186. Solving Problems with Real Numbers
Learning Intentions
- Choose appropriate rational or irrational values for a given problem.
- Apply real number operations to Solve simple problems.
- Interpret answers involving rational and irrational numbers in context.
Pre-requisite Summary
- Know that rational numbers can be written as fractions, terminating decimals or recurring decimals.
- Know that irrational numbers cannot be written exactly as fractions, including square roots of non-perfect squares and
. - Know how to add, subtract, multiply and divide rational numbers.
- Know that operations involving irrational numbers can sometimes be left in exact form, such as
or . - Know that exact answers are often preferred in mathematics, while decimal approximations are useful for interpreting measurements.
- Know that context determines whether an answer should be rounded and what units should be included.
Worked Examples
Worked Example 1
Choose the most appropriate value for the missing length.
A square has an area of
a)
b)
c)
Worked Example 2
Choose whether a rational or irrational value is more appropriate.
A circular garden has radius
Should the answer Use
Worked Example 3
Solve the exact perimeter of the shape.
A rectangle has side lengths
Worked Example 4
Find the exact area of the shape.
A rectangle has side lengths
Worked Example 5
A square has an area of
a) Write the exact side length.
b) Estimate the side length to one decimal place.
c) Explain what the decimal approximation means in context.
Worked Example 6
A walking track is
a) Estimate the length of the track to one decimal place.
b) Decide whether the exact form or decimal form is more useful for a sign at the track entrance.
c) Explain your choice.
Problems
Problem 1
Choose the most appropriate value for the missing length.
A square has an area of
a)
b)
c)
Problem 2
Choose whether a rational or irrational value is more appropriate.
A circular pond has radius
Should the answer use
Problem 3
Find the exact perimeter of the shape.
A rectangle has side lengths
Problem 4
Find the exact area of the shape.
A rectangle has side lengths
Problem 5
A square has an area of
a) Write the exact side length.
b) Estimate the side length to one decimal place.
c) Explain what the decimal approximation means in context.
Problem 6
A cycling path is
a) Estimate the length of the path to one decimal place.
b) Decide whether the exact form or decimal form is more useful for a public sign.
c) Explain your choice.
Exercises
Understanding and Fluency
Exercise 1.
Choose the most appropriate exact value for each missing side length.
a) A square has area
b) A square has area
c) A square has area
Exercise 2.
Decide whether each exact value is rational or irrational.
a)
b)
c)
d)
Exercise 3.
For each circle, choose whether
a) Radius
b) Diameter
c) Radius
Exercise 4.
Find the exact perimeter of each rectangle.
a) Length
b) Length
c) Length
Exercise 5.
Find the exact area of each rectangle.
a) Length
b) Length
c) Length
Exercise 6.
Estimate each value to one decimal place.
a)
b)
c)
d)
Exercise 7.
For each square, write the exact side length and estimate it to one decimal place.
a) Area
b) Area
c) Area
Exercise 8.
Estimate each length and State whether the answer is rational or irrational.
a)
b)
c)
Exercise 9.
A rectangular garden has side lengths
a) Find the exact perimeter.
b) Estimate the perimeter to one decimal place.
c) State which form is more useful for buying fencing.
Exercise 10.
A square tile has area
a) Write the exact side length.
b) Estimate the side length to one decimal place.
c) Explain why the side length is irrational.
Reasoning
Exercise 11.
A student says the side length of a square with area
Exercise 12.
Explain why
Exercise 13.
A student writes the exact circumference of a circle with radius
Exercise 14.
Decide which answer is more appropriate for measuring the length of a real walking track:
Exercise 15.
A rectangle has side lengths
Problem-solving
Exercise 16.
A square park has an area of
a) Write the exact side length.
b) Estimate the side length to one decimal place.
c) Explain whether a builder should use the exact value or decimal approximation.
Exercise 17.
A circular table has diameter
a) Write the exact circumference in terms of
b) Estimate the circumference Use
c) Explain why the exact and approximate answers are both useful.
Exercise 18.
A rectangular art frame has side lengths
a) Find the exact perimeter.
b) Estimate the perimeter to one decimal place.
c) Interpret the answer in context.
Exercise 19.
A hiking trail has three sections with lengths
a) Find the exact total length.
b) Estimate the total length to one decimal place.
c) Explain why the decimal form is useful in this context.
Exercise 20.
Create your own real-world problem where the exact answer is irrational.
Your problem must include:
- a context
- an exact answer involving
or - a decimal approximation
- a sentence interpreting the answer in context
Potential Misunderstandings
- Students may choose a decimal approximation when an exact irrational value is more appropriate.
- Students may think all real-world measurements must be rational, even though irrational numbers can Describe exact mathematical lengths.
- Students may use
when itself is the required value, or use when is required. - Students may add unlike terms incorrectly, such as writing
. - Students may multiply incorrectly, such as writing
instead of . - Students may forget that perimeter involves addition of side lengths, while area involves multiplication of side lengths.
- Students may give an answer without units, making the interpretation unclear.
- Students may round too early and create inaccurate final answers.
- Students may not distinguish between an exact answer, such as
, and an approximate answer, such as .