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For teachersTopic 2.6 · Lesson kit

Comparing Boolean Expressions

  • 2 class periods
  • 45 minutes each
  • Objectives 2.6.A, 2.6.B
  • 6 downloads with keys

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At a glance

Pacing
2 class periods of 45 minutes
Learning objectives
2.6.A2.6.B
Essential knowledge
2.6.A.12.6.A.22.6.B.12.6.B.22.6.B.3
Materials
Projector and the Same or Equal? slide deckWhiteboard and markersMini whiteboards or scrap paperIDE on every computer (optional demonstrations)Printed Same or Equal? puzzle worksheet (answer key for you) and exit ticket

45 minutes a day

Agenda

Day 145 minutes

  • Warm-up
  • Direct instruction
  • Guided practice
  • Closure
  1. 0:00–0:055 minWarm-uptwo signs at the pool (slide 2)
  2. 0:05–0:2015 minDirect instructionequivalent expressions and truth tables (slides 3–7)
  3. 0:20–0:3515 minDirect instructionDe Morgan's laws (slides 9–12)
  4. 0:35–0:438 minGuided practiceequivalent or not, push the ! inside (slides 8 and 13)
  5. 0:43–0:452 minClosurewhat is a counterexample?

Day 245 minutes

  • Warm-up
  • Direct instruction
  • Guided practice
  • Closure
  • Exit ticket
  1. 0:00–0:055 minWarm-upsame text, same object?
  2. 0:05–0:2015 minDirect instruction== on objects, aliases, equals (slides 14–17)
  3. 0:20–0:288 minDirect instructionnull and the null check (slide 18)
  4. 0:28–0:357 minGuided practicewhat prints (slide 19)
  5. 0:35–0:405 minClosurevocabulary and sentence starter (slides 20–21)
  6. 0:40–0:455 minExit ticket

Overview

Day 1 is comparing Boolean expressions: two expressions are equivalent when they have the same value in every case, so one counterexample disproves it and a complete truth table proves it (2.6.A.1), and De Morgan’s laws rewrite !(a && b) as !a || !b and !(a || b) as !a && !b (2.6.A.2). Day 2 is comparing objects: == and != on references ask whether two variables refer to the same object, aliases share one object, and null means no object (2.6.B.1, 2.6.B.2); equals is the method a class uses to decide when two objects count as equivalent, and String compares characters (2.6.B.3).

Students know !, &&, and || and short-circuit evaluation (Topic 2.5), references and null (Topics 1.13 and 1.14), and equals on String (Topic 1.15). Stay inside the topic boundary: no loops, and no writing an equals method of your own (classes and methods come in Unit 3). Truth tables have two variables on Day 1; mention that three variables have eight rows.

Teacher note

The slides are the Same or Equal? deck (21 slides, with speaker notes); slide numbers below refer to it. Day 1 uses slides 1–13 and Day 2 uses slides 14–21. The Same or Equal? puzzle worksheet (10 questions, about 35 minutes) is split across the two nights: questions 1, 4, 5, and 8 after Day 1; questions 2, 3, 6, 7, 9, and 10 after Day 2.

Files whose names end in .snippet.java hold only the statements that go inside main. All files are in the topic’s examples/ folder. Everything here is optional to run; the slides carry the examples.

  • Day 1 demonstrations: FreezingCheck, PassingCheck, WrongNegation, DeMorganTable, NotBothTable, ImpliesTable, CounterExample, OrVersusNotEqual, TeenCheck, TeenWrong, StoreHoursWrong, StoreHoursFixed, MissingParens.
  • Day 2 demonstrations: SameObject, AliasTrace, ReassignAlias, EqualsVsDoubleEquals, PasswordWrong, PasswordFixed, NullCheck, NullGuard, EmptyNotNull, EqualsOnNull, EqualsNullArgument, LiteralFirstEquals, GuardWrongOrder.
  • Printed only (exit ticket): ExitRefs.

Day 1 · Warm-up: Two signs at the pool

Show slide 2. Sign 1 says “swim unless it is raining or cold”, and sign 2 says “swim if it is not raining or not cold”. Students fill the four rows: with raining and cold both yes, neither sign says swim; with raining yes and cold no, sign 1 says no and sign 2 says yes; with raining no and cold yes, sign 1 says no and sign 2 says yes; with neither, both say swim. Ask: do the two signs always agree? No, so they are not saying the same thing, and today’s lesson is how to tell.

