Geometric Problem Solving
Problem solving in mathematics is critical to understanding geometric concepts and also making connections with other mathematics concepts.
The following activities cause students to use critical thinking and problem solving skills - often referred to as inquiry-based teaching and learning.
Inquiry Math Activity
Materials – 27 one inch snap cubes for each student
Procedures – Part 1
Give each student 27 snap cubes
Ask them to determine how many different ways can they can join three cubes face-to-face? (These are called “tri-cubes”)
Tell them that if a tri-cube can be flipped or repositioned (reflected, rotated) in such a way that it is exactly like a tri-cube already made, it is not different from the other one.
Notes
There are only two different tri-cubes – rectangular and non-rectangular.
Have students retain the non-rectangular tri-cube.
Procedures – Part 2
Ask students to find all possible non-rectangular tetra-cubes (4 unit cubes joined face-to-face).
Note
There are only six different tetra cubes – see picture (the picture also contains the one non-rectangular tri-cube).
Tetra-Cubes
Discussion
Ask the following questions:
Are any of the pieces reflections of each other? (If you put a mirror next to one piece, will you see the other in the mirror?)
Which pieces can be placed so that they are only one unit high?
Which pieces must occupy space that is 2 units high?
Which pieces have a line of symmetry on a given face?
Procedures – Part 3
Ask students to find the surface area and volume of each of the six tetra-cubes and the non-rectangular tri-cube – recording their data in a data table.
Discussion
Ask the following questions:
Did the pieces with the same volume have the same surface area?
Did the pieces with the same surface area have the same volume?
Procedures – Part 4
Ask to make a SOMA Cube by connecting their non-rectangular tri-cube and 6 tetra-cubes into a 3 by 3 cube without altering the shape of the tri-cube or tetra-cubes.
Ask students to record and share how they created a SOMA Cube with the class.
Note
There are 214 different ways.
Extension
Ask students to create other shapes using the non-rectangular tri-cube and 6 tetra-cubes without altering the shape of the tri-cube or tetra-cubes. (See picture for examples)
Soma Cube Shapes
David R. Wetzel, Ph.D.
Tags: geometric shapes, geometry, inquiry math, Math, problem solving, soma cubes
Obsolete Education Trends Through 2020
There are many lists going around about what the next decade will bring in K-12 education, especially focusing on those things that will become obsolete.
Well, I decided to create my own list of 5 things that should be obsolete in K-12 education by 2020.
Homework
The United States continues to fall behind most Asian and some European countries in science and math (Third International Science and Math Study, 2007).
One interesting fact that jumps out about these countries that continually outscores the United States – little or no homework.
Homework is a staple in this country and often parents think their children’s teachers are not very good unless they send home a regular amount of homework.
The biggest problem here is that students who do not understand in school, will not understand at home.
Also, parents are forced into the position of teaching their child something that they probably no little about – especially in science and math.
My definition of homework is repetitious math problems and memorization of science terms or the rote memorization of math and science facts.
Computer Labs
This is one of the biggest wastes of technology resources in schools today. Computers belong in the classroom! I have been in schools that have a computer connected to sewing machines, so students can use computerized patterns for sewing.
I have also been in schools that have three or four computer labs, with one set of 4 or 5 computers for teachers to check out for their classrooms.
Computers belong in a science and math classroom for direct integration in science lab investigations and math problem solving situations.
The best approach for teaching science and math is through project-based investigations, case studies, and problem-based learning situations.
Standardized Tests
SAT and ACT tests do not carry the weight they use to for college admission.
All standardized testing associated with the No Child Left Behind Act (NCLB) is pretty much worthless, only good for politicians and determining a student’s ability to pass a test on a given day.
Standardized testing and NCLB have transformed many schools into testing machines that take the inquisitiveness out science and math by turning our children into bored, underachieving test taking robots.
Treating Teachers as Non-Professionals
Teachers are professionals, just like other adults who require a college degree to be eligible for that career field.
