The First Four Standards - Grades 7-8

Overview

Seventh- and eighth-graders continue to improve their mastery of problem solving, communication, mathematical connections, and reasoning. They are increasingly able to apply these skills in more formal and abstract situations.

Students in grades 7-8 need to have many opportunities to practice problem solving, reflecting upon their thinking and explaining their solutions. They should have developed a considerable repertoire of problem-solving strategies by this time, including working backwards, writing an equation, looking for patterns, making a diagram, solving a simpler problem, and using concrete objects to represent the problem situation. Students should be able to apply the steps of understanding the problem, making a plan, carrying out the plan, and looking back. (See the K-12 Overview at the beginning of this chapter for a discussion of these steps.)

Students in grades 7 and 8 need many opportunities to communicate mathematically. Explaining and justifying one's work orally and in writing leads to deeper understanding of concepts and principles. Through discussion, students reach agreement on the meanings of words and recognize the importance of having common definitions. The need for mathematical symbols arises from the exploration of concepts, and these concepts must be firmly connected to the symbols that represent them through frequent and explicit discussion of the relationships between concepts and symbols. Students must also be encouraged to construct connections among concepts, procedures, and approaches by using questions that require more elaborate communication skills. For example, students might give examples of: a rectangle with four congruent sides, a parallelogram with four right angles, a trapezoid with two equal angles, a number between 1/3 and 1/2, a number with a repeating decimal representation, a net for a cube, or an equation for a line that passes through the point (-1,2). (See Curriculum and Evaluation Standards for School Mathematics, p. 80.) [A net is a flat shape which when folded along indicated lines will produce a three-dimensional object; for example, six identical squares joined in the shape of a cross can be folded to forma cube. Tabs added to the net facilitate attaching appropriate edges so that the shape remains three-dimensional.]

Many students in the middle grades have in the past viewed mathematics as a collection of isolated skills to be memorized and later forgotten. These students must broaden their perspective, viewing mathematics as an integrated whole and acknowledging its relevance in and out of school. They must improve their understanding of mathematical connections. Students must understand, for example, that translations on the number line are fundamentally the same as adding numbers. They should connect the various interpretations of fractions to measurement, ratios, and algebra. They should link Pascal's triangle with counting, exponents, number patterns, algebra, geometric patterns, probability, and number theory. (See Standard 14, Discrete Mathematics.) Such connections can be enhanced by using technology and by exploring the same mathematical ideas in varied contexts. Some of the most important connections for students in grades 7 and 8 include proportional relationships (see Standard 6, Number Sense) and the relationship between data in tables and their algebraic and graphical representations. (See Standard 11, Patterns; Standard 12, Probability and Statistics; and Standard 13, Algebra).

Students in these grades also need many opportunities to discuss the connections between mathematics and other disciplines and the real world. For example, students concerned about traffic at a nearby intersection might design a study to collect data about the situation. Their study might involve first deciding what "traffic" is, how to count it, how to record the data, what the data means, how to remedy the situation, who is responsible for dealing with this type of situation, and how to convince that person that change is needed. Their study might end with a letter to the town council, recommending changes needed at the intersection.

In conjunction with studying about maps in social studies, for example, students might study scaling and its relationship to similarity, ratio, and proportion. Measurement situations arise in social studies, science, home economics, industrial technology, and physical education. Weather forecasting, scientific experiments, advertising claims, chance events, and economic trends offer more opportunities to relate mathematics to the real world, often through the use of statistics and probability. Connections between mathematics and science are plentiful:

• Students use symmetry and perspective to create original artistic work.

• Students might learn about scientific notation for large numbers by studying the solar system or for small numbers by studying viruses and bacteria.

• Different scientific contexts, such as electrical charges or forces in opposite directions, might help students develop multiple representations for integers.

• Students might plot earthquake locations by using a compass and protractor.

• The study of geometric shapes might be related to the study of crystals.

