Turning English into algebra: the phrases that signal addition, subtraction, multiplication and division; the two operations whose order the words can reverse; the difference between a phrase and a sentence; and building a complete equation or inequality from a described situation.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 1 — Connections to Algebra
Translating Words into Mathematical Symbols
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 30-35 — the lesson these objectives are drawn from
Warm-up
You translate English into arithmetic constantly. The only new part is that one of the numbers is now a letter.
Discussion prompt
Write down what 5 less than 12 is. Then write down, in symbols, exactly what you did — and notice which number you wrote first.
Hint: Say the calculation aloud and compare the order of the words with the order of your symbols.
Answer:
\[ 5 \text{ less than } 12 \;\rightarrow\; 12 - 5 = 7 \]
You wrote the twelve first, even though the five was spoken first. You have been reversing that phrase correctly for years without noticing. This lesson simply makes the reversal explicit, because once one of the numbers becomes a letter the mistake stops being obvious.
Concept
Translating a sentence into algebra is two separate jobs. The noun phrases become expressions, and the verb becomes the symbol that joins them. Do them in that order and even a long sentence becomes manageable.
translate — To rewrite words as mathematical symbols. Phrases become expressions; sentences become equations or inequalities.
A phrase has no verb, so it has nothing to join and produces an expression rather than a statement.
Figure (svg): Two columns contrasting phrases, which become expressions, with sentences, which become equations or inequalities
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 30-31
Section
Section 1
Concept
English has several ways of asking for an addition. Because addition gives the same answer in either order, none of these phrasings can be translated backwards by accident.
Sum, more than, plus and increased by all mean the same thing, and 6 plus x is the same number as x plus 6.
| Phrase | Expression |
|---|---|
| the sum of 6 and a number | 6 + x |
| 8 more than a number | x + 8 |
| a number plus 5 | x + 5 |
| a number increased by 7 | x + 7 |
Figure (svg): Four addition phrases paired with the expressions they translate to, all involving x
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 30-30 — Example 1, Translate Addition Phrases
Picture it
Cover the right-hand column and translate each phrase, then cover the left and read each expression back into English.
Figure (svg): Four addition phrases paired with the expressions they translate to, all involving x
Notice that the second row writes the number first even though the phrase says eight first. That is allowed here precisely because addition is order-free — and it is exactly what is not allowed in the next section.
Worked example
This is Example 1 from the textbook. Let x represent the number in each case.
\[ \text{Translate: the sum of } 6 \text{ and a number; } 8 \text{ more than a number; a number plus } 5; \text{ a number increased by } 7. \]
Sum signals addition
Why: The sum of two things is what you get by adding them, so the two quantities are joined by a plus sign.
\[ 6 + x \]
More than also signals addition
Why: Eight more than a number means the number with eight added on.
\[ x + 8 \]
Plus is the most direct signal there is
Why: The word plus is simply the spoken name of the sign.
\[ x + 5 \]
Increased by signals addition too
Why: Anything that describes growth by a fixed amount is an addition.
\[ x + 7 \]
Figure (svg): The solution to Worked example four addition phrases shown as a ladder of expressions, one row per algebraic move
\[ 6 + x, \quad x + 8, \quad x + 5, \quad x + 7 \]
Verify: test each translation at a specific number
Why: Take the number to be 10. The four phrases should give sixteen, eighteen, fifteen and seventeen, and the four expressions do exactly that. Substituting a concrete value is the fastest way to confirm that a translation says what the English said.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 30-30
Sorting
Every phrase below signals exactly one of the four operations.
Sort into buckets
Sort each phrase by the operation it calls for.
Two of the four columns hold order-free operations and two hold order-sensitive ones. That split, rather than the vocabulary, is what makes some translations risky.
Worked example
Guided Practice 1 and 2. One of these reverses and one does not.
\[ \text{Translate: } 11 \text{ more than a number; a number decreased by } 10. \]
Eleven more than a number is an addition
Why: More than means added on, and addition may be written in either order.
\[ x + 11 \]
Decreased by ten is a subtraction
Why: Decreased by describes shrinking, so ten is being taken away from the number.
\[ x - 10 \]
Check which quantity is being reduced
Why: The number is what decreases, so the number goes first and the ten is subtracted from it.
\[ x - 10\text{ not } 10 - x \]
Notice the difference in risk between the two
Why: The first cannot be written backwards; the second can, and reading it carefully is what prevents it.