Day 1 · Direct instruction: Equivalent expressions and truth tables

  1. Goals (slide 3). Goals 1–3 are Day 1; goals 4 and 5 are Day 2.
  2. Same value in every case (slide 5; 2.6.A.1). Two Boolean expressions are equivalent when they evaluate to the same value in all cases. One case where they differ, a counterexample, proves they are not. Test the value right on the boundary: with temp 32, !(temp < 32) is true and temp > 32 is false, so they are not equivalent; temp >= 32 is true and matches. The opposite of < is >=, not >.
  3. A truth table proves it (slide 6; 2.6.A.1). List every combination of values. If two columns match in every row, the expressions are equivalent. Count the rows: 4 for two variables, 8 for three. Walk the table for !(a || b) and !a && !b: they match in all four rows.
  4. Three matching rows prove nothing (slide 7). a && b and a == b agree in rows 1–3 and differ in row 4, so they are not equivalent.

Misconception: It matched for my test values, so they are equivalent

Students test one or two values, see a match, and stop. Matching cases prove nothing; a single mismatch proves they differ, and a complete truth table is the only way to prove equivalence. Boundary values are where look-alikes split apart. This is the Fair Question on worksheet question 5, and the warning below worksheet question 8.

Misconception: The opposite of x < 5 is x > 5

Students negate a comparison by flipping the symbol and forget the boundary. The opposite of < is >=, of > is <=, and of == is !=. Use WrongNegation or PassingCheck to show the failing value (the boundary itself). This is the Think First prediction on worksheet question 4.

Day 1 · Direct instruction: De Morgan’s laws

  1. Negate each part, flip the operator (slide 10; 2.6.A.2). !(a && b) is the same as !a || !b, and !(a || b) is the same as !a && !b. Put the table of opposite comparisons on the board: < and >=, > and <=, == and !=.
  2. Rewrite it without the ! (slide 11). !(level > 3 || lives == 0): flip || to &&, then negate each part, giving level <= 3 && lives != 0. The check at the bottom shows both are true for level 2 and lives 3.
  3. Half a De Morgan is wrong (slide 12). With x 1 and y -1, !(x > 0 && y > 0) is true, but x <= 0 && y <= 0 is false. The correct rewrite is x <= 0 || y <= 0.

Misconception: Negate the parts, but keep the operator

Students negate each comparison and leave && or || alone, or flip the operator and forget to negate the parts. Both steps are required. Teach the two-step routine on slide 11 and have students check a rewrite with one test case before they trust it. This is worksheet question 8.

Day 1 · Guided practice: Equivalent or not, push the ! inside

  1. Equivalent, or not? (slide 8). Pairs on mini whiteboards: 1, !(n > 0) and n <= 0, are equivalent. 2, !(n == 3) and n > 3, are not: n 1 is a counterexample (true and false). 3, !(done || late) and !done && !late, are equivalent. 4, !(ready && !tired) and !ready || tired, are equivalent (the ! on tired cancels).
  2. Push the ! inside (slide 13). Students rewrite with no ! in front of the parentheses: 1, !isOpen || !hasRoom. 2, age < 16 && !hasPermit. 3, count == 0 || total <= 100.

Teacher note

Run slide 8 first, before slide 10 if possible, so students lean on counterexamples. In pairs, ask each pair to find a counterexample for every “not equivalent” answer before showing the class.

Day 1 independent practice and closure

Puzzle worksheet questions 1, 4, 5, and 8 are homework: the truth table, the match-up, the snippet pool (age >= 13 && age <= 19), and the one-token fix (change && to ||). For closure, ask: what is a counterexample, and why is one enough?

Day 2 · Warm-up: Same text, same object?

Show two pictures on the board: two sticky notes that both say “pond”, and one sticky note with two arrows pointing to it. Ask which is “the same.” Take answers, then say that Java has both meanings, and it uses == for one and equals for the other.

Day 2 · Direct instruction: == on objects, aliases, equals

  1. == asks: the same object? (slide 15; 2.6.B.1). new String("pond") twice makes two objects, so home == away is false. After away = home, both variables refer to the same object, so home == away is true. Two variables that refer to the same object are aliases. Draw the before and after diagram on the board.
  2. Never compare text with == (slide 16). typed == saved with two separate objects displays Try again, even though the characters match; typed.equals(saved) displays Welcome.
  3. Each class decides what equal means (slide 17; 2.6.B.3). a == b asks about the same object, a != b asks about different objects, and a.equals(b) asks whether the contents are the same. A class defines its own equals method, usually by comparing attributes; String compares the characters. Students do not write equals in this topic.

