# Math Resources: K-12 Supplementary Collections

Our collection of Math Resources has multiple pages designed for teaching within specific grade bands.  You'll also find valuable collections of support and enrichment resources that will benefit all learners.

Math Resources (Page 1): Elementary and Middle Levels: Basic Mathematics and Skills Development, plus apps for mobile devices

Math Resources (Page 2): Middle, Secondary, Post-Secondary Subject Specific Resources, including apps for mobile devices: Algebra and Pre-Algebra, Geometry, Statistics, Probability, Trigonometry, Precalculus and Calculus

Math Resources (Page 4): Enrichment and Extra Help: Math Contests, Competitions, Challenges, and Camps; Study Skills and Homework Help, plus tips for parent involvement; and Dictionaries, Glossaries, Reference Sheets, and Math Encyclopedias

Math Resources (Page 5): Lesson Plans and Worksheets, Design and Manage Your Own Lesson Plans, and Publishers' Textbook Support Sites

Math Resources (Page 6): Resources for Podcasts and Vodcasts, the Flipped Learning Model, and for Whiteboards, including tips on using whiteboards

## Miscellaneous Math Collections

### Fields of Mathematics Interactive Mind Map

Use the Fields of Mathematics Interactive Mind Map by Antonio Gutierrez of GoGeometry.com to introduce your learners to the broad field of mathematics.  This mind map elaborates on multiple categories within seven fields: the mathematics of quantity, space, change, structure, foundations and philosophy, discrete mathematics, and applied mathematics.  This is a great find!

#### Introduce your students to how math has influenced modern life.

Math Matters includes a brief look at how mathematics has influenced modern life.  It highlights the top twenty technological advances of the twentieth century, and explores the role of mathematics in bringing these innovations to reality.  Consider electrification, highways, electronics, water supply and distribution, household appliances, petroleum and petrochemicals, airplanes, spacecraft, high performance materials, air conditioning and refrigeration, nuclear technology, radio and TV, computers, imagining, telephones, the internet, health technologies, agricultural mechanization, laser and fiber optics, and automobiles.  The events and historical influencers are online and a downloadable poster is available.

BrainPOP produces educational animated movies and is award winning.  K-12 movies address science, technology, health, math, and special topics, including online safety.  The online math movies include topics in algebra, geometry, data analysis/probability, problem solving, measurement, and numbers/operators.  There are homework help, puzzles, activities, prizes, and more for registered users.

Cut the Knot Interactive Mathematics Miscellany and Puzzles was developed by Dr. A. Bogomolny, former Associate Professor of Mathematics at the University of Iowa turned software developer. It contains numerous games and puzzles on mathematics including arithmetic, algebra, geometry, probability, and calculus. There are pages for interesting math facts, proofs, analog gadgets (use of compass, scissors, rulers, etc.), mathematics as a language, and more. This site has some highly interesting intellectual material on mathematics, and links to other sites.

FactMonster.com is an award winning site that features an atlas, almanac, dictionary, encyclopedia, and individualized homework help.  Students can get facts on a range of subjects including math, the world and news, U.S., science, sports, people, and more.  There are games and quizzes also.  Among press releases is the March 2001 recognition in the NCTM News Bulletin, WebBytes, which highlighted the site as a reference source.

LearnAlberta.ca provides multimedia online resources for K-12 in several content areas, accessible for all learners.  Don't miss the use of audio, video, animations, and interactivity in the mathematics course materials.

Math Mnemonics from OnlineMathLearning.com contains math mnemonics that help students remember math conventions, algorithms, definitions, techniques, and formulas. These are categorized by pre-algebra, algebra, geometry, and trigonometry.  [Note Education World has additional math mnemonics.]

Math Power is a must see web site that provides information about basic math, algebra, study skills, math anxiety, and learning styles.  Although the site addresses the needs of community college adult learners, it has appeal for middle school and secondary math students and teachers.

Nick's Mathematical Puzzles are for students grades 9 and upward.  Nick Hobson addresses geometry, probability, number theory, algebra, calculus, and logic. Hints are provided, along with answers, fully worked solutions, and links to related mathematical topics. Many of the puzzles are elementary in their statement, yet challenging. There are 160 puzzles in this collection.  See the index for titles of those.

A key feature of the site is the detailed exposition, from first principles, of the puzzle solutions. Some of the puzzles are used to showcase particular mathematical concepts. Further references are provided with many of the solutions.  The site is designed with accessibility in mind.

OnlineMathLearning.com is a real find!  It contains math help and learning resources for arithmetic, algebra, geometry, statistics, probability, set theory, trigonometry, matrices, SAT and ACT test preparation, worksheets, games, trivia, and links to related resources.  Of particular value are the videos that accompany various topics.  Resources are identified for Common Core math standards.

Platonic Realms for secondary and post-secondary students features a "must-see" interactive mathematics encyclopedia, which can be browsed at elementary and advanced levels.  Topics include basic mathematics, algebra, analysis, biography, calculus, discrete math, history, economics, geometry, graph theory, number theory, statistics, trigonometry, and math quotes.  There is an extensive link library to instructional resources, K-12 and university math departments, institutes, journals, societies, business/commercial math sites, and selected personal pages.  View the mathematics art of M.C. Escher.

Teaching Treasures offers online interactive practice in K-10 mathematics topics with helpful explanations: addition, subtraction, multiplication, division, algebra, measurement, problem solving, shapes, space, reading a clock, and more.

Tecmath on YouTube contains multiple videos, each about 10 minutes or less, on a variety of math topics: geometry, trig, statistics, puzzles, problem-solving tricks and short cuts, fractions, and more.  The collection is extensive and also includes playlists.

TES Teaching Resources contain an extensive library of free teaching resources in multiple areas created and uploaded by teachers in the TES community. Classroom materials include lesson plans, videos, PowerPoints, games, puzzles, worksheets, tutorials, interactives, and whiteboard resources.  Math resources are easiest found by accessing the collection of primary and secondary resources, then narrow your search by a grade level or subject.  This site is based in the UK and highly rated--do have a look.

U.S. Department of Education is a comprehensive site for preK-12 and higher education resources, policy information, grants and funding, research and statistics, and more.

## Practical Applications-- Math in Careers, Everyday Life, and Across the Curriculum

### A Well-rounded Education Support of Practical Applications in the Study of Mathematics

On March 13, 2010, the Obama administration released its blueprint for revising the Elementary and Secondary Education Act (ESEA). One of those proposals calls for a complete and well-rounded education with support for "strong instructional practices in arts, foreign languages, history and civics, financial literacy, environmental education, and other subjects" (p. 19).  The research outlined in the proposal for A Complete Education (2010) noted:

One strategy for ensuring a well-rounded education is the integration of instruction in English-language arts and mathematics with other subject areas that are not currently covered by accountability testing, particularly social studies and the arts (Gunzenhauser, 2003; Manzo, 2005; Meyer, 2005; Pinzur, 2004; Rabkin and Redmond, 2005; Vogler, 2003; von Zastrow and Janc, 2004). In this approach, content from science, social studies, foreign languages, or the arts is used during the instruction of mathematics, reading, or writing. (p. 19)

Hence, readers will note that the resources that follow would support this proposal.

#### Do you need some ideas on getting your students interested in task-based learning, problem-based or inquiry-based learning?

Succeeding with Inquiry in Science and Math Classrooms by Jeff Marshall (2013) includes the need for inquiry in your classroom, details of inquiry instruction (e.g., the framework, exemplars, how inquiry appears in units of study), and advice on how to be successful with this method.  Real-world lesson plans are provided to illustrate effective inquiry-based instruction.

emergent math is a blog that has some great ideas for introducing students to inquiry-based learning.  Begin your exploration with A Problem Based Learning Starter Kit, posted October 30, 2013.  You will also find a set of Common Core Problem Based Curriculum Maps for grades 3-11, algebra 1, geometry, and algebra 2.

Origami, the art of folding two-dimensional paper into three-dimensional objects, can be seen throughout the world.  YouTube has several videos illustrating the value of this art.  For example, view 11 Levels of Origami: Easy to Complex as explained by origami artist and physicist Robert Lang.  Also of interest is How NASA Engineers Use Origami To Design Future Spacecraft.

Get some classroom ideas for using origami at the Origami Resource Center/.  It's easy to connect origami to math and geometry, in particular.

Mathigon: Mathematical Origami includes colorful examples of the Platonic Solids, Archimedean Solids, Stars and Compounds, and Beautiful Origami.  Many come with folding instructions and associated net.

Math occurs in so many careers and in daily life.  According to the U.S. Bureau of Labor Statistics (2020), "Employment in math occupations is projected to grow 27 percent from 2019 to 2029, much faster than the average for all occupations, and will add about 56,100 jobs. Growth is anticipated as businesses and government agencies continue to emphasize the use of big data, which math occupations analyze."  Here are some examples for you:

• People who work in agriculture try to maximize produce and would need to know about mixture problems.  What's the right amount of seed to use, or fertilizer, pesticides, water?

• Accountants/Bankers/and Financial businesses use spreadsheet packages. Who's writing the formulas? This is an algebra skill. Profit/loss translates to knowledge of percents and decimals.  There are algebraic equations involving time and interest rates for calculating mortgage payments.

• Airport air traffic controllers, pilots/aviators, and aviation mechanics use math.  The air traffic controller needs to determine which plane should land first on a runway based on how fast the plane is traveling and how far it is away from the airport, and they need to determine which can take off first.  They need to know about wind speed and direction, and at what altitude a plane should fly.  Aviators need knowledge of the science of air pressure, drag, lift and velocity, and using formulas to calculate aircraft speed.  Pilots also need to know a lot of math including basic arithmetic, algebra, trigonometry, geometry, calculus, physics, and so on.  As examples, it is much more than determining how long a trip will take given distance to a destination and speed of the plane.  They must consider how much fuel their planes need for a trip, so they need to also consider weight of passengers and what else the plane is carrying, and weight of the fuel.  When in flight, they must know when to start to decend a plane, and how fast to decend and at what angle of descent.  The aviation mechanic uses math tools to ultimately ensure the plane is safe for flight.  For more information, read What Kind of Math do Pilots Use?.