This starts in house, where teachers are given the respect they deserve by administrators and central IT gurus, who belief that only they should have the ability to load instructional software on computers. Also, that teachers are professional enough to have the ability to access online programs and websites that support teaching science and math, without begging for permission.
Textbooks
Anyone who has viewed science or math textbook understands that most are monumental wastes of money. The need for textbooks is perpetuated by two things – parents who think they are needed and publishing companies that need to make a buck.
School systems spend enormous amounts of their budget yearly purchasing textbooks that are out of date the moment they are printed.
Schools typically keep science textbooks for six years or more and the basic understanding of science is changing every day. Also, these textbooks are overrun with colorful, useless information that only serves to distract and confuse students.
Math textbooks are only designed for rote memorization of facts and repetitiously boring math problems. Just like science textbooks, they are overrun with colorful useless information that only serves to distract and confuse students.
The best approach for teaching science and math is through project-based investigations, case studies, and problem-based learning situations.
Other
Now it is your opportunity to add things you would like to see phased out of science and math education by 2020.
David R. Wetzel, Ph.D.
Tags: computer labs, homework, Math, obsolete education, Science, standardized tests, textbooks
Nature of Science
Students need opportunities to experience the Nature of Science experiences that replicate the way scientists search for answers to scientific questions.
There is no one scientific method process used by every scientist.
The Scientific Method, as taught in schools, is a false process and a oversimplification of the way scientists actually work, along with missing the point of how to conduct scientific investigations.
A real understanding of the Nature of Science comes from student personal experience, using simple problem-solving processes that can be discussed and analyzed.
Using the Nature of Science as process for investigating problems helps eliminate many common misconceptions students have about science facts.
For example:
Students move away from the misconception that there is only one answer to any scientific investigation.
Which students internalize from working on “canned labs” found in textbooks that also perpetuate the myth of the Scientific Method as the only process for solving science problems.
Scientific Process Principle Concept
Scientific knowledge is fundamentally uncertain.
Related Concepts
- Science is uncertain because it is a human activity.
- Science explanations seem less certain when they are based on indirect information.
- Scientific uncertainty can be reduced through collaboration.
Scientific Investigation
Students use the Science Process Skills as they complete this investigation into the Nature of Science.
Mystery Boxes
Materials:
- Small Boxes – enough boxes to make four different sets and each box set should be the same shape, size, and color. (cardboard jewelry or gift boxes work well).
- Marbles
- Glue
- Scissors
- Cardstock or Cardboard for making partitions in each box
Preliminary Steps:
- Prepare 4 sets of sealed boxes – each set should have enough boxes so that each student group gets a one box from the set. (for example: 6 student groups = each set will have 6 boxes)
- For each box – glue 2 partitions exactly the same way and place a marble in the same location.
Procedures:
- Place students in groups.
- Give each group box from each set
- Tell the students that each box has a marble and they need to determine the location of the marble in each box.
- Give students time to investigate their boxes (about 5 minutes). Do not answer their questions; let them develop their own questions.
- Ask student groups to share their findings.
- Record each groups findings on the board or a piece of newsprint for each group.
- Ask each group to make a drawing or drawings of where they think the marble is in each box and share with the class.
- Ask the class to look a the findings for each group and see if there are nay patterns or similarities they can use to find the location of the marbles.
- Ask each group to write down the process they used to find the location of the marbles. Then have each group share their process with the class.
Student Actions:
They will probably:
- determine that all boxes of the same color will have the marbles in the same location and each colored box is different from the others.
- try to justify what they did, which enhances the discussion.
- ask about the correctness of their reports, in the form of ‘did we get it right?’
- admit to uncertainties about indirect information (something they couldn’t see).
- indicate that they expect confirmation of their answers from teachers.
- recognize a clear statement of the problem, hypotheses (possible inner configurations), experimental testing (tilting, wile listening, feeling for movements), reporting/publishing (showing diagram on board), etc.
