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Alaska Department of Education & Early Development

Alaska 2000

KEY ELEMENT
Collect, organize, analyze, interpret, represent, and formulate questions about data. Make reasonable and useful predictions about the certainty, uncertainty, or impossibility of an event (Statistics and Probability).

STATISTICS AND PROBABILITY FOR ALL
The field of statistics is the process of collecting, displaying, interpreting, and critiquing data. Statistics incorporates the concepts of probability to determine whether or not the results of a survey or experiment are the result of chance or are the result of a cause/effect relationship. Probability and statistics concepts fall within the domain of general mathematical and scientific literacy. These are skills required for effective life-long decisionmaking in democratic society.

BIG IDEAS IN STATISTICS AND PROBABILITY

• Possibly, probably, and certainly
• Predicting and determining probabilities
• Theoretical probability
• Experimental probability
• Simple and compound events
• Independent and dependent events
• Data collection, organization, and graphing
• Central tendency, variance, and correlation
• Analysis and inference

Benchmarks
For example, at these levels a student would be able to:

Statistics/Probability

Level 1 (ages 8-10) Make and use picture, bar, and line graphs. Formulate and solve problems of collecting and analyzing data. Identify mathematically fair and unfair games. Construct real, pictorial, and symbolic graphs. Collect, organize, describe, and interpret real-life data using charts and graphs. Organize and classify attributes using Venn diagrams.

Level 2 (ages 12-14) Model situations by constructing a sample space to determine probabilities. Determine the total number of outcomes of an experiment by using the counting principal or by constructing tree diagrams for simple and compound events. Compare experimental results with mathematical expectations. Classify events by level of certainty. Design a survey, collect and analyze the data, calculate the measure of central tendency, and make predictions about the larger universe based on the sample.

Level 3 (ages 16-18) Determine probabilities and predict outcomes from real-world data. Understand and critique sampling strategies. Utilize measures of dispersion and correlation when making interpretations. Use curve-fitting to predict from data. Design statistical experiments and interpret the outcomes. Extrapolate and interpolate data. Distinguish experimental from survey data. Use simulations to estimate probabilities or generate sampling distributions.

Alaska 2000

Math Content Standard B: Problem Solving All Alaska students will understand, select and use a variety of problem solving strategies.

Key Elements

Students who meet this standard will:

1. Use computational methods and appropriate technology as problem-solving tools;
2. Use problem-solving to investigate and understand mathematical content;
3. Formulate mathematical problems that arise from everyday situations;
4. Develop and apply strategies to solve a variety of problems;
5. Check the results against mathematical rules;
6. Use common sense to interpret results;
7. Apply what was learned to new situations; and
8. Be able to use mathematics with confidence.

Alaska 2000

Math Content Standard B: Problem Solving All Alaska students will understand, select and use a variety of problem solving strategies.

Benchmarks

For example, at these levels a student would be able to:

Level 1 (ages 8-10) Use several problem-solving strategies such as guess and check, drawing a picture, making a model, working backwards, comparing with previous experiences. Create and solve mathematical problems of his/her own design. Create and solve mathematical problems of his/her own design. Identify appropriate operation required by a problem. Work cooperatively and collaboratively to solve problems.

Level 2 (ages 12-14) Solve a given mathematical problem and explain the steps used to find the solution. Explain why the solution is reasonable. Recognize that real problems may have many, one, or no mathematical solutions. Solve multi-step problems using mathematical operations with real numbers using probability, geometry and statistics. Select appropriate strategies such as guess and check, solve a simpler problem, find patterns, work backwards, model, or use technology to solve problems. Formulate questions from given sets of graphical, written, or oral information.

Level 3 (ages 16-18) Apply principles, concepts, and strategies from various strands of mathematics to solve problems that originate within the discipline of mathematics or in the real world. Recognize and formulate problems from situations within and outside mathematics. Apply the process of mathematical modeling to real practical problems. Be aware of available and emerging technologies and apply the appropriate technology to a problem-solving situation. Evaluate the role of various criteria in determining the optimal solution to a problem.

PROBLEM SOLVING FOR ALL
Mathematics literacy targets effective identification, reasoning, persistence, and success at solving problems in real life situations. Mathematics should be presented as a tool to solving problems, identifying patterns, and predicting similar situations. Therefore, instruction in mathematics should be initiated as a tool for solving real problems for students. Through problems students can discover algorithms and problem solving strategies that can be applied throughout their education and lives.

BIG IDEAS IN PROBLEM SOLVING

• Heuristics for problem solving
• Multiple ways to solve the same problem
• Diverse perspectives on the nature of the problem; effectiveness of cooperation
• Using estimations and reasonableness to identify effective strategies
• Determining relevant and irrelevant information

Alaska 2000

Math Content Standard C: Communication All Alaska students will understand, form and use appropriate methods to define and explain mathematical relationships.