Misconception: == compares the text of two Strings

Students used == on numbers, so they expect it to compare contents. For objects it compares references, and two separate objects with the same characters give false. The fix is equals. Worksheet question 2 (answer A, not B) and question 10 return to this.

Misconception: Reassigning one alias changes the other variable too

After b = c, students think a follows b. A reassignment changes only that variable’s reference. a still refers to its original object. This is the lie in worksheet question 3 (claim 3) and the code tiles in question 6.

Teacher note

Use new String("…") in demonstrations, as the guide does, so two equal Strings are really separate objects. String literals can share one object, so == can give a surprising true; do not go into that now.

Day 2 · Direct instruction: Null and the null check

Show slide 18 (2.6.B.2). x == null is true when x refers to no object. Check it before calling a method, or the program throws a NullPointerException. pick != null && pick.equals("blue") is safe because of short-circuit evaluation (Topic 2.5); with pick as null, the right side never runs. An empty string "" is not null: it is a real object with no characters, so pick.length() gives 0.

Misconception: An empty string is null

Students use the two words as if they meant the same thing. null means no object at all, so calling length() on it crashes; "" is a real String with length 0. Worksheet question 7, snippets 1 and 3, shows the contrast.

Misconception: The null check can go after the method call

Students write c.length() > 2 || c == null. The left side runs first and crashes. Same rule as the guard in Topic 2.5: the null check must be on the left. Use GuardWrongOrder to show the exception.

Day 2 · Guided practice: What prints

Show slide 19 with a as new String("hop"), b = a, and c as null. Students write each line’s output on mini whiteboards. A, a == b, is true (aliases). B, a.equals(new String("hop")), is true (same characters). C, c == null || c.length() > 2, is true, because the left side is true and the call is skipped. D, c.length() > 2 || c == null, crashes with a NullPointerException, because c.length() runs first. Ask what small change would make D safe.

Day 2 · Independent practice: Independent practice

Puzzle worksheet questions 2, 3, 6, 7, 9, and 10 are homework: the JVM trace on == and equals, two truths and a lie about aliases, the code tiles (alias, then a new object), you’re the compiler (null checks, and n == null for an int, which does not compile), who am I, and the wrap-up sentence.

Day 2 · Closure: Closure and exit ticket

Show slide 20 (vocabulary), then slide 21 and have students finish the sentence starter. Give the exit ticket (about five minutes). Question 1 checks De Morgan’s laws (2.6.A.2), question 2 checks ==, aliases, and equals (2.6.B.1, 2.6.B.3), and question 3 checks proving or disproving equivalence (2.6.A.1). Re-teach the weakest one at the start of Topic 2.7.

Exit ticket

Question 1 · Multiple choice

Which expression is equivalent to !(a > 4 || b == 0) for every pair of int values?

  1. a <= 4 || b != 0
  2. a <= 4 && b != 0
  3. a < 4 && b != 0
  4. a > 4 && b == 0
Check your reasoning

B. De Morgan’s law: !(p || q) is !p && !q. Negate each part (a > 4 becomes a <= 4, b == 0 becomes b != 0) and flip || to &&. A negates the parts but keeps ||. C uses a < 4, but the opposite of > is <=; they differ when a is 4. D is the original with the operator flipped and the parts not negated.

Question 2 · Tracing

Write the exact output.

String x = new String("ant");
String y = x;
String z = new String("ant");
System.out.println(x == y);
System.out.println(x == z);
System.out.println(x.equals(z));
z = y;
System.out.println(x == z);
Open in IDE
Check your reasoning
true
false
true
true

y = x makes y an alias for the same object, so x == y is true. z is a second, separate object, so x == z is false even though the characters match, and x.equals(z) is true because equals compares the characters. After z = y, z refers to the same object as x, so x == z is true.

A complete answer has all four lines, and a good one explains that == compares references and equals compares characters.

Question 3 · Short answer

Is !(p && q) equivalent to !p && !q? Prove it or disprove it.

Check your reasoning

Model answer: No, they are not equivalent. One counterexample is enough: let p be true and q be false. Then !(p && q) is !(false), which is true, but !p && !q is false && true, which is false. The correct De Morgan rewrite is !p || !q.

A complete answer says “not equivalent” and gives a specific pair of values (p true with q false, or p false with q true) with the value of both expressions. A full four-row truth table also earns full credit.