Capítulo 4. Resultados y discusión
5. Conclusiones
CCSS: Geometry: 8.G
Mathematical Practices: 1, 2, 5, 7
Both choices require that students have an understanding of the formula for the volume of a prism, but Choice 2 also requires that students have either a formal or informal understanding of the formula for the volume of a cylinder. A student who chooses Choice 1 could look for groups of three numbers that multiply to produce 200, for example, 10 by 20 by 1, or 4 by 10 by 5. A student who chooses Choice 2 might recognize that a prism with a base area of 20 cm2 and a height of
10 cm would have the desired volume. They might then choose a cylinder with a slightly larger base and similar height, fi guring that the cylinder needs to be a bit wider to make up for the fact that a circle takes up less space than a square of the same width.
Both groups of students could be asked:
•• Can you be sure of the dimensions of the original prism? Why not? •• Is there a maximum height it could be? A minimum height? Why not? •• Is it helpful to know volume formulas to help you solve the problem? How?
P
arallel T
a
sk
s for Grades
6–8
What are the volume and surface area of the prism?
Choice 1: Choice 2: 2" 4" 10" 7" 3" 8" CCSS: Geometry: 8.G Mathematical Practices: 1, 2, 5
To complete Choice 2, students not only need to know what volume and sur- face area mean and how to calculate them, but they also need to be able to use the Pythagorean theorem to determine the side length of the side rectangles. For Choice 1, they need to know how to calculate surface area and volume of rectan- gular shapes only. The volume and surface area for Choice 1 are 80 cubic inches and 136 square inches; the volume and surface area for Choice 2 are 84 cubic inches and 128.4 square inches.
Whichever choice was selected, a teacher could ask: •• Which face has the greatest area?
•• How do you know?
•• Why is it hard to tell which has the greatest volume by just looking at the pictures?
•• Does a shape that is taller always have a greater volume? A greater surface area?
•• How did you calculate the volume? Th e surface area?
Compare the surface areas and volumes of the two shapes.
Choice 1: Choice 2: 7" 10" 7" 10" 7" 7" 10" 7" 10" 10" CCSS: Geometry: 8.G Mathematical Practices: 1, 2, 5
In both cases, students need to know how to calculate surface area and volume, but in one case they must be able to deal with cylinders and in the other case only with rectangular prisms. Many students will assume that the volumes are equal in Choice 1 because they see 7 and 10 on both shapes, but this is not the case. They may be less sure of how the surface areas compare.
In fact, for Choice 1, the volumes are 175 cubic inches for the short cylinder compared to 122.5 cubic inches for the tall one, and the surface areas are 120 square inches compared to 94.5 square inches. For Choice 2, the volumes are 490 cubic inches and 700 cubic inches, respectively, and the surface areas are 378 square inches and 480 square inches, respectively.
Whichever choice was selected, a teacher could ask:
•• Is it possible for the volumes to be the same but the surface areas to be diff erent?
•• Is it possible for the surface areas to be the same but the volumes to be diff erent?
•• Are either the same in your pair of shapes? •• How did you calculate the volumes? •• How did you calculate the surface areas?
SUMMING UP
The seven big ideas that underpin work in Measurement were explored in this chapter through nearly 50 examples of open questions and parallel tasks, as well as variations of them. The instructional examples provided were de- signed to support differentiated instruction for students at different developmental levels, targeting three separate grade bands: pre-K– grade 2, grades 3–5, and grades 6–8.
Measurement is a strand that links Num- ber and Operations with Geometry. Students experiencing weakness in either of those strands might struggle in Measurement. Because Mea- surement is a very practical part of the math- ematics that is taught, it is important to maxi- mize student success with this strand through differentiating instruction to meet students where they are developmentally.
The examples presented in this chapter only scratch the surface of possible questions and tasks that can be used to differentiate instruction in Measurement. Other questions and tasks can be created by, for example, changing the type of measurement in an example question or task, changing the object to which a measurement is com- pared, or changing the units that are proposed to be used. A form such as the one shown here can serve as a convenient template for creating your own open ques- tions 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.
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Algebra
DIFFERENTIATED LEARNING
activities in algebra are derived from applying the NCTM process standards of problem solving, reasoning and proof, communi- cating, connecting, and representing to content goals of the NCTM Algebra Stan- dard, including•• understanding patterns, relations, and functions
•• representing and analyzing mathematical situations and structures using algebraic symbols
•• using mathematical models to represent and understand quantitative relationships
•• analyzing change in various contexts
TOPICS
Before beginning the task of differentiating student learning in algebra, it is useful for teachers to have a good sense of how the topics in the strand develop over the grade bands. It is particularly important in algebra because the concept of what constitutes algebraic thinking has broadened in recent years. In fact, the Common Core State Standards use the heading “Operations & Algebraic Thinking” starting in kindergarten.
With the notion of algebra representing general relationships and change, stu- dents use algebra in the early grades as they relate numbers additively and solve simple addition and subtraction equations, later as they relate numbers multipli- catively and solve simple multiplication and division equations, and even later as they use symbolism to describe relationships between numbers and solve more complex equations.
The Common Core State Standards (2010) can be helpful in making teachers aware of where students’ learning is situated in relation to what learning in algebra typically precedes and succeeds the work in a particular grade band.
Prekindergarten–Grade 2
Within this grade band, students identify, describe, and extend simple number and shape patterns. They use strategies involving patterns to help them learn addition and subtraction facts, and they skip count as a prelude to later work on multiplica- tive relationships.
Grades 3–5
Within this grade band, students continue to identify, describe, and extend number patterns. They also begin to more explicitly think about pattern rules, graph pat- terns and relate patterns.
Grades 6–8
Within this grade band, students move into more traditional algebra. They use expressions, equations, and formulas to correspond to numerical and real-life situ- ations. They evaluate expressions involving variables and use variables more regu- larly, recognizing that two different expressions might be equivalent. They solve simple equations and use tables of values to uncover relationships and solve prob- lems. They also analyze and solve pairs of simultaneous linear equations, relate linear equations to types of problem situations, and begin to consider the notion of function in a more formal way.