 4:** Consider simpler, similar problems. - **Hint 5:** *Consider equivalent problems with simpler solutions.* - **Hint 6:** Approximations can be useful. - **Hint 7:** *Try alternative patterns of thought.* - **Hint 8:** Do not spin your wheels. --- ## Example: Fundraising tickets (1 of 2) Tickets for a fundraising event were priced at $10 for children and $20 for adults. Shauna worked the first shift at the box office, selling a total of $130 worth of tickets. However, she did not keep a careful count of how many tickets she sold for children and adults. How many tickets of each type (child and adult) did she sell? **Solution:** Try trial and error. Suppose Shauna sold just one $10 child ticket. In that case, she would have sold $130 - $10 = $120 worth of adult tickets. Because the adult tickets cost $20 apiece, this means she would have sold $120 `\(\div\)` ($20 per adult ticket) = 6 adult tickets. We have found an answer to the question: Shauna could have collected $130 by selling 1 child and 6 adult tickets. But it is not the only answer, as we can see by testing other values. --- ## Example: Fundraising tickets (2 of 2) **Solution: (Continued)** In fact, there are seven possible answers to the question. In addition to the answer we've already found, other possible answers are - 3 child tickets and 5 adult tickets - 5 child tickets and 4 adult tickets; - 7 child tickets and 3 adult tickets; - 9 child tickets and 2 adult tickets; - 11 child tickets and 1 adult ticket; - 13 child tickets with 0 adult tickets. Without further information, we do not know which combination represents the actual ticket sales. --- ## Example: Thickness of a piece of paper How thick is a single sheet of paper? **Solution:** Measuring the thickness of a single piece of paper is tricky. Instead we might first measure a stack of paper, and then scale the thickness a single sheet. Printer papers are usually sold in reams. A ream of papers normally contains 500 sheets. We could estimate that a ream of paper is about 2 inches thick. Scaling it down, we get an estimation of the thickness a piece of paper: $$ \frac{2 \text { inches }}{\text { ream }} \cdot \frac{1 \text { ream }}{500 \text { sheets }}=0.004 \text{ inches per sheet}. $$ .footmark[ Source: [Math in Society by David Lippman](https://tinyurl.com/y4mxuw97) ] --- ## Example: Party Decorations (1 of 3) In a room that has 10 large cylindrical posts, you plan to decorate the posts by wrapping 8 turns of ribbon around each post. The posts are 8 feet high and have a circumference of 6 feet. How much ribbon is needed? **Solution:** We can convert the problem into a sampler one by changing the viewpoint.Imagine that post is a hollow cylinder and cut down into a flat rectangle. The width of the rectangle is the 6-foot circumference of the post, and its length is the 8-foot length of the post. The ribbon appears in the rectangle as eight equal-length diagonal segments with ends on the 8-foot-long sides of the rectangle. The length of one segment can be found using the Pythagorean theorem $$ \text{Hypotenuse}=\sqrt{\text{Leg}^2+\text{Leg}^2} $$ --- ## Example: Party Decorations (2 of 3) .center[  ] --- ## Example: Party Decorations (3 of 3) The height of each triangle is 1/8 of the length of the rectangle, that is `\(8\text{ ft}\div 8=1\text{ ft}\)`. The base is 6 feet. Therefore, `$$\text{Hypotenuse}=\sqrt{6^2+1^2}\approx 6.1 \text{ ft}.$$` One each post, there will be 8 such segments. So the ribbon needed for one post is approximately `$$8\times 6.1 \text{ ft}=48.8 \text{ ft}.$$` For 10 posts, we need `$$10\times 48.8 \text{ ft}=488 \text{ ft}.$$` --- ## Example: Possible squares on the chessboard (1 of 3) Find the total number of possible squares on the chessboard by looking for a pattern.  **Solution:** Start with the largest possible square: There is only one way to make an `\(8\times 8\)` square. --- ## Example: Possible squares on the chessboard (2 of 3) **Solution: (Continued)** Now, look for the number of ways to make a `\(7\times 7\)` square. Find the total number of possible squares on the chessboard by looking for a pattern. There are only four ways.  --- ## Example: Possible squares on the chessboard (3 of 3) If you continue looking at `\(6 \times 6\)`, then `\(5 \times 5\)` squares, and so on, you will see the perfect square pattern as indicated in the following table for this chessboard problem: | Square | Number | |:-----------:|:------:| | `\(8\times8\)` | 1 | | `\(7\times7\)` | 4 | | `\(6\times6\)` | 9 | | `\(5\times5\)` | 16 | | `\(4\times4\)` | 25 | | `\(3\times3\)` | 36 | | `\(2\times2\)` | 49 | | `\(1\times1\)` | 64 | | Total | 204 | --- ## Practice: Multiple possible solutions A toll collector on a highway receives $5 for cars and $3 for buses. In 10 minutes, she collected $40. How many cars and buses passed through the toll booth during that period? *List all possible solutions.* --- ## Practice: Heart beats Estimate how many times does your heart beat in a day? --- ## Practice: Coiling problem Ten turns of a wire are wrapped around a pipe with a length of 20 centimeters and a circumference of 4 centimeters. What is the length of the wire? --- ## Practice: Pattern about numbers of blocks .iframecontainer[ <iframe src="https://www.myopenmath.com/embedq2.php?id=343676&amp;seed=2023&amp;showansafter" width="100%" height="400px" data-external="1"></iframe> ] .footmark[ <a href="https://www.myopenmath.com/embedq2.php?id=343676&seed=2023&showansafter" target="_blank">Click here to open the practice in a new window</a> ] --- ## Practice: Sum of a sequence of numbers .iframecontainer[ <iframe src="https://www.myopenmath.com/embedq2.php?id=677767&amp;seed=2021&amp;showansafter" width="100%" height="400px" data-external="1"></iframe> ] .footmark[ <a href="https://www.myopenmath.com/embedq2.php?id=677767&seed=2021&showansafter" target="_blank">Click here to open the practice in a new window</a> ] --- ## Practice: Solve a riddle using math “Brothers and sisters I have none, but that man’s father is my father’s son.” Who is “that man”?