A student hands you a paper with 8 + 2 × 5, and the answer underneath is 50. The work is neat, the multiplication is correct, the addition is correct, and the answer is still wrong. You have seen this paper before, in different handwriting, from different students, in different years: the student did the operations in the order they appeared, left to right, trusting that reading order was the same as math order.
This guide starts there, with the mistakes themselves. Order of operations is one of the few topics where the wrong answer usually tells you exactly what the student was thinking, and reading those errors lets you fix the thinking behind them instead of just marking the answer wrong.
This guide walks through the five mistakes that show up on almost every order-of-operations paper in grades 5 through 8, explains why each one keeps happening, and shows a teaching routine that catches them early. It also lays out a six-day plan, ways to use leveled practice in a mixed classroom, and quick checks that tell you whether the fix actually stuck.
The Five Mistakes That Show Up on Almost Every Paper
Ask a class to solve a handful of multi-step expressions and sort the wrong answers. Most fall into five groups, each pointing to a different misunderstanding that needs a different response.
Mistake 1: Solving strictly left to right
This is the most common error of all. Children have spent years reading left to right, and they bring that habit to math. Give them 8 + 2 × 5 and they add 8 and 2 to get 10, then multiply by 5 to get 50. Give them 20 − 4 × 3 and they subtract first, then multiply 16 by 3. The arithmetic is fine; the order is the problem.
The fix starts with one question: "Which part of this expression is the strongest?" Multiplication and division act first. For a week, use a warm-up where students only point to the operation that goes first, without solving anything. Separating the decision from the computation is what breaks the habit.
Mistake 2: Treating the acronym as a strict sequence
Students who have learned PEMDAS often read it as a strict order: parentheses, then exponents, then multiplication, then division, then addition, then subtraction, each one fully before the next. That reading creates two new errors. A student sees 12 ÷ 3 × 2 and divides first, then multiplies, which happens to be correct here, but the same student sees 12 × 3 ÷ 2 and insists on dividing before multiplying because D comes after M in the acronym. Or a student sees 9 − 4 + 3 and adds before subtracting because A comes before S.
Multiplication and division are partners, solved left to right as they appear, and addition and subtraction are partners too. The acronym is a memory aid for the groups, not a ladder with six rungs. Write the partners together on the board, M and D on one line, A and S on another, then try 10 − 3 + 2, where adding first gives 5 and the correct answer is 9. One surprising result does more than a week of reminders.
Mistake 3: Ignoring or misreading grouping symbols
Grouping symbols are the first thing students are supposed to handle, and they are also the easiest to skip. A student looks at 3 × (4 + 5) and multiplies 3 by 4, then adds 5, because the parentheses felt like decoration. Another student sees nested grouping, such as 2 × [3 + (4 − 1)], and works the brackets before the parentheses inside them, or works the outside first and the inside last.
The cure is a physical habit: draw a box around the innermost parentheses before writing anything else, then start inside and work outward. A student who can point to the innermost group and say "this first" rarely skips it. Brackets and braces in later grades are the same idea with different shapes.
Mistake 4: Exponents that arrive too late, or never at all
Exponents are new for many fifth and sixth graders, and they are easy to forget because they are small. A student solving 2 + 3² writes 5², adding first and then squaring the result. Another reads 3² as 3 × 2 and writes 6. A third handles the exponent correctly but only after doing the multiplication beside it, as if the little number were less important than the big ones.
The fix is to make the exponent visible: rewrite it as a repeated multiplication on the very first line, so 3² becomes 3 × 3 before anything else happens. That turns a forgotten symbol into a number the student can see, and it clears up the confusion between 3² and 3 × 2, since 3 × 3 is plainly 9 and not 6. Once students are comfortable they can stop writing it out, but keep the rewrite for the hardest problems.
Mistake 5: Losing track of negative numbers
By seventh and eighth grade, negative numbers enter the picture, and a new family of errors arrives with them. A student sees 5 − (−2) and writes 3, treating the two minus signs as one subtraction. Another sees −3² and writes 9, squaring the negative instead of squaring 3 and keeping the sign. A third drops a negative sign halfway through a long problem and never notices, because nothing in the work flags the loss.
