The five-stage plan for word problems: write a verbal model in words, assign labels with units to every quantity, translate into an algebraic model, solve and answer the original question, and check that the answer is reasonable rather than merely arithmetically correct.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 1 — Connections to Algebra
A Problem Solving Plan Using Models
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 1.6 A Problem Solving Plan Using Models §1.6, pp. 36-41 — the lesson these objectives are drawn from
Warm-up
You already solve word problems. The plan just writes down what you were doing in your head.
Discussion prompt
Four friends split a 32 dollar bill evenly. Work out each share, then describe the steps you took — including any step you did without noticing.
Hint: One of your steps was deciding what operation to use, and you almost certainly did that before doing any arithmetic.
Answer:
\[ 32 \div 4 = 8 \text{ dollars each} \]
You decided the relationship first — the total equals the number of friends times each share — and only then divided. That decision is the verbal model, and it is the part of the work this lesson makes visible. On an easy problem you can hold it in your head; on a hard one you cannot, and that is exactly when it needs writing down.
Concept
Writing algebraic expressions, equations or inequalities that represent real situations is called modeling. It happens in two stages: first a verbal model using words, then an algebraic model translating those words into symbols.
verbal model — A statement of the relationship in a problem, written in words rather than symbols, which is then translated into an algebraic model.
The verbal model is what you can check by reading. The algebraic model is what you can solve.
Figure (svg): A verbal model in words above an algebraic model in symbols, with each phrase connected to the symbol it became
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 36-36
Section
Section 1
Concept
The plan has five stages, and three of them happen before any algebra is written. That proportion is honest about where the difficulty in a word problem actually lies.
Figure (svg): The five stages of the problem solving plan shown as a chain: verbal model, labels, algebraic model, solve, check
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 37-37 — the Problem Solving Plan Using Models box
Picture it
Each stage feeds the next, and skipping one breaks the chain rather than saving time.
Figure (svg): The five stages of the problem solving plan shown as a chain: verbal model, labels, algebraic model, solve, check
Notice that only one of the five stages involves solving anything. Word problems feel hard because four fifths of the work is decisions rather than technique — and decisions get easier when they are written down.
Worked example
Example 1 from the textbook. You order several two-dollar plates; the bill is 25.20 including 1.20 of tax.
\[ \text{Find the number of plates } p \text{ if } \; 2p = 25.20 - 1.20. \]
Write the verbal model
Why: Cost per plate times number of plates equals the amount of the bill minus the tax. Notice the tax is subtracted, because it was added after the food was priced.
Assign labels with units
Why: Cost per plate is 2 dollars; number of plates is p plates; the bill is 25.20 dollars; the tax is 1.20 dollars.
Write the algebraic model
Why: Replace each labelled phrase by its symbol, keeping the structure of the sentence.
\[ 2 p = 25.20 - 1.20 \]
Solve
Why: The right side simplifies to 24, and two times what number gives twenty-four? Twelve.
\[ 2 p = 24, p = 12 \]
Check that the answer is reasonable
Why: Twelve small plates shared among a group at a restaurant is entirely plausible.
\[ 12\text{ plates} \]
Figure (svg): A bar model of a restaurant bill: twelve two-dollar plates totalling twenty-four dollars, plus one dollar twenty of tax, making twenty-five twenty
\[ 2p = 24 \;\rightarrow\; p = 12 \text{ plates} \]
Verify: rebuild the bill from the answer
Why: Twelve plates at two dollars is twenty-four dollars, and adding the 1.20 of tax gives 25.20, which is exactly the bill stated in the problem. Rebuilding the original figure from the answer is a stronger check than re-solving, because it tests the model as well as the arithmetic.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 36-36
Ranking
Five stages, one correct sequence.
Put in order
Why: The verbal model comes first because everything else is built on it. Labels come next, because the translation needs to know which symbol stands for what. Only then is there an algebraic model to solve, and the reasonableness check comes last because it needs an answer to judge. Note that three of the five stages precede any algebra.
Worked example
The same situation, with one sentence altered. Watch where the change lands in the model.
\[ \text{Suppose the } 25.20 \text{ bill had no tax at all. How many plates then?} \]
Rewrite the verbal model
Why: Cost per plate times number of plates equals the amount of the bill. The tax phrase disappears entirely.
Update the labels
Why: Three quantities now instead of four, and only one still carries a letter.
