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Mathematics Curriculum Framework
Achieving Mathematical Power  January 1996
Geometry and measurement help us represent in an orderly fashion what we see in our world. Whether we are cooking or cartooning, shopping or shipping, painting a canvas or a wall, designing space craft for NASA or for preschool, we continually bump up against these mathematical organizers. Lifelong learners should know and understand these interconnected and symbiotic mathematical domains.
Before students begin school, they already have developed a knowledge of and a curiosity about the physical and spatial world around them. They are exploring size, shape, and position when they buy new sneakers, or select fruit at the farmer's market, or play Simon Says. Teachers can use this natural curiosity and realworld experience to begin helping students build their own mathematical foundation in geometry and measurement. Geometry and measurement experiences that are presented in problemsolving situations and incorporated into number patterns and other mathematical ideas increase their understanding of mathematics in realworld contexts.
In the early grades, informal and intuitive studies of geometry and measurement are appropriate for tapping students' natural curiosity and engaging them. Students usually measure objects with nonstandard units, such as shoes or hands, or strides. These informal, often gross motor, explorations of spatial sense, size, and shape make young learners' mathematical experiences personal and fun. Through them, students discover for themselves the necessity of standard units.
As students explore concepts and ideas with the aid of manipulatives such as blocks, geoboards, balance scales, and measuring tools they develop a personal and sensory understanding of mathematical ideas.
Continuing the study of geometry and measurement in the middle grades allows students to see for themselves the dynamic role that mathematics plays in the environment. As students use two and threedimensional models to investigate patterns and to develop spatial skills, they uncover some of the practical uses of mathematics. Ways in which mathematics is used in a wide variety of careers become believable.
Geometry and measurement often spark a renewed interest in mathematics for those students who have been turned off for some reason, or heretofore have felt unsuccessful with mathematics learning. Investigating problems that involve geometry and measurement broadens all students' mathematical understanding and engages them as they explore mathematical ideas.
As the study of geometry and measurement becomes more formal in high school, students become better equipped for mathematical argumentation. They should have opportunities to develop and defend their conclusions in verbal debates and written essays with emphasis on traditional methods of proof. Instead of emphasizing only twocolumn proofs, students should also be given opportunities to justify their own conclusions with less formal, but nonetheless convincing, arguments. Students' spatial reasoning and visualization skills should be enhanced. The study of geometry should make full use of all available technology.
Handson, interactive investigations, using nonstandard and standard units, help adult basic education students develop an understanding of the many measurable attributes of physical objects. Measurement sense including length, time, temperature, capacity, weight, mass, area, volume, and angle will benefit from this approach. This realistic approach helps build an accessible measurement vocabulary and a meaningful comprehension of what it means to measure.
PreK4 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 describe, model, draw, and classify shapes.
 investigate and predict the results of combining, subdividing, and changing shapes.
 develop spatial sense.
 use geometric ideas to develop numerical ideas.
Examples of Student Learning
 Place a shape inside a covered box. Pass it around. Have each student feel, then describe some of the characteristics of the shape. Record. When everyone has contributed, name the shape. Review the characteristics and see which characteristics always meet the criteria for the shape.
 Students design quilt blocks by drawing contiguous geometric shapes on graph paper and shading or coloring them. As an extension, they transfer their design to cotton fabric and assemble a quilt block, which can be finished as a pot holder, or combined with other students' blocks to form a geometric wall hanging for the classroom. A field trip to the New England Quilt Museum in Lowell enhances the historic aspect of the activity.
 Students work in groups with the traditional seven piece tangram, trying to see whether they can make each of the tangram shapes (triangle, square, parallelogram) using 1, 2, 3,...7 pieces. They record their findings, using drawings and charts, and give a presentation to the class that includes their reasoning to justify their findings concerning shapes that may be impossible to form. This activity can be extended to include work with fractions and ratios by assigning unit values to different shapes. Students can make their own tangram puzzles. Connections with literature and other cultures can be made by exploring the historical background of the tangram puzzle.
PreK4 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 demonstrate the attributes of length, capacity, weight, area, volume, time, temperature, and angle.
 use the process of measuring and the concepts related to units of measurement.
 make and use estimates of measurement.
Examples of Student Learning
 Working in small groups, students make boxes from pieces of paper and explore relationships. They have available to them a myriad of materials and manipulatives, including grid paper for graphing and cutting, cubes or beans or rice or candy bits, graduated cylinders, tape, and scissors. Students try to find the box with the largest volume, and the smallest volume. Is this the same as the box that will hold the most or least number of cubes, beans, rice or candy? As they work with the materials, students explore different relationships as they convince others that their box holds the most.
