Operations and Algebraic Thinking for RIT under 215: writing numerical expressions with one operation and with two, and evaluating them with the order of operations.
Subject: NWEA MAP Growth Math · 62 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
MAP Growth · Operations and Algebraic Thinking · RIT under 215
Turning words into numerical expressions, and numerical expressions into one number
Objectives
Three MAP skills sit in this deck, and they build on each other in order.
The habit worth stealing: decide what the words are asking for before you touch a single number.
Section
Part 1
Concept
A numerical expression is numbers joined by operations. It has no equals sign, and it is not finished.
Figure (svg): Two cards side by side: an expression card reading 6 plus 4 with no equals sign, and an answer card reading 10
Numerical expression — Numbers and operation signs written together, with no equals sign — for example 6 + 4.
Evaluating is the act of running the recipe until only one number is left.
Definition probe
MAP items are picky about this difference, so sort these before going further.
Sort into buckets
Drop each one into the right box.
Intuition
Every expression hides a tree. The operation at the top happens last, and whatever sits lower happens first.
Figure (svg): An expression tree for three times the sum of four and five, with the addition grouped underneath the multiplication
If you can draw the tree, you can always write the parentheses correctly.
Warm-up
Retrieve before you are taught — it makes the teaching stick harder.
Discussion prompt
Write down every math word you know that means add, and every word you know that means subtract. Aim for three of each.
Hint: Think about the words a word problem would use, not the symbols.
Answer:
Add: sum, plus, total, altogether, combined, increased by, more than.
Subtract: difference, minus, less than, fewer, decreased by, take away, remains.
Two of these are dangerous: less than and more than flip the order of the numbers. That trap gets its own slide.
Notation
Knowing the names makes the MAP wording readable.
Annotate
On: \( 3 \times (4 + 5) \)
MAP names an expression by its last operation. That single fact answers a surprising number of items.
Socratic
A fair question from a student who can already get the answer in their head.
Discussion prompt
If you can work out the answer mentally, why would a test ask you to write the expression instead?
Hint: Think about what happens when the numbers get large or the problem gets long.
Answer:
Because the expression is the part that scales. Mental arithmetic runs out at bigger numbers; a correctly written expression never does.
And because the expression is the part that can be checked. If your answer is wrong, the expression shows where the thinking went wrong.
Pattern
Every one of these items yields to the same three questions, asked in this order.
Question three is the one students skip, and it is the one the two-operation items are testing.
Discrimination
Sort by operation only. Do not compute anything.
Sort into buckets
Put each phrase under its operation.
Section
Part 2
Concept
With a single operation there is only one thing to get right: which operation, and in which order the numbers go.
Figure (svg): An array of four rows of six dots, showing four groups of six as a multiplication picture
| phrase | expression |
|---|---|
| the sum of 14 and 9 | 14 + 9 |
| the product of 6 and 7 | 6 x 7 |
| 25 divided by 5 | 25 / 5 |
| the difference of 30 and 12 | 30 - 12 |
Worked example
A classroom has 7 tables. Each table seats 4 students. Write an expression for the number of students.
Name what is being combined
Why: 7 tables, and 4 students at each one.
Find the operation
Why: Equal groups repeated is multiplication.
\[ 7 \times 4 \]
Verify: the picture matches
Why: Seven groups of four dots is the same as four dots counted seven times, so the expression fits the situation.
Intuition
Comparison words become multiplication when one quantity is a whole number of copies of the other.
Figure (svg): A bar model with a short bar of eight and a bar twice as long of sixteen
Bars first, expression second. The bars make the operation obvious.
Translation
Every pair here uses the same two numbers, so only the wording separates them.
Match the pairs
Why: Sum means add, product means multiply, divided by keeps the stated order, and 3 less than 9 means start at 9 and take 3 away.
Prediction
This is the single most missed wording on one-operation items.
Predict first
Write an expression for: 4 less than 15.
Correct: 15 - 4
Read it as: from 15, take 4. The 15 is the starting point even though it is spoken second.
Why: The phrase 4 less than 15 means start at 15 and come down by 4. The number spoken first is the one being subtracted, which reverses the order you read it in.
Trap
4 less than 15 becomes 4 - 15, copying the numbers down in the order they were spoken.
4 less than 15 becomes 15 - 4. The phrase names the amount removed first and the starting point second.
The same reversal happens with subtracted from: 6 subtracted from 20 is 20 - 6.
Concept
It is a quirk of English, not of mathematics. The words less than describe a comparison, and comparisons name the small thing first.