• Area and volume might be considered in conjunction with looking at the number of fish in a pond. Why would each measure be important to consider? How could the area of a pond be found? How could the volume be found?

• Students might investigate levers, gears, or pulleys, looking for numerical patterns and generating equations that describe the relationships found.

Students need frequent opportunities to explore and discuss many of these types of relationships in order todevelop an appreciation for the power of mathematics.

Students in grades 7 and 8 experience rapid improvements in their ability to use mathematical reasoning. Special attention should be paid at these grades to proportional and spatial reasoning and reasoning from graphs. Students need to experience both inductive reasoning (making conjectures, predicting outcomes, looking for patterns, and making generalizations) and deductive reasoning (using logical arguments, justifying answers). Experiences with inductive reasoning need to include situations in which incorrect generalizations based on too few examples are tested. Instructional activities that address two specific types of deductive reasoning are appropriate at these grades. Class reasoning involves applying generalizations to a specific situation: for example, all even numbers are divisible by 2; x is an even number; so x is divisible by 2. Conditional reasoning involves using if-then statements. These types of reasoning need to be used on a regular basis in class discussions, assignments, and tests in order to help students become familiar with valid reasoning patterns. Students should also be asked to justify and explain solutions to the satisfaction of their peers.

In grades 7 and 8, students apply their problem-solving, communication, and reasoning skills in an increasingly diverse set of situations as they develop better understanding of the connections within mathematics and between mathematics and other subjects and the real world. In doing so, they learn new mathematical concepts and apply familiar mathematics to new situations.

References

National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, VA, 1989.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.

Standards: In addition to the First Four Standards, this vignette highlights Standards 5 (Tools and Technology), 7 (Geometry), 9 (Measurement), and 15 (Building Blocks of Calculus).

The problem: A heterogeneously grouped class of middle school students reported to the computer lab for their mathematics class. Students had used Geometer's Sketchpad software (Key Curriculum Press) frequently during the year. They had been studying proportions in everyday and geometric situations during several previous class periods. They had also defined similar figures as those generated by an expansion or a contraction. The teacher presented this task to the class: Draw several similar figures. What do you notice about the ratios of the measures of corresponding sides? What do you notice about the measures of the corresponding angles? Record your results and write your conclusions.

The discussion: The teacher asked students to begin brainstorming to generate ideas about the problem. As ideas were suggested, the teacher wrote them on the chalkboard. Ideas included: Use simple figures to make the problem easier. Use the dilate transformation. Use measure and length on Sketchpad to find the lengths of the sides. Make more than two figures because the problem says several. What is corresponding? Do we have to write complete sentences? Use measure and angle on Sketchpad to find the measures of the angles. After all the ideas were recorded, the class discussed each one. Some were questions that could be answered by students, by rereading the problem or by checking glossaries in math books. Those answers were recorded next to the appropriate questions. Some of the ideas were eliminated because they were either incorrect or off-task. The teacher then had students restate the problem and, using the list, talk about the steps they would follow to solve the problem.

Solving the problem: Students joined their regular lab partners at the computers and began the investigation. As students worked, some requested teacher help, asking if their work was "finished yet" and if their results were "good enough." The teacher directed those students to reread the problem and the list of ideas developed during the discussion, and determine for themselves which parts of the problem were completed adequately and which were left to be done. Students worked for two class periods, completing the task by preparing diagrams, ratios, and written explanations. The teacher provided each pair with a transparency on which they were to record their diagrams and ratios to share with the class.

Summary: The summary discussion began with a volunteer restating the problem. Another volunteer described the steps she used to solve the problem. Others interjected during the description to tell how their methods differed. Then students presented their findings, using their transparencies and their written conclusions. After lengthy discussion, students wrote a general class conclusion that similar figures have corresponding angles with equal measures and have corresponding sides whose measures form equal ratios.