Key Elements

Students who meet this standard will:

1. Express and represent mathematical ideas using oral and written presentations, physical materials, pictures, graphs, charts, and algebraic expressions;
2. Relate mathematical terms to everyday language;
3. Develop, test, and defend mathematical hypotheses;
4. Clarify mathematical ideas through discussion with others.

Alaska 2000

Math Content Standard C: Communication All Alaska students will understand, form and use appropriate methods to define and explain mathematical relationships.

Benchmarks
For example, at these levels a student would be able to:

Level 1 (ages 8-10) Relate models, pictures, and diagrams to mathematical ideas. Relate everyday language to mathematical language and symbols. Realize that representing, discussing, reading and listening to mathematics are a vital part of learning and using mathematics. Prepare and deliver a mathematical presentation.

Level 2 (ages 12-14) Understand and appreciate the value of standard mathematical notation and its role in the development of mathematical ideas. Contribute to a group solution. Express, discuss and justify strategies and processes in oral and written form; model mathematical situations with concrete objects. Use calculators or appropriate technology to store, retrieve and communicate information. Write and discuss ideas to interpret and formulate solutions, including making predictions and conjectures. Use mathematical vocabulary and symbols in communicating concepts and interpret information described in graphs or charts.

Level 3 (ages 16-18) Translate a real-world problem into standard mathematical notation. Record in a journal everyday experiences involving mathematics. Read and understand publications which include mathematically related materials. Appreciate the economy of mathematical symbolism and its role in the development of mathematics.

COMMUNICATION IN MATHEMATICS FOR ALL
Mathematics is a multidimensional language that requires the communicating group to simultaneously interpret between symbols, words, and models. Each of these communication tools is more or less useful depending upon the context of the mathematical problem, the culture of the communicators, and the complexity and specificity of the problem. Often an idea can be validated in one communication mode more easily than in others. Instruction should allow students to use the communication modality that best explains their reasoning.

BIG IDEAS IN COMMUNICATION OF MATHEMATICS

• Oral, written, concrete, pictorial, graphical, algebraic and technological communication modes
• Creating mathematical ideas through formulating questions, conjectures, definitions, and generalization about data and problems
• Refining and clarifying mathematical ideas through individual and group reflections
• Connecting the reasoning between procedures and concepts

Alaska 2000

Math Content Standard D: Reasoning All Alaska students will use logic and reason to solve problems.

Key Elements

Students who meet this standard will:

1. Analyze situations;
2. Draw logical conclusions;
3. Use models, known facts and relationships to explain their reasoning;
4. Verify conclusions, judge the validity of arguments, and construct valid arguments (Deductive Reasoning); and
5. Recognize patterns and form mathematical propositions (Inductive Reasoning).
   

Alaska 2000

Math Content Standard D: Reasoning All Alaska students will use logic and reason to solve problems.

Benchmarks
For example, at these levels a student would be able to:

Level 1 (ages 8-10) Recognize that mathematics makes sense. Use patterns and relationships to analyze mathematical situations. Justify solution to a problem using a manipulative or other model. Given a pattern of numbers, predict the next two numbers in the sequence. Justify the logic used to arrive at a solution.

Level 2 (ages 12-14) Express the pros and cons of an issue and defend an appropriate solution based upon a given set of criteria. Develop and apply deductive or inductive reasoning to solve problems. Draw diagrams to aid in the solution of problems. Solve logic problems. Conjecture, collect data, represent it graphically, interpret the information presented, and validate the conclusions to test mathematical ideas.

Level 3 (ages 16-18) Justify choice of a trigonometric or geometric method for determining the distance between two points in space, such as in navigation or planetary exploration. Find two examples of statistics that support opposing sides of the same issue and explain the differing interpretation.

MATHEMATICAL REASONING FOR ALL
Mathematical literacy involves a willingness to engage in a problem without knowing the answer in advance. For many students this requires a step into an insecure territory in which they must enter into uncertainty to gain access to success. Successful experiences with the habits of mind that support higher order critical thinking are crucial at an early age in order to develop a willingness to work with this uncertainty in more complex problems with larger stakes.

BIG IDEAS IN REASONING

• Non-algorithmic and algorithmic reasoning
• Multiple solutions; multiple criteria for correctness
• Uncertainty dominates
• Requires sustained effort
• Interdisciplinary usefulness of these habits of mind
• Valuing debate and critique

Alaska 2000

Math Content Standard E: Connections All Alaska students will apply mathematical concepts and processes to situations within and outide school.