Figure (svg): The solution to Worked example guided practice on addition and subtraction shown as a ladder of expressions, one row per algebraic move
\[ x + 11 \qquad x - 10 \]
Verify: substitute 25 into both
Why: Eleven more than twenty-five is thirty-six, and the first expression gives thirty-six. Twenty-five decreased by ten is fifteen, and the second gives fifteen. Had the second been written backwards it would have given negative fifteen, which is visibly not a decrease of the original number.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 30-30
Trap
\[ \text{8 more than a number} \;\rightarrow\; x + 8 \;\text{ or }\; 8 + x \]
Conclude that word order never matters, because it did not matter here
Why: Two correct translations in different orders is strong evidence for a rule that is only true for addition.
\[ \text{4 less than a number} \;\rightarrow\; 4 - y \quad \text{(wrong)} \]
The habit formed on addition is carried straight into subtraction, where it produces the wrong sign every single time.
\[ \text{8 more than a number} = x + 8 = 8 + x \]
Note that order is free only because addition itself is order-free
Why: The freedom belongs to the operation, not to the translating.
\[ \text{4 less than a number} = y - 4 \neq 4 - y \]
Ask which operation you are translating before you decide whether order is negotiable. For addition and multiplication it is; for subtraction and division it never is.
Translation
Four phrasings of the same operation. Let x be the number.
Match the pairs
Why: All four are additions, so any of them could equally be written the other way round without changing its value. The differences are purely in the English: sum, more than, plus and increased by are four ways of asking for the same operation, and recognising all four is what stops an unfamiliar phrasing from stopping you.
Elimination
Three of these phrases call for a plus sign and one does not.
Eliminate the wrong options
Which phrase does NOT translate to an addition?
Survives elimination: D
Why: Less than signals a subtraction, and it also reverses the spoken order: four less than a number is the number minus four. It is the odd one out twice over — different operation, and different word order from what you hear — which is why it appears in almost every test on this topic.
Faded example
The phrase and the variable are given. Supply the expression and its value.
Fill in the blanks
\text11 \;\rightarrow\; x + 36 \qquad \text___ x = 25: \; ___
Why: More than signals addition, so eleven is added to the number, and at twenty-five the expression gives thirty-six. Substituting a concrete number after translating is the check that the translation says what the English said — and here it plainly does, since eleven more than twenty-five really is thirty-six.
Section
Section 2
Concept
Order is important for subtraction. Most phrases put the quantities in the order you write them, but the phrase less than reverses what you hear, and it is the single most common translation mistake in the chapter.
Four less than a number means the number minus four, not four minus the number.
| Phrase | Expression |
|---|---|
| the difference between 5 and a number | 5 - y |
| 7 minus a number | 7 - y |
| a number decreased by 9 | y - 9 |
| 4 less than a number | y - 4 |
Figure (svg): Four subtraction phrases paired with their expressions, highlighting that 4 less than a number is y minus 4 and not 4 minus y
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 30-30 — Example 2, Translate Subtraction Phrases, and the Reading Algebra note
Picture it
The colour changes at the third row, where the number stops being second and starts being first.
Figure (svg): Four subtraction phrases paired with their expressions, highlighting that 4 less than a number is y minus 4 and not 4 minus y
Difference between and minus put the spoken first quantity first. Decreased by and less than put the number first, whatever was spoken first. Only the last of those four contradicts the spoken order.
Worked example
Example 2 from the textbook. Let y represent the number.
\[ \text{Translate: the difference between } 5 \text{ and a number; } 7 \text{ minus a number; a number decreased by } 9; \; 4 \text{ less than a number.} \]
Difference between 5 and a number keeps the spoken order
Why: The phrase names the five first, and it is the quantity being reduced.
\[ 5 - y \]
Seven minus a number also keeps the order
Why: Minus is the spoken name of the sign, so the words map straight onto symbols.
\[ 7 - y \]
A number decreased by 9 puts the number first
Why: The number is what is shrinking, and nine is the amount it shrinks by.
\[ y - 9 \]
4 less than a number reverses what you hear
Why: The four is the amount taken away, and the number is what it is taken from — so the number is written first despite being spoken second.
\[ y - 4 \]
Figure (svg): The solution to Worked example four subtraction phrases shown as a ladder of expressions, one row per algebraic move
\[ 5 - y, \quad 7 - y, \quad y - 9, \quad y - 4 \]
Verify: substitute 10 into all four and sanity-check the signs
Why: With the number equal to ten the four expressions give negative five, negative three, one and six. The first two are negative because ten is larger than five and seven, which is correct — taking a larger number from a smaller one has to land below zero.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 30-30
Discrimination
For each phrase, decide whether the algebra puts the quantities in the same order as the English.
