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Preceding Pages
- Looking Ahead
- Part A: Beliefs About Best Practices for Teaching and Learning Mathematics and Science
- Part B: Beyond Beliefs: The Guiding Principles
- Structure for the Discussion of Guiding Principles
- Guiding Principles
- Structure for the Discussion of Guiding Principles
- Organization for Discussion of Each Guiding Principle
- 1. Students Understand the Nature of Mathematics and Science
- 2. Students communicate effectively in mathematics and science.
- 3. Students reason effectively in mathematics and science.
- 4. Students are problem-solvers in mathematics and science.
- 5. Students understand their roles in the natural world.
- 6. Students understand historical and societal implications of mathematics and science.
Current Page
- Guiding Principle #7. Students attain and apply essential knowledge and skills of mathematics and science.
- Instructional Implications
- Content Standards
- Part 1. Mathematics
- A. Students understand and demonstrate number sense.
- B. Students understand and demonstrate computation skills.
- C. Students understand and apply concepts of data analysis.
- D. Students understand and apply concepts of probability.
- E. Students understand and apply concepts from geometry.
- F. Students understand and demonstrate measurement skills.
- G. Students understand that mathematics is the science of patterns, relationships and functions.
- H. Students understand and apply algebraic concepts.
- I. Students understand and apply concepts in discrete mathematics.
Following Pages
- Guiding Principles
- 7. Students attain and apply essential knowledge and skills of mathematics and science.
- Content Standards
- References
This Guiding Principle deals with the essential knowledge and skills of science and mathematics. These knowledge and skills should not be viewed or delivered as isolated facts and skills. Essential knowledge combined with problem-solving, reasoning, communication and technological skills develops the mathematical and scientific power Maine students require.
The Content Standards in this Guiding Principle apply to all students, who must develop these essential skills and gain these important understandings in both mathematics and science to become informed, literate and capable citizens. To achieve these goals, the formal study of mathematics and science is necessary at every level of a child's school career.
Instructional Implications Providing conditions in which students can construct personal mathematical and scientific knowledge requires a look at the teacher's role, the student's role and the classroom environment.
In such conditions, the teacher strives to connect new concepts to the knowledge and experiences students already have acquired so that the new ideas can be assimilated more easily. Learning activities connect to the world of the child, to daily events and to other disciplines as much as possible. The teacher provides everyday activities incorporating problem-solving, reasoning and communicating. Learners are actively involved in learning and doing mathematics and science, using materials such as calculators, measuring devices, computers, manipulatives and tools designed to extend the senses.
The classroom environment is one where students are able to take risks and make mistakes in the pursuit of knowledge. Acquisition and application of knowledge and skills are treated as equally important, and all students enjoy equitable opportunities to learn. The teacher makes informed curricular decisions based on continuous assessment so that the key concepts of a core curriculum are addressed.
Geometry is the study of the spatial world and its symmetries. One-, two-, and three-dimensional figures are examined in terms of the attributes size, shape, location, dimension and perspective as well as the relationship between and within these attributes.
PART 1: MATHEMATICS
==========================================Content Standard A. Students understand and demonstrate number sense.
________________________________________________________Numbers are used to describe and interpret phenomena. Building a sense of number relationship is essential for developing the ability to deal with natural numbers, the real number system and the complex number system.
Number sense involves understanding the meaning of numbers, relationships among numbers, the magnitude of numbers and the effects of operations on numbers. Skilled estimation is also an important component of number sense. The use of hands-on experiences, physical materials, tools such as calculators and computers, and connections to other mathematics helps students develop number sense.