Standards: In addition to the First Four Standards, this vignette highlights Standards 7 (Geometry), 8 (Numerical Operations), 9 (Measurement), 11 (Patterns), and 15 (Building Blocks of Calculus).

The problem: Students in this multi-aged middle school class had been working for several days on a project that gave them opportunities to find the surface area and volume of objects constructed with Cuisenaire rods and tacky putty. Today they were to begin an investigation of the relationship between the surface area and the volume of 3-dimensional objects enlarged with scale factors from 2 to 8. Students, working in groups of two or three, were instructed to build a "dog" using 1 yellow (5cm in length) and

5 red (2cm) rods. The yellow rod represented the dog's body, and a red rod represented each leg and the head. Then students calculated the surface area and volume of the dog. (Note that all rods have cross-section 1cm x 1cm.) Students were then instructed to build a "double dog" that was twice as big in each of the three dimensions. After several false starts, students constructed a dog with 4 orange (10cm) rods and 20 purple (4cm) rods. The 4 orange rods represented the dog's body, and 4 purple rods represented each leg and the head. Then students found and recorded the surface area and volume of this double dog. Each group was to build one other dog, using an assigned scale factor of 3, 4, 5, 6, 7, or 8 times the original dog. Their challenge was to determine how the scale factors were related to the surface area and volume of the enlarged dogs.

The discussion: Students asked questions to clarify the problem, mostly checking the task by saying You mean, if my scale factor is 7, my dog's body has to be 35 by 7 by 7? The teacher encouraged other students to confirm or correct each question. Then students focused on how they were to construct the dogs. Many began to think about how many rods they would need to complete the construction and decided that for some of the scale factors, they just would not have enough rods. The teacher then pointed out the centimeter grid paper, scissors, and tape at the front of the room and suggested that the students could build the dogs with those supplies.

Solving the problem: Students worked for two days building their enlarged dogs. Part of the challenge was to lay out a pattern for each part of the dog on a sheet of paper so that it could be folded and then put together with as few taped edges as possible. Those students who were given scale factors of 3 or 4 found the task rather simple and were able to count the surface area and volume without much trouble. Those with greater scale factors such as 7 or 8 had a more difficult task, but were able to complete it successfully.

Summary: The scale factor, surface area, and volume of each dog was listed on the chalkboard. Students discussed the various methods that different groups used to construct the dogs and to find the surface area and volume. The teacher challenged students to find the pattern in the chart, to apply the rule to scale factors other than those already used, and to generalize the rule to a scale factor of n. Students made liberal use of calculators, using a guess and check strategy to find the pattern. Students were able to verbalize that the surface area was the scale factor squared times the original surface area and that the volume was the scale factor cubed times the original volume.

The cumulative progress indicators for grade 8 for each of the First Four Standards appear in boldface type below the standard. Each indicator is followed by a brief discussion of how the preceding grade level vignettes might address the indicator in the classroom in grades 7 and 8. The Introduction to this Framework contains three vignettes describing lessons for grades 5-8 which also illustrate the indicators for the First Four Standards; these are entitled The Powers of the Knight, Short-circuiting Trenton, and Mathematics at Work.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 will be such that all students:

4^*. Pose, explore, and solve a variety of problems, including non-routine problems and open-ended problems with several solutions and/or solution strategies.

• The students in Sketching Similarities explore a problem involving geometry in which they choose their own figures to investigate. The students in Rod Dogs explore and solve a non-routine problem.

5^*. Construct, explain, justify, and apply a variety of problem-solving strategies in both cooperative and independent learning environments.

• In both vignettes, students use the strategy of looking for patterns. The students in Sketching Similarities work in pairs and use inductive reasoning to find a pattern. The groups of students in Rod Dogs use concrete materials as their primary strategy, with some using rods and others using grid paper.

6^*. Verify the correctness and reasonableness of results and interpret them in the context of the problems being solved.

• In both vignettes, the students verify their results and interpret them through a class discussion following their work in pairs or groups.