• Art and Design of any kind (think people in marketing, too) require visualization. Many of the pleasing designs use "The Golden Ratio, Phi, 1.618." See: https://www.goldennumber.net/ and https://mathworld.wolfram.com/GoldenRatio.html

• Architects use sophisticated software to build online virtual models of buildings, houses, and so on. They certainly need visualization skills--those spatial aspects students study in math, which they might practice using online virtual manipulatives. Certainly architects need understanding of geometric and algebraic concepts for making physical models of the real thing.  Scale drawings rely on accurate use of fractions and decimals.

• Artificial Intelligence careers rely on mathematics.  Sakshi Gupta at Springboard (2022, March 7) noted 10 examples, including machine learning engineers, data scientists, business intelligence developers, big data engineers/architects, software engineers, data analysts, robotics engineers, and natural language processing engineers.

• Athletes can improve their performance by knowing about mathematics (e.g., as it relates to motion, vectors, geometric shapes, nutrition/health, torque, etc.)

• Auto industry: How about all the math used in the auto industry? Car designers use math. Cars contain sophisticated computers. Even now, crash tests are done with simulations.

• Biologists use statistics and probability.

• Builders need to measure accurately to cut those boards, pour those floors (think proportions in getting cement to the right consistency--too much water and the structure breaks). That's where knowledge of fractions and decimals comes in.  They would need to know about areas, volumes, slopes, angle measures, distance, ratios, and so on.  They need skills to calculate how much of a product to buy for the job they are doing, and then its total cost.

• Climatologists use "mathematical skills in collecting climate data, investigating climate indicators, and making predictions regarding climate patterns. They may use computer models to study how Earth's climate changes with time, and sophisticated computer software programs that assist them in modeling the Earth's climate and check that data against known information. They conduct research to determine if humans are affecting Earth's present and future climate." Source: WeUseMath.org: Climatologist from Brigham Young University.

• Cooks, even in the big restaurants, need measurement skills, and proportions for converting recipes for larger or smaller quantities. But there are software packages to help with that and databases for storing those recipes and spreadsheet packages for inventories, profit/loss, and so on.  You need to understand formulas in algebra to design those spreadsheets.

• Cybersecurity is a top career field in high demand.

• Data analysts are almost everywhere and are employed in a number of careers, such as traditional technology firms, consulting, banking/finance, credit bureaus, the health industry, retail businesses, education, e-commerce and so on.  Many of those have a math and science background.  Learn more in Data Analyst: Career Path & Qualifications by Greg Depersio (2020).

• Engineers play a role in the design and modification of products we use.  A civil engineer, for example, would be concerned about the amount of weight a road or bridge can handle under a variety of weather conditions.  Algebra would help with those calculations in their design.

• Fitness trainers can use algebra to help determine a ratio of diet to exercise for achieving a desired weight and maintaining it.

• Forensic scientists analyze evidence from crime scenes for a court of law.  Taking precise measurements (e.g., distance, weight, volume, temperature, etc.) is an essential skill, as is proportional analysis, probability, and trigonometry.  Read more in What Math is Used in Forensic Science?

• Geologists and geoscientists use math models to study the earth: daily weather patterns, earthquake predictions, how glaciers are changing, the best building sites, where flooding might occur, where best to perform archeological and paleontological digs, and so on.

• Math and Science Educators and others in STEM fields.

• Medical doctors study the human body and must be able to properly prescribe doses of medicines, and detect illnesses.  A heart doctor, for example, would rely on an algebraic equation involving heart rate and oxygen levels in the body to determine cardiac output.

• Manufacturing jobs require math skills: "[T]oday's manufacturing work requires strong math skills — not just adding and subtracting, but a good grasp of fractions, decimals and basic trigonometry. And job applicants who want to go into manufacturing often don't have what it takes" (Boodhoo, 2012, para. 2).  Learners who are interested in manufacturing careers might be interested in a glossary of 95 manufacturing terms provided by Propel PLM Inc.

• Police apply math concepts to reconstruct vehicle crashes.  Read how they do this in Building Mathematical Skills and Community Relationships Through Crash Reconstruction by Kelly Remijan (2017, November 22) in ASCD Express.

• Robotics: Robots play a significant role in today's society, including for handling hazardous materials, welding components, performing surgery, or vacuuming carpets.  They are everywhere.  There are self-operating robotic machines.  They are used in factories, mining operations, military, healthcare, and even in the arts and sculpture.  The field is growing and involves applying mathematics (e.g., algebra and geometry) in the development.  Read What to Study for a Career in Robotics, a blog post by Alex Owen-Hill (2020, March 31).

• Sports of all kinds keep track of player statistics.  View Real-Life Math | Baseball on how an assistant general manager of a major league baseball team uses math.

• Stock brokers: Anyone interested in the stock market? Positive and negative numbers are used to indicate what's happening on the market. Gain of so many points, or loss of so many points. Graphing programs (tech part) can be used to create the visual displays. Of course, knowledge of data analysis is needed.

• Software developers

• Space science industry--it's not just about becoming an astronaut.  Florida Tech discusses Career Options for the Future Space Scientist.

• Surveyors use measurement tools and GPS systems.

• Tradesmen of all kinds use math to estimate job costs, and to deal with the technical aspects of their jobs.

• People in the military use simulations for training. Programmers need math skills to design those environments.  Angles of elevation, range, trajectories are useful to locate and hit targets.  You need math to determine how much food, supplies, ammunition, trucks, equipment that troops will need and statistical estimations to determine the rate at which those will be used up.  See how math is used in the army.

• Virtual Reality--think of virtual environments like Second Life -- who programs those? Everything in Second Life is designed with basic prims, which are geometric shapes that are placed and stretched to get whatever you see in the virtual worlds.  Virtual reality technology is being used in medicine, engineering, and architecture to help solve problems, and it's used in the entertainment industry.  For more on virtual reality in careers, view the the series of three short videos from the Stem Career Lab, located in Ohio.  "Students will hear how virtual reality modeling is being used in a number of disciplines and applications. Students will learn about career paths into the Virtual Reality field and how the math and science they learned in high school will help them later in their professional lives."

The list goes on and on.  Animators, computer game designers, economists, special effects designers, professional photographers all use math.

To find books on this topic, enter the search phrase: mathematics and careers.

The developers of Tableau, software for data visualization, stated: "Every STEM field benefits from understanding data—and so do fields in government, finance, marketing, history, consumer goods, service industries, education, sports, and so on" (Why data visualization is important for any career section).  You'll find a data visualization beginner's guide with a definition, examples, and learning resources.  Learn to create your own.  Learn about historical examples and theory from books, access blogs on data visualization, get information on data visualization tools and software, free training videos, and more.  Tableau Public is a free version to publicly share data visualizations online.

#### Why not go on a virtual or real field trip or invite a professional to your classroom to talk about how they use math?

Field trips are a great way for students to explore the use of mathematics in careers, everyday life, in other subjects or to engage more deeply in a concept, such as the history of mathematics, number sense; patterns, functions, and algebraic thinking; estimation, measurement, and computations; spatial sense and geometric concepts; and various math activities (cryptology, harvest time math, and puzzles).  Create a project in which students investigate this theme.  Connect these activities to learning outcomes/standards.  Need ideas?  Visit the following resources:

Discovery Education Virtual Field Trips also includes on-demand virtual field trips in multiple categories.  Per the site: "Each no-cost Virtual Field Trip comes with a companion guide packed with standards-aligned, hands-on learning activities!"

Math Learning Field Trip Ideas for Homeschoolers posted at Dreambox Learning with field trip and lesson ideas for architecture, sports, museums, factory tours, and agriculture.

Field Trips Motivate Mathematicians posted at Teachers Network--consider the local bank, grocery store, restaurant, dairy, quilt shop, or train station.

#### Do you need to explore more careers in math and get ideas on professionals you might invite?

In 101 Careers in Mathematics (3rd Edition) by Andrew Sterrett (2014), you'll actually find more than 101 careers described.  Per the book's description, "Each of the jobs presented shows real people in real jobs. Their individual histories demonstrate how the study of mathematics was useful in landing well-paying jobs in predictable places such as IBM, AT&T, and American Airlines, and in surprising places such as FedEx Corporation, L.L. Bean, and Perdue Farms, Inc. You will also learn about job opportunities in the Federal Government as well as exciting careers in the arts, sculpture, music, and television. There are really no limits to what you can do if you are well prepared in mathematics."  Also see 101 Careers in Mathematics (4th Edition) by Deanna Haunsperger and Robert Thompson (2019).  Both books were published by the American Mathematical Society.

Resources at TechGuide.org include essays of interest, such as A Career with Numbers: A Complete Guide by John O'Donovan (2022, July 14), a K-12 STEM Guide, and Women in Tech.  The site also contains a section on STEM careers.

The STEM Careers Coalition provides a Careers Portal with "Career Profile videos and student activations focusing on the STEM skills, knowledge, and interests needed to pursue exciting careers in the STEM industry."

To find books on this topic, enter the search phrase: math in the real world.

### Introduce career pathways early in the school experience.

Don't wait until high school (or even after graduating) to start exploring career interests.  Students might not be aware of the many options available to them.  According to a 2017 report from the Office of the Washington State Auditor: Schools should "Improve career guidance given to students, and provide it in a classroom setting in the 7th or 8th grade.  Introducing students to the many careers and jobs available after high school could help them choose occupations that pay well but may not require a four-year degree." (p. 3)

American Mathematical Society (AMS) provides resources to answer the question, "What do mathematicians do?"  This is an excellent place to begin discussion on the topic of "Where am I going to ever use this stuff?" The AMS also has multiple resources for building a career in mathematics within its section on The Profession.  It produces Mathematical Moments, a series of one-page posters that promote an appreciation and understanding of the role that math plays in science, nature, technology, and human culture.  Each poster comes with a related resource.  Some feature podcast interviews with experts in the field.  There is a section for high school students.  The AMS is also a partner sponsoring The Big Math Network, which promotes careers in business, industry, and government to students and departments of the mathematical sciences.