Performance Indicators Primary
1. Demonstrate an understanding of what numbers mean (e.g., that the number 7 stands for a group of objects). (M-A1)
2. Interpret the many uses of numbers (e.g., prices, recipes, measurement, directions in play). (M-A2)
3. Order, compare, read, group, and apply place value concepts to numbers up to 1000 (M-A3)
4. Determine the reasonableness of results when working with quantities. (M-A4)
Intermediate
1. Read, compare, order, classify, and explain whole numbers up to one million. (M-A1)
2. Read, compare, order, classify, and explain simple fractions through tenths. (M-A2)
3. Demonstrate knowledge of the meaning of decimals and integers and an understanding of how they may be used through hands-on experiences, the use of physical materials and connections to other mathematics. (M-A3)
Middle
1. Use numbers in a variety of equivalent and interchangeable forms (integer, fraction, decimal, percent, exponential, and scientific notation) in problem-solving. (M-A1)
2. Demonstrate understanding of the relationships among the basic arithmetic operations on different types of numbers. (M-A2)
3. Apply concepts of ratios, proportions, percents, and number theory (e.g., primes, factors, and multiples) in practical and other mathematical situations. (M-A3)
4. Represent numerical relationships in graphs, tables, and charts. (M-A4)
5. Apply number operations in situations that require rational numbers.
Secondary
1. Describe the structure of the real number system and identify its appropriate applications and limitations. (M-A1)
2. Explain what complex numbers ( real and imaginary) mean and describe some of their many uses through hands-on experiences, the use of physical materials, and connections to other mathematics. (M-A2)
_______________________________________________________
Snapshot Eighth-graders working with partners have been presented with the following problem. The national debt is close to two trillion dollars. Create an image that describes that amount of money in terms anyone can understand. Students brainstorm ways to describe this number, determine the kind of information that would be most useful, and find the information with the help of their teacher, librarian, or computer teacher. Calculators are available. After the presentations of images, the class discusses the implications.
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Content Standard B. Students understand and demonstrate computation skills.
________________________________________________________Proficiency in computational skills is essential to problem-solving and other mathematical activities. Estimating, evaluating reasonableness of answers and obtaining accuracy in calculations are included in this proficiency. Computation can be done mentally, by using manipulatives, paper and pencil, calculators or computers. Understanding relationships in operations allows students greater facility with mental computation and provides context for meaning. Computational skill promotes efficient and confident learners.
Performance Indicators Primary
1. Use and apply estimation with quantities, measurements, computations, and problem-solving. (M-B1)
2. Use multiple strategies in solving problems involving addition and subtraction of whole numbers. (M-B2)
3. Show understanding of addition and subtraction by using a variety of materials, strategies, and symbols. (M-B3)
4. Describe a wide range of situations that can be expressed by one expression.
5. Explore relationships among operations and strategies that give meaning to operations.
Intermediate
1. Solve multi-step, real life problems using the four operations with whole numbers. (M-B1)
2. Solve real life problems involving addition and subtraction of simple fractions. (M-B2)
3. Demonstrate and explain the problem solving process using appropriate tools and technology and defend the reasonableness of results. (M-B3)
4. Develop proficiency with the facts and algorithms of the four operations on whole numbers using mental math and a variety of materials, strategies, and technologies. (M-B4)
Middle
1. Compute and model all four operations with whole numbers, fractions, decimals, sets of num,bers, and percents applying the proper order of operations. (M-B1)
2. Create, solve, and justify the solution for multi-step, real life problems including those with ratio and proportion. (M-B1)
Secondary
1. Use various techniques to approximate solutions, determine the reasonableness of answers, and justify results. (M-B1).
2. Explain operations with number systems other than base ten. (M-B2)
3. Model and explain operations with real numbers using a variety of tools and techniques.
_______________________________________________________Snapshots Second-graders have displayed numbers in the tens and hundreds using the base ten materials while checking with their neighbor to verify both displays. Then they have displayed addition on the base ten grid by putting the original number (first addend) on the grid and then putting the second number (second addend) on the grid below the first.
Through movement of the base ten materials, children push the second group to combine with the first, thus showing an action associated with addition. They read sums from the grids, and do examples individually, checking sums with partners in supportive groups. They add numbers where no regrouping is needed, then move to regrouping when there are suddenly too many tens or ones. The need to transform ten ones to one ten or ten tens to one hundred builds a meaningful basis for paper and pencil algorithm work.