Write the algebraic model
Why: The subtraction is gone from the right side.
\[ 2 p = 25.20 \]
Solve and judge
Why: Two times what gives 25.20? Twelve point six — which is not a whole number of plates.
\[ p = 12.6 \]
Figure (svg): The solution to Worked example what changes if the tax is included differently shown as a ladder of expressions, one row per algebraic move
\[ 2p = 25.20 \;\rightarrow\; p = 12.6 \text{ plates} \]
Verify: ask whether the answer is reasonable rather than whether it is correct
Why: The arithmetic is right and the answer is still wrong, because you cannot order six tenths of a plate. That impossibility is evidence that the tax really was included in the bill, which is what the original problem said. A reasonableness check can tell you that a model is wrong even when every calculation in it is right.
Trap
\[ 2p + 1.20 = 25.20 \]
Read the problem once, spot the numbers, and start writing an equation
Why: The numbers are the visible part of a word problem, so they attract attention first.
This particular equation happens to be equivalent to the right one, but it was arrived at by luck. On a problem with four quantities and two relationships the same approach produces something unreadable and unfixable.
Cost per plate times number of plates equals the amount of the bill minus the tax.
\[ 2p = 25.20 - 1.20 \]
Write the sentence in words first, then translate it
Why: A wrong sentence can be spotted by reading it. A wrong equation has to be debugged.
The verbal model is the thing you can show to somebody who does not know algebra and ask whether it sounds right.
Sorting
Each item below is a piece of work from a solved problem. Place it in the right stage.
Sort into buckets
Sort each item into the stage of the plan it belongs to.
Four items out of six come from stages other than the algebra. That ratio is the real content of this lesson.
Elimination
You order two-dollar plates. The bill is 25.20 and includes 1.20 of tax.
Eliminate the wrong options
Which verbal model is correct?
Survives elimination: A
Why: The bill includes the tax, so the money spent on food is the bill with the tax removed, and that amount is the price per plate times the number of plates. Two of the wrong options fail a units check before any arithmetic: you cannot add dollars to plates, and you cannot set an amount of money equal to a count.
Socratic
The verbal model is extra writing. It has to earn its place.
Discussion prompt
Give two specific advantages of writing a verbal model before the algebraic one, at least one of which is about finding a mistake. Then describe a kind of problem where you would be willing to skip it.
Hint: Think about who can check a sentence in words compared with who can check an equation.
Answer:
First, a sentence in words can be checked by reading. If the sentence says the bill minus the tax when it should say plus, you can hear that it is wrong; an equation with a minus sign in the wrong place looks exactly as convincing as the right one.
Second, it separates modelling errors from algebra errors. If the answer is wrong you can look at the verbal model and decide immediately whether the problem was understanding the situation or manipulating the symbols — two different weaknesses needing two different kinds of practice.
Skipping it is defensible on a problem with one quantity and one operation, where the sentence and the equation have the same length. It stops being defensible the moment there are three or more quantities, which is where the marks are.
Section
Section 2
Concept
Assigning labels means listing each quantity in the verbal model with the symbol or value that represents it, and the unit it is measured in. Exactly one label should carry a letter — that letter is what you are solving for.
The unit column is not decoration: it is what makes the algebraic model checkable before you solve it.
| Quantity | Symbol or value | Unit |
|---|---|---|
| Cost per plate | 2 | dollars |
| Number of plates | p | plates |
| Amount of bill | 25.20 | dollars |
| Tax | 1.20 | dollars |
Figure (svg): A labels table listing each quantity, its symbol and its unit
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 36-36 — the LABELS block of Example 1
Picture it
Four quantities, and only one of them unknown.
Figure (svg): A labels table listing each quantity, its symbol and its unit
If two rows carry letters you have two unknowns and will need two equations, which is Chapter 7. If no row carries a letter you have not identified what the question is asking for.
Worked example
Example 2 from the textbook. A football field is 53 yards by 120; a soccer field of the same area is 60 yards wide.
\[ \text{Assign labels for the four quantities in the two-fields problem.} \]
Width of soccer field is 60 yards
Why: Given directly in the problem, so it is a value rather than a letter.
\[ 60\text{ yards} \]
Length of soccer field is x yards
Why: This is what the question asks for, so it is the one quantity that gets a letter.
Width of football field is 53 yards
Why: Given directly.
\[ 53\text{ yards} \]
Length of football field is 120 yards
Why: Given directly.
\[ 120\text{ yards} \]
Check the count of letters
Why: Exactly one label carries a letter, so one equation will be enough.
Figure (svg): The solution to Worked example labels for the two fields shown as a ladder of expressions, one row per algebraic move
\[ 60, \; x, \; 53, \; 120 \quad \text{all measured in yards} \]
Verify: check that all four units agree
Why: Every quantity is a length in yards, so multiplying two of them gives square yards on both sides of the eventual equation. Had one field been measured in feet, the model would have been wrong even with perfect algebra, and the labels table is where that would have been caught.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 37-37
Matching
Four quantities from the two problems in this lesson.
