At a glance. One lesson of about 60 minutes. It assumes rational and irrational numbers and repeating decimals from Year 9. Students use ∈, ∉, ⊂, n(A) and ∅, read and write set-builder notation, and place any number in the nested sets ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ.
Calculator: no calculator. None is needed. Every question is about exact form and counting, and a calculator's decimal for √12 or 22/7 can mislead: a long decimal on a screen does not show whether it repeats.
What students will learn
Know
- ℕ = {0, 1, 2, …}, ℤ, ℚ (p/q with q ≠ 0) and ℝ, and ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ.
- The meaning of ∈, ∉, ⊆, ⊂, n(A) and ∅.
- An irrational number is a real number that is not rational.
Understand
- The sets are nested, so a number belongs to its smallest set and every set outside it.
- Set-builder notation is a rule, and the same rule describes different sets in ℕ, ℤ and ℝ.
- A root sign does not decide rational or irrational; the number under it does.
Do
- Classify a number into every set it belongs to.
- List a set given in set-builder notation and find n(A), including the empty set.
- Write a repeating decimal as a fraction to show it is rational.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: set notation, the number sets and set-builder notation in familiar and less familiar forms (questions 1 to 9).
- Criterion B. Investigating patterns: the second figure. Ask students to predict n(A) before pressing ℕ, ℤ or ℝ, and to describe in general when a set-builder rule gives the empty set.
- Criterion C. Communicating: reading set-builder notation aloud in words, and using ∈ and ⊂ correctly (question 4, the warning card and the worked example).
- Criterion D. Applying mathematics in real-life contexts: the hook's temperature, shares, tile diagonal and table area, sorted by the kind of number each one is.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “Where will √9 land?” and take a show of hands for ℕ and for irrational before anyone moves the slider. Many will choose irrational.
In the second figure, keep a = −2 and b = 3 and press ℕ, ℤ and ℝ in turn: the rule is the same, but the set changes. Then ask for limits that make the set empty.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: sort the five numbers from the day. Collect predictions for √2 and π and the reasons; do not settle them yet. |
| 10 min | First figure and key idea: the nested sets and the symbols. Settle the hook. |
| 10 min | Second figure, the warning card, the worked example; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to √9 in the first figure. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 0.375 (it ends); −2, −1, 0, 1; irrational. |
|---|---|
| Hook | −4 = −4/1, 12 = 12/1 and 1/4 are rational; √2 and π are irrational. |
| Worked example | E = {0, 1, 2, 3, 4}, n(E) = 5, and 2.5 ∉ E. |
| Faded example | F = {0, 1, 2, 3}, n(F) = 4. |
| 1. Which number is irrational? | C. A thinks a root sign means irrational (√25 = 5); B forgets that a repeating decimal is rational (1/6); D confuses 22/7 with π. |
| 2. Counting a set from set-builder notation | 10. −6 to 3 inclusive: 6 negatives, 0, and 1, 2, 3. |
| 3. Rational or irrational | 4. −7, 0.25, 3/8 and √36 = 6; √2 and π are irrational. |
| 4. Element or subset | B. A lists a number not in A; C uses ⊂ for an element; D reads n(A) as the largest element. |
| 5. A condition on x² | 9. −4 ≤ x ≤ 4. |
| 6. A repeating decimal is in ℚ | 27. 0.185185… = 185/999 = 5/27. |
| 7. Set-builder with an inequality to solve | 7. x < 7 with ℕ starting at 0. |
| 8. How the number sets are related | C. A and B reverse the subsets; D forgets perfect squares. |
| 9. Every set a number belongs to | A. B ignores the sign; C thinks a root sign means irrational; D forgets that the sets are nested. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Thinks a root sign means irrational, forgets repeating decimals are rational, or confuses 22/7 with π. | Ask: can you write it as one integer over another? Try √25 and 0.1666… (= 1/6). |
| 11, 9 or 4 in question 2 | Included 4 despite <, left out −6 despite ≤, or counted only natural numbers. | Ask which end is closed and which is open, then mark both on a number line. |
| 3, 2 or 5 in question 3 | Thinks √36 is irrational, thinks integers are not rational, or counted √2 or π. | Ask for each number written as p/q. If it cannot be, it is irrational. |
| A, C or D in question 4 | Did not check the list, used ⊂ for an element, or read n(A) as the biggest element. | Say each statement aloud in words: “6 is a subset of A” sounds wrong once said. |
| 5, 4 or 33 in question 5 | Forgot that negative integers have positive squares, left out 0, or read the rule as −16 ≤ x ≤ 16. | Make a table of x and x² from −5 to 5. |
| 999 or 200 in question 6 | Did not simplify, or treated the decimal as ending. | Ask what 999 and 185 have in common (37), and whether the decimal stops. |
| 6, 8 or 21 in question 7 | Started ℕ at 1, included the boundary 7, or gave the bound instead of a count. | Check x = 0 and x = 7 in 3x − 1 < 20 directly. |
| A, B or D in question 8 | Reversed a subset, or thinks every root is irrational. | Find one counterexample for each false statement: −5, 1/2, √16. |
| B, C or D in question 9 | Ignored the minus sign, thinks a root sign means irrational, or forgot the sets are nested. | Ask: what is √49? So what is x? Then place x in the first figure's boxes. |
Questions worth asking
- Why does ℕ sit inside ℤ, and not the other way round? Listen for “every natural number is an integer, but −5 is an integer that is not natural”.