American Statistical Association Careers in Statistics: Find answers to what statistics is, what statisticians do, industries that employ statisticians, how to become a statistician, and career prospects.

archKIDecture (architecture for kids) "was established in 1996 to encourage visual literacy and explain math, science and visual arts concepts through the medium of architecture" (Why section).  You'll find projects, lessons, stories, architecture books for kids, and learn about the BuildIt! Traveling Exhibit for kids.

Career Cornerstone Center has numerous resources for careers in mathematics, computing, science, technology, engineering, and health fields, and assistance for career planning.  The student section is particularly helpful as it addresses resources relevant for learners in middle school, high school, college/university, and graduate school.  Highly recommended.

Comparitech includes top online bachelor degree programs in cybersecurity, information about the field, and delves into how to select an appropriate program to fit your needs.

Cybersecurityguide.org contains a list of careers in this field, programs available by state, and degree programs.  Among resources is a section called Cybersecurity Math Guide, which elaborates on the role of math in cybersecurity careers.

Discover Data Science includes a section on Exploring a Career with Numbers, with links to popular careers such as a data scientist, data analyst, data architect, data engineer, business analyst, marketing analyst, data and analytics manager, business intelligence analyst, data mining specialist, statistician, and more.

Dollars Attached to a Career in Data Science is a 2019 blog post from UC Berkley School of Information. The resource provides statistics on the data science field, such as demand, salary, and number of job openings. It also includes industries in which many data scientists typically work, and highlights some well-known companies that employ data scientists.

Engineering the Future: The Educator's Guide to Building and Construction includes a "collection of lessons, activities, projects, videos, and more, broken down by grade level [K-4, 5-8, 9-12, and When I Grow Up]... to assist educators in teaching young people core areas of study by introducing them to the world of building and construction" (Overview section).  Each grade level group relates building and construction to various STEM fields of study and then provides practical math projects/activities suitable for the grade level.  The activities have the potential to influence career choices.

Examples of Math in Technical Fields is a set of examples provided by the Mathematics Department at the British Columbia Institute of Technology illustrating how topics in algebra, geometry, trigonometry, calculus, statistics and quality control are used in a variety of careers involving technology.  You will find an appropriate math problem and its solution related to the math subject and area of technology you select.  Areas include biomedical engineering, food technology, building technology, chemical sciences, civil and structural engineering, graphics and computer-aided drawing, electronics, environmental health, mechanical engineering, mining technology, nuclear medicine, occupational health, petroleum technology, prosthetics, forestry and wildlife, robotics, and surveying.

Forensics Colleges includes careers for a forensic scientist including what they do, potential specialties, colleges that offer forensic degrees, the career outlook and salary information, and commentary on the work environment.

Get Real Math and Science Videos from the NEW (NorthEast Wisconsin) Manufacturing Alliance "showcase over 80 common core skills used in the real world. The videos, produced at manufacturing companies, serve as a capstone after a skill is learned in school. Math and science skills featured are 3rd grade through high school. The lesson plans are created by math and science teachers" (Description in Thank You section).  You can search videos by title, category, or keyword.  For example, what does math have to do with automation or dog food or medical towels?  What do canisters, coffee filters, cookies, coupons, ramps, robots, weld inspections, or waves have to do with math? Find out about these and many more examples in videos and lessons at this site.

Get the Math is a multimedia project about algebra in the real world, which is for middle and high school learners.  Begin by viewing videos that show how professionals working in fashion,  videogame design, music production, basketball, the restaurant business, and the special effects industry use algebraic thinking. They present challenges connected to their jobs to two teams of teens.  "At that point viewers are encouraged to try the challenges themselves using interactive tools provided on the GET THE MATH website, before returning to the video to see the teams’ solutions.  Students can further explore the same algebra concepts through additional interactive challenges on the website" (About section).  The project is made possible by the Moody's Foundation THIRTEEN in association with WNET New York Public Media.

Knowitall: Career Explorations is just one of the many collections at Knowitall.  It "brings an abundance of career education and job shadowing videos to your computer, tablet or mobile device! These videos provide insight into a wide array of professions, as students plan the pathway that best suits them.  You'll also find Career Clusters, which "connect school work with the knowledge and skills needed for success in college and careers. Each Career Cluster identifies pathways from secondary school to two- and four-year colleges, graduate school, and the workplace. These videos will motivate students to work harder to achieve their career goals."  You can also explore topics, one of which is Science, Technology, Engineering, and Math.  The site features over 8,600 media assets created by the South Carolina ETV and its partners for preK-12.

Mathematical Association of America posts mathematics careers along with related resources.

Math Apprentice answers the question that nearly every student will ask during his or her math career: When are we ever going to use this stuff in the real world?  The site is among several developed by Colleen King, who said "This rich, multimedia site provides students an opportunity to try various professions that use math. Students can be scientists, engineers, computer animators, video game programmers, and more. Math Apprentice provides areas of free exploration as well as specific problems to solve. The site is free of advertising, free of charge, wholesome and safe for children. I would recommend it for upper elementary and middle school students" (personal communication, September 16, 2009).

Math Central is for teachers and students.  Explore careers in mathematics within the section Mathematics with a Human Face.  The section Math Beyond School provides resources for K-12 that address the question: "Where will I ever use this math?"  In addition Math Central contains teaching resources, a database of answers to mathematics questions, a problem of the month, and outreach activities. The site, which is available in English, French, and Spanish, is maintained by faculty and students in Mathematics and Statistics and Mathematics Education at the University of Regina in Canada.

Math for Economists is a free series of 16 YouTube videos featuring lectures by Jason Kronewetter, UC Irvine OpenCourseWare.

Math in the Workplace will definitely help you to answer the question often posed by students, "How am I ever going to use this stuff?"  This project contains real-world math problems with solutions contributed by a variety of businesses in Southwest Idaho.  Problems are aligned with the National Standards for School Mathematics and Idaho State Mathematics Achievement Standards.  They are categorized by strand: Numbers and Operations, Algebra, Geometry, Measurement, Data Analysis and Probability.  The intent of these lessons is to excite students about mathematics, to expose students to professions that employ mathematics, and to demonstrate the relevance of mathematics in solving real-world challenges.  The site is supported by Micron Technology.  Lessons were field tested by teachers.

Math of Finance is a free series of 16 YouTube videos featuring lectures by Donald Saari, UC Irvine OpenCourseWare.

Mathematics at Work brochures "examine how higher-level mathematics is used in today's workplaces."  Produced by Achieve.org, each reflects expectations on college and career readiness promoted in state standards and the Common Core math standards.  You'll find brochures on the aerospace industry, civil engineering, construction, health care, information technology, manufacturing, and semiconductors.

Math@Work is a free series of 22 videos that introduce learners in grades 4 and up to how math is used in various careers.  Each video is about 25-minutes long and is accompanied by lessons to promote discussion and problem-solving.  Examples of episodes include Math Meets Fashion, Math Meets Culinary Arts, Math Meets Homebuilding, and Math Meets Entrepreneurship.

Mathscareers.org.uk "is a resource to provide a single starting point for those wishing to know where studying mathematics can lead. The website has a particular focus on young people between the age of 11 and 19" (Teacher section).  You'll find a range of resources categorized by age level; and resources for undergraduates, graduates and adult learners.  Sections showing how math applies in careers include maths, environment, health and society, business and money, entertainment, science and engineering, and sport.  Highly recommended.

NASA Robotics Career Corner contains profiles of scientists and engineers at NASA involved with robotics.  Along with this, NASA has an entire section about Robotics with K-12 lesson plans, multimedia, an image gallery, and related sites.

National Science Foundation Engineering Classroom Resources contain a list of organizations that provide lessons and resources aimed at K-12 educators, their students, and families.  They illustrate the role of math and science in engineering, include projects for the K-12 classroom, and promote engineering as a career.

PBS LearningMedia has an extensive collection of Real-Life Math videos in which professionals in numerous careers explain how they use math.  For example, view Math in Fashion, Math in Restaurants, Math in Special Effects, Real Estate Appraiser.

Roboblockly offers free K-12 curriculum for math, computer science, and robotics.  The math courses in the prime curriculum are aligned with the Common Core Math standards.  Mathematics with Robitics includes K-8, integrated math 1, algebra 1, geometry, and algebra 2.

Science of NFL Football includes a series of ten videos relating mathematics to the game.  "For each segment in the series, an NSF-supported scientist explains the selected scientific principle, while NFL athletes describe how these principles apply to their respective positions."  Videos address vectors; projectile motion and parabolas; nutrition, hydration, and health; position, velocity, and acceleration; geometric shapes; Newton's first, second, and third laws of motion; torque; and the Pythagorean theorem.

Society for Industrial and Applied Mathematics (SIAM) Careers & Jobs section contains resources to learn about careers in the math sciences and applied math.  Also learn about "WhyDoMath."

SpaceMath@NASA introduces students to the use of mathematics in scientific discoveries.  "SpaceMath products, simple one-page problems featuring a NASA discovery or engineering issue, are designed for direct classroom use by students in grades 3 through 12 using authentic, on-grade-level math topics rooted in real-world science and engineering data. It was developed in 2003 to help NASA missions upgrade their E&PO offerings by explicitly integrating mathematics problems into the science content" (Press Release, 2012).  There are multiple ways to locate math problems.  Search by grade level (3-5, 6-7, Algebra 1, Algebra 2, Geometry, Calculus), by NASA mission, by space topic, or by standards themes (NGSS, CCSS-M).  Problems follow the 5-Es Constructivist Learning Cycle, which "represent five stages of a sequence for teaching and learning: Engage, Explore, Explain, Extend (or Elaborate), and Evaluate."  If you are a NASA fan, also see the free NASA App for Smartphones, Tablets, and Digital Media Players.  Content, includes "images, videos on-demand, NASA Television, mission information, news & feature stories, latest tweets, ISS [International Space Station] sighting opportunities, satellite tracking, Third Rock Radio and much more" (NASA App description section).