Match the pairs
Why: The first is a rate, so its unit is a fraction of two other units, and multiplying it by a number of plates leaves dollars. The last is a product of two lengths, so it carries a second power of the unit, exactly as in Lesson 1.2. Getting these right in the labels table is what makes the algebraic model correct by construction rather than by luck.
Worked example
The same fields, but the football field is now given as 360 feet long instead of 120 yards.
\[ \text{Assign labels when one length is given in feet and the others in yards.} \]
List the quantities with the units as given
Why: Sixty yards, x yards, fifty-three yards, and three hundred and sixty feet.
Notice that one unit does not match
Why: Three of the four are in yards and one is in feet, so the products on the two sides would not be comparable.
Convert before writing the model
Why: Three feet make a yard, so 360 feet is 120 yards.
\[ 360 \text{ft} = 120 \text{yd} \]
Rewrite the labels with one consistent unit
Why: All four quantities are now in yards, and the model can be written.
Figure (svg): The solution to Worked example labels that expose a units problem shown as a ladder of expressions, one row per algebraic move
\[ 360 \text{ feet} = 120 \text{ yards} \]
Verify: check what would have happened without the conversion
Why: Leaving the 360 in would give sixty x equals fifty-three times three hundred and sixty, so x would come out as 318 — three times too large, because the length was counted in units three times smaller. The error would be invisible in the algebra and obvious in the labels table.
Error analysis
The student labelled the plates problem before writing a model. Two rows are wrong.
Annotate
On: \( \begin{aligned} \text{cost per plate} &= 2 \text{ dollars} \\ \text{number of plates} &= p \text{ dollars} \\ \text{amount of bill} &= 25.20 \text{ dollars} \\ \text{tax} &= 1.20 \text{ plates} \end{aligned} \)
The units are the cheapest error detector in the whole plan. They cost one extra column and they catch the mistakes that recalculating never will.
Discrimination
In the fields problem, decide which quantities are known and which is not.
Sort into buckets
Sort each quantity by whether it gets a number or a letter.
Missing information
A question can be perfectly well written and still be unanswerable.
Discussion prompt
A soccer field has the same area as a football field and is 60 yards wide. How long is it? Say exactly what is missing, and explain how the labels table would have made the gap obvious.
Hint: Fill in the labels table for this version and count the letters.
Answer:
The dimensions of the football field are missing, so its area cannot be computed and there is nothing for the soccer field's area to equal.
In the labels table this shows up immediately: the width of the soccer field is 60, its length is x, and both football-field dimensions would also have to be letters. Three letters and one equation is unsolvable, and the count of letters says so before any algebra is attempted. That counting habit is one of the most useful things the labels stage gives you.
Edge cases
A units check is only as good as the units you write.
Discussion prompt
In the plates problem, suppose you labelled the cost per plate simply as dollars rather than dollars per plate. Would the units check still catch an error where you added the cost per plate to the number of plates? Explain what you would and would not catch.
Hint: Work out the unit of the left side under each labelling.
Answer:
You would still catch that one, because adding dollars to plates is impossible under either labelling. What you would lose is the check on the multiplication: dollars per plate times plates gives dollars, and that cancellation is what confirms the product is a money amount.
Labelled merely as dollars, the product of cost and count would appear to be dollars times plates, which is not a unit anybody uses — and you would have no way to tell whether that signalled an error or just a sloppy label. Rates should always be labelled as a fraction of two units, because the cancellation is the whole point.
Section
Section 3
Concept
The translation stage is mechanical once the first two stages are done. Every phrase in the verbal model has a label, and the algebraic model is the verbal model with each phrase replaced by its symbol.
\[ \underbrace{2}_{\text{per plate}} \cdot \underbrace{p}_{\text{plates}} = \underbrace{25.20}_{\text{bill}} - \underbrace{1.20}_{\text{tax}} \]
Writing the words directly above the symbols means a wrong translation can be found by reading rather than by solving.
Figure (svg): A verbal model in words above an algebraic model in symbols, with each phrase connected to the symbol it became
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 36-37 — the ALGEBRAIC MODEL block of Examples 1 and 2
Picture it
Every word sits directly over the symbol it became.
Figure (svg): A verbal model in words above an algebraic model in symbols, with each phrase connected to the symbol it became
Laid out this way, a modelling error is visible as a mismatch between a column's word and its symbol — which is much easier to spot than an error buried in a line of algebra.