7^*. Know when to select and how to use grade-appropriate mathematical tools and methods (including manipulatives, calculators and computers, as well as mental math and paper-and-pencil techniques) as a natural and routine part of the problem-solving process.

• Students in Sketching Similarities use computers. Students in Rod Dogs use manipulatives and calculators as needed in solving their problems.

8^*. Determine, collect, organize, and analyze data needed to solve problems.

• The students in Sketching Similarities decide what data to collect in the initial discussion of the problem. They analyze the collected data in the discussion following their work in pairs. The students in Rod Dogs collect and organize the data as they work in groups. In their summary discussion, they analyze all the data collected to look for patterns and form a generalization.

10. Use discovery-oriented, inquiry-based, and problem-centered approaches to investigate and understand mathematical content appropriate to the middle ages.

• The students in both vignettes use a problem to investigate new mathematical concepts.

11. Recognize, formulate, and solve problems arising from mathematical situations, everyday experiences, - and applications to other disciplines.

• The Sketching Similarities vignette involves examining a problem that arises from a mathematical situation. The Rod Dogs vignette involves a problem that arises from everyday experiences - enlarging a three-dimensional object.

12. Construct and use concrete, pictorial, symbolic, and graphical models to represent problem situations and effectively apply processes of mathematical modeling in mathematics and other areas.

• The students in Sketching Similarities use graphical (computer diagrams) and symbolic (numbers) models in considering their problem. The students in Rod Dogs use concrete materials (rods and grid paper folded to make boxes) and symbolic (numbers and formulas) models.

13. Recognize that there may be multiple ways to solve a problem, weigh their relative merits, and select and use appropriate problem-solving strategies.

• The students in Sketching Similarities use brainstorming to generate ideas about solving the problem, and share their methods with the class. In Rod Dogs, students approach the problem in different ways, and the class discusses the various methods that different groups use to construct the "dogs" and to solve the problem; some students use a guess and check strategy to find the pattern.

14. Persevere in developing alternative problem-solving strategies if initially selected approaches do not work.

• When some students are at a standstill in Rod Dogs, the teacher directs their attention to available manipulatives. In Sketching Similarities, the teacher encourages students toresearch the problem and to make use of the list of ideas generated by the class in their continued attempts to solve the problem.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 will be such that all students:

1^*. Discuss, listen, represent, read, and write as vital activities in their learning and use of mathematics.

• Students in both vignettes discuss, listen, represent the problem situation concretely and symbolically, and read and write about their solutions.

2^*. Identify and explain key mathematical concepts, and model situations using oral, written, concrete, pictorial, and graphical methods.

• Students in Sketching Similarities explain the concept of similarity and model the situation using oral, written, concrete, and pictorial methods. Students in Rod Dogs explain how the scale factor is related to changes in area and volume and use oral, written, concrete, and pictorial methods.

3^*. Represent and communicate mathematical ideas through the use of learning tools such as calculators, computers, and manipulatives.

• Students in Sketching Similarities use computers and manipulatives. Students in Rod Dogs use manipulatives and calculators.

4^*. Engage in mathematical brainstorming and discussions by asking questions, making conjectures, and suggesting strategies for solving problems.

• Students in both vignettes begin their activity by asking questions, making conjectures, and suggesting strategies to be used.

5^*. Explain their own mathematical work to others, and justify their reasoning and conclusions.

• Students in both vignettes share their work with the class and justify their solutions.

6. Identify and explain key mathematical concepts and model situations using geometric and algebraic methods.

• In Sketching Similarities students recognize that similarity of triangles has an algebraic counterpart, that corresponding sides have the same ratio; the activities for the vignettecould be extended by asking the students to use algebra to determine the lengths of the sides. In Rod Dogs the students develop a formula relating the volume and surface area of the enlarged "dogs" to the volume and surface area of the original one.

