B1.2 Describe, compare, and order numbers in the real number system (rational and irrational numbers), separately and in combination, in various contexts.
Activity 1: One Lie, Two Truths (Recognizing the Difference Between a Rational and an Irrational Number)
Introduce the rules of the game which consists of presenting 3 statements, 2 of which will be true and the other false. The student must justify to their partner (or to the other students in the class) why each statement is true or false. The goal is to determine which one is false.
Example
- I am a rational number :
- \(\frac{2}{7}\)
- \({}^ - 1,555\;555\;55 \ldots \)
- 0,457 342 56…
Examples of justifications
Statement a is true since the number is written as a fraction. It is therefore a rational number.
Statement b is true because the decimal part is terminating and repeating. This number is therefore a rational number.
Statement c is false, because the decimal part is non-terminating and non-repeating. It is therefore an irrational number.
- I am a number that can be written as a fraction:
- 6
- 2,387 335 01…
- 0,555 55…
Examples of justifications
Statement a is true, because 6 can be written as \(\frac{6}{1}\).
Statement b is false since its decimal part is non-terminating and non-repeating, so it cannot be written as a fraction.
Statement c is true because the decimal part is repeating (or this number is equivalent to \frac{5}{9}\, which is a fraction).
Activity 2: The Number Battle (Find the Largest Number)
Prepare a set of cards (about 30 cards per student).
Suggestion: On the cards, write numbers ranging from \({}^ - 10\) to 10.
Make sure you have whole numbers, integers (positive and negative), positive and negative fractions, decimal numbers (terminating as well as non-terminating and repeating). Also ensure that you have several equivalent numbers, written differently (for example, \(0.5;\;\frac{1}{2};\;\frac{5}{{10}}).
Rules of the game:
Both students receive the same amount of cards. They must then place them in a deck, face down on the desk (they must not see the value of the number written on the card). At the same time, they turn the card that is on top of their respective deck. The student who has the card with the highest value wins the battle, keeps the 2 cards and places them under their deck.
*A battle occurs when both students turn over 2 cards that have the same value. Each student must then turn over the next 3 cards in their deck and compare who has the highest value on the3rd card. The student who wins this battle then gets to keep all 8 cards turned over.
The student who has the most cards at the end of the game period is the winner (or if a player has all of his opponent's cards).
Activity 3: An Armful of Numbers (Comparing and Ordering Real Numbers)
Materials
- pieces of cardboard
- felt-tip pen
- clothespins
- rope (about five metres)
Cut out cardboard pieces of \(30\;{\rm{cm}}\; \times \;{\rm{30}}\;{\rm{cm}}\).
Write on each card a real number (make sure to have whole numbers), integers (positive and negative), positive and negative fractions, decimal numbers, and zero.
To begin the activity, tie the rope to the ends of the board or to 2 net poles in the gym, as if to create a clothesline.
Distribute a card to each student. Ask them to take turns placing their number on the rope. Ask that the whole process be done in silence (optional). Once all the numbers have been placed, ask the students to analyze the rope. After this analysis, ask them to check that the numbers are in the right place.
- Is the number x to the right of the number y? Why is it so?
- Is the number x greater than the number y? How do you know that?
Question and discuss until all numbers are placed correctly in ascending order on the rope.