~~~~~ Fifth-graders are exploring relationships in multiplication to develop mental computational skills. Using Cuisenaire RodsTM, they make a train that is twenty-four units long with two 12 trains underneath. After describing the first row as 1 of the 24's or 1 x 24, students decide that the second row shows 2 of the 12's or 2 x 12.
Other rows of trains are constructed that show multiplication with a product of 24 under those already made. The results are listed as multiplication examples.
Collaborative groups talk about the relationship they see by looking at how they can move from one fact to the next when products stay the same. Some see that if one factor is halved, the other factor is doubled. They justify these observations with other facts they know: 4 x 7 = 28 = 2 x 14; 12 x 12 = 144 and so does 6 x 24 = 144.
Many examples of problems that may seem difficult can be made simpler by mentally doubling and halving (e.g., 15 x 12 becomes 30 x 6). Partners generate such hard-easy problems until they understand what is happening and gain greater facility in mental computation.
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Content Standard C. Students understand and apply concepts of data analysis.
________________________________________________________In contemporary life we are faced with massive amounts of information which must be selected, sorted and analyzed to reach conclusions. Sound decision making requires us to collect data effectively, organize data, discover patterns, summarize trends, make inferences, draw conclusions and make predictions. Appropriate computer and calculator technology can help with data analysis and representation. The ethical use of statistical results is a paramount concern in the Information Age.
Performance Indicators Primary
1. Formulate and solve problems by collecting, arranging, and interpreting data. (M-C1)
2. Make tallies and graphs of information gathered from immediate surroundings. (M-C2)
Intermediate
1. Make generalizations and draw conclusions using various types of graphs, charts, and tables. (M-C1)
2. Read and interpret displays of data. (M-C2)
Middle
1. Organize and analyze data using mean, median, mode, and range. (M-C1)
2. Assemble data and use matrices to formulate and solve problems (M-C2)
3. Construct inferences and convincing arguments based on data (M-C3)
4. Use a variety of organizers (graphs, stem and leaf, chart ....) to organize data that they have generated.
Secondary
1. Determine and evaluate the effect of variables on the results of data collection. (M-C1)
2. Construct, model, predict, and draw conclusions from charts, tables and graphs that summarize data from practical situations. (M-C2)
3. Demonstrate an understanding of concepts of standard deviation and correlation and how they relate to data analysis. (M-C3)
4. Demonstrate an understanding of the idea of random sampling and recognition of its role in statistical claims and designs for data collection. (M-C4)
5. Revise studies to improve their validity (e.g., in terms of better sampling, better controls, or better data analysis techniques). (M-C5)
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Snapshots Kindergartners are exploring the woods behind their school by making observations of the plants and animals that live there. In field notebooks, students draw pictures of the organisms they find. Resource books and adults are there to help children label or identify findings for later drawing from a source book. There are quiet times when all sit still and listen, trying to locate the origin of the sounds.
At the end of the woods walk, the class lists their findings on a chart. Pictures are added to the chart before the next meeting. Field books in hand, they add a small square after each item in the plant category and each item in the animal category where they have made an observation. Children look at the resulting graph and make quantitative observations in a sharing session with their peers. They write about what they see in the graph.
~~~~~ A school department is contemplating whether the local political climate might be favorable toward a proposal for building a new school. The old schools have fallen into disrepair, and the school populations are rising. The superintendent asks high school statistics students for their help in conducting a community survey. Their task is to help the superintendent develop survey questions, determine the size of the sample necessary for the survey, gather a stratified random sample, compile and analyze the results of the survey, and make recommendations. Students make a presentation to the school board relating their results, analysis and recommendations. The school board uses the report in considering future plans.
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Content StandardD. Students understand and apply concepts of probability.
________________________________________________________Probability is the study of uncertainty. Understanding the basic principles of probability makes us informed consumers of information. People need to understand the uncertainty and limitations involved in drawing conclusions from a set of data.