Worked example
Example 2 from the textbook, using all five stages.
\[ \text{A football field is } 53 \text{ by } 120 \text{ yards. A soccer field of the same area is } 60 \text{ yards wide. Find its length.} \]
Write the verbal model
Why: The width of the soccer field times its length equals the width of the football field times its length — because the two areas are equal.
Assign labels
Why: Sixty yards, x yards, fifty-three yards and one hundred and twenty yards.
Write the algebraic model
Why: Each phrase becomes its symbol, and the equal sign comes from the words the same area.
\[ 60 x = 53 \cdot 120 \]
Simplify and solve
Why: Fifty-three times one hundred and twenty is six thousand three hundred and sixty, and sixty times what gives that? One hundred and six.
\[ 60 x = 6360, x = 106 \]
Check that the answer is reasonable
Why: A soccer pitch a little over a hundred yards long is about right, and it is longer than it is wide as expected.
\[ 106\text{ yards} \]
Figure (svg): Two rectangles of equal area, a football field 53 by 120 yards and a soccer field 60 yards wide by an unknown length
\[ 60x = 53 \cdot 120 = 6360 \;\rightarrow\; x = 106 \text{ yards} \]
Verify: compute both areas from the answer
Why: The soccer field is sixty by one hundred and six, giving 6360 square yards, and the football field is fifty-three by one hundred and twenty, also 6360. The two areas match, which is exactly what the verbal model claimed, so both the model and the arithmetic hold up.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 37-37
Translation
Four verbal models, four equations. Let x be the unknown in each.
Match the pairs
Why: In each case the structure of the sentence survives into the symbols: the word times becomes a multiplication in the same position, plus becomes addition and equals becomes an equal sign. Reading the equation back out loud should reproduce the sentence, and if it does not, the translation has changed the meaning of the model.
Worked example
Guided Practice 1. The first garden is 5 metres by 16; the second is 8 metres wide.
\[ \text{Find the length of the second garden if the two areas are equal.} \]
Write the verbal model
Why: Width of the second garden times its length equals width of the first times its length.
Assign labels
Why: Eight metres, x metres, five metres and sixteen metres.
Write the algebraic model
Why: Each phrase becomes its symbol.
\[ 8 x = 5 \cdot 16 \]
Simplify and solve
Why: Five times sixteen is eighty, and eight times what gives eighty? Ten.
\[ 8 x = 80, x = 10 \]
Figure (svg): The solution to Worked example two gardens of equal area shown as a ladder of expressions, one row per algebraic move
\[ 8x = 5 \cdot 16 = 80 \;\rightarrow\; x = 10 \text{ metres} \]
Verify: compare the two shapes
Why: The first garden is 5 by 16 and the second is 8 by 10, and both have an area of 80 square metres. The second garden is wider and shorter, which is what you should expect when the area is held fixed and the width is increased — a genuine reasonableness check rather than a repeat of the arithmetic.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 37-37
Trap
Width of soccer field times length of soccer field equals width of football field times length of football field.
\[ 60 + x = 53 + 120 \]
Write the numbers in the order they appear and join them with whatever operation comes to mind
Why: Once the labels exist, the translation feels like a formality and gets rushed.
This gives x equal to 113, which is close enough to the right answer to look plausible and is produced by a model that says nothing about area at all.
\[ 60 \cdot x = 53 \cdot 120 \]
Copy the structure of the verbal model, not just its numbers
Why: The word times in the sentence has to become a multiplication sign in the symbols, in the same position.
Reading the algebraic model back into words is the check: sixty times x equals fifty-three times one hundred and twenty says exactly what the sentence said. Sixty plus x does not.
Elimination
The verbal model is: width of the second garden times its length equals width of the first times its length. The first garden is 5 by 16 and the second is 8 wide.
Eliminate the wrong options
Which algebraic model is correct?
Survives elimination: A
Why: Equal areas means the two products are equal, so eight times the unknown length equals five times sixteen, giving x equal to ten. Option C is worth noting separately: it fails a units check, since square metres cannot equal metres, and that failure is visible before a single digit is computed.
Fill the middle
The verbal model is given and half the symbols are in place.
Fill in the blanks
\text120 \cdot \text6360 = \text___ \cdot \text___ \;\rightarrow\; 60 \cdot x = 53 \cdot ___ \;\rightarrow\; 60x = ___
Why: The football field's length is 120 yards, and multiplying it by the width of 53 gives 6360 square yards. That single number is the whole of the right-hand side, and once it is computed the equation is a one-step mental solve. Simplifying the known side before solving is almost always worth doing.