Sort into buckets
Sort each phrase by whether translating it reverses the spoken order.
Worked example
The clearest way to feel the difference is to translate both readings and evaluate them.
\[ \text{Compare } \; 4 \text{ less than } y \; \text{ with } \; 4 \text{ minus } y, \text{ at } y = 10. \]
Translate the first phrase
Why: Less than reverses the order, so the four is taken from the number.
\[ y - 4 \]
Translate the second phrase
Why: Minus keeps the order, so the number is taken from the four.
\[ 4 - y \]
Substitute 10 into each
Why: Ten minus four, and four minus ten.
\[ 6\text{ and } -6 \]
Compare the results
Why: The two answers are opposites, which is what reversing a subtraction always does.
\[ 6\text{ against } -6 \]
Figure (svg): Two panels showing 4 less than a number as y minus 4 and the incorrect reading 4 minus y, evaluated at y equal to 10
\[ y - 4 = 6 \qquad 4 - y = -6 \]
Verify: check that the two results really are opposites
Why: Six and negative six sum to zero, which confirms they are the same distance from zero on opposite sides. That is the general fact: reversing a subtraction always negates the answer, so a reversal never produces a merely slightly wrong number.
Error analysis
The student translated four phrases, letting n be the number. Two of the four are wrong.
Annotate
On: \( \begin{aligned} \text{6 less than a number} &\;\rightarrow\; 6 - n \\ \text{a number decreased by 3} &\;\rightarrow\; n - 3 \\ \text{the difference between 9 and a number} &\;\rightarrow\; n - 9 \\ \text{a number minus 2} &\;\rightarrow\; n - 2 \end{aligned} \)
There is only one phrase in the whole lesson that reverses, and it is less than. Everything else can be translated left to right as spoken.
Translation
Four phrasings, four expressions. Let y be the number.
Match the pairs
Why: The first two keep the spoken order because the fixed number is named first and it is the quantity being reduced. The third names the number first because the number is what decreases. The fourth is the reversal: the four is spoken first but written second, since it is the amount being taken from the number rather than the thing being reduced.
Prediction
Reversing a subtraction is never a small error. Predict how large it is.
Predict first
The correct expression is y minus 4 and a student writes 4 minus y. At y equal to 30, how far apart are the two answers?
Correct: 52 apart — 26 against negative 26.
\[ y - 4 \big|_{y=30} = 26 \qquad 4 - y \big|_{y=30} = -26 \qquad \text{gap } = 52 \]
Why: Thirty minus four is twenty-six, and four minus thirty is negative twenty-six. The two answers are opposites, so the gap between them is twice the correct answer, which grows as the number grows. A reversal is never a near miss, and the larger the numbers involved the more obviously wrong it becomes.
Socratic
The reversal is not arbitrary. It comes from what the English is describing.
Discussion prompt
Explain why the phrase less than names its two quantities in the opposite order from the symbols. Use the everyday meaning of the phrase in your explanation, and give an example from outside mathematics where the same reversal happens.
Hint: Think about what less than describes when you say someone is two years younger.
Answer:
Less than describes a comparison, and it names the gap before it names what the gap is measured from. Four less than a number is describing the number, and telling you to go four below it — so the number is the starting point and the four is the step, even though the four was said first.
The same reversal shows up in everyday speech: two years younger than Sam means Sam's age minus two, not two minus Sam's age. Nobody is confused by that sentence, and reading the algebraic phrase the same way is what makes the translation reliable.
Section
Section 3
Concept
Product, times and multiplied by all signal multiplication, and because multiplication is order-free none of them can be written backwards. Division is different: quotient and divided by both signal division, but you have to decide which quantity goes underneath the bar.
The word quotient names the numerator first, and the phrase divided by names the numerator first as well.
| Phrase | Expression |
|---|---|
| the product of 9 and a number | 9n |
| 10 times a number | 10n |
| a number multiplied by 3 | 3n |
| one fourth of a number | n over 4 |
| the quotient of a number and 6 | n over 6 |
| 7 divided by a number | 7 over n |
Figure (svg): Six multiplication and division phrases paired with their expressions
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 31-31 — Example 3 and the Vocabulary Tip on quotient
Picture it
The last two rows use the same two quantities in opposite positions.
Figure (svg): Six multiplication and division phrases paired with their expressions
The quotient of a number and 6 puts the number on top. Seven divided by a number puts the number underneath. In both cases the quantity named first ends up on top, which is the rule worth remembering.
