PARALLEL TASKS FOR PREKINDERGARTEN–GRADE
Choice 2: Use as many transformations as you wish to move shape A to position B.
B
A
CCSS: Geometry: 8.G
Mathematical Practices: 1, 2, 5
Both tasks require students to use transformations. Choice 1 stipulates the number of transformations to be used, which adds to the complexity of the prob- lem. Three refl ections can be used to accomplish the fi rst task; the fi rst moves one point on the original shape to its image, and the next two together result in the required turn: B A a b c
Whichever task was performed, in follow-up discussion students could be asked how they recognized which transformations to try and how they tested their predictions. Further, they could be asked to describe, using appropriate mathemati- cal language, the transformations they used.
SUMMING UP
The four big ideas that underpin work in Ge- ometry were explored in this chapter through 57 examples of open questions and parallel tasks, as well as variations of them. The in- structional examples provided were designed to support differentiated instruction for stu- dents at different developmental levels, target- ing three separate grade bands: pre-K–grade 2, grades 3–5, and grades 6–8.
Sometimes a student struggling with Num- ber and Operations performs well in Geome- try or vice versa. Rather than assuming that certain students need scaffolding and others do not based on their performance in Number and Operations, teachers can use differenti- ated questions and tasks, as suggested, to allow students to work at an appropriate level.
The examples presented in this chapter only scratch the surface of possible questions and tasks that can be used to differentiate instruction in Geometry. Other questions and tasks might be created by building shapes by putting together different numbers of shapes than suggested in the examples, by applying different transformations than in the example situations, or by requiring different construction conditions for shapes than those in the examples. A form such as the one shown here can serve as a convenient template for creating your own open questions and parallel tasks. The Appendix includes a full-size blank form and tips for using it to design customized teaching materials.
MY OWN QUESTIONS AND TASKS
Lesson Goal: Grade Level: _____ Standard(s) Addressed:
Underlying Big Idea(s): Open Question(s): Parallel Tasks: Choice 1: Choice 2:
Principles to Keep in Mind:
•• All open questions must allow for correct responses at a variety of levels.
•• Parallel tasks need to be created with variations that allow struggling students to be successful and profi cient students to be challenged.
•• Questions and tasks should be constructed in such a way that will allow all students to participate together in follow-up discussions.
111
Measurement
DIFFERENTIATED LEARNING
activities in measurement are derived from applying the NCTM process standards of problem solving, reasoning and proof, communicating, connecting, and representing to content goals of the NCTM Mea- surement Standard, including•• understanding measurable attributes of objects and the units, systems, and processes of measurement
•• applying appropriate techniques, tools, and formulas to determine measurements
TOPICS
Although students typically learn length before other measurements, there should be the same sort of development in teaching almost any type of measurement: moving from comparisons based on a particular attribute, to the use of nonstan- dard and then standard units, and then often to the use of formulas as a shortcut for measurement calculation. The Common Core State Standards (2010), which suggest what mathematical content should be the focus of each grade level, are use- ful for seeing which measurement attributes are usually considered at which grade levels. These guidelines can help make a teacher aware of where students’ learning is situated in relation to typical earlier and later learning in measurement.
Prekindergarten–Grade 2
Within this grade band, students focus on measurement of length, initially compar- ing lengths either directly or indirectly and later using nonstandard and nonmetric and metric units to measure length. When using units, they begin to recognize that a larger unit results in a smaller numerical value for the measurement. There is also some comparison and ordering of objects in terms of other attributes.
Grades 3–5
Within this grade band, students begin to use fractions of units in measuring length and measure perimeter. They learn what area means and begin to measure
area. They begin to develop and use formulas for areas of simple shapes, such as rectangles, and use standard units for measuring and describing areas.
Students also learn about volume of 3-D objects and learn to calculate and estimate volumes of simple objects.
Grades 6–8
Within this grade band, students continue to solve problems involving length, area, and volume. They develop and use formulas for surface areas and volumes of prisms, cylinders, cones, and spheres and for areas and circumferences of circles.
Students in this grade band begin to recognize the relationships between mea- surements of similar shapes. They also use the Pythagorean theorem to simplify the measurement of lengths of the sides of right triangles.
You will notice that the links to the common core standards are all under the heading of geometry for Grades 6–8. The items in this chapter focus on measure- ment aspects of that geometry.