Key Elements

Students who meet this standard will:

1. Explore problems and describe results using graphical, numerical, physical, algebraic, and verbal mathematical models or representations;
2. Use mathematics in daily life;
3. Use mathematics in other curriculum areas.

Alaska 2000

Math Content Standard E: Connections All Alaska students will apply mathematical concepts and processes to situations within and outide school.

Benchmarks

For example, at these levels a student would be able to:

Level 1 (ages 8-10) See the applications of numeracy concepts and skills to all objects in their environment, not just mathematics manipulatives. Envision similar problems in different contexts that a recent solution would address. Identify when they are applying attribute recognition, comparisons, and analysis in all situations, whether or not numbers are utilized. Identify when they are making decisions based upon probabilities. Use simple mathematical tools and concepts such as a number line to perform everyday tasks such as measuring or dividing quantities. Identify patterns in nature. Make change.

Level 2 (ages 12-14) Understand the development of our numeration system in relation to earlier cultures. Express the results of a survey in a variety of numerical and graphical formats and interpret the contexts from which such data could arise. (Calculate the cost of real life expenses. Identify numerical patterns and symmetry in nature, visual art and music. Analyze turnout, patterns of voting, outcomes, money spent prior to the contest, and number of political parties involved in local elections.) Write about relationships between mathematics and real life. Recognize the relationship between timelines and number lines, scale representations and measurements/proportions.

Level 3 (ages 16-18) By using appropriate technology and curve-fitting techniques, determine the best equation to model a wind-chill chart. Graph daily times, rates, and position of competitors in a multi-day race such as the Iditarod, the Americaís Cup, or the Tour de France. List in a given day all instances where their understanding of mathematics has enriched his/her lives or empowered him/her. Recognize when a model can be modified from one context to address a solution in another context. Recognize how mathematics changes to respond to changing societal needs.

MATHEMATICAL CONNECTIONS FOR ALL
Curriculum priorities should address the unifying ideas of mathematics. These ideas tie together all of the disciplines and explain how mathematical tools and reasoning can be utilized in most aspects of life.

BIG IDEAS IN MATHEMATICAL CONNECTIONS

• Quantification
• Patterns
• Representation
• Proportional relationships
• Multiple modeling
• Variation
• Algorithmic thinking
• Mathematical argumentation

Alaska Content Standards in Science

Alaska 2000

Science Content Standard A: Content of Science All Alaska students will understand scientific facts, concepts, principles, and theories.

Key Elements

Students who meet this standard will:

1. Understand the models describing the nature of molecules, atoms and sub-atomic particles, their relation to the structure of atoms and sub-atomic particles, and their relation to the structure and behavior of matter (Structure of Matter);
   
2. Know about the physical, chemical and nuclear changes and interactions that result in observable changes and interactions in the properties of matter (Changes and Interactions of Matter);
   
3. Understand the models describing composition, age and size of our universe, galaxy and solar system. Know that our universe is constantly moving and changing (Universe);
   
4. Understand observable natural events such as tides, weather, seasons and moon phases in terms of the structure and motion of the earth (Earth);
   
5. Understand the strength and effects of forces such as gravity and electromagnetic radiation (Forces of Nature);
   
6. Understand that natural forces cause different types of motion. Describe the relationship of these forces and changes in motion (Motion);
   
7. Understand how the earth changes because of plate tectonics, earthquakes, volcanoes, erosion and deposition, and living things (Processes that Shape the Earth);
   
8. Understand the scientific principles and models that:
   1. describe the nature of physical, chemical and nuclear reactions;
   2. state that whenever energy is reduced in one place, it is increased somewhere else by the same amount;
   3. state that whenever there is a transformation of energy, some is spent in ways that make it unavailable for use (Energy Transformations);

   
   

9. Know about the transfers and transformations of matter and energy that link living things and their physical environment, from molecules to ecosystems (Flow of Matter and Energy);
   
10. Know that living things are made up mostly of cells and that all life processes occur in these basic units (Cells);
    
11. Know that similar features are passed on by genes through reproduction (Heredity)
    
12. Distinguish the patterns of similarity and differences in the living world in order to understand the diversity of life. Understand the theories that describe the importance of diversity for species and ecosystems (Diversity);
    
13. Understand the theory of natural selection as an explanation for evidence of changes in life forms over time (Evolution and Natural Selection);
    
14. Understand the interdependence between living things and their environment. Know that the living environment consists of individuals, populations, and communities. Recognize that a small change in a part of the environment may affect the whole (Interdependence);
    
15. Use science to understand and describe the local environment (Local Knowledge)
    
16. Understand basic concepts about the theory of relativity that changed our view of the universe by uniting matter and energy and linking time with space (Relativity)
    

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Last modified on: Sat, Jun 15, 1996.