Worked example
Example 3 from the textbook. Let n represent the number.
\[ \text{Translate the six phrases in the table, letting } n \text{ be the number.} \]
The product of 9 and a number is a multiplication
Why: Product names the result of multiplying, and a number written against a letter is a product.
\[ 9 n \]
Ten times a number and a number multiplied by 3 are also multiplications
Why: Both are order-free, so the number may be written in front in each case.
\[ 10 n\text{ and } 3 n \]
One fourth of a number is a division by 4
Why: Taking a fraction of something means multiplying by that fraction, which is the same as dividing by its denominator.
\[ n\text{ over } 4 \]
The quotient of a number and 6 puts the number on top
Why: Quotient names the numerator first, so the number is being divided by six.
\[ n\text{ over } 6 \]
Seven divided by a number puts the number underneath
Why: The quantity named before the words divided by is the one on top.
\[ 7\text{ over } n \]
Figure (svg): The solution to Worked example six multiplication and division phrases shown as a ladder of expressions, one row per algebraic move
\[ 9n, \quad 10n, \quad 3n, \quad \tfrac{n}{4}, \quad \tfrac{n}{6}, \quad \tfrac{7}{n} \]
Verify: substitute n equal to 2 and check the sizes
Why: The three products give 18, 20 and 6, all larger than two, as multiplying by a number above one should. The three quotients give one half, one third and three and a half — the first two smaller than two, and the last one larger because a small denominator makes a big quotient. Every size behaves as its operation predicts.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 31-31
Translation
Six phrases, six expressions. Let n be the number.
Match the pairs
Why: The last two are the pair worth studying: both are divisions involving a number and a fixed value, and they put the number in opposite positions. The rule that settles both is the same — whichever quantity is named first goes on top — and one fourth of a number follows it too, since taking a fourth of something is dividing it by four.
Worked example
Guided Practice 3 and 4, with the risky one made explicit.
\[ \text{Translate: the quotient of } 8 \text{ and a number; the product of } 2 \text{ and a number.} \]
The quotient of 8 and a number puts the 8 on top
Why: Quotient names its numerator first, and here that is the eight.
\[ 8\text{ over } x \]
Contrast with the quotient of a number and 8
Why: Reversing the two nouns swaps the positions, giving the number on top instead.
\[ x\text{ over } 8 \]
The product of 2 and a number is a multiplication
Why: Order is free, so this is simply two against the letter.
\[ 2 x \]
Note which of the two carried a risk
Why: Only the division could have been written backwards, because only division cares about order.
Figure (svg): The solution to Worked example the two divisions side by side shown as a ladder of expressions, one row per algebraic move
\[ \tfrac{8}{x} \qquad 2x \]
Verify: substitute x equal to 4 into both divisions
Why: The quotient of eight and four is two; the quotient of four and eight is one half. Two very different numbers from the same pair of quantities, which is why the order in a division has to be decided rather than assumed.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 31-31
Trap
\[ \text{the quotient of a number and 6} \;\rightarrow\; \tfrac{6}{n} \]
Write the two quantities in whichever order feels natural, since division only involves two numbers
Why: With multiplication just behind you, order feels negotiable.
\[ \text{At } n = 2: \quad \tfrac{6}{2} = 3 \quad \text{but the correct value is } \tfrac{2}{6} = \tfrac{1}{3} \]
Three against one third. Reversing a division does not shift the answer slightly — it replaces it with its reciprocal.
\[ \text{the quotient of a number and 6} = \tfrac{n}{6} \]
Put the quantity named first on top
Why: Both quotient of A and B, and A divided by B, name the numerator first. One rule covers every division phrase in the lesson.
\[ \text{At } n = 2: \quad \tfrac{2}{6} = \tfrac{1}{3} \]
A quick size check confirms it: dividing a number by six should make it smaller, and one third is smaller than two while three is not.
Elimination
All four expressions involve the number 4 and a variable n.
Eliminate the wrong options
Which expression has the value one half when n is 4?
Survives elimination: A
Why: Four over eight is one half. The second option is its reciprocal, which is what you get whenever the two quantities in a division are swapped, and noticing that a wrong answer is the reciprocal of the right one is a reliable sign that the order was reversed rather than the arithmetic botched.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Operation | Can it be written in either order? | Example phrase |
|---|---|---|
| Addition | Yes | the sum of 6 and a number |
| Subtraction | No | 4 less than a number |
| Multiplication | Yes | the product of 9 and a number |
| Division | No | the quotient of a number and 6 |
The two order-sensitive operations are exactly the two that Lesson 2.4 and Lesson 2.8 will rewrite as addition and multiplication, precisely so that this problem goes away.
Edge cases
Some substitutions hide a reversed division. Find one that does not.
