Problem Solving s2

Problem Solving This chapter attempts to exercise and develop the students’ problem solving ability within the relatively simple framework of percents, rates, and proportional reasoning. These types of problems are ones that regularly come up in everyday life, and the familiar mathematics is a nice springboard for exploring more complex problem solving.

One of the focuses of this chapter is the idea that real life problems are rarely “nice” like math textbook problems. In real life, you usually come up with a question you want to answer rather than being handed one. You are never told what type of problem it is, what formulas to use, or exactly the measurements and data you’ll need to answer the question. Instead, you usually either have to look up information, or weed out unnecessary information.

It is challenging to replicate these situations within the confines of text. The problems at the end of the exercise set attempt to touch on these skills, but it will be better to do problems in-class to build these skills. One format that I’ve had good success with is the “three acts” approach advocated by Dan Meyer1. In this approach, you bait students with a video or photo, one that hopefully begs a question (like the bottle cap covered car photo in the exercises that begs “how many bottle caps is that?”). Then you ask the students to figure out what information they need to answer the question. Finally, it’s great if we view a video or see a photo that verifies their mathematical answer and opens a conversation about measurement error.

In a bit more detail, the idea is:  Act 1: Show the lead-in video/photo (or a portion of the full video in some cases).  Ask “what’s the first question that comes into your head?” Come to a consensus on a question to explore. If possible, go with what the students are interested in, rather than what you originally had in mind.  Ask students to estimate the answer. Ask for values they know are too low and too high. This builds buy-in (now they want to see if they’re right).  Act 2: Ask them what information they need to answer the question. Provide them the information, make them estimate it, or give them physical objects they can measure.  Act 3: When available, view the rest of the video or another photo to see the answer. Compare the results to the calculated values and to the initial estimates.

Penny circle Act 1: http://vimeo.com/44558245 Potential question: How many pennies will fit in the larger circle? Act 2: None. Have them estimate the size of the first circle using a penny width as the unit, perhaps. Or tell them a penny is 19 mm in diameter. Act 3: None. Filling a cylinder with sand

1 See http://blog.mrmeyer.com/?p=10285, and his TED talk http://www.ted.com/talks/dan_meyer_math _curriculum_makeover.html about the general “be less helpful” approach. Act 1: http://vimeo.com/38699414 Potential question: How many scoops to fill it? Act 2: Scoop is ½ cup. Vase is 14.5 cm in diameter, 24 cm tall. Act 3: http://vimeo.com/38699554

Hot Coffee Full 3-act is available here: http://mrmeyer.com/threeacts/hotcoffee/ with edited videos. The information and similar Youtube videos are provided below. Act 1: http://www.youtube.com/watch?v=bYvHl8CQu6M and http://www.youtube.com/watch?v=bDqokWcFyjc Potential question: Do they break the record? How many cups of coffee is that? Act 2: The cup is 7 ft in diameter and 7 ft tall. Previous record: http://www.youtube.com/watch?v=K8l1mAgiiwo. Rate it’s filling: 2.1 gallons/minute from each of the hoses. 1 cubic foot is about 7.5 gallons. Act 3: End of http://www.youtube.com/watch?v=XMLR6TivHSI

Filling an octagonal tank Full 3-act is available here: http://mrmeyer.com/threeacts/watertank/ The information and similar streaming videos are provided below. Act 1: http://vimeo.com/34763479, or start this video and pause it after 30 seconds or so: http://vimeo.com/9552977 Potential question: How long will it take for the tank to fill? Act 2: The octagon has side length 12.4cm. The distance across is 29.9cm. The tank is 36.5 cm tall. The hose could fill 16oz in 8.46 seconds (or 8.7 seconds, depending on the video) http://vimeo.com/9659647. Or, have students estimate the dimensions (perhaps letting them figure out they can compare to the 2x4 the hose hanger is attached to). 1 fl oz = 29.6 mL = 29.6 cm3. Act 3: http://vimeo.com/9632608

Leaking faucet Act 1: http://www.youtube.com/watch?v=_5-ouXC4-Pg Potential question: How much water is wasted in a year? Or: How long will it take to drip a cup of water? Act 2: Measuring the flow rate: http://www.youtube.com/watch?v=fffICPaBCD4 . Act 3: May not need one if your question is how much water is wasted in a year. This video shows how long it takes to drip 1 cup: http://www.youtube.com/watch? v=Ys9BWUaA0nk

Car atlas Full 3-act is available here: http://threeacts.mrmeyer.com/carcaravan/ The information with alternate links is provided below. Act 1: http://www.flickr.com/photos/53380495@N02/4993931189/ Potential question: How many cars were used to make that installation? Act 2: Width of the installation: 300cm. Dimensions of a sample toy car: http://www.flickr.com/photos/41126256@N03/7755812426 and http://www.flickr.com/photos/41126256@N03/7755826758 Act 3: http://davidtwaller.com/Gallery.aspx 2,500 toy cars were used.

Post-it Car Act 1: http://imgur.com/NUBzS Act 2: Dimensions of a post-it: http://www.amazon.com/Post-Original-Inches-Canary- Yellow/dp/B000078UWA. The car appears to be a Lexus: http://www.lexus.com/models/IS/detailed_specifications.html Act 3: Apparently it took 42 pads. Each pad is 100 sheets, so about 4200 post-its altogether: (warning: some foul language in this link) http://www.reddit.com/r/pics/comments/109wlk/so_i_get_off_work_today_and_t his_is_what_i_come/c6bmqvb

Another focus in the chapter is estimation skills. The site http://www.estimation180.com/ provides a great set of estimation questions, many of which build off of previous estimations.

Weighted voting: A video overview of Electoral college: http://www.youtube.com/watch?v=ok_VQ8I7g6I

Crypto: A collection of calculation apps for modular arithmetic, RSA key generation, and more. Handy for students to use, and handy for generating usable numbers for questions. http://www.opentextbookstore.com/mathinsociety/apps/

A nice example of how frequency analysis is used to break encryption codes. http://en.wikipedia.org/wiki/Frequency_analysis

A nice 20 minute video on breaking the Engima code, Alan Turing, and the beginnings of modern computing: http://www.youtube.com/watch?v=5nK_ft0Lf1s

A video intro Diffie-Hellman key exchange: http://www.youtube.com/watch? v=YEBfamv-_do (the actual explanation of DH starts about 2:15 in) A nice video intro to RSA: http://www.youtube.com/watch?v=wXB-V_Keiu8

Appt: From the Census Bureau, current apportionment maps and Excel files with census counts: http://www.census.gov/population/apportionment/data/2010_apportionment_results.html

More on history: http://www.census.gov/population/apportionment/about/history.html

Fractals: A much more extensive coverage of fractals can be found on the Fractal Geometry site. This site includes links to several Java software programs for exploring fractals.

The Mandelbrot Explorer site, provides more details on the Mandelbrot set, including a nice visualization of Mandelbrot sequences.