Socratic
The best check on a translation costs about five seconds.
Discussion prompt
Describe how you would check an algebraic model without solving it or substituting any numbers. Apply your method to the model 60x equals 53 times 120 and say what it confirms.
Hint: The check should involve your mouth rather than your pencil.
Answer:
Read the equation aloud in words and compare it with the verbal model. Sixty times x equals fifty-three times one hundred and twenty reads as the width of the soccer field times its length equals the width of the football field times its length, which is exactly the sentence written at the start.
This confirms the structure — that both sides are products and that they are set equal — without any arithmetic. It cannot confirm that the numbers were copied correctly, so it complements a units check rather than replacing one. Together the two catch almost every modelling error there is.
Section
Section 4
Concept
Solving the algebraic model produces a number. The stage is not finished until that number has been turned back into a statement about the situation, with its unit attached.
The question asked how many plates, so the answer is twelve plates and not merely twelve.
Figure (svg): A bar model of a restaurant bill: twelve two-dollar plates totalling twenty-four dollars, plus one dollar twenty of tax, making twenty-five twenty
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 36-37
Picture it
Every quantity in the picture appears in the labels table.
Figure (svg): A bar model of a restaurant bill: twelve two-dollar plates totalling twenty-four dollars, plus one dollar twenty of tax, making twenty-five twenty
Twelve plates costing twenty-four dollars, plus one twenty of tax, giving the 25.20 the problem started from. The answer closes the loop back to the original numbers.
Worked example
Finishing the plates problem properly. The algebra takes one line; the answer takes one more.
\[ \text{Solve } 2p = 24 \text{ and answer the original question.} \]
Solve the equation
Why: Two times what number gives twenty-four? Twelve.
\[ p = 12 \]
Look up what p stood for
Why: The labels table says p is the number of plates, measured in plates.
\[ p =\text{ plates} \]
Attach the unit
Why: Twelve plates, not twelve dollars and not just twelve.
\[ 12\text{ plates} \]
Write the answer as a sentence
Why: Your group ordered twelve plates of food, costing twenty-four dollars.
Figure (svg): The solution to Worked example solve and answer in a sentence shown as a ladder of expressions, one row per algebraic move
\[ p = 12 \text{ plates, costing } 24 \text{ dollars} \]
Verify: rebuild the original bill
Why: Twelve plates at two dollars each is twenty-four dollars, and adding the tax of 1.20 gives 25.20 — the figure the problem opened with. Getting back to a number that was given rather than one you computed is the strongest form of check available.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 36-36
Elimination
The model 2p equals 24 has been solved and p is 12.
Eliminate the wrong options
Which of these properly answers the question how many plates did you order?
Survives elimination: A
Why: The answer names the quantity, gives its value and carries its unit, and it could be read by someone who has never seen the algebra. Options C and D are both real quantities from the problem attached to the wrong question, which is the most common way a correct calculation still loses the final mark.
Worked example
The same model, but the question changes. Watch how much of the work is reusable.
\[ \text{Using the same bill, how much was spent on food alone, and what fraction of the bill was tax?} \]
Reuse the model and the solution
Why: The number of plates is still twelve, and the food cost is still twenty-four dollars.
\[ 24\text{ dollars on food} \]
Read the first question's answer straight off the model
Why: The quantity asked for is the right side of the algebraic model, which was computed on the way.
\[ 24\text{ dollars} \]
Set up the second question
Why: The fraction of the bill that was tax is the tax divided by the whole bill.
\[ \frac{1.20}{25.20} \]
Compute and express it usefully
Why: One twenty over twenty-five twenty is about 0.048, which is a little under five percent.
\[ \text{about } 4.8 \% \]
Figure (svg): The solution to Worked example when the question asks for something else shown as a ladder of expressions, one row per algebraic move
\[ \text{food} = 24 \text{ dollars} \qquad \tfrac{1.20}{25.20} \approx 0.048 \]
Verify: check the percentage against a rough estimate
Why: One dollar twenty out of about twenty-five dollars is roughly one in twenty, which is five percent, and 4.8 is just under that. Sales tax rates in that range are entirely ordinary, so the answer is reasonable as well as correct.
Trap
\[ 2p = 24 \;\rightarrow\; p = 12 \]
Write p equals 12, circle it, and move on
Why: The algebra is finished, and the letter has a value, so the work feels complete.
The question asked how many plates you ordered. A line of algebra ending in a letter is not an answer to that, and on a marked paper it usually does not score the final mark.
\[ 2p = 24 \;\rightarrow\; p = 12 \]
Your group ordered 12 plates of food, costing 24 dollars.