Discussion prompt
A student writes 6 over n when the correct expression is n over 6. Find a value of n where the two expressions agree, and explain why testing at that value would fail to catch the error. Then give a value that exposes it clearly.
Hint: Look for a number that is its own reciprocal after the division.
Answer:
\[ \text{At } n = 6: \quad \tfrac{6}{6} = 1 = \tfrac{6}{6} \]
At n equal to six the two expressions both give one, so a check at that value proves nothing. This is a general hazard: a test value that happens to make the two quantities equal cannot distinguish an expression from its reversal.
\[ \text{At } n = 2: \quad \tfrac{2}{6} = \tfrac{1}{3} \quad \text{against} \quad \tfrac{6}{2} = 3 \]
Two is a much better test value, because the two quantities are different and the results differ by a factor of nine. When checking a translation, deliberately avoid values that make the numbers in the expression coincide.
Section
Section 4
Concept
In English a phrase has no verb and a sentence has one. That grammatical difference carries straight into algebra: a phrase becomes an expression, and a sentence becomes an equation or an inequality.
Figure (svg): Five English verbs paired with the relation symbols they translate to
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 31-31 — the Translating Sentences box and Example 4
Picture it
Translating is two jobs, and this figure is the second one.
Figure (svg): Five English verbs paired with the relation symbols they translate to
Do the nouns first and the verb last. Once both halves are expressions, the verb tells you exactly which symbol goes between them.
Worked example
Example 4 from the textbook. One becomes an equation and one becomes an inequality.
\[ \text{Translate: the sum of a number } x \text{ and } 12 \text{ is } 16; \text{ the quotient of } 15 \text{ and a number } x \text{ is less than } 3. \]
Split the first sentence at its verb
Why: The noun phrase is the sum of a number and twelve; the verb is is; the second noun phrase is sixteen.
\[ x + 12 |\text{ is } | 16 \]
Translate each half and join with an equal sign
Why: The word is on its own becomes an equal sign.
\[ x + 12 = 16 \]
Split the second sentence at its verb
Why: The noun phrase is the quotient of fifteen and a number; the verb is is less than; the second noun phrase is three.
\[ \frac{15}{x} |\text{ is less than } | 3 \]
Translate each half and join with the correct symbol
Why: Quotient names its numerator first, so the fifteen is on top, and is less than becomes the strict symbol.
\[ \frac{15}{x} < 3 \]
Figure (svg): The solution to Worked example two sentences into symbols shown as a ladder of expressions, one row per algebraic move
\[ x + 12 = 16 \qquad \tfrac{15}{x} < 3 \]
Verify: test each statement at a value
Why: In the first, x equal to 4 gives sixteen equals sixteen, so 4 is a solution. In the second, x equal to 10 gives one and a half, which is less than three, so the statement is true there. Both translations behave like statements — they can be tested — which confirms they came from sentences rather than phrases.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 31-31
Sorting
Decide whether each item would become an expression or a statement.
Sort into buckets
Sort each item by what it translates into.
Each sentence in the right column is its neighbour from the left column plus a verb and one more quantity. That is the whole difference, and it is the difference between something you evaluate and something you solve.
Worked example
Guided Practice 5 and 6. Notice how little of the work is arithmetic.
\[ \text{Translate: the product of } 5 \text{ and a number } x \text{ is } 25; \; 10 \text{ times a number } x \text{ is at least } 50. \]
Translate the first noun phrase of each sentence
Why: Product of five and a number is 5x; ten times a number is 10x. Both are order-free multiplications.
\[ 5 x\text{ and } 10 x \]
Identify the verb in each sentence
Why: The first says is; the second says is greater than or equal to.
\[ =\text{ and } \ge \]
Translate the second noun phrase of each
Why: Both are plain numbers, so they need no work.
\[ 25\text{ and } 50 \]
Assemble both statements
Why: Left expression, relation symbol, right expression.
\[ 5 x = 25\text{ and } 10 x \ge 50 \]
Figure (svg): The solution to Worked example an equation and an inequality from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 5x = 25 \qquad 10x \geq 50 \]
Verify: find a value satisfying each
Why: For the equation, x equal to 5 gives twenty-five, so it is the solution. For the inequality, x equal to 5 gives exactly fifty, which the bar permits, and every larger value passes too. The equation pinned down one value and the inequality described a range, which is exactly the difference between the two symbols.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 31-31
Trap
\[ \text{The sum of a number and 12 is 16} \;\rightarrow\; x + 12 \]
Translate the noun phrase and stop
Why: The first half of the sentence looks like a complete piece of algebra, so the work feels done.