STEM Career Lab, located in Ohio, provides a collection of videos on multiple STEM careers in which STEM professionals share "their educational pathway, what it is they love about their jobs, and how they really do use that math and science they learned in high school."  Each video comes with a viewing guide that can be used as a pre- and post-assessment, and ideas for discussion and teaching the math and science concepts presented in the videos.

STEM Careers Coalition includes STEM career profiles and real-world classroom activities and lesson plans that "showcase the STEM skills and knowledge that lead to career success," per the website.

STEM Fundamentals for Kids is a guide that includes links to over 150 courses, resources for each subject of the STEM curriculum, handpicked websites, activities and age-appropriate contests, and YouTube channels addressing STEM.  Resources are organized by age group for easy access.  You'll also find information about specific careers in computer science.

The Futures Channel features a library of 140 STEM videos in which diverse professionals talk about the use of math and science in what they do. They are 2 to 6 minutes long.  You'll also find 200 lesson plans with activities that you can filter by keyword, topic, subject, grade level, and Common Core standard.  Some resources are free; however, full access is subscription based.

The Math You Need, When You Need It is a series of math tutorials for students in introductory geosciences.  The beauty of this site is that the practice problems with solutions illustrate real-life examples of the concepts that those in the geosciences would encounter.  Instructor pages are provided.  Units contain an online quiz.  Topics include density, graphing, hypsometric curves, rates, rearranging equations, slope and topographic maps, trigonometry, unit conversions  The site is a project of SERC, the Science Education Resource Center at Carleton College, supported by the National Science Foundation.

TryEngineering.org: Engineering is a broad field with many disciplines.  This resource for students ages 8-18 is provided by IEEE, IBM and the New York Hall of Science.  It provides resources for exploring engineering as a career, related games, lesson plans allowing teachers and students to apply engineering principles in the classroom, information about finding a university offering engineering degrees, and a Q&A section from actual engineers and undergraduates pursuing engineering degrees.

Unique Jobs for Math Majors from Maryville University Online in Missouri contains a list of careers that use mathematics and their descriptions.  Several also include potential salaries.

WeUseMath.org helps to answer the question "When will I use math?"  "This website describes the importance of mathematics and many rewarding career opportunities available to students who study mathematics" (Site description). There are sections containing multiple Careers in Math and Careers Using Math. Math Tidbits elaborates on interesting "did you know" snippets about people, numbers, and more.  There's a Blog, and Teacher Resources.  It's a valuable site sponsored by Brigham Young University Mathematics Department.

To find books on this topic, enter the search phrase: math for the trades.

### Help students to gain financial literacy.

National Standards for Financial Literacy from the Council for Economic Education (updated 2021) provide a framework for teaching personal finance in K-12.  The standards address earning income, buying goods and services, saving, using credit, financial investing,and protecting and insuring.  Per the description, "Each of these six standards includes benchmarks outlining what a student should be able to understand and examples at the 4th, 8th and 12th grade levels of how the student might demonstrate this understanding.  The benchmarks also emphasize decision-making skills by explicitly relating planning and goal setting, financial decision making, and assessing outcomes to each standard."

From their literature review on financial literacy, Kasman, Heuberger, and Hammond (2018) of The Brookings Institution observed "there are overall low levels of financial literacy among American youth, with large numbers of students unprepared to navigate the many financial decisions that they will encounter during their lifetimes; this has serious, deleterious consequences for individuals, and implications for the U.S. economy" (p. 1).  Unfortunately, K-12 public schools typically don't address financial literacy until the end of grade 11 or grade 12.  In their Recommendations for Improving Youth Financial Literacy, they noted "Early, formal financial education can be “preventative,” acting as a barrier against and corrective for detrimental misconceptions and habits before an individual is faced with substantive financial decisions" (p. 2).  Their research confirms that it's never too early to start teaching kids about finances and money management.  Parents are often the first ones to begin this process.

Walter Andal (2016) does this, too, in Finance 101 for Kids: Money Lessons Children Cannot Afford to Miss.  While meant for middle school learners, anyone can benefit from this informative and entertaining book.  In ten chapters, he explains what finance is, how money began, how to earn money, the power of money, credit and how to deal with it, how to save money.  He relates money to our economy introducing supply/demand, inflation, unemployment, the stock market, and money around the world.  He concludes with his perspective of sharing money with the community (e.g., donations to charities).

In his 2021 book, Finance 102 for Kids: Practical Money Lessons Children Cannot Afford to Miss, Andal provides 30 lessons that address spending, cost and price, stretching your money's worth, media and social influence, credit cards, saving and investing, protecting your money, property, and reputation; and money and life principles.

This section on Math in Everyday Life includes several resources that address financial literacy.

ABCya.com: Learn to Count Money is an interactive game among several offered for preK-5 learners.  Per its description at the site: "Children practice counting money by clicking and dragging bills and coins into a box. A correct answer earns a fish for the fish bowl. The activity progressively becomes more difficult as children progress. There are two levels: Beginner & Expert. Beginner allows students to count money using any amount of coins and bills. Expert requires students to use the least amount of coins and bills."

Actuarial Foundation's Math Academy is a series of free downloadable booklets with hands-on activities dealing with real life applications of math for students in grades 3-8.  Each was designed by teachers who wanted a fun learning environment to engage students.  Explorations address probability (Are you game?), patterns and functions (Can you see it in nature?), data analysis and statistics (Play Ball!), fractions, decimals, percents (Dining Out!), combinatorics (Let's Go the the Mall!), and more.

Biz Kid\$ "is a national financial literacy initiative based on the Emmy Award-winning public television series where kids teach kids about money and business." You'll find the video episodes by season with their descriptions, lesson plans for middle and high school, an online course, and engaging games on business.  The lesson plans are based on national standards for financial literacy and entrepreneurship education.

Cha-Ching Money Smart Kids is a free program, which "aims to empower youth across the country with critical 21st century skills by beginning high-quality financial literacy education in elementary school."  K-6 students will benefit from four storylines and the Cha-Ching band in 18 engaging episodes: Earn, Spend, Save, Donate.  Each video also includes a classroom activity.  There is also a contest.  This resource was developed in partnership by Discovery Education and the Jackson Charitable Foundation.

HOT!: Crash Course Statistics is series of videos posted on YouTube that takes a closer look at how statistics play a significant role in everyday life "from batting averages and insurance rates to weather forecasting and smart assistants" and more.  The first of those was posted in January 2018.  You'll find episodes involving topics such as mathematical thinking, measures of central tendency and spread, data visualization, correlation, controlled experiments, sampling methods and bias in surveys.

DFI KidsPage (Department of Financial Institutions from the State of Wisconsin) presents an overview of the history of money, savings options, financial institutions, and investing.  The end of each section includes a short quiz.  There is also a glossary of key financial terms and links to other valuable financial literacy resources for educators, parents, and students.

Economics for Kids: Fun Learning Games for Business, Math, and Money from Rider University contains an extensive list of games for K-12 students on these topics that also address business and entrepreneurship, money management and personal finance.  Activities, quizzes, questions and printables are included.

Everfi K-12 Financial Education Courses for Kids provide virtual interactive learning experiences in game-based envionments.  Courses are designated by grade level and are free.  For example, one high school course delves into topics such as managing personal finances in the real world, applying for financial aid, establishing credit and investing.  Another for grades 9-12 introduces the basics of investing.  Middle school learners can take on a role of mayor of a town to help its citizens with their financial problems.  There's a course for grades 7-10 in which students start a food truck business to learn what it means to be an entrepreneur.  Students in grades 4-6 explore a virtual world to learn about the ins-and-outs of finances and the difference between needs and wants.  Courses are aligned to standards, enable progress tracking, and are scenario-based to put learning in context.

FDIC: Teacher Online Resource Center provides "tools to help you teach financial education including lesson plans, videos, and other resources."  You'll also find sections called Money Smart for Young People and a Youth Banking Resource Center.  Per the site, the "Federal Deposit Insurance Corporation (FDIC) and the Consumer Financial Protection Bureau (CFPB) are working together to make it easier for schools to bring financial education into the classroom by developing a one-stop-shop for educators."

Financial Fables from the Federal Reserve Bank of Kansas City "are entertaining stories that combine reading, economics and personal finance into life lessons that feature "money morals."  Through quick, interactive online tales, students can learn useful financial management skills that will better prepare them for the future. Created for students ranging from preschool to fifth grade, Financial Fables teach students about savings, money management, and how to set short- and long-term financial goals. Each fable is designed around the concept that it’s never too early to start learning about financial responsibility." (Website description)

Financial Football developed by Visa is a sports-themed game that helps students to learn about money management: saving and spending, budgeting, and wise use of credit.  Players move down the football field to score touchdowns by answering multiple choice questions.  For use in the classroom, each module comes with an overview of the concept, goals and objectives, teacher notes, discussion, and the activity questions.  There are three levels of difficulty, available in English and Spanish.  The game is primarily for ages 11 and up.  For those who prefer soccer, there is also a Financial Soccer game on the same topics.

Financial Life Cycle Education Corp. (FiCycle) provides a project-based curriculum, which helps high school students to "acquire the algebra, probability, and statistics skills needed to make sound decisions at every stage of their financial life cycle" (Home page description).  The course is aligned to Common Core math standards and Jump Start standards for personal finance, which are listed.  Five units with the associated math address financial statements, earning interest, regular payments, insurance and expected value, and stocks and risk.  There are also three additional alternative units.  Teachers have access to free lessons covering topics ranging from stock market trends to the difference between wealth and cash.

FinEDge is a research-based, comprehensive financial education curriculum produced by the University of Chicago Financial Education Initiative.  Per the program overview, it is a one-semester course designed for high schoolers.  "The goal is that finEDge students leave the program with an integrated, self-reinforcing foundation of productive financial decision-making skills, attitudes, and knowledge."