The students have now experienced the Nature of Science, along with uncertainty of what they are observing, which can be reduced through collaboration. Also, that indirect information compounds the uncertainty.
Finally – if the students did not discover the right answer then share the location of the marble in each box with them.
Any comments, suggestions, or other examples of the Nature of Science scientific investigations?
Readers who found this helpful may want to read:
Science Case Studies and Problem Based Learning
20 Questions to Ask Students in Science Projects
David R. Wetzel, Ph.D.
Tags: nature of science, science investigations, science process process, scientific method
Pythagorean Theorem
The Pythagorean Theorem is used any time you have a right triangle in which you know the length of two sides and want to find the third side.
The formula is: a2 + b2 = c2
There are many strategies for getting students to learn how to solve a problem related to this theorem. Many of these problems are boring, procedural driven, and lacking in connection to real world applications.
In the view of many this is often necessary because they do not have the time to teach it any other way. Curriculum, testing, and No Child Left Behind pressures cause these views.
However, there are other strategies to teach the same concept and yet make it more interesting for students. The critical aspect of this strategy is to make connections to their world.
Students always ask:
- When will I every use the Pythagorean Theorem?
- Why do need to learn this? It is boring!
The answer is, in any kind of job that deals with triangles. For example, carpenters, engineers, architects, construction workers, those who measure and mark land, artists, and designers of many sorts need to know it.
Problem Solving Application
The following are problem is more interesting and designed to find the length of a2 and b2, when only c2 is known.
Shopping for a New TV
Jacob and Denise, brother and sister, are trying to get their parents to buy them a new TV. Their parents have agreed; however, they are limited to a size 32 inch TV.
Now they must go shopping to find the one they want. At their local electronics store, they discover that 42 inch TVs come in a lot shapes and sizes.
This does not seem to make since, until the salesman tells that 42 inches is diagonal measurement that is the normal way of sizing TVs.
They decide to compare the difference between a 32 inch standard TV and 32 inch flat screen HD TV. Jacob and Denise both agreed that they should buy the TV with greatest viewing area.
After looking a both TV types, they predicted that the flat screen HD TV had the greatest viewing area.
How would you determine which TV has the greatest viewing area?
Answer
TV Problem: the key to this problem is the aspect ratio of the TVs. The Standard TV has an aspect ratio of 4:3, while the Flat Screen HD TV has an aspect ratio of 21:9.
First they must find out what each unit equals (for example in the Standard TV is 4 units wide and 3 units high - aspect ration 4:3) in the aspect ratio for each TV by using the Pythagorean Theorem.
Next they must find the area of each TV using A = w x h.
Standard TV Area = 491.56 in2
Flat Screen HD TV Area = 370.44 in2
Jacob and Denise now must decide which TV they really want, because their prediction was wrong.
——
This problem not only makes connections to something that students are actually interested in, they also make connections to other math concepts. Pythagorean Theorem now has a real world application to students and they will see its value.
Readers who liked this post may also be interested in:
Three Project-Based Learning Math Activities
David R. Wetzel, Ph.D.
Tags: finding the area, geometry, Math, math formulas, problem solving, pythagorean theorem, triangles
A Million Grains of Rice
We need to make changes in the way we teach students math. A comparison of state standardized test scores show that test scores a leveling off or are flat State Mathematics Comparisons 2000 - 2009 (National Center for Education Statistics).
During the early part of the decade our students made great strides in improving their test scores. This appears to be due to an increased emphasis on math in schools; a dependence on rote memorization of facts and direct instruction.
Now there needs to be another reform in the way students learn math by using project-based teaching and learning strategies. This integrates problem-solving which is a critical skill that students need in mathematics.
The use of reading strategies that incorporate project-based learning, helps set a situation in which math is learned. These circumstances help students understand that math is more relevant to their own world – making math personal.
Math Lesson Integrating Reading
Use the book “How Much is a Million” by David M. Schwartz.
Questions
Ask Students:
- How much a million grains of rice really is and what they think it looks like?
- How long it would take to count to a million?