An expression cannot be true or false and cannot be solved. The sentence made a claim, and the translation threw it away.
\[ \text{The sum of a number and 12 is 16} \;\rightarrow\; x + 12 = 16 \]
Find the verb and translate it too
Why: Every sentence has a verb, and in a mathematical sentence the verb is the relation symbol.
A quick test: can the thing you wrote be true or false? If not, you translated a phrase when you were given a sentence, and the verb is missing.
Translation
Four verbs, four relation symbols. Two of them include their boundary.
Match the pairs
Why: Is on its own is the equal sign. Is less than is strict, with no bar. At least and at most both include their boundary, so both take a bar, and they point in opposite directions: at least means the quantity may not go below the value, and at most means it may not go above it. Those two phrases appear in almost every real constraint you will meet.
Fill the middle
Each line has one half translated. Complete it.
Fill in the blanks
\text= x \text16 12 \text___ 16 \;\rightarrow\; x + 12 \;___\; ___
Why: The verb is becomes an equal sign, and the noun phrase after it is the plain number sixteen. Splitting a sentence at its verb turns one hard translation into two easy ones plus a symbol, and it works on sentences far longer than this one.
Socratic
It would be simpler if everything just became symbols. The distinction earns its keep.
Discussion prompt
Explain what you can do with an equation that you cannot do with an expression, and what you can do with an expression that makes no sense for an equation. Give one concrete example of each.
Hint: Think about the words evaluate and solve.
Answer:
An equation can be solved and can be checked, because it makes a claim that is true for some values and false for others. Asking whether 4 is a solution of x plus 12 equals 16 is a sensible question with a yes-or-no answer.
An expression can be evaluated, which an equation cannot be. Asking for the value of x plus 12 when x is 4 makes sense and gives 16; asking for the value of x plus 12 equals 16 does not, because a claim does not have a numerical value. Keeping the two straight is why Chapter 3 can talk about solving without ambiguity.
Section
Section 5
Concept
A real problem does not hand you a phrase to translate. You have to decide what the unknown is, name it, and only then write the statement. Writing the definition down as a sentence is what makes the rest of the translation possible.
Skipping the middle step is why long word problems feel impossible: there is nothing written down to translate.
Figure (svg): A bar model of a phone bill: a fixed connection charge plus a per-minute rate times the number of minutes
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 32-33
Picture it
Something fixed, plus something per unit, equalling a total.
Figure (svg): A bar model of a phone bill: a fixed connection charge plus a per-minute rate times the number of minutes
Once you can see this shape, the translation stops being about vocabulary and becomes about identifying which number plays which role.
Worked example
The situation from the start of the lesson. A long-distance call costs 60 cents to connect plus 8 cents a minute, and the call cost 1 dollar 96.
\[ \text{Write an equation for the length of the call in minutes.} \]
Decide what is unknown and name it
Why: The question asks how long the call was, so the unknown is a number of minutes.
\[ \text{let } m =\text{ minutes} \]
Say the relationship in English first
Why: The connection charge plus the per-minute charge times the number of minutes equals the total cost.
Translate each noun phrase
Why: The connection charge is 0.60; the per-minute charge times the minutes is 0.08m; the total is 1.96.
\[ 0.60, 0.08 m, 1.96 \]
Join them with the symbol the verb calls for
Why: The verb equals becomes an equal sign.
\[ 0.60 + 0.08 m = 1.96 \]
Figure (svg): The solution to Worked example the length of a phone call shown as a ladder of expressions, one row per algebraic move
\[ 0.60 + 0.08m = 1.96 \]
Verify: check the units of every term
Why: Dollars plus dollars per minute times minutes gives dollars plus dollars, which matches the dollars on the right. Had the rate been left in cents while the total was in dollars, the units would have disagreed and the equation would have been wrong before any solving started.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 32-32
Elimination
A call costs 60 cents to connect plus 8 cents a minute, and the total was 1 dollar 96. Let m be the minutes.
Eliminate the wrong options
Which equation is correct?
Survives elimination: A
Why: The 60 cents is paid once so it is added on, and the 8 cents is paid per minute so it multiplies m. Option C is worth noticing separately: the arithmetic error is not in the structure but in the units, and a units check catches it before any solving happens.