- Is √2 × √2 rational? Yes: it is 2. Two irrationals can multiply to a rational, which is a good reason to check rather than assume.
- Why can the set {x | x ∈ ℝ, 0 ≤ x < 1} not be listed? Between any two real numbers there is another one, so no list could ever be complete.
- How can you tell from its decimal that a number is rational? It ends or repeats. A decimal that goes on forever without repeating is irrational.
Differentiation
A way in for anyone stuck
- Give a large printed copy of the nested boxes from the first figure and have students write numbers into them before using symbols.
- Read every set-builder rule aloud as “the set of x such that…” before listing anything.
- Draw a number line for every listing question, marking a filled dot for ≤ and an open dot for <.
Stretch
- Prove that √2 is irrational: suppose √2 = p/q in lowest terms, square both sides and show that p and q are both even.
- Find n(A) for A = {x | x ∈ ℤ, |x| < 10, x is a multiple of 3}, then write a set-builder rule for the set {1, 4, 9, 16, 25}.
Students learning in English as an additional language
Set notation is a language of its own, so give each symbol its spoken form: ∈ “is an element of”, ⊆ “is a subset of”, ⊂ “is a proper subset of”, | “such that”. “Rational” comes from “ratio”: a number that is a ratio of two integers.
| Word | Plain meaning | Example |
|---|---|---|
| set | a collection of things, called its elements, written in curly brackets | {1, 2, 3, 6} |
| element | one member of a set | 5 is an element of {1, 3, 5, 7} |
| subset | a set whose elements are all inside another set, written ⊆; ⊂ means a subset that is not the whole set | {1, 3} ⊆ {1, 3, 5} |
| empty set | the set with no elements, written ∅ | {x | x ∈ ℕ, x < 0} = ∅ |
| integer | a whole number, positive, negative or zero | −3, 0, 7 |
| rational number | a number that can be written as one integer over another, not over 0 | −2/3, 0.25, 5 |
| irrational number | a real number that cannot be written as a fraction of two integers | √5 and π |
Sentence starters for explaining aloud or in writing:
- … is in ℚ because it can be written as …
- … is not in ℕ because …
- The set of x such that … is {…}, so n(A) = …
Two things not to say
“A square root is irrational.” √9, √25 and √36 are whole numbers; say “the square root of a number that is not a perfect square is irrational”.
“Natural numbers are the counting numbers 1, 2, 3.” In IB mathematics ℕ includes 0, and the exam answers depend on it.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ListingList {x | x ∈ ℤ, −3 < x ≤ 1} and find n.
{−2, −1, 0, 1}, n = 4. - ClassifyingGive every set that 0.6 belongs to.
ℚ and ℝ (0.6 = 3/5).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA lift in a Mumbai tower serves floors −2 (car park) to 14. Write the set of floors in set-builder notation and find n.
{x | x ∈ ℤ, −2 ≤ x ≤ 14}, n = 17. - Earlier work (Year 9)Write 0.4545… as a fraction in simplest form.
45/99 = 5/11.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs the sum of a rational number and an irrational number ever rational? Explain.
Never: if r + i = s with r, s rational, then i = s − r would be rational. - Working backwardsn(A) = 6 for A = {x | x ∈ ℤ, −2 ≤ x < b}. Find b.
A = {−2, …, 3}, so b = 4.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Draw the nested boxes on the board and give each pair of students cards with numbers to place.