These errors are about sign sense, not order, and they surface here because these problems are long enough for a sign to get lost. Slow the work down at these points: ask students to say the sign out loud when they copy each new line, since "negative three, squared" is not the same as "three squared, then negative." When an answer looks wrong, check first whether a sign changed in the middle.
Why These Mistakes Keep Coming Back
If the same five errors appear year after year, it is worth asking why. Order of operations asks students to do two things at once: remember a sequence and carry out arithmetic without losing their place. A student concentrating on 7 × 8 has no attention left for remembering that the exponent comes first.
The rules also feel arbitrary, because students are told that multiplication comes first without being told why. A short explanation helps: multiplication is repeated addition, so 2 + 3 × 4 means 2 plus three groups of 4, which is 2 + 12. The order is a shared agreement that keeps everyone reading an expression the same way, not a trick.
The third reason is practice that moves too fast. If a student makes the same error on problem three that they made on problem one, the rest of the page just rehearses the mistake. That is why this guide slows down, checks each step, and puts error-spotting before speed: five problems right with the steps written out beats fifteen wrong in a hurry.
The Step-Down Method: One Step at a Time
The Step-Down Method is a simple idea: solve one operation at a time, and write the new expression on the line below after each step. Nothing is done in your head. Every step leaves a record, so when the answer is wrong you can walk back down the page and find exactly where it went wrong.
Take the expression 6 + 2 × (5 − 3)². Solved one step at a time, the work looks like this:
- Parentheses first. 5 − 3 is 2, so the expression becomes 6 + 2 × 2².
- Exponents next. 2² is 4, so the expression becomes 6 + 2 × 4.
- Multiplication next. 2 × 4 is 8, so the expression becomes 6 + 8.
- Addition last. 6 + 8 is 14.
Each line has exactly one change, and the rest is copied down untouched. That copying is not busywork; it is the skill. Students who copy carefully keep their signs and numbers intact.
A checklist beside every problem makes the method stick. Before writing the next line, the student checks grouping, then exponents, then multiplication and division left to right, then addition and subtraction left to right. The checklist turns remembering the order into reading it off the page.
Teach it with a worked example on display for the first few days. Solve one problem slowly at the board, narrating each decision, then have the class solve the next one with you, one line at a time, before anyone works alone. The Order of Operations Worksheets includes a worked step-down example and a step-by-step solution model that you can keep posted while students practice, along with reference guides for both PEMDAS and GEMDAS so the wording matches what your class already uses.
Turning Mistakes Into Lessons: Find the Mistake
Once students can name the five mistakes, give them the job of finding them. A Find the Mistake problem shows a solved expression with one error already inside it, and the student has to spot the error, explain what went wrong, and solve the problem correctly. It is a small change with a large effect: instead of worrying about their own answer, students get to play detective, and the error becomes something interesting rather than something shameful.
Run it as a partner task. One student reads the worked solution out loud, line by line, while the other checks each line against the order. When they disagree, they defend their reading with the checklist, not with "I think so." Four error-analysis problems make a good session. Ask each pair to report the name of the mistake they found, using the language from the five-mistake list. By the third problem, you will hear "they added before multiplying" instead of "this one is wrong," and that shift in language is the shift in understanding.
A Six-Day Plan That Fits a Real Week
Six days is enough to teach the topic properly. Here is one way to pace it, built around the 6-day study plan that comes with the practice set.
Day 1: Parentheses and the left-to-right habit. Start with expressions that have only the four operations plus parentheses. The only new idea is that grouping comes first. End with a quick sort: students mark the operation that goes first in six expressions, without solving.
Day 2: Multiplication and division before addition and subtraction. Use expressions like 8 + 2 × 5, and let students predict the wrong answer first, then find the right one. Prediction makes the correction memorable.
Day 3: Exponents. Introduce the exponent as repeated multiplication, and have students rewrite every exponent on the first line. Add brackets as a second kind of grouping symbol.
Day 4: Nested grouping and multi-step expressions. Work inside out, one layer at a time, keeping expressions short enough to finish in four or five lines.
Day 5: Error analysis and Make It True. Spend the day on the four Find the Mistake problems and the Make It True equations. Both ask students to think about the order instead of just following it.
Day 6: Review and check. Mixed practice across all three levels, then an exit ticket. Anything still shaky becomes small-group work the following week, not a reason to repeat the unit.