- How long they think it would take to count one million grains of rice?
- How much a million grains of rice would weigh?
- How they would go about counting a million grains of rice?
Procedures
Students (in groups):
- Count the rice grains in a cup of rice (150–200 grains).
- Weigh their cups of rice – subtracting the weight of the cup.
- Records their data in a data table for the whole class to see (e.g., draw a chart on the board or poster paper, or enter the data into a spreadsheet).
- Estimate the total number of grains of rice that have been counted so far and the total weight.
- Add the estimates to the data.
- Add up the total number of grains and compare the actual number to the estimates, along with the weight.
- Which estimates are close?
More Questions
Hold up a 2 pound package of rice.
Ask Students:
- How many more cups would they need to count in order to reach 1 million grains?
- To estimate how many grains of rice are in the package?
- To estimate the number of 2 pound packages of rice needed to equal one million grains of rice?
- To calculate the number of 2 pound packages needed. (The answer is around 31 packages)?
After stacking the 31 packages rice the students can visualize how many grains of rice it takes to make a million. They also have concrete evidence of what one million grains of rice looks like.
Extension
Try using this same activity with other math trade books, such as:
“If You Hopped Like a Frog: by David M. Schwartz and illustrated by James Warhola – ratio and proportions.
“One Grain of Rice” by Demi - villager in a developing country trying to feed her village.
This lesson makes the necessary connections that students need to make between project-based learning, problem solving, numbers and operation, measurement, data analysis and probability, reasoning and proof, communication, and representation.
These are skills necessary to help change math and help students score better on standardized testing.
Resource
Lesson adapted from Reading and Writing Mathematics – Introduction
David R. Wetzel, Ph.D.
Tags: literacy in math, Math, problem solving, project based learning, reading in math
Science Nature Journals
A nature journal allows students to make observations and connections about the natural world in which they live.
As they develop their own nature journals, students develop a concrete understanding of what is going on in the part of nature they are studying.
One example is the impact of global warming on animals which live in a specific ecosystem.
Nature journals let students build upon their experience, remember a certain order of recorded events, and link data with events to develop relationships.
Students make close observations of nature to determine patterns and motions, along with considering the weather, sky, sounds, and temperature changes as they reflect upon nature’s disposition.
They use the science process skills as they observe, analyze, and communicate their findings.
Nature journals assist students with learning how to write about science as they follow these guidelines:
- Begin writing and often. Do not be too critical or edit writing at the beginning to let the writing flow naturally.
- Write as if writing a letter to yourself, close friend, or family member by creating a narrative account.
- Complete sentences are optional; the important point is to record pictures, observations, and data.
- Draw pictures (if a paper journal) or incorporate photographs (if an online journal).
Nature Journals and Technology
Developing a class nature journal using a free Wiki allows teachers and students to view and make comments.
Using Wikis to engage students by integrating technology provides them with their own online journal page in the class Wiki to update and maintain.
Advantages of using a Wiki for nature journals include:
- Students have the ability to embed links to online resources to support their journals.
- Students can upload images into their Wiki page.
- Students can work collaboratively to complete a group journal.
- Students can access and update their nature journal from home.
- Teachers can monitor their students’ progress at their convenience.
- Teachers can limit access to class Wikis, which eliminate concern for open access.
Any other ideas for Specific Science Journals?
David R. Wetzel, Ph.D.
Tags: journals, nature, Science, writing in science
Math Problem Based Learning
Project-based learning or Problem-Based Learning (PBL) is one of the best teaching strategies for engaging students in realistic learning activities. Students are not only interested, they are also learning math in the process.
Why?
Because their minds are engaged, critical thinking is taking place!
This is often referred to as critical thinking, minds-on, or inquiry-based teaching and learning.
PBL activities are designed to answer a real-world question or solve real-world problem. A good PBL problem provokes students to struggle with central concepts and principles in math.
These problems reflect the types of learning and work people do in the everyday world outside the classroom.
