From Two Columns to a Paragraph

A two-column proof shows: given $3x + 5 = 20$, prove $x = 5$. Its rows are (1) $3x + 5 = 20$, given; (2) $3x = 15$, subtraction property of equality; (3) $x = 5$, division property of equality. Rewrite it as a paragraph proof.

Turn the first row into an opening sentence: Row 1 is the given fact, so the paragraph starts: *We are given that $3x + 5 = 20$.* = Opening sentence written

Join row 2 with its reason: *Subtracting 5 from both sides gives $3x = 15$ by the subtraction property of equality.* = $3x = 15$

Join row 3 with its reason: *Dividing both sides by 3 gives $x = 5$ by the division property of equality.* = $x = 5$

State the conclusion: *Therefore $x = 5$.* The word therefore marks the conclusion, the last row of the table. = Paragraph proof complete

Answer: We are given that $3x + 5 = 20$. Subtracting 5 from both sides gives $3x = 15$ by the subtraction property of equality. Dividing both sides by 3 gives $x = 5$ by the division property of equality. Therefore $x = 5$. QED

The Sum of Two Odd Integers Is Even

Write a paragraph proof: if $a$ and $b$ are odd integers, then $a + b$ is even.

Name the objects with separate letters: Let $a$ and $b$ be odd integers. Since they are odd, $a = 2m + 1$ and $b = 2n + 1$ for some integers $m$ and $n$. = $a = 2m + 1$, $b = 2n + 1$

Add and simplify: So $a + b = (2m + 1) + (2n + 1) = 2m + 2n + 2$. = $a + b = 2m + 2n + 2$

Factor out 2: It follows that $a + b = 2(m + n + 1)$. = $a + b = 2(m + n + 1)$

Give the reason for the conclusion: Because $m + n + 1$ is an integer (a sum of integers), $a + b$ is 2 times an integer, so it is even by the definition of an even number. = $a + b$ is even

Answer: Let $a$ and $b$ be odd integers. Since they are odd, $a = 2m + 1$ and $b = 2n + 1$ for some integers $m$ and $n$. Then $a + b = 2m + 2n + 2 = 2(m + n + 1)$. Because $m + n + 1$ is an integer, $a + b$ is even by the definition of an even number. QED

Three Consecutive Integers

Prove in a paragraph: the sum of any three consecutive integers is divisible by 3. Then check the result with $14 + 15 + 16$.

Name the three integers: Let the smallest integer be $n$. Since the integers are consecutive, the other two are $n + 1$ and $n + 2$. = $n$, $n + 1$, $n + 2$

Add them: Their sum is $n + (n + 1) + (n + 2) = 3n + 3$. = $3n + 3$

Factor and conclude: So the sum equals $3(n + 1)$. Because $n + 1$ is an integer, the sum is 3 times an integer, therefore it is divisible by 3. = Sum $= 3(n + 1)$

Check with an example (not part of the proof): For $14, 15, 16$ we have $n = 14$, so the sum should be $3 \times 15 = 45$. Indeed $14 + 15 + 16 = 45$. = $45 = 3 \times 15$

Answer: Let the integers be $n$, $n + 1$ and $n + 2$. Their sum is $3n + 3 = 3(n + 1)$, and since $n + 1$ is an integer, the sum is divisible by 3. Bonus fact from the proof: the sum is always 3 times the middle number. QED

Vertical Angles Are Equal

Two lines intersect and form four angles, $\angle 1$, $\angle 2$, $\angle 3$, $\angle 4$ in order around the point, so $\angle 1$ and $\angle 3$ are vertical angles. Write a paragraph proof that $\angle 1 = \angle 3$ (the lesson Two-Column Proofs proves the same theorem as a table).

Use the first linear pair: Since $\angle 1$ and $\angle 2$ together form a straight line, they are a linear pair, so $\angle 1 + \angle 2 = 180^\circ$. = $\angle 1 + \angle 2 = 180^\circ$

Use the second linear pair: Likewise $\angle 2$ and $\angle 3$ form a straight line, so $\angle 2 + \angle 3 = 180^\circ$. = $\angle 2 + \angle 3 = 180^\circ$

Combine the two equations: Both sums equal $180^\circ$, so by substitution $\angle 1 + \angle 2 = \angle 2 + \angle 3$. = $\angle 1 + \angle 2 = \angle 2 + \angle 3$

Subtract and conclude: Subtracting $\angle 2$ from both sides (subtraction property of equality) gives $\angle 1 = \angle 3$. Check: if $\angle 1 = 65^\circ$, then $\angle 2 = 115^\circ$ and $\angle 3 = 180^\circ - 115^\circ = 65^\circ$. = $\angle 1 = \angle 3$

