Extended MYP Mathematics - Being Specific
Being Specific
Authors: Rose Harrison, Clara Huizink, Aidan Sproat-Clements, Marlene Torres-Skoumal
Publisher: Oxford University Press
Publication Year: 2021
Framework: Content in the print book has been reviewed by the IB to ensure it fully aligns with the revised MYP framework.
ATL Cluster(s): Organization, communication, critical thinking
Key Concept: Form
Statement of Inquiry: Representing numbers in different forms to simplify them can help understand human-made systems.
Global Context: Globalization and sustainability
Related Concepts: Simplification, representation, quantity, approximation
Exploration: Exploring different ways of measuring human-made systems
E1.1 Laws of Exponents and Rational Exponents
Objectives:
Evaluating numerical expressions with positive or negative fractional exponents.
Writing numerical expressions with fractional exponents as radicals.
Using the rules of indices to simplify expressions that contain radicals and/or fractional exponents.
Critical thinking to analyse complex concepts and projects to create new understanding
Inquiry Questions:
How do you evaluate a number with a fractional exponent?
How do you use the rules of indices to simplify expressions with fractional exponents?
How are the rules of indices and the rules of radicals related?
How do the rules of indices help simplify radical expressions?
Can expressions with decimal exponents be simplified?
Is simpler always better?
For any positive real number x, and integer n, where n \neq 0, \sqrt[n]{x} = x^{\frac{1}{n}}.
Example: Evaluating numerical expressions with fractional exponents.
25^{\frac{1}{2}} = \sqrt{25} = 5
\sqrt[3]{8} = 8^{\frac{1}{3}} = 2
81^{\frac{1}{4}} = \sqrt[4]{81} = 3
Example: Using the multiplicative rule for exponents
(x^a)^b = x^{ab}
For any positive real number x, and integers m and n, where n \neq 0, x^{\frac{m}{n}} = (x^{\frac{1}{n}})^m = \sqrt[n]{x^m}.
Example:
Simplify 16^{\frac{3}{4}} without using a calculator.
16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3
= (2)^3 = 8
The base x must be a positive number because the even root of a negative number does not exist.
For any x \neq 0, \frac{a^{-n}}{b^{-m}} = \frac{b^m}{a^n}
Example:
\frac{27^{\frac{2}{3}}}{8^{\frac{2}{3}}} = (\frac{27}{8})^{\frac{2}{3}} = (\frac{3}{2})^2 = \frac{9}{4}
When radicals have the same radicand but different indices, write them with fractional exponents to simplify them.
\sqrt[n]{a} \times \sqrt[m]{a} = a^{\frac{1}{n}} \times a^{\frac{1}{m}} = a^{\frac{n+m}{nm}}
\frac{\sqrt[n]{a}}{\sqrt[m]{a}} = \frac{a^{\frac{1}{n}}}{a^{\frac{1}{m}}}= a^{\frac{1}{n} - \frac{1}{m}} = a^{\frac{m-n}{nm}}
When radicals have different radicands, but one is a power of the other, write them as exponents with the same base to simplify them.
When radicals have different radicands, but one is not a power of the other, then they can not be simplified into one radical.
A rational number is a number that can be written as a fraction.
An irrational number cannot be written as a fraction (e.g., \pi).
Summary
For any positive real number x, and integer n, where n \neq 0, \sqrt[n]{x} = x^{\frac{1}{n}}.
For any positive real number x, and integers m and n, where n \neq 0, x^{\frac{m}{n}} = (x^{\frac{1}{n}})^m = (x^m)^{\frac{1}{n}} = \sqrt[n]{x^m}.
For a, b \geq 0, \sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}.
For a \geq 0, b > 0, \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}.
For a \geq 0, n > 0, \sqrt[n]{a^n} = a.
E1.2 Upper and Lower Bounds
Objectives:
Finding and using upper and lower bounds for data given to a specified accuracy.
Performing calculations with limits of accuracy.
Critical thinking by finding and recognizing unstated assumptions and bias in given information.
Inquiry Questions:
How can you use upper and lower bounds?
How accurate are approximated numbers?
How do limits of accuracy affect your results when you perform mathematical operations?
When does the choice of rounding up or rounding down depend on the situation?
When does rounding give an impossible answer?
Is it always acceptable to round?
Are some approximations better than others?
Every measurement made is approximated to a certain degree of accuracy.
The midpoint between two possible values is the lower bound, and the midpoint between the other two values is the upper bound.
All continuous measurements are rounded to a certain degree of accuracy, which allows you to determine a range of values before the rounding has occurred.
To determine the upper and lower bounds:
Add half of the specified accuracy onto the value to find the upper bound.
Subtract half of the specified accuracy from the value to find the lower bound.
To maximize the result of calculations:
Addition ➜ upper bound + upper bound
Subtraction ➜ upper bound − lower bound
Multiplication ➜ upper bound × upper bound
Division ➜ upper bound ÷ lower bound
To minimize the result of calculations:
Addition ➜ lower bound + lower bound
Subtraction ➜ lower bound – upper bound
Multiplication ➜ lower bound × lower bound
Division ➜ lower bound ÷ upper bound
It is accepted that the exact measurements can lie anywhere between the limits of accuracy.