Convert the value of the letter back into a statement about the situation
Why: The labels table records what the letter meant and what unit it carries; the answer sentence uses both.
One extra line, and it is the line that shows you understood the problem rather than merely the equation.
Prediction
A good model tells you what happens when the situation shifts.
Predict first
If the plates cost 3 dollars each instead of 2, and the bill and tax are unchanged, how many plates were ordered?
Correct: 8 plates.
\[ 3p = 25.20 - 1.20 = 24 \;\rightarrow\; p = 8 \text{ plates} \]
Why: The food cost is still 25.20 minus 1.20, which is 24 dollars, and at three dollars a plate that buys eight plates. Only one number in the model changed, and because the model was written with labels rather than with the numbers baked in, answering the new question took one division rather than a fresh start.
Faded example
The model is written. Supply the value and the unit.
Fill in the blanks
8x = 5 \cdot 16 = 80 \;\rightarrow\; x = 10 \textmetres ___
Why: Eight times ten is eighty, so the second garden is ten metres long. The unit comes from the labels table rather than from the equation — the algebra produces a bare number every time, and it is the labels that say what kind of quantity that number is.
Socratic
The unit is the difference between a calculation and a communication.
Discussion prompt
Imagine handing your working to somebody who has not read the problem. Say what they could and could not work out from the line p equals 12 alone, and how the answer sentence changes that.
Hint: Think about what information lives in the labels table and nowhere else.
Answer:
From p equals 12 alone they learn that some quantity has the value twelve. They cannot tell whether it is plates, dollars, minutes or metres, whether it is the thing the problem asked for, or whether it is an intermediate step on the way to something else.
The answer sentence carries all three: the quantity, its value and its unit. That is why examiners award a mark for it separately, and it is also why the labels stage is worth the writing — the sentence at the end is assembled directly from it.
Section
Section 5
Concept
The final stage asks whether the answer makes sense in the situation, not whether the calculation was performed accurately. It is the only stage that can catch a wrong model, because a wrong model can be solved perfectly.
A negative length, a fractional number of plates or an answer a thousand times too large all signal a modelling error rather than an arithmetic one.
Figure (svg): Three candidate answers to the plates problem judged against common sense
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 37-37 — the CHECK stage of the plan
Picture it
Each of these could come out of an arithmetically flawless calculation.
Figure (svg): Three candidate answers to the plates problem judged against common sense
Nothing about the arithmetic distinguishes twelve from twelve hundred; only knowledge of the situation does. That knowledge is what the check stage is for.
Worked example
A student models the plates problem as 2p equals 25.20 plus 1.20 and solves it correctly.
\[ \text{Solve } 2p = 25.20 + 1.20 \text{ and judge the answer.} \]
Solve the model as written
Why: The right side is 26.40, and two times what gives 26.40? Thirteen point two.
\[ p = 13.2 \]
Check the kind of quantity
Why: Plates are counted in whole numbers, and 13.2 plates is not something a restaurant can serve.
Diagnose the cause
Why: A fractional count usually means a quantity was added when it should have been subtracted, or a rate was applied to the wrong total.
Correct the model and re-solve
Why: The tax is already inside the bill, so it is subtracted rather than added.
\[ 2 p = 24, p = 12 \]
Figure (svg): The solution to Worked example catching a wrong model by its answer shown as a ladder of expressions, one row per algebraic move
\[ 2p = 26.40 \;\rightarrow\; p = 13.2 \quad \text{impossible, so the model is wrong} \]
Verify: confirm that the corrected answer passes every test
Why: Twelve is a whole number, it is a plausible order for a group, and rebuilding the bill from it gives exactly 25.20. All three checks pass, which is what a correct model looks like from the outside.
Sorting
Judge each answer against its question, ignoring how it was obtained.
Sort into buckets
Sort each answer by whether it could possibly be right.
Notice that all three failures point at the model rather than the arithmetic. That is the general pattern: implausible answers come from wrong sentences, not from wrong sums.
Worked example
A student solves the fields problem but mixes up which quantities multiply.
\[ \text{A student gets } x = 6360 \text{ yards for the soccer field's length. Judge it.} \]
Check the kind of quantity
Why: A length in yards is the right kind of thing, so this test passes.
Check the size against the problem
Why: The football field is 120 yards long, and the two fields have equal areas, so the soccer field cannot be fifty times longer.
Diagnose the cause
Why: Six thousand three hundred and sixty is the area, so the student stopped one step early and reported the product instead of the length.
