-Patterns-

Students recognize similarities and generalize patterns, use patterns to create models and make predictions; describe the nature of patterns and relationships; and construct representations of mathematical relationships.

The learner will be able to:

• skip count by 2's, 3's, 4's, 5's-12's.

• describe the patterns and relationships between addition, subtraction, multiplication, and division.

• identify the rule for a pattern.

• use patterns to make predictions and provide reasonable explanations.

• describe the patterns in a 100's chart.

• explore more advanced patterns.

• develop and use the strategy "look for a pattern".

• use the constant function on a calculator to develop the concept of multiple.

Students describe the relationships among variables, predict what will happen to one variable as another variable is changed, analyze natural variation and sources of variability, and compare patterns of change.

The learner will be able to:

• identify, analyze, and describe change using a table to record and then identify the pattern when playing "What's my rule?" or using an input/output machine.
• explore how the element of chance makes any set of data subject to variation.
• plot coordinate graphs.
• continue a growing pattern.
• use knowledge of variability and change to make and defend their conjectures and predictions and to solve problems-such as explaining a prediction for the next term.

Students develop spatial sense, use shape as an analytic and descriptive tool, identify characteristics and define shapes, identify properties and describe relationships among shapes.

The learner will be able to:

• create models of 2 and 3 dimensional shapes using clay, straws, or paperclips and explore their uses.
• understand the following terms: plane, solid, lines and angles, linear, line, ray, segment, parallel, and perpendicular.
• use specific language to describe shapes.  Ex. The opposite sides of a rectangle are parallel, 6 faces of a cube are congruent.
• explore attributes of quadrilaterals (square, rectangle, parallelogram, rhombus, trapezoid).
• classify shapes by an attribute such as:  has corners, one curved edge, no curved edge.
• compare two shapes on a geoboard by describing length.
• identify congruent shapes.
• explore attributes of triangles.
• construct a shapes on the geoboard and then record the shape on dot paper.
• use 4", 6", 8" straws to construct as many triangles as possible and them record results in a table.
• use tangram pieces to explore sliding, flipping, and turning.
• explore rotations with tangrams.
• continue to discover symmetrical properties by folding and reflective devices.
• draw, trace, and use models to illustrate concepts of parallel and perpendicular lines.
• cut boxes to show what 2-dimensional shapes come together to form the original 3-D shape.

-Position-

Students identify location relative to other objects, and describe the effects of transformations (e.g., sliding, flipping, turning, enlarging, reducing) on an object.

The learner will be able to:

• give positions of pegs on the geoboard as inside or outside of the geoboard.
• create patterns or designs on coordinate girds using number pairs.
• decide which letters of the alphabet have lines or rotational symmetry.
• find a specific point on a map using ordered pairs.
• locate cities on a map by directional base: N, S, E, W.
• find and describe objects that reequidistant to, parallel to, a perpendicular to each other.
• explore what happens when an object is enlarged or reduced.
• list steps necessary to ravel from the classroom to....

Students compare attributes of two objects, or of one object with a standard (unit), and analyze situations to determine what measurement (s) should be made and to what level of precision.

The learner will be able to:

• discover why there is a need for a standard unit of measurement.
• decide the best way to make change for money.
• use bus, airline, train schedules, TV, and movie schedules to explore the idea of elapsed time.
• cover a grid with a tracing of their foot and find out how many units are inside/outside the tracing.
• model a scale drawing of a room at home or school.
• explore enlarging or reducing pictures using a square grid.
• read labels to determine volume and weight.
• simulate purchases to determine change due.
• analyze date and make decisions.

Students collect and explore data, organize data into a useful form and develop skill in presenting and reading data displayed in different forms.

The learner will be able to:

• raise questions related to their interests and activities which can be answered by collecting, organizing, and presenting data.
• will organize and present data using different formats such as: tallies, tables, picture graphs, bar graphs, circle graphs, line graphs, coordinate graphing, tree diagrams.
• find examples of data presentation (e.g., newspaper, baseball card).
• identify what data need to be collected to answer a question or solve a problem, and suggest strategies for collecting and presenting their data.
• use current data-analysis activities to surface new questions and explorations.

Students examine data and describe characteristics of the distribution; and they relate data to the situation from which they arose and use data to answer questions convincingly and persuasively.

The learner will be able to:

• describe and explain data representations. (a bar graph from the newspaper, Top 5 from TFK, Science and Social Studies Texts).
• provide an appropriate title for a graph.
• write a summary about the results of a survey.
• use data to defend conclusions and to convince others.
• collect and discuss data displays from print materials, such as newspapers, magazines, and identifying the source of the data.
• discuss what data represents.
• different representations of the same set of data can communicate different information about data, such as comparing the ways different groups of students displayed the same set of data.
• compare and contrast different sets of data.
• consider how different representations of the same set of data can be used to communicate different information about data.

Students draw defensible inferences about unknown outcomes; make predictions and identify the degree of confidence they have in their prediction.

The learner will be able to:

• explore how the element of chance makes any set of data subject to variation.
• explore the concept of randomness such as drawing students' names at random and collecting data to see how often each student's name is drawn.
• determine whether events are likely, unlikely, or equally likely to occur based on the data collected.
• make and justify predictions made from analyzing data.
• solve data-analysis problems using an investigative approach which encourages: 1) raising questions and brainstorming, 2) understanding the problem, 3) gathering and exploring data, 4) describing, interpreting and analyzing data making inferences and predictions, 5) making and implementing decisions reflecting back.

Students experience counting and measuring activities to develop intuitive sense about numbers; develop understanding about properties of numbers; understand the need for and the existence of different sets of numbers; and investigate properties of special numbers.

