A. NUMBERS AND NUMBER SENSE
Students will understand and demonstrate a sense of what numbers mean and how they are used. Numbers are used to describe and interpret phenomena. Building a sense of number relationships is essential for developing the ability to deal with any set of numbers. Number sense involves understanding the meaning of numbers, relationships among numbers, relative number magnitudes, and the effects of operations on numbers. Skilled estimation is also an important component of number sense.
MIDDLE GRADES 5-8
1. Use numbers in a variety of equivalent and interchangeable forms (e.g., integer, fraction, decimal, percent, exponential, and scientific notation) in problem-solving.
2. Demonstrate understanding of the relationships among the basic arithmetic operations on different types of numbers.
3. Apply concepts of ratios, proportions, percents, and number theory (e.g., primes, factors, and multiples) in practical and other mathematical situations.
4. Represent numerical relationships in graphs, tables, and charts.
EXAMPLES
Use integers to write a play-by-play description of a game (e.g., football, soccer, or golf).
Respond to the following in a journal: How can you get a smaller answer when you multiply? How can you get a larger answer when you divide?
Estimate a 15% tip for a meal costing $38.60.
B. COMPUTATION
Students will understand and demonstrate computation skills. Understanding the fundamental operations of addition, subtraction, multiplication, and division is central to knowing mathematics. 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. Understanding relationships in operations allows students greater facility with mental computation. Computational skill promotes efficient and confident learners.
MIDDLE GRADES 5-8
1. Compute and model all four operations with whole numbers, fractions, decimals, sets of numbers, and percents, applying the proper order of operations.
2. Create, solve, and justify the solution for multi-step, real-life problems including those with ratio and proportion.
EXAMPLES
Show that there must have been at least one misprint in a newspaper report on an election that reads:
- Yes votes 13,657 (42%)
- No votes 186,491 (58%)
and suggest two specific places a misprint may have occurred.
On a number line, name the point located midway between 1/4 and 6.
C. DATA ANALYSIS AND STATISTICS
Students will understand and apply concepts of data analysis. We are faced with massive quantities of information which must be selected, sorted, and analyzed to reach conclusions. Sound decision making requires the ability to collect data effectively, organize data, discover patterns, summarize trends, make inferences, draw conclusions, and make predictions. The ethical use of statistics is a paramount concern in the Information Age.
MIDDLE GRADES 5-8
1. Organize and analyze data using mean, median, mode, and range.
2. Assemble data and use matrices to formulate and solve problems.
3. Construct inferences and convincing arguments based on data.
EXAMPLES
Conduct an experiment to determine the effects of fertilizer on plant growth, recording and analyzing information on charts and graphs.
Using the height of students in the room, calculate the mean, median, mode, and range.
D. PROBABILITY
Students will understand and apply concepts of probability. Probability is the study of uncertainty. Informed consumers of information understand the basic principles of probability. People need to understand the uncertainties and limitations involved when drawing conclusions from data.
MIDDLE GRADES 5-8
1. Find the probability of simple events and make predictions by applying the theories of probability.
2. Explain the idea that probability can be represented as a fraction between and including zero and one.
3. Use simulations to estimate probabilities.
4. Find all possible combinations and arrangements involving a limited number of variables.
EXAMPLES
Develop and analyze games of chance for a school carnival.
Determine how many license plates are possible if the first two symbols are letters and the last four are numbers.
E. GEOMETRY
Students will understand and apply concepts from geometry. Geometry is the study of the spatial world and its symmetries. The ideas of geometry are used to describe, interpret, represent, and change the spatial world in which we live. The understanding and development of spatial and visual skills strengthens problem-solving abilities.
MIDDLE GRADES 5-8
1. Compare, classify, and draw two dimensional shapes and three dimensional figures.
2. Apply geometric properties to represent and solve real-life problems involving regular and irregular shapes.
3. Use a coordinate system to define and locate position.
4. Use the appropriate geometric tools and measurements to draw and construct two and three dimensional figures.
EXAMPLES
Collect magazine pictures of different styles of architecture and identify all the geometric figures and relationships seen in each building.