Students in a second grade class explore 3dimensional figures made from wood by the grandfather of one of the students. First they observe and feel the differences, then verbally compare the objects in cooperative groups. Students are asked to construct shapes in any way they wish. The shapes to be constructed are cubes, cylinders, triangular prisms, rectangular prisms, and spheres. Students become creative in solving the problem of how to construct a sphere. Some students use a partially inflated balloon, others make wads of aluminum foil. Still others overlap and connect metal rings for holding looseleaf paper.
As a culminating activity, the many shapes are labeled and hung together to make mobiles, which are hung in the hallway for everyone to see.
Grades 58 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 identify, describe, compare, and classify geometric figures.
 explore and describe the properties of points, lines, and planes.
 visualize and draw geometric figures.
 explore and describe transformations of geometric figures.
 represent and solve problems, using geometric models.
 apply geometric properties and relationships.
 develop and explain the concept of [[pi]].
 develop and explain the concept of the Pythagorean theorem.
Examples of Student Learning
 Students build shapes with toothpicks and 1 cm clay balls. They make geopanes by hanging the shapes on thread and dipping them into a mixture of water and soap. Beforehand, students predict what they think will happen and describe what does happen. This illustration of surface tension encourages discovery of the relationship of vertices, edges, and faces of 3dimensional figures and an understanding of Euler's theorem.
 Students build a scale model of their school. They sketch elevations of the school building, estimate and measure the dimensions of the sides, determine the most costeffective materials for construction of the model, and write a presentation describing their model. The model is displayed and the presentation is delivered to school and local communities.
 Students make Möbius strips to model a continuous onesided surface. They write about what they observe and why they believe it is that way.
Grades 58 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 select appropriate units and tools to measure to the degree of accuracy required in a particular situation.
 describe the meaning of perimeter, area, volume, angle measure, capacity, density, weight, and mass.
 develop and describe the concepts of rates and other derived and indirect measurements.
 develop and apply formulas and procedures for determining measures to solve problems.
Examples of Student Learning
 Monuments found in Massachusetts provide the basis for exploration of indirect measurement and geometric constructions. At home, students peruse collections of post cards and photographs to find those with shadows cast by tourists and monuments. At school, groups of students cull the photographs and use them to derive the heights of different monuments. Students use measurement to construct clay and sand replicas of pyramids or obelisks are constructed and measured. Formulas are used to derive volume. A connection with art is reinforced with a trip to the Daniel Chester French house museum called Chesterwood in Great Barrington. It is a tribute to one of the most notable monument sculptors of our region.
 Students construct containers, explore surface area, and use popcorn to explore volume relationships. They sketch the container they believe to hold the most popcorn. They use surface area to find the most economical container to hold a specific volume.
To explore the constant relationship between the diameter of a circle and its circumference, a class of sixth graders form a straight line in a snowy field, with the teacher in the middle. As the line rotates around the teacher, a circle is formed by footprints in the snow. The class lines up around the circumference and predicts how many times their length will fit around the circle. By marking off start and finish points as the class "diameter" advances along the circumference, students discover that a little more than three diameters fill the circumference. They hypothesize whether this would be different for a larger or smaller class. Later in the art room, they test their ideas by using potato prints and units to measure the diameter and circumference of different size circles.
Grades 910 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 interpret and draw threedimensional objects.
 represent problem situations with geometric models and apply properties of figures.
 classify figures in terms of congruence and similarity and apply these relationships.
 deduce properties of, and relationships between, figures from given assumptions.
 develop and defend conclusions.
 formulate counter examples.
 construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction.
Examples of Student Learning
 Students examine Ascending and Descending by Maurits Escher. They focus on the staircase in the picture and individually in writing, describe relationships between the points, lines, and planes that portray the appearance desired by the artist. Then they make their own geometric "Escher" drawings and describe the geometry they used to make them.
 Students use congruent right triangles cut from a square to prove the Pythagorean theorem both geometrically and algebraically.
 Students generate and solve problems involving the shortest paths for interconnecting three or more points. They use geometric constructions (both compass/straightedge and computer programs) to locate possible points, and find distances and angle measures. The students generalize their results and form conjectures regarding the location of new points in the network that will provide the solution.
Grades 910 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 translate between synthetic and coordinate representations.
 deduce properties of figures, using transformations and coordinates.
 identify congruent and similar figures, using transformations.
 develop and explain geometric interpretations and applications of slope.