Performance Indicators Primary
1. Use concepts of chance and record outcomes of simple events. (M-D1)
Intermediate
1. Explain the concept of chance in predicting outcomes (M-D1)
2. Estimate probability from a sample of observed outcomes and simulations. (M-D2)
Middle
1. Find the probability of simple events and make predictions by applying the theories of probability. (M-D1)
2. Explain the idea that probability can be represented as a fraction between and including zero and one. (M-D2)
3. Use simulations to estimate probabilities. (M-D3)
4. Find all possible combinations and arrangements involving a limited number of variables. (M-D4)
Secondary
1. Find the probability of compound events and make predictions by applying probability theory. (M-D1)
2. Create and interpret probability distributions. (M-D2)
3. Use experimental or theoretical probability to represent and solve problems involving uncertainty.
_______________________________________________________
Snapshot Seventh-graders are involved in an activity where groups need to determine whether a game is fair or not. Groups of four play the paper, rock, scissors game with one serving as recorder (Marilyn Burns, About Teaching Mathematics: A K-8 Resource). All players make a fist and on the count of four, show paper (four fingers), scissors (two fingers), or rock (by showing a fist). There are three ways to score. Player A gets a point if all players show the same sign. Player B gets a point if only two players show the same sign. Player C gets a point if all players show different signs.
Members of the groups decide who will be player A, B, and C, and play the game 20 times. The winning points are tallied. If the game is not fair (which it is not), groups work to make it fair, testing out their game a reasonable number of times.
Every member must understand their group's game and be able to share it with the class. Each group shares their new game and explains why they see this as fair. Possible outcomes of events are studied and each group uses these ideas to revisit their fair game. Adaptations can be made to their games.
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Content Standard E. Students understand and apply concepts from geometry.
________________________________________________________
Primary
1. Describe, model, draw, construct, and classify 2D shapes and selected 3D figures. (M-E1)
2. Investigate and predict results of combining, dividing, and changing shapes using manipulatives. (M-E2)
3. Use positional words to describe the relationship of two or more objects (e.g., over, under, beside, to the left). (M-E2)
Intermediate
1. Describe, model, and classify shapes and figures using applicable properties. (M-E1)
2. Experiment with shapes and figures to make generalizations regarding congruency, symmetry, and similarity. (M-E2)
3. Use transformations such as slides, flips, and rotations. (M-E3)
4. Use the properties of shapes and figures to describe the physical world. (M-E4)
5. Investigate and predict the results of combining, subdividing, and transforming two and three dimensional figures.
6. Examine applications such as perimeter, area, surface area, volume and their interconnections.
Middle
1. Compare, classify, and draw two dimensional shapes and three dimensional figures. (M-E1)
2. Apply geometric properties to represent and solve real life problems involving regular and irregular shapes. (M-E2)
3. Use a coordinate system to define and locate position. (M-E3)
4. Use the appropriate geometric tools and measurements to draw, measure and construct two- and three- dimensional figures (straightedge, compass, protractor, mirror and computer). (M-E4)
5. Reproduce geometric shapes in reduced and enlarged scales and on the coordinate plane.
6. Examine applications such as surface area, volume, capacity, tessellations, golden ratio and the Fibonacci sequence.
Secondary
1. Draw coordinate representations of geometric figures and their transformations ( changes in position and size). (M-E1)
2. Use inductive and deductive reasoning to explore and determine the properties of and relationships among geometric figures. (M-E2)
3. Apply trigonometry to problem situations involving triangles and periodic phenomena. (M-E3)
4. Construct, draw and interpret two- and three-dimensional objects using geometrical tools, manipulatives and computers for problemsolving and simulation.
5. Apply the ideas of transformations to similarity and congruence of figures
6. Use different geometric systems to model different aspects of reality.
_______________________________________________________
Snapshots Sixth-graders are beginning a study of transformations by enlarging, shrinking, stretching and shearing. They imagine a fish that has been created on a 15 cm x 12 cm grid with grid marks every 3 cm. There will be 20 squares, each 3 cm x 3 cm.
