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.
ELEMENTARY GRADES Pre-K-2
1. Demonstrate an understanding of what numbers mean (e.g., that the number 7 stands for a group of objects).
2. Understand the many uses of numbers (e.g., prices, recipes, measurement, directions in play).
3. Order, compare, read, group, and apply place value concepts to numbers up to 1,000.
4. Determine reasonableness of results when working with quantities.
EXAMPLES
Show that 6 is larger than 3, using beans in a cup.
Explain different ways to make 263, using hundreds, tens, and ones.
ELEMENTARY GRADES 3-4
1. Read, compare, order, classify, and explain whole numbers up to one million.
2. Read, compare, order, classify, and explain simple fractions through tenths.
3. Demonstrate knowledge of the meaning of decimals and integers and an understanding of how they may be used.
EXAMPLES
Using pattern blocks, represent equivalent fractions, such as 1/3, 2/6, 4/12
Show how three pizzas can be shared equally by four people.
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.
ELEMENTARY GRADES Pre-K-2
1. Use and apply estimation with quantities, measurements, computations, and problem-solving.
2. Use multiple strategies in solving problems involving addition and subtraction of whole numbers.
3. Show understanding of addition and subtraction by using a variety of materials, strategies, and symbols.
EXAMPLES
Estimate reasonably and count accurately the number of seeds in a container.
Solve a problem such as: we have 24 seats and 19 children. How many seats will be empty after all the children are in their chairs?
Show different coin combinations to make 75¢.
ELEMENTARY GRADES 3-4
1. Solve multi-step, real-life problems using the four operations with whole numbers.
2. Solve real-life problems involving addition and subtraction of simple fractions.
3. Demonstrate and explain the problem-solving process using appropriate tools and technology and defend the reasonableness of results.
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.
EXAMPLES
Solve problems such as finding the number of rectangular arrangements for 36 like objects.
Organize a budget for a project.
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.
ELEMENTARY GRADES Pre-K-2
1. Formulate and solve problems by collecting, arranging, and interpreting data.
2. Make tallies and graphs of information gathered from immediate surroundings.
EXAMPLE
Make a graph to represent the number of family members for students in the class and use this graph to determine the amount of refreshments needed for a family-night presentation.
ELEMENTARY GRADES 3-4
1. Make generalizations and draw conclusions using various types of graphs, charts, and tables.
2. Read and interpret displays of data.
EXAMPLE
Predict the number of buttons per student in the room. Collect data on the number of buttons for each student in the room, display the data on a line plot, and analyze it to determine the average number of buttons per student.
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.
ELEMENTARY GRADES Pre-K-2
1. Use concepts of chance and record outcomes of simple events.
EXAMPLE
Investigate the possible and likely outcomes when rolling a number cube.
ELEMENTARY GRADES 3-4
1. Explain the concept of chance in predicting outcomes.
2. Estimate probability from a sample of observed outcomes and simulations.
EXAMPLE
Determine the combinations that could occur when four coins are tossed, and predict which of the combinations would occur most often. Conduct an experiment to show how close the predictions are to actual events.
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.
ELEMENTARY GRADES Pre-K-2
1. Describe, model, and classify 2D shapes and selected 3D figures.
2. Investigate and predict the results of combining, dividing, and changing 2D shapes.
3. Use positional words to describe the relationship of two or more objects (e.g., over, under, beside, to the left).
EXAMPLES
Find all the shapes you can make with five squares of the same size if the sides touch completely.
Create symmetrical designs with pattern blocks and indicate lines of symmetry.
ELEMENTARY GRADES 3-4
1. Describe, model, and classify shapes and figures using applicable properties.
2. Experiment with shapes and figures to make generalizations regarding congruency, symmetry, and similarity.
3. Use transformations such as slides, flips, and rotations.
4. Use the properties of shapes and figures to describe the physical world.
EXAMPLES
Design a nine patch quilt in which each patch is a nine-inch square. Squares, triangles, rectangles, and/or parallelograms can be used for the design of each square.
Create symmetrical designs with pattern blocks and indicate the lines of symmetry.
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.