Examples of Student Learning
 Students fold a rectangular sheet of construction paper once diagonally and cut it to make a square. They open the square, then fold it into fourths, representing four quadrants. With their paper open and the diagonal fold in the lower left corner, students use colored chalk to draw any graph they wish in the upper two quadrants. They fold over their square, along the horizontal fold, press and rub with their palm. When students open it, they describe what is represented.
 Students investigate slope by developing plans for increasing handicap access to buildings by investigating state and local codes for ramps and problems that occur in their construction.
Students in Ms. Rivers ninth grade class are studying many topics of mathematics combined with other disciplines through the study of tessellations. They explore the work of the Dutch artist M.C. Escher, and historical examples of tilings, such as the Islamic tiling found in the Alhambra.
They begin by exploring polyominoes (using one, two, three, four, five, and, for those who want a challenge, six squares), looking for patterns; learning about reflections, translations and rotations; and deepening their spatial visualization skills.
Students use many different materials and media to arrive at and present their findings. Some students write and use a Logo program, while others do their work on grid paper, and others build models and trace their results. As results are presented, new questions are raised regarding functional relationships, with students trying to find either an explicit or recursive expression for the relationships. They use graphing calculators to try to answer their questions. After the presentations, students play a computer game involving shapes made up of four squares with new understanding of the mathematical connections.
Grades 1112 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 deduce properties of, and relationships between, figures from given assumptions.
 use vectors, phase shift, maxima, minima, inflection points, and precise mathematical descriptions of symmetries to locate and describe objects in their orientation.
Example of Student Learning
 Topology and networks are explored by means of a manipulative designed by the teacher. With the aid of the vocationaltechnical school, two pieces of Plexiglas are anchored with three randomly placed nuts and bolts. When the Plexiglas is dipped in a soap solution, a network is revealed, demonstrating a Steiner point. This manipulative can be used on an overhead or placed atop a map to illustrate networking problems such as those addressed by the airlines.
Grades 1112 Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 apply transformations and coordinates when solving problems
Example of Student Learning
 Students graph separately several functions on grid paper. Using a mirror, they reflect each function, in the line y = x, and sketch their results. Students list some values of x and fill in a table by reading the y coordinates from the graph. They try to derive the relationship between the x and y coordinates of the new graph. They write the new function and compare it with the original. After students have done several pairs of functions, they should see that the inverse of a function can be found by reflecting about the line y = x.
ABE Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 use geometry to describe the physical world.
 apply geometric properties and relationships to concrete situations.
 visualize and represent geometric figures.
 identify, describe, compare, and classify geometric figures.
 relate geometric ideas to number and measurement ideas, including the concepts of perimeter area, volume, angle measure, capacity, weight, and mass.
 explore and describe transformations of geometric figures.
 represent and solve problems, using geometric models.
 use a variety of technologies to study geometry and spatial sense.
Example of Student Learning
 Each student works on every aspect of designing and making a quilt square. They design a pattern, buy the fabric, cut out their pieces, and sew the square. Students explore relationships between shapes and angles, find a range of challenging levels of geometric complexity, and solve the many problems that arise as they transform their abstract design into a concrete product. Finally, the quilt squares are assembled into a lap quilt, which can be donated to a local hospital.
ABE Learning Standards
Students engage in problem solving, communicating, reasoning, and connecting to:
 make and use exact and estimated measurements to describe and compare phenomena.
 select appropriate units and tools to measure to the degree of accuracy required.
 use systems of measurement.
Example of Student Learning
 Students use an assortment of measurement tools, including rulers, thermometers, gas gauges, and speedometers. They learn to read and interpret both labeled and unleveled calibrations for tools used in a variety of workplace settings.
At one Massachusetts community college, a teacher collected measuring cups, pint containers, empty quartsize milk cartons, and gallon jugs to set up a liquid measurement lab in her classroom.
Students poured water and discovered dropfordrop that eight ounces filled one cup. They poured cups of water into a pint carton and saw for themselves that one pint equaled two cups. It was messy, and it took more class time than asking students to memorize a table of measures.
Some students said after class that for the first time they could see what a pint was, that they understood measuring, and could picture equivalencies in their heads. They felt that this idea was a good one and hoped to do more labs. One 16yearold boy, who has lived in foster homes most of his life, said that he had never used a measuring cup before.
When Roberta came to class the next day, she told the class what she had learned. "I never realized that when I gave my son eight ounces of formula that he was drinking a whole cup of formula."
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Last Updated: January 1, 1996