Worked example
The same situation with a budget instead of an exact bill.
\[ \text{You have } 2 \text{ dollars. Write an inequality for the calls you can afford.} \]
Keep the same variable and definition
Why: The unknown is still the number of minutes, and it still means the same thing.
\[ \text{let } m =\text{ minutes} \]
Say the relationship in English
Why: The cost of the call must be at most the two dollars you have.
\[ \cos t\text{ is at most } 2 \]
Translate the cost expression
Why: It is unchanged from the previous example: the fixed charge plus the rate times the minutes.
\[ 0.60 + 0.08 m \]
Translate the verb
Why: At most includes the boundary, so the symbol carries a bar.
\[ 0.60 + 0.08 m \le 2 \]
Figure (svg): The solution to Worked example a sentence that becomes an inequality shown as a ladder of expressions, one row per algebraic move
\[ 0.60 + 0.08m \leq 2 \]
Verify: test a value you can reason about
Why: At m equal to 17 the cost is 0.60 plus 1.36, which is 1.96, and that is at most two — so a seventeen-minute call is affordable. At m equal to 18 the cost is 2.04, which is not, so the answer sits between them. An inequality describing a range is exactly right for a question about what you can afford.
Trap
\[ 0.60 + 0.08x = 1.96 \]
Start writing the equation as soon as the numbers are clear, and decide later what x stands for
Why: The numbers are the visible part of the problem, so they get attention first.
Is x the minutes, or the cost, or the number of calls? Without a definition the equation cannot be checked, and its answer cannot be reported in words.
\[ \text{Let } m \text{ be the length of the call in minutes.} \]
\[ 0.60 + 0.08m = 1.96 \]
Write the definition as a full sentence before the equation
Why: The definition fixes the units, which is what makes the units check possible and what turns the final number into an answer.
One extra line, written before the algebra, and it is the line that lets you say seventeen minutes rather than just seventeen.
Missing information
A question can be perfectly well written and still be unanswerable.
Discussion prompt
A long-distance call costs 8 cents a minute. Write an equation for the length of a call that cost 1 dollar 96. Say what is missing compared with the earlier version, and what answer you would get if you wrongly assumed it away.
Hint: Compare this description with the one in the worked example.
Answer:
The connection charge is missing. Without it you cannot tell whether the whole 1.96 was spent on minutes or whether some of it went on a fixed fee.
\[ \text{assuming no fee: } 0.08m = 1.96 \Rightarrow m = 24.5 \text{ minutes} \]
\[ \text{with the 60 cent fee: } 0.60 + 0.08m = 1.96 \Rightarrow m = 17 \text{ minutes} \]
Assuming the fee away overstates the call by seven and a half minutes. A missing fixed term is one of the easiest things to overlook in a word problem and one of the most damaging, because it distorts every answer rather than just one.
Translation
Four descriptions. Let n be the unknown in each.
Match the pairs
Why: The four differ in only two ways: whether a fixed fee is present, and whether the verb is totalling or a limit. Those two decisions determine the whole statement, and making them explicitly — rather than reading the sentence once and writing something plausible — is what turns word problems from guesswork into a procedure.
Socratic
You know what the letter means while you are writing it. The definition is for later.
Discussion prompt
Give two things that the written definition of a variable makes possible, at least one of which happens after the algebra is finished. Use the phone-call problem as your example.
Hint: One of them is about checking; the other is about reporting.
Answer:
First, it makes the units check possible. Once m is defined as a number of minutes, the term 0.08m is dollars per minute times minutes, which is dollars, and that has to match the dollars on the other side. Without the definition there is nothing to check against.
Second, it makes the final answer sayable. Solving gives 17, which is not an answer to anything on its own. The definition converts it into seventeen minutes, which is what the question actually asked for. Lesson 1.6 builds this into a formal five-step plan, and the labelling step is exactly this one.
Comparison
Fill the blanks from memory before you scroll back. The last column is the one that decides where the risk is.
Comparison matrix
| Operation | Words that signal it | Order matters? |
|---|---|---|
| Addition | sum, more than, plus, increased by | No |
| Subtraction | difference, minus, decreased by, less than | Yes |
| Multiplication | product, times, multiplied by, of | No |
| Division | quotient, divided by, per | Yes |
Half the operations are safe and half are not. Every mistake worth worrying about in this lesson lives in the two rows that say yes.
Pattern
Whether you are handed a phrase, a sentence, or a whole paragraph, the same five moves cover it.
Step one is the step people skip, and skipping it removes the possibility of doing steps four and five properly.
OpenStax Elementary Algebra 2e, §1.2 Use the Language of Algebra §1.2
Check
Subtraction order. Read the phrase twice before you choose.
Check your understanding
Which expression means 6 less than a number n?
Answer: A
Why: Less than names the amount being taken away before naming what it is taken from, so the number goes first and the six is subtracted from it. At n equal to 20 the phrase means fourteen, and only this expression gives fourteen.