Display data with an accurately drawn and divided pie chart.
F. MEASUREMENT
Students will 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 (e.g., length, volume, minutes), units, and tools enhance the ability to describe and understand the world.
MIDDLE GRADES 5-8
1. Demonstrate the structure and use of systems of measurement.
2. Develop and use concepts that can be measured directly, or indirectly (e.g., the concept of rate).
3. Demonstrate an understanding of length, area, volume, and the corresponding units, square units, and cubic units of measure.
EXAMPLES
Calculate the rate of speed of a moving object after measuring the distance traveled and the elapsed time.
Examine areas that can be enclosed using 24 feet of fencing and figuring out the maximum area.
Calculate the volume and surface areas of cones and pyramids.
G. PATTERNS, RELATIONSHIPS, FUNCTIONS
Students will understand that mathematics is the science of patterns, relationships, and functions. 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.
MIDDLE GRADES 5-8
1. Describe and represent relationships with tables, graphs, and equations.
2. Analyze relationships to explain how a change in one quantity can result in a change in another.
3. Use patterns and multiple representations to solve problems.
EXAMPLES
Collect data on the cost of first class postage stamps for a one hundred year period of time and predict future costs for such stamps.
Determine the units digit (one's place) of (3)78.
H. ALGEBRA CONCEPTS
Students will 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 not all information is known. Taught within the context of mathematical and practical applications, the concept of functions is a unifying theme for algebraic concepts.
MIDDLE GRADES 5-8
1. Use the concepts of variables and expressions.
2. Solve linear equations using concrete, informal, and formal methods which apply the order of operations.
3. Analyze tables and graphs to identify properties and relationships in a practical context.
4. Use graphs to represent two-variable equations.
5. Demonstrate an understanding of inequalities and non-linear equations.
6. Find solutions for unknown quantities in linear equations and in simple equations and inequalities.
EXAMPLES
Study the steepness of wheelchair ramps and stairs.
Solve for x: 3x - 5 = 23 - x.
I. DISCRETE MATHEMATICS
Students will understand and apply concepts in discrete mathematics. Discrete mathematics studies discrete processes (e.g., all possible bus routes in a school district). This study includes the exploration of diagrams, networks, and flowcharts that students construct to model situations or use for planning, scheduling, and decision making. Three main concerns of discrete mathematics are: existence (Is there a solution?), counting (How many solutions are there?), and efficiency (What is the best solution?).
MIDDLE GRADES 5-8
1. Create and use networks to explain practical situations or solve problems.
2. Identify patterns in the world and express these patterns with rules.
EXAMPLE
Use graphs and matrices to determine delivery routes from Augusta to other major cities in Maine with a combination of one way and round-trip routes.
J. MATHEMATICAL REASONING
Students will understand and apply concepts of mathematical reasoning. Reasoning is fundamental to the knowing and doing of mathematics. To give more students access to mathematics as a powerful way of making sense of the world, it is essential that an emphasis on reasoning pervade all mathematics. Students need a great deal of time and many experiences to develop their ability to construct valid arguments in problem settings and to evaluate the arguments of others.
MIDDLE GRADES 5-8
1. Support reasoning by using models, known facts, properties, and relationships.
2. Demonstrate that multiple paths to a conclusion may exist.
EXAMPLE
Prepare proposals for a fixed-height bridge and a drawbridge. Make recommendations after considering total cost, steepness of incline, traffic patterns, time of construction, etc.
K. MATHEMATICAL COMMUNICATION
Students will reflect upon and clarify their understanding of mathematical ideas and relationships. Communication plays a key role in helping make important connections among physical, pictorial, graphic, symbolic, verbal, and mental representations of mathematical ideas. Providing individual and collaborative opportunities for discussions about issues, people, and the cultural implications of mathematics reinforce student understanding of the connection between mathematics and our society.
MIDDLE GRADES 5-8
1. Translate relationships into algebraic notation.
2. Use statistics, tables, and graphs to communicate ideas and information in convincing presentations and analyze presentations of others for bias or deceptive presentation.