184.108.40.206 Knowledge & Understanding
Describe how technological problems and advances often create a demand for new scientific knowledge, improved mathematics and new technologies.
Determine and use appropriate safety procedures, tools, computers and measurement instruments in science and engineering contexts.
For example: Consideration of chemical and biological hazards in the lab.
Select and use appropriate numeric, symbolic, pictorial, or graphical representation to communicate scientific ideas, procedures and experimental results.
Relate the reliability of data to consistency of results, identify sources of error, and suggest ways to improve data collection and analysis.
For example: Use statistical analysis or error analysis to make judgments about the validity of results.
Demonstrate how unit consistency and dimensional analysis can guide the calculation of quantitative solutions and verification of results.
Analyze the strengths and limitations of physical, conceptual, mathematical and computer models used by scientists and engineers.
MN Standard in lay terms:
Science, Technology, Mathematics and Engineering work together and rely on each other in order to solve problems in the world. The connections include statistical analysis, modeling, safety and technology solutions among others.
Big Idea: Critical attributes or essential understanding of the standard.
In order to solve problems using science, technology, mathematics and engineering these skills must be intertwined and used together. The problems often create the demand for the new technology and the new technology opens the door to solve more problems. This includes a variety of mathematical analysis including statistics, graphing and significant figures among others. It also requires appropriate safety concerns and the reliability and creation of reliable models including computer, physical, conceptual, mathematical and computer.
MN Standard Benchmarks:
220.127.116.11.1 Describe how technological problems and advances often create a demand for new scientific knowledge, improved mathematics and new technologies.
18.104.22.168.2 Determine and use appropriate safety procedures, tools, computers and measurement instruments in science and engineering contexts.
22.214.171.124.3 Select and use appropriate numeric, symbolic, pictorial or graphical representation to communicate scientific ideas, procedures and experimental results.
126.96.36.199.4 Relate the reliability of data to consistency of results, identify sources of error, and suggest ways to improve data collection and analysis.
188.8.131.52.5 Demonstrate how unit consistency and dimensional analysis can guide the calculation of quantitative solutions and verification of results.
184.108.40.206.6 Analyze the strengths and limitations of physical, conceptual, mathematical and computer models used by scientists and engineers.
A quote, cartoon or video clip link directly related to the standard.
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Skills necessary to become independent inquirers about the natural world.
The dispositions to use the skills, abilities, and attitudes associated with science.
the end of the 12th grade, students should know that
In some cases the more of something there is, the more rapidly it may change (as the number of births is proportional to the size of the population). 9B/H1a
Sometimes the rate of change of something depends on how much there is of something else (as the rate of change of speed is proportional to the amount of force acting). 9B/H1b
Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. 9B/H2a
Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time. 9B/H2b
Any mathematical model, graphic or algebraic, is limited in how well it can represent how the world works. The usefulness of a mathematical model for predicting may be limited by uncertainties in measurements, by neglect of some important influences, or by requiring too much computation. 9B/H3
Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another. 9B/H4
When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes by more than one, and sometimes not at all. 9B/H5
Computer modeling explores the logical consequences of a set of instructions and a set of data. The instructions and data input of a computer model try to represent the real world so the computer can show what would actually happen. In this way, computers assist people in making decisions by simulating the consequences of different possible decisions. 8E/H1
A mathematical model uses rules and relationships to describe and predict objects and events in the real world. 11B/H1a*
A mathematical model may give insight about how something really works or may fit observations very well without any intuitive meaning. 11B/H1b
Computers have greatly improved the power and use of mathematical models by performing computations that are very long, very complicated, or repetitive. Therefore, computers can reveal the consequences of applying complex rules or of changing the rules. The graphic capabilities of computers make them useful in the design and simulated testing of devices and structures and in the simulation of complicated processes. 11B/H2*
The usefulness of a model can be tested by comparing its predictions to actual observations in the real world. But a close match does not necessarily mean that other models would not work equally well or better. 11B/H3*
Often, a mathematical model may fit a phenomenon over a small range of conditions (such as temperature or time), but it may not fit well over a wider range. 11B/H4** (SFAA)
The behavior of a physical model cannot ever be expected to represent the full-scale phenomenon with complete accuracy, not even in the limited set of characteristics being studied. The inappropriateness of a model may be related to differences between the model and what is being modeled. 11B/H5** (SFAA)
Mathematics Core Standard
S-ID 1-4 Summarize, represent, and interpret data on a single count or measurement variable.
S-ID 5-6 Summarize, represnt, and interpret data on two categorical and quantitative variables.
S-ID 7-9 Interpret linear models
S-IC 1-2 Undertand and evaluate random processes underlying statistical experiments
SIC 3-6 Make inferences and justfy conclusions from sampe surveys, experiments, and observational studies.