Figure (svg): Two number lines showing ten minus four landing on six and four minus ten landing on negative six
Addition and multiplication do not care about order, so the trap only bites on subtraction and division.
Faded example
The numbers are placed for you; only the operation is missing.
Fill in the blanks
\text\times \;\Rightarrow\; 8 ___ 5
Why: Product is the answer to a multiplication, so the operation between the two numbers is a multiplication sign.
Check
Solve it on paper before you click.
Check your understanding
A baker makes 9 trays of rolls with 12 rolls on each tray. Which expression gives the total number of rolls?
Answer: B
Why: Nine equal groups of twelve is repeated equal groups, which is multiplication. The expression is 9 times 12.
Error analysis
A student wrote an expression for: 7 fewer than 30.
Annotate
On: \( 7 - 30 \)
Whenever the phrase ends with than, expect the order to reverse.
Explain it to yourself
Explaining it yourself is what moves it from recognised to known.
Discussion prompt
In your own words, when does the order of the two numbers matter, and when does it not?
Hint: Try each operation with the numbers 9 and 3 both ways round.
Answer:
Order does not matter for addition and multiplication: 9 + 3 and 3 + 9 are the same number, and so are 9 times 3 and 3 times 9.
Order does matter for subtraction and division. Those are the only two you have to think about.
Real world
Expressions are how a spreadsheet, a recipe and a receipt all work.
Discussion prompt
A receipt shows 3 drinks at 2 dollars each. What expression would a till use for that line, and why does it store the expression rather than just the total?
Answer:
The till stores 3 times 2. Keeping the expression means that if the price changes, the total recalculates itself.
This is exactly why writing the expression matters more than writing the answer.
Section
Part 3
Concept
With two operations you must show which one waits. That is what parentheses are for.
Figure (svg): Two expressions compared: three times four plus five with parentheses giving twenty seven, and without parentheses giving seventeen
If the words say one thing must be finished before the other starts, that part goes inside parentheses.
Worked example
Write an expression for: three times the sum of four and five.
Find the last operation
Why: The phrase is three times something, so the final operation is a multiplication.
Find what must happen first
Why: The something is the sum of four and five, so that addition has to finish first.
Group the part that goes first
Why: Wrap the addition in parentheses so it cannot be skipped.
\[ 3 \times (4 + 5) \]
Verify: by evaluating
Why: The sum is 9, and 3 times 9 is 27. Without the parentheses it would have been 12 plus 5, which is 17 — a different number, so the parentheses are doing real work.
Intuition
The tree from Part 1 is the fastest way to check a two-operation expression.
Figure (svg): An expression tree for three times the sum of four and five, with the addition grouped underneath the multiplication
Read it bottom up: the addition finishes, then the multiplication uses its result.
Anomaly
These two phrases differ by one small word, and the answers differ by ten.
\[ 3 \times (4 + 5) \quad \text{versus} \quad 3 \times 4 + 5 \]
Predict first
Which phrase produces the expression on the right, with no parentheses?
Correct: the sum of three times four, and five
Naming the last operation first is the reliable way to tell these two phrases apart.
Why: The right-hand expression finishes with an addition, so the phrase must name a sum as the last operation. Its parts are three times four, and five.
Comparison
Fill the blanks. The pattern will announce itself.
Comparison matrix
| expression | what happens first | value |
|---|---|---|
| 3 x (4 + 5) | the addition | 27 |
| 3 x 4 + 5 | the multiplication | 17 |
| (20 - 8) / 4 | the subtraction | 3 |
| 20 - 8 / 4 | the division | 18 |
Parentheses are only needed when the operation you want first is not the one the ladder would do first anyway.
Ranking
Build the expression for: the sum of 6 and 4, all divided by 5.
Put in order
Why: Naming the last operation first tells you the overall shape, and everything else is then slotted inside it. The finished expression is the quantity 6 plus 4, in parentheses, divided by 5.
Trap
For the sum of 6 and 4, divided by 5, a student writes 6 + 4 / 5 and calls it done.
The words group the sum, so it must be written (6 + 4) / 5, which is 10 divided by 5, or 2.
Without the parentheses the ladder divides first: 4 divided by 5, then add 6. That is not what the words asked for.
Fill the middle
The phrase is: four times the difference of ten and six.
Fill in the blanks
4 \times ( 10 - 6 )
Why: The difference has to finish before the multiplication can use it, so the subtraction is wrapped in parentheses. The value is 4 times 4, which is 16.