ELEMENTARY GRADES Pre-K-2
1. Estimate and measure length, time, temperature, weight, and capacity.
2. Identify and give the value of different coins.
3. Select standard and non-standard tools for determining length, time, temperature, weight, and capacity, and use them to solve everyday problems.
EXAMPLE
Put five objects, such as books, rocks, or pumpkins, in rank order by weight, first by estimating and then by measuring exactly.
ELEMENTARY GRADES 3-4
1. Solve and justify solutions to real-life problems involving the measurement of time, length, area, perimeter, weight, temperature, mass, capacity, and volume.
2. Select measuring tools and units of measurement that are appropriate for what is being measured.
EXAMPLES
Find all of the rectangular areas, measured in whole square inches, for a particular perimeter, such as 24 inches, and see if the same pattern of shapes holds for another perimeter.
Determine what time it is necessary to leave the house for the fifteen minute walk to the bus stop and 1-1/4 hour bus ride to the 7:30 concert. Explain the answer.
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.
ELEMENTARY GRADES Pre-K-2
1. Recognize, describe, extend, copy, and create a wide variety of patterns.
2. Explore the use of variables and open sentences to describe relationships.
3. Represent and describe both geometric and numeric relationships.
EXAMPLE
Recognize and describe the inverse relationship between addition and subtraction, e.g. 3+__=7, __+3=7, 7-3=__, and 7-__=3.
Show how the letters "aab, aab, ...," can represent the pattern "metal, metal, plastic, ...,", "leaf, leaf, rock, ...," or many other patterns.
ELEMENTARY GRADES 3-4
1. Use the patterns of numbers, geometry, and a variety of graphs to solve a problem.
2. Use variables and open sentences to express relationships.
EXAMPLES
Find, make, and describe linear patterns on the 99-chart, for example 4, 14, 24, 34.
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.
ELEMENTARY GRADES Pre-K-2
1. Make drawings for problem situations and mathematical expressions in which there is an unknown, using a variety of tools and approaches.
2. Use language and symbols to express numerical and other relationships.
EXAMPLE
Show all the ways to make 10 (e.g., 2 + x = 10, 3 + x = 10, and so forth) by using blocks or other objects to demonstrate the mathematical statements.
ELEMENTARY GRADES 3-4
1. Develop and evaluate simple formulas in problem-solving contexts.
2. Find replacements for variables that make simple number sentences true.
EXAMPLE
Plot points on a coordinate graph according to the convention that (x,y) refers to the intersection of a given vertical line and a given horizontal line.
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?).
ELEMENTARY GRADES Pre-K-2
1. Classify sets of objects into two or more groups using their attributes.
2. Create and use an organized list to determine possible outcomes or solve problems.
EXAMPLES
Sort sets of tiles by color, size, and shape.
Determine the possible arrangements for a triple ice cream cone given three flavors of ice cream.
ELEMENTARY GRADES 3-4
1. Create and use organized lists, tree diagrams, Venn diagrams, and networks.
2. Give examples of infinite and finite solutions.
EXAMPLE
Use a Venn diagram to record the correlation between students who wore a sweater to school and students who walked to school.
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.
ELEMENTARY GRADES Pre-K-2
1. Describe a simple argument's strengths and weaknesses.
2. Distinguish between "important" and "unimportant" mathematical information.
EXAMPLE
Solve a simple word problem by finding and using the relevant information (Jack has 3 red marbles, 2 green marbles, and 4 toy cars. How many marbles does he have? 3+2=5.)
ELEMENTARY GRADES 3-4
1. Demonstrate an understanding that support for a claim should be based on evidence of various types (e.g., from logical processes, from measurement, or from observation and experimentation).
EXAMPLE
Using data from the classroom and the entire school, prepare a report, including graphs, charts, and diagrams, on the optimal number and location of recycling containers.
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.
ELEMENTARY GRADES Pre-K-2
1. Use numerals and symbols (>, <, =, +, -) to report numerical data and relationships.
ELEMENTARY GRADES 3-4
1. Use simple tables and graphs to communicate ideas and information in presentations in a concise and clear manner.