Mathematics 9-
Number (Number Concepts)

This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Extensions
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

In order to prepare students to develop a number sense of powers with integral exponents and variable and rational bases, it is expected that students will:

• give examples of situations where answers would involve the positive (principal) square root or both positive and negative square roots of a number
• illustrate power, base, coefficient, and exponent using rational numbers or variables as bases or coefficients
  

SUGGESTED EXTENSIONS

To extend students' understanding of number concepts, they could:

• give examples of numbers that satisfy the conditions of natural, whole, integral, and rational numbers, and show that these numbers comprise the rational number system
• describe, orally and in writing, whether or not a number is rational
• explain and apply the exponent laws for powers with integral exponents
  
• determine the value of powers with integral exponents, using the exponent laws
  

SUGGESTED INSTRUCTIONAL STRATEGIES

As students extend their knowledge of numbers to include the integral exponents, they become exposed to more difficult yet "real-world" equations and formulae.

• Give examples of situations in which answers would involve the positive (principal) square root or both positive and negative square roots of a number. Pose as a problem: Why does the equation x² = 9 have two solutions [± 3], but the area of a square with an area of 9cm² is found using the principal square root [3cm]?
• Ask students to consider the concept in an historical context, noting the difficulty that having two answers posed for mathematics.
• Ask students to use a mathematics glossary or encyclopedia (print or electronic) to define the terms power, base, coefficient, and exponent.
• Use cubes and diagrams to represent and explain the difference between two numbers (e.g., 3² and 2³.)
• Ask students to practise translating formulae with powers into formulae without powers and vice versa.
• Have the class brainstorm examples of common misconceptions or errors that occur when applying the exponent laws. For example:

  

• Have students work in co-operative learning groups to develop the rules of exponents and create visual representations of these roles. Display their work.
  

SUGGESTED ASSESSMENT STRATEGIES

By demonstrating an understanding of exponents, students show they possess the basic knowledge needed to use mathematics to solve problems. Assessment focusses on the understanding of meaning and procedures.

Question

• Ask students to illustrate the difference between 2³ and 3². Do their diagrams clearly differentiate between a 2-D and a 3-D representation?
• To determine if students understand the difference between the square root of a number and the principal square root, ask questions such as:
  • What numbers squared equal 25? Students should recognize that both 5 and -5, when squared, equal 25.
  • Students should recognize has one solution, 3. The equation x² - 9 = 0 has two solutions, ± 3.
  • Do students know when it is appropriate to use only the principal (positive) square root?

Collect

• Have students demonstrate their understanding of the meaning of negative exponents by giving them a pattern of powers such as 2⁴, 2³, 2², 2¹, 2⁰, 2^-1, 2^-2, 2^-3, 2^-4. Have them predict the rules x⁰ = 1, x⁰ and justify their predictions to the rest of the class. Provide feedback to the students to help them clarify their thinking. To what extent do students:
  • predict rules correctly
  • clearly support their predictions
  • continue the pattern when asked
• After students have studied exponent laws for powers with integral exponents, give them problems in which the solutions contain errors. Ask students to use their knowledge of the laws to identify the errors, describe them, and make corrections where appropriate.
  

Print Materials

• Interactions (Level 9)
• MATHPOWER 9, Western Edition

Multimedia

• Hot Dog Stand: The Works
• The Learning Equation Mathematics 9 (TLE)
• Math Tools
• Mathematics 9 (Distance Education Package)
• Minds on Math 9, Revised Edition
• Understanding Math Series
  

© Copyright 2001. All Rights Reserved. BC MOE Standards Department.
Maintained by: Mathematics Coordinator
Revised: September 1, 2001

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