Next they imagine that each of these squares has been increased to 5 cm x 5 cm and that the drawn fish has enlarged proportionately. Then they imagine that each square has been decreased to 1 cm x 1 cm and the resulting fish shrinks. Finally, they imagine that the squares in the original grid have been stretched so that they are 5 cm x 3 cm with height remaining the same and length increasing. The teacher asks, "What happens to the fish?"
~~~~ A shear is created when the rectangular grid transforms to a non-rectangular parallelogram in which the height remains the same. Students create drawings on grids that simulate these changes, examine the effects of these transformations and compare these to changes in perspectives or viewing angles.
Learners then investigate the world around them to determine where these perceived effects can be seen. Each student writes a review of findings, with drawings and outside research.
Groups of high
school geometry students are studying reflections using mirrors
and lasers. They place the laser on the floor, turn on the beam,
and place a mirror in its path, being careful not to look directly
at the source of the beam or the mirror. They draw the incoming
and reflected paths of the laser beam with reference to the mirror.
Using protractors, the students measure the angles created by
the incident and reflected beam lines, noting that they are equal.
They repeat the process using a different incident angle, collecting
the same data. The class discusses the implications of their
findings on such objects as pool tables, miniature golf holes,
or satellite communications.
After further investigation, each student designs a miniature
golf hole in which reflections are necessary for a hole in one.
Students must show mathematically that the hole works as designed,
but also discuss the difficulties that would arise when actually
trying to play for a hole in one.
==========================================
F. Students understand
and demonstrate measurement skills.
________________________________________________________
Measurement is valuable as an integrating skill throughout the
curriculum and in everyday life. The use of estimation is vital
in determining the reasonableness of measurement. Measurement
attributes, units and tools provide opportunities to describe
and understand the world around us.
Primary
1. Estimate and measure length, time, temperature, weight, and capacity. (M-F1)
2. Identify and give the value of different coins. (M-F2)
3. Select standard and nonstandard tools for determining length, time, temperature, weight, and capacity, and use them to solve everyday problems. (M-F3)
Intermediate
1. Solve and justify solutions to real life problems involving the measurement of time, length, area, perimeter, weight, temperature, mass, capacity, and volume. (M-F1)
2. Select measuring tools and units of measurement that are appropriate for what is being measured. (M-F2)
3. Determine quantities such as area and volume using standard and non-standard units of measure.
Middle
1. Demonstrate the structure and use of systems of measurements. (M-F1)
2. Develop and use concepts that can be measured directly, or indirectly (e.g., the concept of rate). (M-F2)
3. Demonstrate an understanding of length, area, volume, and the corresponding units, square units, and cubic units of measure. (M-F3)
4. Develop formulas and procedures for using measurements to solve problems.
Secondary
Relationships are central to mathematical understanding. A study of patterns often reveals regularity, indicating the presence of a mathematical relationship. Studying relationships allows students to make generalizations and predictions about phenomena and occurrences in the world around them.
1. Use measurement tools and units appropriately and recognize limitations in the precision of the measurement tools. (M-F1)
2. Derive and use formulas for area, surface area, and volume of many types of figures. (M-F2)
3. Use dimensional analysis to help solve problems.
_______________________________________________________
Snapshot Fifth-graders are studying the Pilgrims' voyage on the Mayflower. To understand the hardships of the journey and its ramifications, they obtain data on the number of travelers and the size of the Mayflower's hold and sleeping quarters. Through scale drawings, three-dimensional models, and actual measurements that are simulated on the playground, they calculate the amount of sleeping space and cargo space per person or family.
After brainstorming how this problem may be solved, the class splits into pairs who work to determine the amount of cargo each family can bring. With that knowledge, students list what families might bring and then justify their decisions. A final report shows their measurement and scale work, their calculations and their justification of solutions.
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Content Standard G. Students understand that mathematics is the science of patterns, relations and functions.