S-CP 1-6 Understand independence and conditional probability and use them to interpret data
S-CP 6-9 Use the rules of probability t compute probabilities of compound evens in a uniform proability model.
Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text.
For literacy in History/Social Studies
(1)Cite specific textual evidence to support analysis of primary and secondary sources, attending to such features as the data and origin of the information.
(7) Integrate quantitative or technical analysis (e.g. charts, research data) with qualitative analysis in print or digital text.
For literacy in Science and Technical Subjects
(3) Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks: analyze the specific results based on explanations in the text.
(8) Assess the extent to which the reasoning and evidence in a text support the author's claim or a recommendation for solving a scientific or technical problem.
Write arguments to support claims in an analysis of substantive topics or texts, using valid reasoning and relevant and sufficient evidence.
Students of all ages often interpret graphs of situations as literal pictures rather than as symbolic representations of situations.
Many students interpret distance/time graphs as the paths of actual journeys.
In addition, students confound the slope of a graph with the maximum or the minimum value and do not know that the slope of a graph is a measure of rate. They have trouble with the notions of interval scale and coordinates even after traditional instruction in algebra. For example, some students think it is legitimate to construct different scales for the positive and the negative parts of the axes.
Alternatively, students think that the scales on the X and Y axes must be identical, even if that obscures the relationship.
Students read graphs point by point and ignore their global features.
Students of all ages often do not view the equal sign of equations as a symbol of equivalence between the left and the right side of the equation, but rather interpret it as a sign to begin calculating.
Faced with no correlation of antecedent and outcome, high school students only sometimes draw the conclusion that the variable has no effect on the outcome. A basic problem appears to be understanding the distinction between a variable making no difference and a variable that is correlated with the outcome in the opposite way than the students initially conceived.
Extensive research points to several misconceptions about probabilistic reasoning that are similar at all age levels. One common misconception is the idea of representativeness, according to which an event is believed to be probable to the extent that it is "typical". For example, many people believe that after a run of heads in coin tossing, tails should be more likely to come up. Another common error is estimating the likelihood of events based on how easily instances of it can be brought to mind.
It is now the 3rd day of chemistry class and the group is about to start their first laboratory experience. They inter the lab prepared to measure a variety of liquids and mass a variety of solids. They are confronted with a variety of equipment including graduated cylindars, balances, beakers, pipettes and more. The students are then asked the all important question; how far can you actually estimate a value given a particular piece of measuring equipment. They quickly agree that measureng half way between the numbers is easy but it would be far less accurate to estimate to the tenths portion of the distance.
Students are then given a problem in measurement. They are to measure the volume of 10 small volumes of liquid and estimate to the greatest possible degree the amount. with as many decimal places as they possibly can. They are to combine the 10 liquids and then measure again. Is the number the same as when they were measured individually and added together. Probably not. Students are then introduced to the concept of significant figures and why they are important in science. The students measure again, using significant figures this time.
Safety should be covered during the first days of school. It is important that students and parents are informed of the safety rules and procedures. Having parents and students sign a safety form/contract is recommended. Graphing should be learned and done by hand. If students do not go through the motions of visualizing and then making a logical representation of their data which shows trends and relationships, they may not learn how to do this or to understand what they have done. Although computer programs are convenient can be too easy and take the element of visualization and logical representation of the immediate picture.
Statistical analysis as it is affected by sample size and trials is an important discussion to have.
Dimensional analysis is usually formally introduced in chemistry classes. However, all classes can involve a level of understanding of dimensional analysis. It enhances the experimental design and brings an element of logic to the plan.
Selected activities, labs, lessons, problems, etc., for each standard (adhere to copyright!).
220.127.116.11.1 Space shuttle spin-offs
Students explore the technology of the space program. What spin-offs of the space program do we now have because of it. What were the modifications needed to take equipment into space? Has that influenced any products that we have today?
Safety is an important aspect of teaching science. Safety instruction specific to each content area - life, earth and the physical sciences - must be part of a student's learning experience. Safety contracts, safety assessments (ex: Flinn Safety Test) and content-related safety instruction are highly recommended. In biology, for example, the use of safety training including "Sterile technique" in handling bacterial samples is required and highly important in the study of biology. A safety agreement is always a good idea at the beginning of any study in biology including classes, projects, mentorships, workshops, etc. Any class with a field experiences - biology or earth science most likely - would have safety concerns regarding outdoor experiences. Chemistry and physics would have extensive lab safety issues.