The learner will be able to:

• investigate the base-ten numeration system using tens frames and hundreds charts to recognize the quantity of numbers, and the calculator to discover number patterns using the constant feature.
• develop place-value concepts by using base-ten blocks to show trading for regrouping purposes and representing a quantity using a place-value holder.
• apply their understanding to solve problems such as: 1) talking about and recognizing uses of numbers in the environment by reading bottles, boxes, clocks, license plates, meter sticks, money values, phone numbers, road signs, rulers, scales, watches. 2) working with problems like "If I have 2 bags of candy bars with 14 candy bars in each, will I have enough fro everyone in our class?  Too many?  How many is not enough?  How many am I short?

Students recognize that numbers are used in different ways like counting, measuring, estimating and ordering; understand and produce multiple representations of a number; and translate among equivalent representation.

The learner will be able to:

• represent whole numbers, fractions, and decimals in concrete, pictorial forms by using the ten frame or hundreds chart to display a given quantity.
• use fraction pie charts to shade in a given quantity and show relationships between fraction families.
• use decimal mats or graph paper to shade in a given quantity.
• demonstrate how many ways they can represent 35 by using base-ten blocks, counters, money values, and answering, "How do you know you have found all possible combinations?"
• demonstrate how fractions can be equivalent (e.g. 1/2 = 3/6) using concrete materials.
• use strategies for estimation like front end, rounding, and compatible numbers and then evaluating the reasonableness of the answer.
• mentally estimate sums and the corresponding differences and then decide if the calculation is an overestimate or underestimate.
• select the appropriate information from a word problem and solve it.

Students investigate relationships such as equality, inequality, inverses, factors and multiple; and represent and compare very large and very small numbers.

The learner will be able to:

• represent, compare, and order whole numbers to 10,000 using words and symbols (e.g. "equals" (=), "less than" (<) or "greater than" (>).
• add and subtract using the decimal notation and symbols.
• find and recognize simple fractions.
• develop strategies to classify numbers as even or odd.
• explore concepts of factors and multiples.
• use repeated addition, count by multiples, and make arrays to demonstrate and understanding of factors and multiples.
• know multiplication facts to 9 x 9.

Students understand and use various types of operations (e.g., addition, subtraction, multiplication, and division) to solve problems.

The learner will be able to:

• model operations with concrete objects, connecting the manipulative model to a symbolic/recorded action.
• relate models to standard expressions and algorithms.
• explain their reasoning for the selection of method and thinking.
• know which operation to perform in a given situation.

Students analyze problems to determine an appropriate process for solution and they use algebraic notations to model or represent problems.

The learner will be able to:

• model different meanings/uses for variables.
• create pictorial representations.
• construct tables of input and output.
• use a balance scale to write algebraic sentences requiring equivalent weights like: m large paper clips weight the same as y small paper clips.
• create and solve simple open sentences (e.g., number sentences with missing numbers or operational signs).

 Students develop and understanding of the notion of certainty and of probability as a measure of the degree of likelihood that can be assigned to a given event based on the knowledge available; and, they make critical judgments about claims that are made in probabilistic situations.

The learner will be able to:

• explore and discuss everyday experiences (e.g., weather, sports, and games) which involve chance.
• explore ways to model the probability of an event occurring.
• list the possible outcomes of a simple event (e.g., tossing a coin, pulling colored marbles out of a bag) and predict whether outcomes are certain, likely, unlikely, or impossible.
• explore counting problems and experiments which involve recording outcomes of events.
• develop strategies for recording outcomes of events - such as pictures and tree diagrams.
• conduct simple probability experiments where they can discuss possibilities, make predictions, experiment, and then compare results with the expected outcome.
• use two-spinner activities to introduce combining outcomes.
• use an investigative approach to probability which engages them in: 1) recording and studying possible outcomes, 2) examining results to see if they make sense, 3) searching for reasonable explanations for outcomes, 4) exploring probability as ratio or fraction, 5) exploring how conditions affect the outcome.
• experiment with methods which generate random outcomes in order to develop a feel for randomness.
• look at a spinner or toss a coin, and tell whether the situation seems fair or unfair, whether outcomes are equally likely or whether outcomes should, or should, not, occur an equal number of times.
• discuss situations where results do not "come our right" and explore how the element of chance makes any set of data subject to variation.
• make spinners that model certain probabilities.
• investigate the likelihood of a event and use data as the basis for making probability statements - such as predicting what types of books children most frequently check out of the library.
• use results of probability experiments to predict future events.

Students investigate practical situations such as scheduling routing sequencing, networking organizing and classifying; and analyze ideas like recurrence relations, induction and algorithm design.

The learner will be able to:

• explore a variety of problems which involve counting and arranging objects.
• explore simple permutation, where the order of objects is important, and combinations, where order is not important.
• use diagrams (arrows, Venn diagrams) to represent relationships.
• explore situations (e.g., networks, relationships, routes, circuits) which can be modeled using vertices connected by edges.
• connect vertex-edge graphs to familiar experiences such as planning trips or shortest paths, planning bus routes.
• explore patterns activities which repeat a procedure over and over to develop a sequence.
• explore pattern activities which involve trying to describe what comes next by looking at previous steps.
• sequence events.
• reassemble short stories/nursery rhymes/cartoons that have been cut apart.
• develop and use a variety of strategies and approaches to solve problems and explain the approaches that others use.
• look for multiple solutions to a problem.
• discuss whether there is a best solution.
• justify their thinking as a way to help clarify their reasoning such as asking questions like:  Why?  How do you know?  What makes you think that?
• coloring maps/drawings with fewest colors so regions sharing boundaries do not use the same color (minimize conflicts).