Demonstrate Understanding of Irrational Numbers in Both Radical (Including Mixed Radical)

FP10.2

Demonstrate understanding of irrational numbers in both radical (including mixed radical) and exponent forms through:

• representing

• identifying

• simplifying

• ordering

• relating to rational numbers

• applying exponent laws.

[C, CN, ME, PS, R, V]

a. Sort, with justification, a set of numbers into rational and irrational numbers.

b. Create and explain a pattern that describes the decimal form of an irrational number (e.g., write the digits from 0 to 9 in order, then put two of each digit – 0011223344 … – followed by three of each digit and so on).

c. Approximate the value of a given irrational number and explain the strategy used.

d. Order a set of Real numbers, including rational and irrational numbers, on a number line and explain the strategies used.

e. Express a radical as a mixed radical in simplest form (limited to numerical radicands).

f. Express a mixed radical as an entire radical (limited to numerical radicands).

g. Explain, using examples, how changing the value of the index of a radical impacts the value of the radical.

h. Represent, such as through the use of a graphic organizer, the relationships among the subsets of the Real numbers: natural, whole, integer, rational, and irrational.

i. Analyze patterns to generalize why ,

j. Analyze patterns to generalize why ,and when n is an even integer.

k. Extend and apply the exponent laws to powers with rational exponents (limited to expressions with rational and variable bases and integral and rational exponents):

l. Analyze simplifications of expressions involving radicals and/or powers for errors.

m. Express powers with rational exponents as radicals and vice versa.

n. Create a representation that conveys the relationship between powers, rational numbers, and irrational numbers.

Level / Scale / Descriptor / Indicators / Student-Friendly Language
Pre-Requisite Knowledge / I understand and can work with rational numbers.
1 / There is a partial understanding of some of the simpler details and processes.
Prior knowledge is understood. /

Knowledge and Comprehension

/ Sort, with justification, a set of numbers into rational and irrational numbers.
Create and explain a pattern that describes the decimal form of an irrational number (e.g., write the digits from 0 to 9 in order, then put two of each digit – 0011223344 … – followed by three of each digit and so on).
Order a set of Real numbers, including rational and irrational numbers, on a number line and explain the strategies used.
Represent, such as through the use of a graphic organizer, the relationships among the subsets of the Real numbers: natural, whole, integer, rational, and irrational. / I know the difference between rational and irrational numbers. I can sort them and create examples of them.
I can use patterns to write irrational numbers.
Ex/ 2.112123123412345…
I can organise the Real number systems (Natural, Whole, Integer, Rational and Irrational) in a diagram.
I can place real numbers in the correct order on a number line.
2 / No major errors or omissions regarding the simpler details or processes, but assistance may be required with the complex processes. /

Applying and Analysing

/ Approximate the value of a given irrational number and explain the strategy used.
Express a radical as a mixed radical in simplest form (limited to numerical radicands).
Express a mixed radical as an entire radical (limited to numerical radicands).

Analyze patterns to generalize why , / I can approximate the value of an irrational number and explain the process.
Ex/ is closer to than so value is closer to 2 than 3…
I can convert mixed radicals to entire radicals and vice versa.
I can express rational numbers as powers with negative exponents.
3 / No major errors or omissions regarding any of the information and/or processes that were explicitly taught.
This is the target level for proficiency. /

Evaluating and Creating

/ Analyze patterns to generalize why ,and when n is an even integer.
Explain, using examples, how changing the value of the index of a radical impacts the value of the radical.
Extend and apply the exponent laws to powers with rational exponents (limited to expressions with rational and variable bases and integral and rational exponents):

Analyze simplifications of expressions involving radicals and/or powers for errors.
Express powers with rational exponents as radicals and vice versa.
Create a representation that conveys the relationship between powers, rational numbers, and irrational numbers. / I can analyse patterns to generalize rules about writing radicals as powers and vice versa. I can use my rules to express radicals as powers and vice versa.
I understand what the index of a radical is and how it affects the value of the radical.
I can use laws of exponents with positive and negative rational exponents.
I can simplify radicals expressed in either form and analyze simplifications for errors.
I can represent my understanding of the relationship between powers, rational numbers and irrational numbers.
4 / In addition to level 3 performance, in-depth inferences and applications go beyond what was explicitly taught. /

Students successful at level 4 will receive supplementary comments specific to their achievement.

/ All Real Numbers are either Rational or Irrational. Are there numbers that are not Real? Research to find examples.