Finish the missing step
Why: Divide the area by the soccer field's width of 60 to get the length.
\[ \frac{6360}{60} = 106 \]
Figure (svg): The solution to Worked example an answer of the wrong size shown as a ladder of expressions, one row per algebraic move
\[ 6360 \text{ square yards} \div 60 \text{ yards} = 106 \text{ yards} \]
Verify: check the unit of the abandoned answer
Why: Six thousand three hundred and sixty is a product of two lengths, so it carries square yards. The question asked for a length in yards, and the unit alone says the answer is the wrong kind of quantity — which the size check confirms independently.
Trap
\[ 2p = 26.40 \;\rightarrow\; p = 13.2 \]
Re-do the division twice, confirm 13.2 each time, and conclude the answer is right
Why: Repeating a calculation feels like checking, and it does confirm something — just not the thing that was wrong.
The arithmetic was never in doubt. Recalculating a wrong model as many times as you like will keep producing the same wrong answer.
\[ p = 13.2 \text{ plates} \]
Ask whether the answer could be true of the situation, independently of how it was obtained
Why: A fractional count of plates is impossible regardless of what equation produced it.
Then go back and look at the model, not the arithmetic. The error is always upstream of a check that fails on grounds of plausibility.
Elimination
A model is solved correctly and gives a length of negative 8 metres.
Eliminate the wrong options
What is the most useful conclusion?
Survives elimination: A
Why: An impossible answer produced by correct arithmetic is evidence about the model, because that is the only remaining place for the error to live. The usual causes are a subtraction written in the wrong order or two quantities set equal that describe different things, and both are found by rereading the verbal model rather than the algebra.
Estimation
A rough answer computed first turns the check into a comparison.
Predict first
A football field is 53 by 120 yards. A soccer field of the same area is 60 yards wide. Roughly how long is it?
Correct: A bit over 100 yards.
\[ 60x = 53 \cdot 120 = 6360 \;\rightarrow\; x = 106 \text{ yards} \]
Why: The soccer field is slightly wider than the football field, 60 against 53, so at equal area it must be slightly shorter than 120 — a little over 100. Working that out before solving means the exact answer of 106 arrives already confirmed, and any answer far from the estimate announces itself immediately.
Socratic
The plan ends with a check, and it is not the check you did in Lesson 1.4.
Discussion prompt
Explain the difference between substituting an answer back into an equation and checking that an answer is reasonable. Give one error that each catches and the other misses.
Hint: One of them takes the equation on trust.
Answer:
Substituting back tests the arithmetic against the equation you wrote. It catches a division done wrongly, and it misses everything about whether that equation described the situation — a model with the tax added instead of subtracted will pass substitution perfectly.
A reasonableness check tests the answer against the situation. It catches a wrong model, since a wrong model usually produces an answer of the wrong size or the wrong kind, and it misses small arithmetic slips that leave the answer plausible. The two tests are complementary, which is why the plan includes one and Lesson 1.4 taught the other.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Stage | What you write | What it catches |
|---|---|---|
| Verbal model | a sentence in words | a misunderstanding of the situation |
| Labels | each quantity, its symbol and its unit | mismatched units and missing information |
| Algebraic model | an equation or inequality | a translation that changed the meaning |
| Solve | a value and an answer sentence | arithmetic slips |
| Check | a judgement about plausibility | a wrong model solved correctly |
Each stage catches a different kind of error, which is why the plan is worth following even when the algebra is easy. Skipping a stage does not just save writing — it removes a specific safety net.
Pattern
Whether the problem is about restaurants, playing fields or phone bills, the same five moves cover it.
One letter in the labels table means one equation will do. Two letters means you have either missed some information or reached a Chapter 7 problem.
OpenStax Elementary Algebra 2e, §3.1 Use a Problem-Solving Strategy §3.1
Check
The verbal model. Read the situation twice before you choose.
Check your understanding
You buy several 3 dollar notebooks. The total is 19 dollars including 1 dollar of tax. Which verbal model is correct?
Answer: A
Why: The tax is already part of the 19 dollars, so the money spent on notebooks is the total with the tax removed — 18 dollars, which buys six notebooks at three dollars each. The verbal model has to subtract the tax rather than add it.
Check
Labels and units. Think about what kind of quantity each one is.
Check your understanding
In the notebook problem, what unit does the number of notebooks carry?
Answer: A
Why: It is a count of notebooks, so its unit is notebooks. That matters because the cost per notebook is measured in dollars per notebook, and multiplying the two cancels the notebooks and leaves dollars — which is what has to appear on the other side of the model.
Check
The reasonableness check. Judge the answer, not the arithmetic.