________________________________________________________
Primary
1. Recognize, describe, extend, copy, and create a wide variety of patterns. (M-G1)
2. Explore the use of variables and open sentences to describe relationships. (M-G2)
Intermediate
1. Use the patterns of numbers, geometry, and a variety of graphs to solve a problem. (M-G1)
2. Use variables and open sentences to express relationships. (M-G2)
Middle
1. Describe and represent relationships with tables, graphs and equations. (M-G1)
2. Analyze relationships to explain how a change in one quantity results in a change in another. (M-G2)
3. Use patterns and multiple representations to solve problems. (M-G3)
Secondary
1. Create a graph to represent a real-life situation and draw inferences from it. (M-G1)
2. Translate and solve a real life problem using symbolic language. (M-G2)
3. Model phenomena using a variety of functions (linear, quadratic, exponential, trigonometric etc.). (M-G3)
4. Identify a variety of situations explained by the same type of function.(M-G4)
________________________________________________________
Snapshots Fourth-graders have been exploring the many patterns of a ninety-nine chart. They are asked to put their finger on a number and record it in the first column of a table. They then move one row down and find a new number and record that in the second column of the table.
Students do this with other numbers, following the "one down rule," filling in the table. They are asked to talk with others in their group about what they see, drawing a conclusion that the second number is ten more than the first. The teacher asks, "Suppose the first number is 'YUMMY,' what will the next one be? " and notes the response "YUMMY + 10." The Yummy becomes Y and the relationship is described as the ten plus relationship or Y + 10.
Students find another such relationship on the chart, making a table, finding data and stating the relationship in terms of the unknown. They share it with another member of their group. The relationships are then shared with the class, and other students try to decipher the rules for movement. N - 19 is seen as a move up two rows and to the right one column.
High school
students are studying the effect of the coefficients on the graphs
of polynomial functions. Beginning with the familiar slope-intercept
equation of a line through the origin, they record the effect
changes in the coefficient have on the graph. Using graphing
calculators or graphing software, they continue their investigation
through quadratic, cubic and fourth degree monomials. They analyze
the data for questions such as "Does the coefficient have
the same effect in second, third, and fourth degree monomials?"
Discussing different patterns, the students begin to predict
the effects of fifth or sixth degree monomials in an effort to
generalize effects on any monomial. The investigation is then
expanded to look for patterns using polynomials.
==========================================
H. Students understand
and apply algebraic concepts.
________________________________________________________
Algebra and analytic thinking are fundamental tools for working
in and thinking about mathematics. These tools provide ways to
generalize and predict problem solutions when all information
is not known. Taught within the context of mathematical and practical
applications, the concept of functions must be a unifying theme
for algebraic concepts.
Primary
1. Make drawings for problem situations and mathematical expressions in which there is an unknown, using a variety of tools and approaches. (M-H1)
2. Use language and symbols to express numerical and other relationships. (M-H2)
Intermediate
1. Develop and evaluate simple formulas in problem-solving contexts. (M-H1)
2. Find replacements for variables that make simple number sentences true. (M-H2)
Middle
1. Use the concepts of variable, expressions and equation. (M-H1)
2. Solve linear equations using concrete, informal and formal methods which apply the order of operations. (M-H2)
3. Analyze tables and graphs to identify properties and relationships in a practical context. (M-H3)
4. Use graphs to represent two-variable equations. (M-H4)
5. Demonstrate an understanding of inequalities and non-linear equations. (M-H5)
6. Find solutions for unknown quantities in linear equations and in simple equations and inequalities. (M-H6)
Secondary
Discrete mathematics studies sets of discrete, separate objects (such as the set of integers), rather than sets of non-discrete objects (such as the set of real numbers). This study includes such topics as counting techniques, sets, relations, functions, logic and reasoning, patterns (iteration and recursion), algorithms, induction and mathematical modeling. Probability, networks, graph theory, social decision making and matrices are also included. Three main themes of discrete mathematics are existence (Is there a solution?), counting (How many solutions are there?) and efficiency (What is the best solution?).