Check
Solve it on paper before you click.
Check your understanding
Which expression means: five more than the product of 3 and 8?
Answer: A
Why: The last operation named is the addition of five, and the thing it is added to is the product of 3 and 8. Because multiplication already happens before addition, no parentheses are needed.
Step zero
Step zero is the step that prevents most two-operation errors.
Discussion prompt
You are given: twice the sum of a number of apples and a number of pears. What is the very first thing you should decide, before writing any symbols?
Hint: Do not start at the left-hand end of the sentence.
Answer:
Decide which operation is last. Here the phrase is twice something, so the last operation is a multiplication by 2.
Everything else then has to fit inside that shape, which forces the sum into parentheses.
Explain it
A classmate writes 2 x 3 + 4 for twice the sum of 3 and 4.
Discussion prompt
Explain, without just giving the right answer, how they can tell their expression does not match the words.
Answer:
Ask them to name the last operation in their expression. Theirs finishes with an addition; the phrase finishes with a doubling.
That mismatch is the signal. The fix is to group the sum: 2 times the quantity 3 plus 4.
Missing information
Not every question can be answered.
Discussion prompt
A shop sells notebooks and pens. Write an expression for the total cost of 3 notebooks and 2 pens. What information do you still need?
Answer:
You need the price of one notebook and the price of one pen. Without them the expression can only be written with words, not numbers.
With prices n and p it would be 3 times n, plus 2 times p — two multiplications joined by an addition.
Section
Part 4
Concept
Evaluating means running the recipe. The only question is the order, and the order is fixed by convention.
Figure (svg): A three rung ladder listing grouping symbols first, then multiply and divide, then add and subtract
Notice that multiply and divide share a rung, and add and subtract share a rung. Within a rung you work left to right.
Worked example
Evaluate: 20 minus 3 times 4.
Scan for grouping symbols
Why: There are none, so drop to the next rung.
Multiply and divide first
Why: 3 times 4 is 12, so the line becomes 20 minus 12.
Then add and subtract
Why: 20 minus 12 is 8.
\[ 20 - 3 \times 4 = 20 - 12 = 8 \]
Verify: against the wrong route
Why: Going left to right would give 17 times 4, which is 68. The ladder exists precisely so that two people do not get 8 and 68 from the same line.
Intuition
Rewrite the entire expression on every line, changing exactly one thing.
Figure (svg): A three step trace evaluating twenty minus three times four, ending at eight
Students who lose marks here almost always lost a number, not a rule.
Pattern
Each frame performs exactly one operation.
Step through it
At which step would going left to right first have gone wrong?
The line shrinking by one operation per step is the visual signature of a correct evaluation.
Worked example
Evaluate: (6 + 2) times 5 minus 4.
Figure (svg): The expression six plus two in brackets, times five, minus four, with the bracket highlighted as the first step
Grouping symbols first
Why: 6 + 2 is 8, so the line becomes 8 times 5 minus 4.
Multiply before subtracting
Why: 8 times 5 is 40, giving 40 minus 4.
Finish on the bottom rung
Why: 40 minus 4 is 36.
\[ (6 + 2) \times 5 - 4 = 8 \times 5 - 4 = 40 - 4 = 36 \]
Verify: the parentheses mattered
Why: Without them, 6 plus 2 times 5 minus 4 would be 6 plus 10 minus 4, which is 12. The grouping changed the answer, so it was not decorative.
Elimination
Evaluate 12 + 6 / 3 by elimination rather than by computing blindly.
Eliminate the wrong options
Which value is correct for 12 plus 6 divided by 3?
Survives elimination: c2
Why: Division sits above addition on the ladder, so 6 divided by 3 is resolved first, giving 2. Then 12 plus 2 is 14.
Trap
For 12 + 6 / 3 a student computes 12 + 6 first, getting 18, then divides by 3 to get 6.
Division outranks addition, so the division happens first: 6 divided by 3 is 2, and 12 plus 2 is 14.
Left to right only applies within a rung, never across rungs.
Error analysis
A student evaluated 30 minus 4 times 5 and wrote 130.
Annotate
On: \( 30 - 4 \times 5 = 26 \times 5 = 130 \)
The size of the error is the clue: an answer that is wildly too big usually means an operation was done out of order.
Check
Solve it on paper before you click.
Check your understanding
What is the value of 4 x (9 - 6) + 2?
Answer: A
Why: Grouping symbols come first, so 9 minus 6 is 3. Then multiplication before addition gives 4 times 3, which is 12, and finally 12 plus 2 is 14.