Three Levels, One Classroom
A fifth-grade class and an eighth-grade class do not need the same problems, and neither does a mixed group in one room. The three levels in the Order of Operations Worksheets let you teach one lesson and differentiate the practice: Level 1 keeps to parentheses with the four operations, Level 2 adds exponents, brackets and multi-step expressions, and Level 3 brings in nested grouping symbols, negative numbers and complex expressions.
Start everyone on the same worked example, then hand out different levels. Students at the same table can be on different levels without it feeling like a public ranking, because the problems look the same on the page: two per page, a large workspace, and the same checklist beside each one. When a student finishes a level confidently, move them up the next day. When a student is stuck, drop them down a level for one session; a confident Level 1 paper teaches more than a frustrated Level 3 paper.
Exit Tickets and Quick Checks
A long worksheet tells you how a student did at the end of the week. An exit ticket tells you how they did today, while there is still time to fix it. Four quick tickets, cut apart and handed out at the end of a lesson, are enough to check one skill at a time: parentheses on Monday, multiplication before addition on Tuesday, exponents on Wednesday, and a mixed problem on Thursday.
Keep the routine small. One problem, three minutes, collected at the door. Sort the tickets into three piles: correct, right answer with messy work, and wrong. The middle pile is the most useful, because a student with the right answer and no written steps is one hard problem away from a wrong one. The next day, start with one ticket problem on the board, solved step by step, and let the class catch the error before you reveal the answer.
Make It True: When Parentheses Change Everything
Make It True turns the usual question around. Instead of solving an expression, students are given an equation that is false and asked to add parentheses until it becomes true. The problem 6 + 2 × 3 = 24 is false as written, but 6 + 2 × 3 with parentheses around 6 + 2 makes it true, and suddenly parentheses are not a rule to memorize but a tool with visible power.
These eight logic-based equations work well as a challenge for early finishers and as a whole-class puzzle on a Friday. They reward the student who experiments: try the parentheses here, check, move them, check again. If two students find different placements that both work, celebrate that. It proves the point better than any explanation.
Supporting Struggling Learners and Stretching Fast Finishers
For students who are still shaky, shrink the problem, not the expectation. Cover up all but the first two operations and ask which comes first. Let them use the reference guide on every problem for a week, and check their work line by line instead of only marking the final answer. A student who writes each step down is a student you can help. For English learners, keep the checklist language the same every day and point to the symbols as you name them.
Fast finishers need harder thinking, not more of the same. Give them the Make It True equations, or ask them to write their own Find the Mistake problem for a partner, complete with a wrong solution that hides a believable error. Writing a good wrong answer takes more understanding than solving a problem correctly, and it keeps them busy in a way that is genuinely useful to the class.
Practice at Home Without Tears
Families can help without knowing the rules themselves. The most useful thing a parent can do is ask the child to explain one line of working out loud: "What did you do first here, and why?" If the child cannot answer, that is the signal to write the steps out more carefully, not to do more problems. A short daily habit works better than a long session before a test: two problems a night, each solved with the checklist beside it, and the answer key used to check the steps, not just the final number.
Frequently Asked Questions
Should I teach PEMDAS or GEMDAS?
Use whichever wording your class already knows. The set comes in a full PEMDAS version and a full GEMDAS version, with a reference guide for each. The order is the same either way; only the letter for grouping changes.
How do I stop students from solving everything left to right?
Separate the decision from the computation. For a week, have students mark the operation that goes first before they solve anything. When choosing comes before calculating, the left-to-right habit has nothing to grab onto.
What if a student gets the right answer but shows no working?
Treat it as unfinished. A right answer without steps gives you no way to know whether the student understood the order or guessed well. Ask them to redo one problem with every step written out.
How much practice do students need each day?
A small number of problems done carefully beats a long page done quickly. Eight problems at one level, each with the checklist completed, is a solid daily session, and the four exit tickets give a daily check without a full worksheet.
How do I help a student who keeps forgetting exponents?
Have them rewrite the exponent as a repeated multiplication on the first line of working, every time, until it becomes automatic. The rewrite makes the exponent visible, and it clears up the confusion between 3² and 3 × 2.
When should students move up a level?
Move a student up when they can solve a full level with the steps written out and explain each choice, not when they get a few right answers by luck.