Check
Division order. Decide which quantity goes underneath.
Check your understanding
Which expression means the quotient of 12 and a number x?
Answer: A
Why: The word quotient names its numerator first, so the twelve goes on top and the number underneath. At x equal to 4 the expression gives three, which is what dividing twelve into four parts produces.
Check
A whole sentence. Split it at the verb before you choose.
Check your understanding
Which statement means 10 times a number x is at least 50?
Answer: A
Why: At least means the quantity may equal the value or exceed it, so the symbol points towards fifty and carries a bar underneath. At x equal to 5 the left side is exactly fifty, and the bar is what makes that case count as satisfying the condition.
Real world
A job advert says the salary is 4000 dollars less than the industry average, and a second says it is at least 4000 dollars above the minimum wage.
Discussion prompt
Let a be the industry average and w the minimum wage, both annual and in dollars. Translate both statements, then say which of them pins down a single salary and which describes a range. Explain what each translation would look like if you got the order or the boundary wrong.
Hint: One of these contains the phrase that reverses, and the other contains a boundary word.
Answer:
\[ \text{first: } s = a - 4000 \qquad \text{second: } s \geq w + 4000 \]
The first is an equation and pins down one salary once the average is known. Written backwards as 4000 minus a it would claim the salary is four thousand dollars less than nothing, which for any realistic average is a large negative number — visibly absurd, which is why substituting a plausible value catches it.
The second is an inequality and describes a range, with the boundary included because of the phrase at least. Dropping the bar would exclude a salary of exactly four thousand above the minimum, which is very likely to be the exact figure offered — boundary cases are the common case whenever a round number is involved.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Does 5 less than a number mean the same as a number less 5?
Correct: Yes, both mean the number minus 5.
\[ 5 \text{ less than } n = n - 5 \qquad n \text{ less } 5 = n - 5 \]
\[ \text{but } 5 \text{ minus } n = 5 - n, \text{ which is different} \]
Why: This is a case where two phrasings that look different agree. Five less than a number reverses what you hear and gives the number minus five; a number less five states the same thing directly. The reason to be careful is not that they differ but that only one of them reverses, so a rule based on word order alone would translate one of them wrongly. Reading for meaning rather than order is what settles both.
Explain it
They can do all four operations and have never had to turn a sentence into symbols.
Discussion prompt
In no more than four sentences, explain how to translate a sentence into algebra. Then give them the one phrase from this lesson that has to be memorised rather than reasoned out, and say what goes wrong if they get it backwards.
Hint: Your method should have a step about the verb.
Answer:
A usable answer: find the verb and split the sentence there. Translate the words on each side into expressions, using the operation words to choose the operations. Then put the symbol the verb calls for in between. Two easy translations and a symbol beats one hard one every time.
The phrase to memorise is less than, because it reverses: four less than a number is the number minus four. Getting it backwards does not give a slightly wrong answer, it gives the exact opposite — six becomes negative six — so it is worth checking every single time it appears.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Recognising operations is fixed by learning the four short word lists, three or four words each. Subtraction order is fixed by memorising the single phrase that reverses and reasoning about the rest. Division order is fixed by one rule: the quantity named first goes on top. Whole situations are fixed by always writing the variable definition as a full sentence before any symbols. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
Divide a page into four boxes, one per operation, and in each box write every word from this lesson that signals that operation. Mark the two boxes whose operation is order-sensitive with a warning symbol of your own choosing. Underneath, write the one phrase that reverses the spoken order, write its correct translation, and write the incorrect one beside it with both evaluated at a number of your choice. At the bottom of the page, write a five-word list of verbs and the relation symbol each becomes. Finally, in the margin, write one situation from your own life and translate it into a complete equation, starting with a full sentence defining your variable.
Your two evaluations of the reversing phrase should be opposites — equal in size and different in sign. If they differ in any other way, one of them is not a reversal of the other.
Recap
Five things, and the second one is the one that appears on every test of this material.
| If the question says | Your first move is |
|---|---|
| Write the phrase as an expression | Find the operation word |
| 4 less than a number | Write the number first, then subtract |
| The quotient of A and B | Put A on top |
| Write the sentence as an equation | Split it at the verb |
| Write an equation for this situation | Define the variable in a full sentence |
Lesson 1.6 turns this translation skill into a complete problem-solving plan, adding two steps around it: labelling every quantity before translating, and checking that the answer is reasonable afterwards.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.5 Translating Words into Mathematical Symbols §1.5, pp. 30-35 — everything on these slides traces back here
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