PBL is typically completed by groups of students working together to solve a problem, as they reflect upon their own ideas, prior knowledge and experiences, and communicate their recommendations based on findings.
Math Project Based Learning Activities
The following are examples of Problem Based Learning activities:
Repainting Tennis Courts
Students:
Determine the total cost of supplies.
The number of gallons of paint to cover all 8 courts if they apply two coats of green paint on each court, along with two coats of white paint on the lines of each court.
The cost of all the paint combined.
The grand total spent on paint and supplies.
Contextual information needed by students include:
Dimensions of a tennis court
Total number of lines, along with line dimensions on a tennis court.
How many square feet does a gallon of exterior paint cover.
Cost of a gallon of exterior paint.
Cost of a combo pack of roller frame, roller cover, and paint tray.
Cost of an appropriately sized paint brush.
Cost of any other materials they feel they need.
Additional Math PBL Activity Themes
Scavenger Hunt
Students:
Locate definitions of Tessellations on the Internet to compare with the real-world examples of tessellations (for example M. C. Escher’s work).
March Madness
Students:
Explore the many different areas of math found in the NCAA basketball tournament.
Use the NCAA bracket to determine which team or teams they want to follow in the tournament.
Find fractions, decimals, and percents; probability statistics; make predictions; and look for patterns within the basketball statistics of the team(s) they selected.
Make connections with and between the math contained within the NCAA tournament.
Explain the importance of mathematics in basketball.
Readers who enjoyed this post may also enjoy:
Math Problem Solving Stories and Case Studies
Challenging Math Strategy for Students
The Math and Science of Junk Mail Project
David R. Wetzel, Ph.D.
Tags: basketball, critical thinking in math, math activities, math lessons, problem based learning, project based learning
Hands-On, Minds-on Science
The science community needs to develop better ways to assess students’ understanding and skills, not just their science factual knowledge.
For many years, education has traditionally been about standing in front of a classroom and giving a great lecture, or just a lecture. What people rarely do is try to work out whether or not the students have actually learned anything.
A student’s ability to regurgitate facts on a multiple choice proves little. Most any student can memorize enough facts to earn a minimal passing score. However, what have they learned – minimal! No wonder most students are turned off by science by the time they complete high school.
Students need to understand scientific literature, along with the ability to design an experiment and interpret data. This ability to make informed decisions not only has its implications in learning science, this ability is transformed to everyday decisions in a real world.
Educating students in science is important because they’re going to be the leaders of the future. These leaders will need to be able to make policy decisions by analyzing scientific evidence, instead of throwing up their hands in confusion or making wrong decisions.
When students understand the scientific process they become informed citizens. They have the ability distinguish between fact and fiction. Whenever scientists disagree, informed citizens realize that they are conducting scientific research and there are many paths. Rarely is there one correct path to an answer in science. Science is like Times Square – there are many paths that will take you and all are correct.
Students need to learn how to critically analyze science information and argue points from evidence, data, or observations. Educators need to get rid of the lecture format and let students do more in class to facilitate their learning.
We have to stop telling and let them work with scientific information. A hands-on, minds-on approach is best. This means to let them design experiments under teacher facilitation so then learn that there are many methods for finding an answer.
Canned laboratory investigations leave little to the scientific imagination. Although they may be hands-on, students do not use a minds-on approach or critical thinking.
Students also need to use the power of technology to conduct their experiments.
Three teaching strategies that are excellent hands-on and minds-on approaches are:
When students are allowed to think and not just memorize they will learn science facts, along with the ability to make connections between facts. This is what assessments should focus on, the ability of students to use critical thinking skills to make scientific connections. This will make them more informed citizens of the world.
Suggestions for additional strategies to make science learning meaningful for students is requested.
David R. Wetzel, Ph.D.
Tags: discrepant events, education programs, hands on and minds on, problem based learning, problem solving, Science, science education
Earth Day and Phone Book Activities
Ever wonder what to do with all those old phone books?