1. Use tables, graphs and spreadsheets to interpret expressions, equations and inequalities. (M-H1)
2. Investigate concepts of variation using equations, graphs and data collection. (M-H2)
3. Formulate and solve equations and inequalities. (M-H3)
4. Analyze and explain situations using symbolic representations (M-H4)
5. Use computer and calculator technology to model situations effectively.
6. Investigate, analyze and model situations using multiple solutions, discussing the usefulness of each solution.
_______________________________________________________
Snapshots Third-graders are measuring the perimeters of rectangular areas inside and outside the classroom. They are compiling all their data and are gathering together to discuss the meaning of their findings. They see that if they know the length and width of the rectangle, they can easily find the perimeter. Some add each dimension twice and others double each dimension and add them together.
The teacher challenges pairs of children to find the length or width of rectangles, given the other dimension and the perimeter, and to describe in writing how they get their answer. They need to verify that their answers are correct. They can draw pictures to help explain their work.
In class discussion, the teacher introduces the notion of a variable when appropriate. One student says, "You take the perimeter, 42 cm, and subtract 2 of the lengths, 24 cm (2 x 12 = 24) and get 18. So two of the widths must be 18. So I think what plus what equals 18." (This can be seen as width plus width = 18 or w + w = 18 or 2 of the widths = 18 or 2 x w = 18.)
~~~~~ Second-year algebra students are investigating the concepts of inverse and direct variation. Using the familiar density-mass-volume relationship, they systematically examine the relationship between two of the three variables. For example, keeping volume constant, they gather and analyze data about density and mass. Working with a partner, students mass metal strips of constant volume and calculate the density of each metal.
The class collects each groups' data and uses graphing calculators to analyze the relationship between density and mass. Looking at the graph, they find an increasing linear relationship indicating direct variation.
Examining the density equation, the students write how they can tell if two variables are directly related. Then they write the type of relationship they would expect to find between density and volume. The process is repeated, and students check their predictions and understanding of variation from the results.
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Content Standard I. Students understand and apply concepts of discrete math.
________________________________________________________
Primary
1. Classify sets of objects into two or more groups using their attributes. (M-I1)
2. Create and use an organized list to determine possible outcomes or solve problems. (M-I2)
Intermediate
1. Create and use organized lists, tree diagrams, Venn diagrams, and networks to generate possible outcomes and to solve problems. (M-I1)
2. Gives examples of infinite and finite solutions. (M-I2)
Middle
1. Create and use networks to explain practical situations or solve problems. (M-I1)
2. Identify patterns in the world and express these patterns with rules. (M-I2)
3. Develop and use counting principles of combinations and permutations.
Secondary
1. Use networks to find solutions to problems (M-I1)
2. Use linear programming to find optimal solutions to a system. (M-I2)
3. Apply strategies from game theory to social problem-solving situations. (M-I3)
4. Use matrices as a tool to interpret and solve problems. (M-I4)
_______________________________________________________
Snapshots With a large blue yarn circle on the floor simulating the bounds of a category (set), second-graders are putting themselves into self-descriptive categories, moving in and out if they belong in the category. Sample categories have been "wearing sneakers," "has pets," and "enjoys walking along the beach."
A red yarn circle has been introduced to overlap the first by about a third; there are now two interlocking circles. Children will now play the same game to enter one circle, both circles, or no circle to show how they fit into given categories. The blue yarn circle is "has a dog" and the red yard circle is "has a cat". Children move into the circles or stay on the outside of the circles (has neither cat nor dog). They discuss what each person's position means.
Eventually a third yard ring is added and the children move into the particular space describing the categories to which they belong. They discuss their positions and the meaning of placement within sets. Subsequent work involves categories of geometry shapes and numbers.
~~~~~ Students in a high school mathematics class are studying directed graphs. They begin by examining the Königsberg Bridge Problem and progress through a series of Euler and Hamiltonian graphs and circuits and the Traveling Salesman Problem. The students are working in supportive groups, checking with a partner when necessary. The class discusses the distinguishing characteristics of directed graphs and circuits. The students then research a real application of these graphs and circuits and make a short presentation to the class.