Answer: Angles 1 and 2 form a linear pair, so $\angle 1 + \angle 2 = 180^\circ$. Angles 2 and 3 also form a linear pair, so $\angle 2 + \angle 3 = 180^\circ$. By substitution $\angle 1 + \angle 2 = \angle 2 + \angle 3$, and subtracting $\angle 2$ gives $\angle 1 = \angle 3$. Therefore vertical angles are equal. QED

Finding the Gap

A student wants to prove: the product of an even integer and any integer is even. The student writes: *Let $a = 2k$ and $b = 2k$. Then $ab = 4k^2 = 2(2k^2)$, so $ab$ is even.* Find the gaps and repair the proof.

Read what the statement really says: The statement is about an even integer $a$ and **any** integer $b$, which may be odd and may be different from $a$. = $b$ can be any integer

Check every sentence: Writing $b = 2k$ assumes two things that were never given: that $b$ is even, and that $b = a$ (same letter $k$). The proof only covers products like $6 \times 6$, not $6 \times 7$. = Gap: $b = 2k$ is not justified

Repair the start: Let $a$ be even and $b$ be any integer. Since $a$ is even, $a = 2k$ for some integer $k$. Nothing is assumed about $b$. = $a = 2k$, $b$ arbitrary

Finish with reasons: Then $ab = 2kb = 2(kb)$. Because $kb$ is an integer (a product of integers), $ab$ is even by definition. Check: $6 \times 7 = 42 = 2 \times 21$. = $ab = 2(kb)$ is even

Answer: The original proof has a gap: it silently assumes that $b$ is even and equal to $a$. Correct paragraph proof: Let $a$ be even and $b$ any integer. Since $a$ is even, $a = 2k$ for some integer $k$. Then $ab = 2(kb)$, and since $kb$ is an integer, $ab$ is even. QED

Mistake: Writing a chain of equations with no words

Why: A list like $a = 2m + 1$, $a + b = 2(m + n + 1)$ does not say where each line comes from or what it proves. The reader has to guess the reasons.

Correct: Attach a reason to every claim (*Since $a$ is odd, $a = 2m + 1$...*) and finish with a sentence that states the conclusion.

Mistake: Using the same letter for two different numbers

Why: Writing $a = 2k + 1$ and $b = 2k + 1$ forces $a = b$, so the proof only covers sums like $7 + 7$.

Correct: Give every object its own variable: $a = 2m + 1$ and $b = 2n + 1$.

Mistake: Proving with examples

Why: $3 + 5 = 8$ and $7 + 9 = 16$ are even, but a few examples can never cover all odd numbers.

Correct: Use variables that stand for every possible case. Examples are useful only as a check after the proof.

Mistake: Using the conclusion as a reason

Why: Writing *Since $a + b$ is even, it equals $2(m + n + 1)$* assumes exactly what you are trying to prove (circular reasoning).

Correct: The words since and because may only point to facts you already have: the given, definitions, properties or earlier sentences.

Calendar Puzzles

Any three days next to each other in a calendar row are consecutive integers, so the paragraph proof from this lesson explains a classic number trick.

Pick 12, 13 and 14: the sum is $39 = 3 \times 13$, three times the middle day.

Scissors and Crossing Roads

Where two straight objects cross, like scissor blades, crossing roads or the beams of a roof truss, the vertical angles theorem lets builders measure one angle and know the opposite one.

If the blades of open scissors make a $35^\circ$ angle, the handles make a $35^\circ$ angle too.

Packing in Pairs

Shops and warehouses often pack items two per box. The proof that an even number times any integer is even tells you when a grid of items splits into pairs with nothing left over.

A tray with 8 rows of 7 cookies holds $56 = 2 \times 28$ cookies, so they fill 28 bags of two.

Q: Is a paragraph proof less rigorous than a two-column proof?

A: No. Both formats contain the same logic. A paragraph proof is only a different layout: every claim still needs a reason, it is simply written inside the sentence instead of in a second column.

Q: How long should a paragraph proof be?

A: Long enough that a classmate could follow every step without asking why. For the proofs in this lesson, 3 to 6 sentences is typical. Simple algebra, like factoring out 2, does not need its own sentence.

Q: Can I use examples in a paragraph proof?

A: Only as a check, clearly separated from the proof. A sentence like *for example, $3 + 5 = 8$* shows the idea, but the proof itself must use variables that cover every case.

Q: Which signal word should I use?

A: Use since or because before a reason, and so, therefore or it follows that before a new claim. Vary them so the paragraph reads naturally, but make sure every claim has a reason somewhere in its sentence.