Handbook of Essential Mathematics is from the Air Force Research Laboratory (2006) at Wright Patterson Air Force Base in Dayton, Ohio.  It's a resource developed as part of their K-12 math, science, and technology outreach program.  This open source online textbook with three sections is "a compendium of mathematical formulas and other useful technical information that will well serve both students and teachers alike from early grades through early college" (p. 3).  It is included in this Math in Everyday Life section because "Section III, “Applications in Personal Finance”, is a small textbook within [this] book where the language of algebra is applied to that everyday financial world affecting all of us throughout our lives from birth to death" (p. 4).

Hands on Banking, provided by Wells Fargo, contains free financial curriculum designed for four age groups: kids in grades 4-5, teens in grades 6-8, young adults ages 15-21, and adults.  Engaging self-paced online courses include instructional guides with lessons and classroom activities.  Students gain real-world skills in money management.

IRS Understanding Taxes Simulations--"Apply what you've learned by putting yourself in the shoes of 20 different taxpayers while you explore the ins and outs of filing tax returns electronically!"  Students and any taxpayer will benefit from these simulations.  You'll also find tax tutorials and a series of assessments to test your understanding.  The IRS also has classroom lesson plans in its Understanding Taxes Teacher Site.

Kids' Money is "an interactive resource for parents, teachers, teens, kids, organizations and international visitors designed to help children develop successful money management habits and become financially responsible adults" per the site's description. Teachers will find financial principles and lesson plans listed by grade level preK-12. Sections include topics in saving, spending, earning, sharing, allowances.  Age appropriate books, news, and a financial terms glossary are among other resources.

Math and Money for Kids is a collection of over 100 games, activities, audiobooks and more gathered by CreditDonkey.com.  Of values is that resources are grouped into categories: How Money Works, Counting Money, Earning Allowances; Buying, Selling, and Borrowing; Saving Money and Why It's Important, Financial Literacy for Kids, and Advanced Concepts: Investing and Checks.

Math Interactives is part of LearnAlberta.ca.  "This multimedia resource includes interactive math activities, print activities, learning strategies, and videos that illustrate how math is used in everyday life. The resource addresses the following mathematics topics: Fractions; Integers; Percentages; Rate/Ratio/Proportion; Square Roots; Exponents; Patterns; Algebra; Linear Equations; Polynomials; Angles; Circles; Surface Area and Volume; Area and Perimeter; Triangles; Pythagoras; Trigonometry; Similarity and Congruence; Transformations; Shape Classification; Data Display and Graphs; Central Tendency and Distribution; and Probability." (Math Interactives Home page section).

Math Matters, Apply It! SIAM, the Society for Industrial and Applied Mathematics, developed this series of pdf documents to increase awareness of applied mathematics.  SIAM indicated these were "created for anyone who wants to know more about the mathematics behind everyday life and the technologies we encounter."  You will learn about the math behind renewable wind energy, immunology, vaccination, earthquake simulations, x-ray crystallography, sand and dune motion, image compression, statistics in sports, supercomputing, speeding up the internet, cardiology and heart attacks, stopping and preventing fires, digital face recognition, digital animation, using DNA, CDs and anti-skip technology, with more to come.  Each is one page and can be used with elementary, middle, and high school learners.

Math Munch aims to help "kids find something mathematical that they love."  It's tagline is "A Weekly Digest of the Mathematical Internet." In weekly posts, the developers "share stories about ordinary people who have extraordinary passions for mathematics.  [They] share resources that students can use to create math of their own.  Perhaps most importantly, [they] share and model an excitement for exploring mathematics and learning new things."  And they "use language and examples that are accessible to students at least as young as fifth grade, and probably younger" (Why Math Munch? section).  You'll also find math games, math art tools, videos, software, and some thoughts for teachers.

Mathigon: Applications of Mathematics help to "Learn about the countless hidden applications which mathematics has in everyday life: from computers to weather prediction, video games, medicine, sports and music.  Explore by industry or subject topic.  Each activity includes additional resources to learn more.

Money Math: Lessons for Life from the U.S. Department of the Treasury is a four-lesson curriculum supplement designed for students, grades 7-9, who will learn math concepts using real-life examples from personal finance. The free downloadable book features a teacher's guide with lesson plans, reproducible activity pages, and teaching tips.  Lessons are titled: The Secret to Becoming a Millionaire, Wallpaper, Math & Taxes, and Spreading the Budget.  Highly recommended.

Next Gen Personal Finance is a free personal finance curriculum and professional development partner helping teachers deliver essential money understanding with an emphasis on real-world applications.  Curriculum options include nine-week, semester, and year long courses that are appropriate for learners in middle school and high school.  There are 12 units featuring "Investing, Taxes, Saving, Behavioral Economics, and Paying for College" (About Us section).  There is also a free Fnancial Algebra Course for grades 9-12 with 10 units, each of which "blends one core personal finance topic with one relevant math concept (e.g. Investing and Exponential Functions)."  Curriculum also includes a collection of math activities with real-world personal finance application problems.

NRICH from the University of Cambridge is devoted to enriching mathematics for learners in grades 1-12 with games, problems, investigations, articles and books.  Resources are free.  This project is part of the the Millennium Mathematics Project, a long-term national math education initiative in the UK for ages 3-19 and the general public.  See also the Plus site, which is their internet magazine with articles, podcasts, news, a blog, puzzles, careers in math, and more.  Plus aims to introduce readers to the beauty and the practical applications of mathematics in science, art and society.  The Plus Careers with Math section contains a library of interviews that explore the careers of people who use math in their jobs every day.

Pathway to Financial Success in Schools from Discover Financial Services and Discovery Education includes comprehensive financial literacy resources for middle and high school learners.  Educators, students and families will appreciate the standards aligned lessons, guides and videos, self-paced modules, and family activities.

PBS Learning Media: Cyberchase: Per its description, Cyberchase is "an Emmy award-winning animated math mystery show featuring a team of curious kids who use their math and problem solving skills to outwit and outsmart the villain Hacker in their quest to save Cyberspace. In addition to segments from the animated portion of the show, this collection of resources includes live-action For Real segments, in which hosts Bianca and Harry explore the show's math topics in everyday life."  You'll find videos on algebra, geometry, data analysis and measurement, numbers and operations, math and health, math and the environment, and more.

PBS NOVA Labs include the free game-based Financial Lab.  It is "designed to help students (and adults!) understand how we humans make financial decisions, and how to develop habits to overcome the behavioral biases that often get in our way. Think of it as a fun, safe place to practice managing your spending, paying off debts, and investing in your future" (About section).  There are three mini-games: Shopportunity Cost, Budget Buster, and Exponential Potential.  An educator guide is provided.  Other NOVA Labs with games in each are available, which teach students age 6 and up a range of STEM subjects.

Practical Money Skills: Visa partnered with leading consumer advocates, educators, and financial institutions to create the Practical Money Skills program so that students of all ages can learn the basics of personal finance.  These free educational resources include personal finance articles, games, lesson plans linked to standards, and more.  Online multimedia games include such titles as Financial Football, Financial Soccer, Smart Money Quiz Show, Roadtrip to Savings, Ed's Bank, and more.  You'll also find financial lesson plans for special needs learners.

PUMAS: Practical Uses of Math and Science, is an online journal for pre-college education with "brief examples showing how math and science topics taught in K-12 classes can be used in interesting settings, including every day life."  It is aimed primarily at classroom teachers for use with their learners.  Examples are available by grade levels K-2, 3-5, 6-8, and 9-12.

Statistics in Schools from the U.S. Census Bureau helps to educate K-12 students about the value and use of statistics in everyday life.  The program provides resources for teaching and learning with real life data.  Common Core standards-aligned, classroom-ready activities in math and history, data visualizations and more are included.  Activities are available by grade band with an approximate time for completion of each.  Teacher and student versions are provided.  This is an excellent resource.

Stukent: Mimic Personal Finance is a simulation at the high-school level that "lets students practice decision-based learning, make real-world financial choices, go beyond theory, and experience in-class consequences."  Students learn about banking, assets and liability, income growth, make financial decisions on unexpected events, and apply budgeting skills in a section called "buying grades."  Lecture videos are included, as well as a curriculum and in-class activities to help teaching.  You can request a quote.

TeachEngineering provides a free digital library with "teacher-tested, standards-based engineering content for K-12 teachers to use in science and math classrooms. Engineering lessons connect real-world experiences with curricular content already taught in K-12 classrooms. ... curricula are hands-on ... and relevant to children's daily lives" (Home Page Welcome).  The site is "a collaborative project between faculty, students and teachers associated with five universities and the American Society for Engineering Education, with NSF National Science Digital Library funding" (About section).  Highly recommended.

TeachFinLit.org contains resources for high school educators to teach financial literacy concepts to their learners.  Topics include income and careers, budgeting and spending, credit and debt, saving and investing, risk management and insurance, and key concepts.  The site is comprehensive and also includes a full curriculum for new teachers.

Teaching Children About Money: Resources for Promoting Financial Literacy by Age from FiscalTiger.com also include age-appropriate activities and apps.  Resources are grouped for toddlers, and elementary, middle, and high school.

TV411.org from the Education Development Center has a series of videos and web lessons involving math skills used in everyday life and in some careers.  They are organized by basic math; ratios, averages, and exponents; fractions, decimals and percents; and geometry.  For example, learn about area and tiling the New York subway, carpenter math, estimating pricing a painting job, reading a graph on a utility bill, ratios in the kitchen, finding the best cell phone plan, planning a trip using a map to figure time and distance, selecting the best unit price when shopping, using football, basketball, music and dance as explained by pros to learn about percentages, decimals, and probability, and fractions; using origami to learn some geometry, and more.

The U.S. Currency Education Program will help everyone learn about the seven denominations of U.S. currency issued by the Federal Reserve Board (\$1, \$2, \$5, \$10, \$20, \$50, \$100), their design and security features, and life cycle.  The site also offers videos to learn more, and a history of U.S. currency.

Lavamind.com has simulations (e.g., Gazillionaire, Zapitalism, & Profitania), each with a higher level in building proficiency with business, math, and economics principles.  A free demo version of Gazillionaire, Zapitalism, and Profitania is available.  Gazillionaire, for example, is a “cross between Monopoly set in outer space and Wall Street in wonderland” with real-world economic tools to “help you determine where supply meets demand as you race to build your trade empire. Six people can play at a time on the same computer or by email over the Internet. The first company to reach a net worth of a billion kubars wins!”  Also see Biztopia, a mobile business simulation game for iOS and Android.