7. Use mathematical language and symbols to represent problem situations, and recognize the economy and power of mathematical symbolism and its role in the development of mathematics.

• Students in both vignettes use mathematical language and symbols to represent their solutions.

8. Analyze, evaluate, and explain mathematical arguments and conclusions presented by others.

• Students in both vignettes analyze, evaluate, and explain their solutions.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 will be such that all students:

1^*. View mathematics as an integrated whole rather than as a series of disconnected topics and rules.

• The Sketching Similarities vignette illustrates the connection between similarity and ratios. The Rod Dogs vignette illustrates the connection between geometry and measurement (surface area, volume, scale factor) and exponents.

2^*. Relate mathematical procedures to their underlying concepts.

• The Sketching Similarities vignette relates the mathematical procedure of setting up ratios for similar figures to the underlying concept of similarity. The Rod Dogs vignette relates the mathematical procedure of taking powers of a number to the underlying concepts of area and volume.

3^*. Use models, calculators, and other mathematical tools to demonstrate the connections among various equivalent graphical, concrete, and verbal representations of mathematical concepts.

• Students in both vignettes use tools (models, computers, calculators) to discover, demonstrate, and verbalize the connections between the concrete geometric models in each problem and specific numerical relationships which result. In Sketching Similarities, they might also apply their conclusions to express and solve numerical and algebraic problemsassociated with the angles and sides of similar triangles. In Rod Dogs, students might use their discoveries regarding the 3-dimensional objects and graph the relationships of scale factor vs. surface area and scale factor vs. volume.

4^*. Explore problems and describe and confirm results using various representations.

• Students in both vignettes explore problems and describe their results in various ways; the representations used are generally concrete and written.

8. Recognize and apply unifying concepts and processes which are woven throughout mathematics.

• The students in Sketching Similarities recognize and apply the concept of proportion. The students in Rod Dogs recognize and apply the concept of powers.

9. Use the process of mathematical modeling in mathematics and other disciplines, and demonstrate understanding of its methodology, strengths, and limitations.

• Students in both vignettes establish a mathematical model for their problem situations and use it to solve their problems. The students in Rod Dogs, in particular, also note the limitations of modeling concretely and the need to move to a more symbolic model.

10. Apply mathematics in their daily lives and in career-based contexts.

• The students in both vignettes apply mathematics to a realistic problem.

11. Recognize situations in other disciplines in which mathematical models may be applicable, and apply appropriate models, mathematical reasoning, and problem solving to those situations.

• Students might investigate how the 2-dimensional relationships in Sketching Similarities and the 3-dimensional results from Rod Dogs could be used in art, architecture, engineering, business, and the building trades.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 will be such that all students:

2^*. Draw logical conclusions and make generalizations.

• Students in both vignettes draw logical conclusions and make generalizations based on the data they collect.

3^*. Use models, known facts, properties, and relationships to explain their thinking.

• Students in both vignettes use models to explain their thinking.

5^*. Analyze mathematical situations by recognizing and using patterns and relationships.

• Students in both vignettes look for patterns to help them solve the problems.

6. Make conjectures based on observation and information, and test mathematical conjectures and arguments.

• Students in both vignettes make conjectures and test them.

7. Justify, in clear and organized form, answers and solution processes in a variety of problems.

• Students in both vignettes justify their answers in writing and orally.

8. Follow and construct logical arguments, and judge their validity.

• In both vignettes, the students could be challenged to explain why they thought that their generalizations would always work.

9. Recognize and use deductive and inductive reasoning in all areas of mathematics.

• The students in both vignettes use inductive reasoning to find a pattern and make a generalization.

10. Utilize mathematical reasoning skills in other disciplines and in their lives.

• Both vignettes illustrate the use of mathematical reasoning in real life.

11. Use reasoning rather than relying on an answer-key to check the correctness of solutions to problems.

• The students base their judgments about correctness of their results on the explanations given in the summary discussion rather than on an answer key.