185. Ordering Real Numbers

Learning Intentions

• Arrange rational and irrational numbers in ascending or descending order.
• Locate approximate positions of irrational numbers on a number line.
• Justify number order Use decimal approximations.

Pre-requisite Summary

• Know that rational numbers include integers, fractions, terminating decimals and recurring decimals.
• Know that irrational numbers cannot be written exactly as fractions and include square roots of non-perfect squares.
• Know common perfect squares such as and .
• Know that if , then for positive numbers.
• Know how to compare decimals by place value.
• Know that numbers further to the right on a number line are greater.

Worked Examples

Worked Example 1

Arrange the following numbers in ascending order.

Worked Example 2

Arrange the following numbers in descending order.

Worked Example 3

Place each irrational number approximately on a number line from to .

a)

b)

c)

Worked Example 4

Place each irrational number approximately on a number line from to .

a)

b)

c)

Worked Example 5

Use decimal approximations to justify which number is larger.

or

Worked Example 6

Use decimal approximations to justify the order of the following numbers.

Problems

Problem 1

Arrange the following numbers in ascending order.

Problem 2

Arrange the following numbers in descending order.

Problem 3

Place each irrational number approximately on a number line from to .

a)

b)

c)

Problem 4

Place each irrational number approximately on a number line from to .

a)

b)

c)

Problem 5

Use decimal approximations to justify which number is larger.

or

Problem 6

Use decimal approximations to justify the order of the following numbers.

Exercises

Understanding and Fluency

Exercise 1.

Arrange the following numbers in ascending order.

a)

b)

c)

Exercise 2.

Arrange the following numbers in descending order.

a)

b)

c)

Exercise 3.

Estimate each irrational number to one decimal place.

a)

b)

c)

d)

Exercise 4.

State the two whole numbers each irrational number lies between.

a)

b)

c)

d)

Exercise 5.

Place each irrational number approximately on a number line from to .

a)

b)

c)

d)

Exercise 6.

Place each irrational number approximately on a number line from to .

a)

b)

c)

d)

Exercise 7.

Use decimal approximations to compare each pair using , or .

a) and

b) and

c) and

d) and

Exercise 8.

Use decimal approximations to arrange each set in ascending order.

a)

b)

c)

Exercise 9.

Use decimal approximations to arrange each set in descending order.

a)

b)

c)

Exercise 10.

Match each irrational number to its approximate position.

a)

b)

c)

d)

Positions:

Reasoning

Exercise 11.

Explain why is greater than but less than .

Exercise 12.

A student says that is less than because is less than . Explain the mistake.

Exercise 13.

Explain why is closer to than to .

Exercise 14.

Without using a calculator, decide whether is greater than . Justify your answer.

Exercise 15.

A student orders the numbers as

because . Explain why this reasoning is incorrect.

Problem-solving

Exercise 16.

A square has an area of . Estimate the side length and place it approximately on a number line from to .

Exercise 17.

Four students give estimates for .

• Asha:
• Ben:
• Cara:
• Dinesh:

Who has the best estimate? Justify your answer.

Exercise 18.

Create a set of four numbers that includes:

• one fraction
• one terminating decimal
• one irrational square root between and
• one irrational square root between and

Then arrange them in descending order.

Exercise 19.

A point is placed between and on a number line. Give one possible irrational square root that could be located at and justify your answer.

Exercise 20.

Arrange the following numbers in ascending order and justify using decimal approximations.

Potential Misunderstandings

• Students may compare and by comparing and instead of estimating the value of .
• Students may think a larger number under the square root symbol always means the final number is larger than nearby decimals or fractions.
• Students may forget to convert fractions to decimals before comparing mixed number types.
• Students may place near on a number line instead of between and .
• Students may place irrational numbers exactly halfway between two whole numbers because the radicand is visually near the middle.
• Students may not Recognise that is closer to than to because is closer to than to .
• Students may round too early when using decimal approximations, causing incorrect ordering.
• Students may give an order without justification, rather than showing approximate decimal values.
• Students may confuse ascending order with descending order.

Next: 186. Solving Problems with Real Numbers