Lemonade Stand: Open your own lemonade stand and run a business for 7, 14, or 30 days.  Make all your own decisions on pricing, inventory, quality, and purchasing supplies and see how much money you can make.  Of course you'll be influenced by the weather.

The Stock Market Game offers a library of learning materials correlated to national voluntary educational standards in math, business education, and economics. This resource has been used at all levels, from fourth grade to college, all across the curriculum. Starting with a virtual cash account of \$100,000, students strive to create the best-performing portfolio using a live trading simulation. The program also reinforces critical thinking, decision-making, cooperation and communication skills, independent research.  Students use real Internet research and news updates, making the simulation an even better mirror of the real marketplace.

How the Market Works is a free simulation game designed to help beginners to understand how the stock market and stock trading works by practicing trading.  After registering, you receive a brokerage account with a virtual \$25,000 in cash and you are able to begin practice trading right away.  The stock simulator performs like a real brokerage account.  To help you get started, there is an Education Center with numerous articles, videos and tutorials, including a teacher guide among the list of Teacher Tools.  This program can be used with middle school learners and above.

Virtual Business Simulations are an engaging way to help high school students "learn key business, marketing, and personal financial concepts including accounting, marketing, management, entrepreneurship, sports marketing and management, financial literacy, hospitality, and more."

Wall Street Survivor is a free stock market simulator.  You can learn about investing and personal finance with free courses.  Practice trading and build a portfolio with a virtual \$100,000.  Registration is free and after you register you can also enter monthly contests and win prizes.

#### Build a Lesson Around Math in Real Life

HOT! TED-Ed Lessons Worth Sharing includes a series of videos on Math in Real Life.  Educators can use them or select a YouTube video to then create a lesson.  You can add multiple choice and open-ended questions, discussion prompts, and additional resources.  Share the lesson with students and track their progress.  Sample lessons are also available.

To find books on this topic, enter the search phrase: math and everyday life.

### Math Across the Curriculum

Artful Maths includes mathematical art lessons, classroom display ideas, origami in lessons and an origami gallery to inspire, and a blog.  Of interest are the variety of images of insects shown in the blog to illustrate math features (e.g., symmetry, hexagons) and the beauty of nature.  This site is definitely worth viewing.

International Society of the Arts, Mathematics, and Architecture (ISAMA) "was founded by Nat Friedman in order to further interdisciplinary education relating the arts, mathematics, and architecture."  Included are links to numerous resources on this relationship and access to the ISAMA publication Hyperseeing, which provides the latest on Math and Art with its "mix of articles, news, reviews of books and exhibits, announcements, and even cartoons."

Masterpieces to Mathematics: Using Art to Teach Fraction, Decimal, and Percent Equivalents by Christopher Scaptura, Jennifer Suh, and Greg Mahaffey (2007, August) is a lesson, which appeared in Mathematics Teaching in the Middle School.  "This article shares how students created their own OpArt (optical art) which was inspired by Ellsworth Kelly, and how they connected that work of art to rational numbers.  By identifying colored portions on a grid, the students recognized fraction, decimal, and percent breakdowns of their own designs" (p. 24).

Mathematics and Art was the 2003 theme for the Mathematics Awareness Month, which is sponsored each year in April by the Joint Policy Board of Mathematics.  Joseph Malkevitch (York College, CUNY) gathered resources to explore this relationship.  He stated, "There are, in fact, many arts (music, dance, painting, architecture, sculpture, etc.) and there is a surprisingly rich association between mathematics and each of the arts. My goal here is to give some pointers concerning these many connections" (Introduction section). You'll learn about mathematical tools for artists, symmetry, mathematical artists and artist mathematicians; polyhedra, tilings, and dissections; origami (the art of paper folding), and more.  For more information, see themes for the Mathematics and Statistics Awareness Month. The Joint Policy Board of Mathematics is a collaborative effort of the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics.

Mathematical Imagery from the American Mathematical Society further explores the relationship between mathematics and art.  The collection of albums of math imagery is extensive from snowflakes, fractals, quilts, woven beads to origami, computer-generated landscapes, tessellations, geometric sculptures, and more.

Mathematics and Geography from TeacherVision. You'll find printables, lessons, and activities for K-12 that connect math and geography. For example, use these resources to teach students how to read a map and navigate with mathematical instruments. You'll find lesson plans to help students learn how to make a budget for trip expenses and how to interpret U.S. Census data. There are also handouts on weather, elevation, and converting measurements of distance.

Math and Children's Literature from Carol Hurst's Children's Literature Site contains a list of books, sample chapters, and articles with strong math themes.

Math and Literature Idea Bank.  This resource is primarily for elementary grades brought to you by MathCats.  Suggested books and how to use them to connect math to literature are presented.

Mathematics & Music from the American Mathematical Society provides videos, podcasts, and articles to show the connections between mathematics and music. Per the AMS: "Counting, rhythm, scales, intervals, patterns, symbols, harmonies, time signatures, overtones, tone, pitch" are connected to mathematics.

Motivate Cross-Curricular Maths Resources is part of the family of activities in the Millenium Mathematics Project from the University of Cambridge.  Math activities link to art and design, citizenship/PSHE, design and technology, economics, geography, history, media studies, music, science, sport.  This is truly a valuable resource.

National Gallery of Art has online interactive classroom lesson units connecting art to curriculum.  In Counting on Art, elementary and middle school learners "explore the paintings of Horace Pippin and Wayne Thiebaud and the mobiles of Alexander Calder to discover and practice math and visual art concepts."  In New Angles on Art, middle and high school learners focus on two artists and an architect. "The interactive lessons focus on the math—lines, angles, two-dimensional shapes and three-dimensional polyhedra, fractions, ratios, and permutations—found in sculptures in the National Gallery Sculpture Garden and in the architecture of the East Building."

Radical Math is for educators interested in integrating issues of social and economic justice into their math classes and curriculum.  You'll find links to access and over 700 lesson plans, articles, graphs, charts, data sets, maps, books, and websites to bring these issues into the classroom.

Where Math Meets Music by Joseph Heimiller takes a look at such things as frequency and sound waves created by musical instruments.  Site might be more appropriate for high school learners and above.

Yummy Math helps teachers bring real-life into math classrooms for learners in grades 3-8 and high school.  You'll find activities that correspond to the NCTM Process Standards and the CCSS Standards for Mathematical Practice, easily found by strand within a grade or genre (e.g., math and food, math and science, math and art, math and social studies, math and entertainment, sports).

To find books on this topic, enter the search phrase: math across the curriculum.

## Problem Solving

### What is Problem-based Learning in Mathematics?

According to Kyeong Ha Roh (2003):

Problem-Based Learning (PBL) describes a learning environment where problems drive the learning. That is, learning begins with a problem to be solved, and the problem is posed is such a way that students need to gain new knowledge before they can solve the problem. Rather than seeking a single correct answer, students interpret the problem, gather needed information, identify possible solutions, evaluate options, and present conclusions. Proponents of mathematical problem solving insist that students become good problem solvers by learning mathematical knowledge heuristically. (Roh, 2003, para. 1)

Paul Zeitz (2007) distinguishes between exercises and problems:

An exercise is a question that tests the student's mastery of a narrowly focused technique, usually one that was recently "covered." Exercises may be hard or easy, but they are never puzzling, for it is always immediately clear how to proceed. Getting the solution may involve hairy technical work, but the path towards solution is always apparent. In contrast, a problem is a question that cannot be answered immediately. Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution. Problems and problem solving are at the heart of mathematics. Research mathematicians do nothing but open-ended problem solving. In industry, being able to solve a poorly defined problem is much more important to an employer than being able to, say, invert a matrix. A computer can do the latter, but not the former.  (Zeitz, 2007, p. x)

Thus, problem-based learning is not the same as solving the exercises that follow sections in math textbooks.  Per David Moursund (2016), "A section in a chapter discusses a quite specific problem-solving topic and gives examples of how to solve that type of problem. Then students are asked to practice solving that type of problem over and over again" (Math Problems versus Math Exercises section).

Moursund (2016) included several possible characteristics of a good problem for a problem-based learning assignment.  In his words:

• The problem is relatively easy to understand.
• The problem can be understood by a wide range of students—that is, by students with diverse math backgrounds.
• The problem is solvable by a variety of methods. Some methods may be more efficient or "elegant" than others.
• A student working on the problem can recognize when progress is occurring.
• It is not easy to look up a solution from readily available resources such as books and the Web.
• Work on solving the problem contributes to increasing math maturity and draws upon a wide range of student backgrounds. (Some Guidelines section)

Phillips Exeter Academy in New Hampshire is one example of a high school that takes a problem-based approach to teaching mathematics.  Their math teaching materials include problem sets for its courses that drive learning.  These latter are well-worth viewing.

### Do you know how to teach your learners to solve problems?

There are many ways to solve problems.  View the Problem-Solving Strategies Mindmap at GoGeometry.com

John Malouff (2018) grouped and explained over 50 problem-solving strategies, many of which can be applied to mathematics problem solving.

George Polya's Problem-Solving Techniques contain details of his four principles that have become a classic for math problem-solving: understand the problem, devise a plan, carry out the plan, and look back.

Powerful Problem Solving: Activities for Sense Making with the Mathematical Practices is by Max Ray and colleagues (2013) at the Math Forum.  The authors provide advice and activities that teachers can use to help students become proficient problem solvers in keeping with the Common Core standards for mathematical practices.

#### What's an effective set-up of a challenging math problem?

In their 2013 study, reviewed by the folks at the Educational Research Newsletter and Webinars, Kara Jackson and colleagues noted The 4 elements of an effective set-up of a challenging math problem:

• Explicitly discuss key contextual factors.
• Point out key mathematical ideas and relationships.
• Develop a common language.
• Maintain the cognitive demand.