Check your understanding
A model for a garden's length is solved correctly and gives negative 4 metres. What is the best next step?
Answer: A
Why: An impossible answer reached by correct arithmetic points at the model, because that is the only place left for the error to be. The usual cause is a subtraction written in the wrong order, which is found by rereading the verbal model rather than by recalculating.
Real world
You are splitting a holiday cost. The flights are 320 dollars total, accommodation is 45 dollars per person per night for 4 nights, and the whole trip must cost each person no more than 500 dollars.
Discussion prompt
Work through all five stages of the plan for the question how many people must go. Write the verbal model and the labels in full before any symbols, then say what the reasonableness check rules out about your answer.
Hint: The flights are shared between everyone, but the accommodation is charged per person.
Answer:
Verbal model: the flight cost divided by the number of people, plus the nightly rate times the number of nights, is at most the budget per person.
\[ \text{labels: } n \text{ people, } 320 \text{ dollars, } 45 \text{ dollars per person per night, } 4 \text{ nights, } 500 \text{ dollars per person} \]
\[ \tfrac{320}{n} + 45 \cdot 4 \leq 500 \;\rightarrow\; \tfrac{320}{n} + 180 \leq 500 \;\rightarrow\; \tfrac{320}{n} \leq 320 \;\rightarrow\; n \geq 1 \]
The algebra says one person is enough, which is correct: at 320 for the flight plus 180 for the rooms, a solo traveller pays exactly 500. The reasonableness check rules out fractional and negative answers, and it also flags that the accommodation term never depends on n — worth noticing, because it means adding people helps only with the flights.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
If an algebraic model is solved perfectly and gives an impossible answer, where is the error?
Correct: In the model, which described the situation wrongly.
\[ 2p = 25.20 + 1.20 \;\rightarrow\; p = 13.2 \quad \text{correct algebra, impossible answer} \]
\[ 2p = 25.20 - 1.20 \;\rightarrow\; p = 12 \quad \text{corrected model} \]
Why: If the solving was correct then the arithmetic is not the problem, and an answer that contradicts the situation cannot simply be accepted. That leaves the model, and in practice the cause is almost always one of three things: a subtraction written in the wrong order, a quantity added when it should have been subtracted, or two quantities set equal that describe different things. All three are found by rereading the verbal model rather than the algebra.
Explain it
They can solve equations and freeze completely when a problem is written in paragraphs.
Discussion prompt
In no more than four sentences, explain what to do first when a word problem looks impossible. Then give them one concrete thing to write down before they touch any algebra, and say what it protects them from.
Hint: The answer to what to do first is not to look for the numbers.
Answer:
A usable answer: do not look for the numbers first. Work out what the question is asking for, then write one sentence in ordinary words describing how the quantities in the problem relate to each other. Only after that sentence exists should any symbols appear.
The concrete thing to write is the labels list: every quantity, its symbol or value, and its unit. It protects them from the most common failure, which is producing a perfectly solved equation that describes a different situation from the one in the problem — and it also tells them, by counting the letters, whether they have enough information to proceed at all.
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: Verbal models are fixed by writing the sentence for five solved problems whose answers you already know, so the sentence is the only thing you are practising. Labels are fixed by always adding the unit column, even when it feels obvious. Translation is fixed by reading the equation back aloud and comparing it with the sentence. Reasonableness is fixed by asking three questions every time — right kind, right size, consistent with a given number. 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
Down the left of a page, write the five stages of the plan as a numbered column. Choose one word problem from this lesson and work it through the full plan in the space to the right, writing the verbal model as a complete English sentence, the labels as a three-column table with units, and the algebraic model directly underneath the verbal one so each phrase sits above its symbol. Finish with an answer sentence carrying its unit. Across the bottom, write three questions you would ask of any answer to decide whether it is reasonable, and beside each one write an example of an answer it would reject.
Your labels table should contain exactly one letter. If it contains two, either the problem gave you something you did not use, or you have written down a Chapter 7 problem by mistake.
Recap
Five things, and three of them happen before you write a single symbol.
| If the question says | Your first move is |
|---|---|
| Use modeling to find | Write the relationship as an English sentence |
| The bill includes tax of | Subtract the tax, do not add it |
| The two areas are equal | Set the two products equal |
| How many did you order | Answer in a sentence, with the unit |
| Your answer looks impossible | Reread the verbal model, not the arithmetic |
Lesson 1.7 turns to data rather than single quantities: organising numbers into tables and graphs, and reading a graph carefully enough to notice when it is misleading.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.6 A Problem Solving Plan Using Models §1.6, pp. 36-41 — everything on these slides traces back here
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