Surprisingly students in high school still have difficulty choosing the right graph for the job. In the first few weeks and, as data is being collected, at the beginning of the year, practice in graphing and choosing types of graphing can be very beneficial. Especially important is that students recognize the difference in use between line, bar and pie charts. There is also research that states that this is best learned through "hand graphing". Computer programs graph any data without giving credence to logic. It is far too easy for students to come up with the wrong conclusion.
18.104.22.168.4 Statistical Analysis and Experimental Error - Roly Poly's
Roly Poly's are tiny invertebrate animals that really like wet moist and dark areas. Testing whether they move by kinesis or taxis is a favorite science experiment which generates simple data. Students run a test in which 5 roly polies are placed on wet paper in petri plates and 5 are placed on dry paper. They are covered with black paper. Every 2 minutes for 20 minutes students check to see how many roly polies are standing still and how many are moving. Therefore 10 sets of data are collected. Students analyze the data and run simple statistical tests to determine whether the roly polies are moving by taxis (would not move in either case) or kinesis (would only move in the dry dishes). The students find the mean, median and mode and then determine the significance of the data. A discussion ensues about sample size and number of trials. Would this data be stronger if it was done for 30 minutes? What if we had 10 roly polies per dish instead of 5? Students then derive their best practice for experimental design, scientific error and its relation to sample size and trial number.
22.214.171.124.5 Significant Figures
Appropriate measuring accuracy is an important discussion in all science classes. This should be emphasized as a matter of "logic" and "common sense" always. Experience usually indicates that the first strong mathematical experience with this in found in chemistry classes. Although it should be referenced and discussed in all science classes.
126.96.36.199.6 Models - "Sparrow Lab".
The Sparrow Lab is a conceptual model which analyzes a population of sparrows on an island. Several assumptions are made in order to make the model work. 1. Each year all parents die - all children survive. 2. Each year all clutches consist of 10 offspring - 5 male and 5 female. At the end of the calculations students graph the results on both classic graph paper and semi-logarithmic graph paper. In a classic result and assuming the assumptions, a straight logarithmic line is produced. A discussion of the accuracy of the assumptions and process ensues. Students usually conclude that the assumptions ultimately cancel each other and the resulting conclusions make for a good model of population growth in an ideal situation.
- Technology - products created based on science that solve problems and/or produce results and/or products.
- Graphing - A pictorial representation of data which shows relationships and trends among the numbers.
- Quantitative - Data based on concrete numbers and measurement.
- Qualitative - Data based on a value determination such as beauty, health etc.
- Models - A representation of a product or system often in 3 dimensions.
- Dimensional analysis - The analysis of numbers in measurement based on the confidence of the values of those numbers.
- Safety Procedures - Scientific procedures and or techniques which decrease the probability of an accident occurring.
Microsoft Excel Programs - Excellent and vital for graphing. However, use caution that students understand the concepts and can visualize the graph by hand before they plug numbers into a program.
Graphing and modeling occur in a variety of classes. The connections are readily seen in mathematics but may be equally applicable in such courses as social studies doing population analyzis and world poverty levels.
1. When do you use a pie chart, a line graph or a bar graph?
Answer: Pie chart is part of a whole, line graph shows linear progress and a bar graph compares disconnected data.
2. What does sample size have to do with experimental error?
Answer: The larger the sample size, the smaller the erorr. As the sample size increases the probability of a significant result increases.
3. Is a graph a conceptual or a mathematical model?
Answer: It may be both.
1. Statistical significance is dependent on what variables? How do we know that we have data which is significant.
Answer: Significance is dependent on both sample size and the standard deviation. A test takes these into consideration when determining whether two sets of data are significantly different. To be significant they must have a probability (p value) of less than .05. (This is the probability of the null hypothesis).
2. How do you ensure that your students truly understand graphing not only in creating them but also in the reading of the graphs of others.
3. How do you ensure that every student knows and understand the safety procedures in your classroom? How do you legally protect yourself from lawsuit when a student does not follow these rules?
When observing a lesson on this standard you may see students analyzing data statistically and graphically. They should be graphing both by hand and then later they can turn to the excel programs on computers. They may be building models either on the computer or physcally and analyzing them for their usefulness and the strength of their product or hypothesis.
Struggling and At-Risk:
Struggling students may need specific examples of these concepts. Although often able to master the concepts they may give up easily when frustrated.
Language may play a role in data communication.
Gifted and Talented students can explore further statistical analysis techniques such as t-tests and chi square.
Students may do a historical analysis of modern conveniences. Where did the ideas come from and what was the advance that created the demand for the new science and/or technology?
Advances of various cultures can be explored and the new scientific knowledge mathematics and technologies can be tracked as to their origin.
Hands on activities with concrete examples will work well with these students. Physical models can be very effective in demonstrating scientific principles.
Parents can enforce safety procedures. A good discussion takes place when the safety sheets are taken home and signed.