The full study by Jackson, Garrison, Wilson, Gibbons, and Shahan was published in July 2013 in the Journal for Research in Mathematics Education: Exploring Relationships Between Setting Up Complex Tasks and Opportunities to Learn in Concluding Whole-Class Discussions in Middle-Grades Mathematical Instruction.  Certainly, results can generalize to all math levels.

#### What's math modeling all about?

The pure mathematics we teach must be linked to real world applications, so that students do not have to ask: Where are we going to use this? In discussing the nature of mathematical modeling, Ang Keng Cheng (2001) noted:

"Mathematical modelling is a process of representing real world problems in mathematical terms in an attempt to find solutions to the problems. A mathematical model can be considered as a simplification or abstraction of a (complex) real world problem or situation into a mathematical form, thereby converting the real world problem into a mathematical problem. The mathematical problem can then be solved using whatever known techniques to obtain a mathematical solution. This solution is then interpreted and translated into real terms" (para. 7).

The steps in modeling may or may not be sequential:

• Identify a problem situation.
• Make a representation of one or more elements of the situation.
• Create a mathematical expression.
• Compare results or predictions from the mathematical model with the real situation.
• Revise the model if needed. (Ellis, 2015, p. 5)

Henry Pollack (Gould, Murray, Sanfratello, 2012) clarified for educators "how mathematical modeling differs from what you already teach, particularly, “problem solving”. Problem solving may not refer to the outside world at all. Even when it does, problem solving usually begins with the idealized real-world situation in mathematical terms, and ends with a mathematical result.  Mathematical modeling, on the other hand, begins in the “unedited” real world, requires problem formulating before problem solving, and once the problem is solved, moves back into the real world where the results are considered in their original context" (p. viii).

Learn more about math modeling in Algebra 1 and its role in the curriculum via the Annenberg Learner: Mathematical Modeling.

MATHmodels.org has contests and problems on math modeling for high school and undergraduate learners. “Mathmodels is COMAP’s new modeling forum. On this site, students and faculty will find a wide range of interesting contemporary modeling problems. Teachers can assign problems. Students can choose to work on problems based on math topic and application area” (About MATHmodels section).  Note: This site has particular relevance, as math modeling is among mathematical practices included in the Common Core State Standards for mathematics.  Read about CCSS high school math modeling.

MathWorks Math Modeling Challenge (M3Challenge) is an applied mathematics competition for high school junior and senior students. Winners receive scholarships totaling thousands of dollars for continuing education. The contest is sponsored by MathWorks and SIAM (Society for Industrial and Applied Mathematics).  Resources include math modeling handbooks, math modeling videos (including teaching math modeling), tips and guidelines for teaching math modeling, and sample problems.

Saltire Software developed Geometry Expressions and has provided GXWeb for free browser-based mathematical modeling.

#### Essential Questions for Modeling and Problem Solving

As you delve into math modeling and problem solving with your learners, consider the following essential questions in mathematics, suggested by Jay McTighe and Grant Wiggins (2013) in their book Essential Questions: Opening Doors to Student Understanding:

• "When and why should we estimate?
• Is there a pattern?
• How does what we measure influence how we measure? How does how we measure influence what we measure (or don't measure)?
• What do good problem solvers do, especially when they get stuck?
• How accurate (precise) does this solution need to be?
• What are the limits of this math model and of mathematical modeling in general?" (p. 2)

#### What kind of math problems are your students solving?

Are you relying on the math problems that follow each section in your textbook?  If so, David Ginsburg (2016) suggested that quality rather than quantity may better serve students when it comes to practice.  He's his "four viable purposes for assigning math problems to students" (para. 4):

1. Learn "new" content.
2. Deepen understanding of previous content.
3. Reveal and resolve misconceptions.
4. Develop procedural fluency--not to be confused with fact fluency. (Ginsburg, 2016)

Art of Problem Solving (AoPS) creates educational materials for students in grades 2-12.  AoPS Online contains a variety of resources for students of mathematics in grades 5-12, including textbooks, online courses, as well as full courses for math contest preparation.  AoPS Beast Academy is an advanced comic-based math program for students in grades 2-5 designed to build problem-solving skills for school and life.  The Art of Problem Solving Foundation also administers the USA Mathematical Talent Search (USAMTS) website.  The USAMTS is a free mathematics competition open to any US middle or high school student.  AoPS also has a set of problem solving textbooks.  The following are among those:

 The Art of Problem Solving, Vol. 1: The Basics (2011) is appropriate for students who are just beginning in math contests.  It contains over 500 problems and exercises taken from past contests, such as MATHCOUNTS and others.  "The emphasis on learning and understanding methods rather than memorizing formulas enables students to solve large classes of problems beyond those presented in the book." Competition Math for Middle School (2011) contains over 700 examples and problems in algebra, counting, probability, number theory, and geometry.  Although written for exceptional students in middle school and teachers looking for challenging advanced math problems, this book could also be used by anyone interested in problems found in national math competitions.  Solutions are included.

A+Click allows students in grades 1-12 to solve challenging math and logic problems, practicing over 1000 math skills.  "The problems include a short description and an illustration to help problem solvers visualize the model. The problems can be solved within one minute and without a calculator. The tests adapt to student ability. To progress to a higher level, you must correctly answer five consecutive questions. The tests concentrate on understanding, spatial reasoning, and problem solving rather than math rules and theorems."  Questions are based on Common Core math standards.  (About section)

Advanced Common Core Math Explorations (ACCME) is a series of six books (2014-2017) by Jerry Burkhart of 5280math.com.  Per their descriptions, each book presents learners "challenging activities that deepen and extend their understanding of concepts from the Common Core State Standards for Mathematics. ... Each activity comes with extensive support for teachers including learning goals, discussion guides, detailed solutions, and suggestions for extending the investigations."  The beauty of these activities, designed for upper-elementary and middle school, is their open-ended nature.  Each book is devoted to a specific topic, which Burkhart describes as follows:

1. ACCME: Numbers and Operations (2014).  Learners "investigate numeration systems of beings from another planet, create and solve stories and problems with extreme numbers, use place value to design their own divisibility tests, and play with a magical number line that multiplies numbers without a calculator."
2. ACCME: Factors and Multiples (2014).  Learners "explore mystifying patterns of colored blocks, analyze paths of pool balls, solve mathematical word puzzles, and unravel a baffling mathematical code."
3. ACCME: Fractions (2014).  Learners "solve puzzles, create stories, and explore fraction-related concepts from the mathematics of ancient Greece to the outer reaches of infinity."
4. ACCME: Measurement and Polygons (2015).  Learners "create and manipulate geometric figures, draw and analyze complex designs, and develop and apply measurement strategies to solve challenging real-world and mathematical problems."
5. ACCME: Ratios, Proportions, and Similarity (2016).  Learners "model the solar system, count the fish in a lake, choose the best gear for a bike ride, solve a middle school's overcrowding problem, and explore the mysteries of Fibonacci numbers and the golden ratio."  "Each problem is introduced with a "teaser" that encourages students to come up with their own questions and observations before receiving the directions." This will "enhance the open-ended, inquiry-based nature of the problems."
6. Advanced Common Core Math Explorations: Probability and Statistics (2017).  Students "learn powerful mathematical ideas about data and chance as they make sense of real-world situations from science, sports, and everyday life."  "Each problem is introduced with a "teaser" that encourages students to come up with their own questions and observations before receiving the directions." This will "enhance the open-ended, inquiry-based nature of the problems."

Citizen Math (formerly Mathalicious) provides supplementary math lessons for grades 6-12, which you can filter by grades 6, 7, 8 and algebra 1, algebra 2.  Lessons relate to real world scenarios and include the standards addressed, lesson guide, student handout, and multimedia presentation.  Lessons ask real open-ended questions requiring students to make sense of problems and to develop their own strategies for solving them. Students must support and justify their own methods and conclusions, and evaluate the validity of others’ arguments. Lessons prompt students to model their findings in a variety of ways: numerically, graphically, algebraically and verbally.  While not free, Citizen Math does provide a sample of free complete lessons to introduce you to this resource.

K-12 Math Problems by Erlina Ronda is written for students.  You'll find problems, solutions, explanations, and common errors.  Problems are categorized into middle school, algebra, geometry, trigonometry, combinatorics, probability, and calculus.

Math and Logic Problems Galore is actually a collection of links to sites with puzzles, math problems, games, logic brain teasers, and more and is well worth a look.  The collection is grouped: Everyday math and logic problems for everyone; math and logic for kids, then teenagers, then mathematicians and logicians.  These are educational in nature and can be used in school settings.

Math.com: The World of Math Online is dedicated to solving math problems at nearly every level of school mathematics, including basic math, algebra, geometry, trigonometry, statistics, calculus and more.  There are online math tools: calculators, graphing utilities, test preparation and study tips, free homework help, a glossary and a math library.  Teacher and parent resources are extensive.

MathPickle.com "is a free online resource of original mathematical puzzles, games and unsolved problems for K-12 teachers. It is supported by the American Institute of Mathematics. ... Puzzles are organized by grade and subject -- each designed for a 45-60 minute period" (About Us section).

MIT BLOSSOMS features a video library with over 100 free math and science lessons primarily for high school students.  "Every lesson is a complete resource that includes video segments, a teacher’s guide, downloadable hand-outs and a list of additional online resources relevant to the topic. Each 50-minute lesson builds on math and science fundamentals by relating abstract concepts to the real world. The lessons intersperse video instruction with planned exercises that engage students in problem solving and critical thinking" (About Us section).  This site is an initiative of MIT's Learning International Networks Consortium.  All lessons are mapped to both the Common Core State Standards and the Next Generation Science Standards.

HOT!: Open Middle for K-12 math, including calculus, contains problems that help learners to develop critical thinking.  They have a "closed" beginning, a "closed" end, but as the title suggests, there is an "open middle" meaning that there are multiple ways to approach and solve the problem.  Problems support the Common Core math standards.  Within each grade level, problems are categorized by standard.

Problem of the Week is produced by The Centre for Education in Mathematics and Computing at the University of Waterloo (Canada).  The problems are presented in appropriate grade bands for students in grades 3 and up.  The Centre also produces a Problem of the Month for upper level high school learners, and a section of Real-World Problems that are being solved by mathematicians.