Estimation
A quick estimate catches an order-of-operations slip before it costs a mark.
Predict first
Roughly how big is 19 + 21 x 3, without computing it exactly?
Correct: a bit over 80
If your exact answer had come out near 120, the estimate would have told you that you added before multiplying.
Why: The multiplication happens first and is about 20 times 3, or 60. Adding roughly 20 more gives about 80. The exact value is 82.
Invariant
One thing never changes while an expression is being evaluated.
Step through it
What quantity is identical on all three lines?
Every legal step preserves the value. That is what makes it a legal step.
Two truths and a lie
One statement below is false. Eliminate it.
Eliminate the wrong options
Which statement about the order of operations is false?
Survives elimination: t2
Why: Multiplication is not ranked above division — they share a rung, so 12 divided by 3 times 2 is worked left to right, giving 8 rather than 2. The other two statements are correct.
Section
Part 5
Concept
A MAP item can hand you words and ask for an expression, or hand you an expression and ask for its value. The same three questions cover both directions.
| given | asked for | what to do |
|---|---|---|
| words | an expression | name the last operation, then group what must finish first |
| an expression | a value | climb the ladder, one operation per line |
| words | a value | write the expression first, then evaluate it |
Never do the third one in your head. Write the expression down, then run it.
Worked example
A class of 6 groups each collects 8 cans, then donates 15 of the total. Write an expression and evaluate it.
Name the last operation
Why: The donation happens at the end, so the last operation is a subtraction.
Find what must finish first
Why: The total collected is 6 groups of 8, a multiplication that has to finish before anything is removed.
Write it with the grouping shown
Why: The multiplication already outranks subtraction, so no parentheses are strictly needed.
\[ 6 \times 8 - 15 \]
Evaluate
Why: 6 times 8 is 48, and 48 minus 15 is 33.
Verify: the answer is sensible
Why: Thirty-three cans is less than the forty-eight collected, which is exactly what donating some should do.
Intuition
The array gives the total, and the subtraction takes a slice off it.
Figure (svg): An array of four rows of six dots, showing four groups of six as a multiplication picture
A picture is the fastest check that the expression matches the story.
Reverse engineer
Here the expression is given and the words are missing.
Fill in the blanks
5 \times (12 - 5) = 35
Why: If five times the bracket is 35, the bracket must be 7. Since the bracket is 12 minus something, that something is 5.
Matching
All four use the same three numbers, so only the structure differs.
Match the pairs
Why: Two of these genuinely share the value 14, which is why the structure has to be read rather than guessed from the numbers involved.
Commit first
Commit to an answer and to how sure you are before you check.
Predict first
What is the value of 18 / (3 + 3)?
Correct: 3
If you were confident and wrong, the cause is almost always skipping the bracket.
Why: The bracket is resolved first, giving 6, and 18 divided by 6 is 3. Dividing by 3 first and then adding would give 9, which is the common slip.
Counterexample
Rules are easier to trust once you have seen them fail.
Discussion prompt
A student claims that parentheses never change the answer. Find an expression that proves them wrong, and one where they happen to be right.
Answer:
They are wrong for 3 times the quantity 4 plus 5, which is 27, versus 3 times 4 plus 5, which is 17.
They are accidentally right for 2 plus the quantity 3 plus 4, which is 9, and 2 plus 3 plus 4, which is also 9 — because addition alone does not care about grouping.
So the honest rule is: parentheses matter whenever the operations are different.
Exit ticket
One question that tells you whether the deck landed.
Predict first
Which expression means: seven less than the product of 4 and 6?
Correct: 4 x 6 - 7
Why: The product of 4 and 6 is the starting quantity, and seven less than it means subtract seven from that product. The phrase less than reverses the spoken order, so the seven comes last.
Check
Solve it on paper before you click.
Check your understanding
A shop has 8 shelves with 15 books each, and 20 books are sold. Which expression gives the number of books left?
Answer: A
Why: The total on the shelves is 8 times 15, which is 120, and the twenty sold are then removed, leaving 100. Multiplication already happens before subtraction, so no parentheses are needed.
Connect it up
Put the deck on one page in your own handwriting.
Draw it
Sketch a flow chart that starts at a worded problem and ends at a single number. Mark the point where you decide the last operation, and the point where you decide whether parentheses are needed.
If you can draw this from memory, you can handle any of the three skills in this deck.
Recap
Three skills, one habit.
If you write the expression down instead of doing it in your head, almost all of these errors disappear.
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