Earth Day is coming and a good earth day activity is to develop creative uses for these phone books, beyond just recycling or throwing them in the garbage.
Phone books have a myriad of uses such as shredding the pages for use as packing material, compost materials, and booster seats.
Phone Book Facts
Despite the increase of Internet based telephone number directories, the production of phone books is increasing.
Here are some facts to get you thinking about how important it is to recycle or reuse your phone books.
- On average, over 600,000 tons of phone books end up in landfills every year.
- There are enough phone books created each year to measure 106,700 miles when lined up end to end. This means they would circle around the earth about 4.28 times!
- About 80 percent of all U.S. paper mills use some recycled material in their manufacturing service. It is estimated that about 200 mills use ONLY recycled material.
- There are more than 7,000 different titles of yellow pages.
- 540 million telephone directories are distributed each year.
Earth Day Activities
Now that you see the current phone book dilemma, a good earth day activity is to find creative uses for all these unwanted phone books.
If just 500 phone books can be kept out of a land fill we could save between 17 and 31 trees, 7,000 gallons of water, 463 gallons of oil, 587 pounds of air pollution, 3.06 cubic yards of landfill space and 4,077 kilowatt hours of energy according to the American Forest & Paper Association.
Example Activities include:
- Earthworm Bedding - shred the white pages and combine them with dirt, which enriches the soil as the pages decompose to support the earthworm habitat. Do not use the yellow pages, because of the chemicals.
- Mulch - tear the pages out a phone book and lay them about 6 - 8 pages thick on top of the soil in a flower garden or among shrubs. Then cover the pages with a thin layer mulch. The pages will act as shield to prevent grass from growing through the mulch. Also the phone book pages will decompose and enrich the soil over time.
- Booster Seat - use fabric to cover phone books, sealing the seams with fabric glue or needle and thread. This keeps the phone books from sliding around as kids wiggle around on them.
- Packing Material - shred phone book pages for use in packaging instead of using packing peanuts. Shredded phone book pages are biodegradable and packing peanuts are not.
Share your ideas for additional uses of these relics of the past.
Additional Resources for Earth Day 2010
Modeling the Earth’s Atmosphere
Global Warming Science Projects
David R. Wetzel, Ph.D.
Tags: climate change, earth day, earth day 2010, earth day activities, phone books, recycling, science projects
Colors of the Rainbow
A common question every parent and teacher has heard from children at some point — Why is the sky blue?
Learning the correct answer is important. Because once children hear an answer several times, right or wrong, this answer becomes embedded in their brain.
Convincing them otherwise, especially if they learned the wrong answer, takes a lot of evidence to undo this misconception.
Why the Sky is Blue? - Common Answers
Children come up with a variety of answers when asked why the sky is blue.
Their answers reflect what they hear from peers, parents, and movies or television.
Here are a few common answers children give:
- The sunlight reflects off the oceans.
- The sky is blue because it is the bottom of space.
- Because of all the water in the sky.
- The sky reflects off the top of clouds which contain water.
The sky appears blue because the molecules of air in the upper atmosphere scatter the blue waves of light more than the other colors as sunlight passes.
This answer will illicit another response from children –- But light is clear because you do not look blue!
It is true light is clear, because of all the colors which make up sunlight. At this point, children will want proof.
This can be proved by using a prism to separate the clear sunlight into the colors of the rainbow – ROYGBIV.
Connecting the prism and wavelengths is important to understanding why the sky is blue.
Children need this evidence to support internalization of the new information they just learned about sunlight and why the sky is blue.
You may have probably predicted the next question children will ask.
This question is — Why isn’t the sky red, yellow, orange, green, etc.?
This is typical and expected from children who are now questioning the evidence being presented to them. They are conducting inquiry-based science.
For more about inquiry-based science visit Understanding Science Inquiry.
David R. Wetzel, Ph.D.
Tags: blue sky, color, home school science resources, inquiry based science, red sky, roygbiv, science of light, sunlight, wavelengths, where do rainbow colors come from