Problems of the Week (PoW) (for NCTM members). NCTM provides weekly challenging problems for students in grade bands 3-12, higher ed, and preK-2 in nine categories: algebra, geometry, discrete mathematics, financial ed, primary, math fundamentals, pre-algebra, trigonometry/calculus, and tPOW (technology problems of the week).

Problems with a Point is designed to complement your existing curriculum with a problem-centered approach to learning mathematics.  According to the Education Development Center in Massachusetts, their goal is to "help students in grades 6-12 learn new mathematical ideas by building on old ones. Each problem or sequence focuses on one mathematical idea and also connects that idea with others. Varying in difficulty and approaches, these problems are useful for teachers, students, parents, math clubs, and home-schoolers. Problems are classified by topic, time required, suggested technology, required mathematical background, and habits of mind that students develop or use as they work. Synopses of the problems are keyword searchable. Answers and solutions are provided, and many problems include hints."

Real World Math: "Students will find downloads for over 30 activities, videos, and instructional tutorials for Google Earth & SketchUp.  The goal is to take the math you learned in class and develop it further with problem solving activities."

Robert Kaplinsky: Lessons with real-world problems for K-8, algebra 1, geometry, and algebra 2.  These also include the Common Core standard.

Thinking Blocks at Math Playground teaches students in grades 1-7 how to visualize and solve math word problems. Using interactive blocks and cubes, they create models that illustrate the underlying math concepts within the problems. Skills include addition, multiplication, fractions, ratios, and there is a modeling tool for students to develop strong problem solving skills.

Visual Patterns contains nearly 500 patterns to encourage mathematical thinking.  Those who try the patterns look for what the pattern would be at the 43 step and then determine an equation associated with the pattern.

### Add some enrichment to remind learners of math in daily life.

Students might not be aware of how much math plays in their daily lives, nor in the entertainment they experience.

Become an avid reader of Plus magazine-- a real find of value!

Plus is an internet magazine which is free for educational and non-commercial purposes.  It aims to introduce readers to the beauty and the practical applications of mathematics.  You'll find "articles, which describe applications of maths to real-world problems, games, and puzzles; reviews of popular maths books and events; a news section, showing how recent news stories were often based on some underlying piece of maths that never made it to the newspapers; a puzzle for you to sharpen your wits against; a lucky dip of mathematical curiosities; and opinions on various maths-related topics and news stories. [Plus also has] a regular interview with someone in a maths-related career, showing the wide range of uses maths gets put to in the real world." (About Plus section)

#### Increase motivation in problem solving--add some humor!

Young learners in grades 3-6 will appreciate Comic-Strip Math: Problem Solving by Dan Greenberg (2010).  They'll laugh as they read each of the 80 comic strips and complete companion mini-story problems.  They'll reinforce skills in place value and rounding, number operations, fractions, decimals, time, money, measurement, geometry, graphs, patterns, rate, ratio, probability and statistics.

#### Add a little dose of Math in the Movies and on TV.

Add some enrichment with Mathematics in Movies, a collection of clips in which mathematics appears gathered by Oliver Knill of the Mathematics Department of Harvard University.  The collection is extensive ranging from Alice in Wonderland to the Simpsons, Shrek the Third, A Walk to Remember, and so much more.  Teachers should preview selections for age appropriateness.  For example, Die Hard is also in the collection and the language might not be suitable for some.

Viewers of the TV series Simpsons might not be aware of all the math one can find in several of the episodes.  There are "over a hundred instances of mathematics ranging from arithmetic to geometry to calculus, many designed to expose and poke fun at innumeracy." The popularity of the Simpsons makes the web site simpsonsmath.com an "ideal source of fun ways to introduce important concepts to students, and to reduce math anxiety and motivate students in courses for non-majors."  Drs. Sarah Greenwald and Andrew Nestler also provide resources for using the site with students.

#### Show your learners the photographs and art involving the beauty of math!  Explore the mystery of mathematics.

Geometry in Art (2010) by Hilton Andrade de Mello of Brazil is an excellent supplementary book for students to explore "how geometric forms have been and continue to be used in the Arts and in Architecture, and also how they appear in Nature."  You'll read analyses of the role of geometry in actual works of art.  In writing this book, which is a free download, the author received contributions from artists around the world.  Ten chapters address Abstraction and Geometry, Planar geometric forms, Polyhedrons, Other spatial figures such as the conics, Composing a canvas with polygons: Tessellations, Perspective, The Golden Ratio, Symmetry and specular image, Geometry and symbolisms, and an Introduction to Informatics and the arts.

Geometry Playground, created by the Exploratorium in San Francisco, will change how you view geometry in nature.  It contains three sections: The Geometry of Seeing with a photo exhibition of the invisible geometry of light; The Geometry of Moving on the arcs, angles, and shapes created when people and things are in motion; and the Geometry of Fitting Things Together.  Each includes hands-on activities for grades K-8.  The Geometry Garden features curiosities found in nature from crystals to seashells to sculptures.  The site was made possible by the National Science Foundation and the Gordon and Betty Moore Foundation.

The Mathematical Association of America hosts the Found Math Gallery, which features interesting photos involving math: the ordinary sidewalk, symmetry in nature, architectural wonders--just look around you.  This will give great inspiration for your students to take their own photos of math in daily life--what a great project idea!

The Natural Museum of American History features 80 of the Mathematical Paintings of Crockett Johnson.  "From 1965 until his death in 1975 Crockett Johnson painted over 100 works relating to mathematics and mathematical physics." Many relate to the geometric figures and concepts that students encounter in their high school math courses, and give great inspiration for projects linking math to art.  The image shown is Crockett Johnson's 1965 Painting: Proof of the Pythagorean Theorem (Euclid).  Image credit from the website.

View The Great Math Mystery (2015, about 50 minutes) from PBS.org.  Per the program description:

"Join NOVA on a mathematical mystery tour—a provocative exploration of math's astonishing power across the centuries. We discover math's signature in the swirl of a nautilus shell, the whirlpool of a galaxy, and the spiral in the center of a sunflower. Math was essential to everything from the first wireless radio transmissions to the prediction and discovery of the Higgs boson and the successful landing of rovers on Mars. Astrophysicist and writer Mario Livio, along with a colorful cast of mathematicians, physicists, and engineers, follow math from Pythagoras to Einstein and beyond. It all leads to the ultimate riddle: Is math a human invention or the discovery of the language of the universe?"

#### Do you want to make your own paper models of polyhedrons?

You can get polyhedron models (i.e., the nets for construction) for several hundred different polyhedrons (e.g., tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron) at https://www.polyhedra.net/en/ then print, cut, fold, and glue your 3D models.  The image shown is called a small stellated dodecahedron.

Discrete Mathematics in the Real World contains short and longer descriptions of everday applications (e.g., computers binary math, networks and routers, web searches, Google maps, area codes, bankruptcy proceedings, cell phone communications, etc.) and interesting research and corporate applications (e.g., medical imaging, logistics, cybersecurity, 3D imaging, etc).  This is a great start to knowing the value of discrete mathematics and its role.

Wikipedia: Discrete Mathematics illustrates where discrete mathematics is found in a variety of topics, such as computer science, logic, set theory, combinatorics, graph theory, probability, number theory, calculus, geometry, topology, game theory, and more.

To find books on this topic, enter the search phrase: mathematics problem solving.

#### References

Boodhoo, N. (2012, July 10). For manufacturing jobs, workers brush up on math. https://www.npr.org/2012/07/10/155837962/for-manufacturing-jobs-workers-brush-up-on-math

Cheng, A. K. (2001). Teaching mathematical modeling in Singapore schools. The Mathematics Educator, 6(1). https://web.archive.org/web/20181125053207/http://math.nie.edu.sg/kcang/TME_paper/teachmod.html

Depersio, G. (2020, September 25). Data analyst: career path & qualifications. https://www.investopedia.com/articles/professionals/121515/data-analyst-career-path-qualifications.asp

Ellis, M. (2015). Mastering the most challenging math standards with rigorous instruction [Whitepaper].  North Billerica, MA: Curriculum Associates.  http://www.casamples.com/downloads/Mastering-Most-Challenging-Math-Standards.pdf

Ginsburg, D. (2016, March 28). Four purposes of math problems. Education Week Teachers. http://blogs.edweek.org/teachers/coach_gs_teaching_tips/2016/03/four_purposes_of_math_problems.html

Gould, H., Murray, D., & Sanfratello, A. (2012). Mathematical modeling handbook (sample). Bedford, MA: The Consortium for Mathematics and Its Applications. https://web.archive.org/web/20220128003936/https://www.comap.com/modelingHB/Modeling_HB_Sample.pdf

Kasman, M., Heuberger, B., & Hammond, R. (2018). Recommendations for improving youth financial literacy education. Washington, DC: The Brookings Institution. https://www.brookings.edu/wp-content/uploads/2018/10/ES_20181001_Financial-Literacy-Recommendations.pdf

Malouff, J. (2018). Over 50 problem solving strategies explained. University of New England Blog. https://blog.une.edu.au/usingpsychology/2018/10/15/over-fifty-problem-solving-strategies-explained/

McTighe, J., & Wiggins, G. (2013). Essential questions: Opening doors to student understanding.  Alexandria, VA: ASCD.

Moursund, D. (2016). Math problem-based learning. http://web.archive.org/web/20210421224319/http://iae-pedia.org/Math_Problem-based_Learning

Roh, K. H. (2003). Problem-based learning in mathematics. ERIC Digest (ED482725). https://files.eric.ed.gov/fulltext/ED482725.pdf

U.S. Bureau of Labor Statistics. (2020). Math occupations. https://www.bls.gov/ooh/math/home.htm

Zeitz, P. (2007). The art and craft of problem solving (2nd edition). http://www.gang.umass.edu/~franz/Paul_Zeitz_The_Art_and_Craft_of_Problem_SolvingBookosorg.pdf

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See related topics:  Math Manipulatives and Standardized Test Preparation.