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Table of contents
How many order of operations problems can a student do in five sessions? Spoiler: more than you imagine.
In Innovamat activities, order of operations are not an isolated objective: they are organized as a short, intensive sequence within a broader trajectory, in which students will return again and again to deepen their understanding.
It is precisely this progression that makes working on order of operations in five sessions much more valuable than teaching a rule and repeating it endlessly. It allows us to go deeper into the content, promote understanding, and create practice opportunities almost without students realizing it. This is everything students work on in these sessions:
- Generate the need to establish the order of operations
- Include powers in order of operations
- Productive practice I
- Streamline the resolution of order operations problems at the same level
- Productive practice II
Students are now ready to start working on order operations problems. Do you want to know how they did in the sessions? So then… let’s enter the classrooms!
We discover order of operations
First day of order of operations. Students already master basic operations and, just in the previous sessions, have finished consolidating powers. This is when we can combine everything into a single expression.
In the first activity, we ask students to solve operations with the calculator. But be careful, the result is different! Why does this happen? It is for this reason that the need arises to establish conventions about the order in which operations should be solved. In this session, students will discover what criteria to follow and why.
And, of course, once that order is established, they will practice to consolidate it. Look at this page from the practice logbook:
Here, the key is that students must master combined operations to solve challenges without mechanically repeating the steps. These activities, in addition to requiring them to practice, also engage them in mathematical processes such as problem-solving, operation optimization, and combinatorics.
Powers and order of operations
In this second session, students revisit order of operations. Following the logic of the forgetting curve, we begin by reviewing what was previously learned to strengthen retention and prevent it from fading, before moving on to deepen understanding.
In this session, we add new operations: we incorporate powers and roots. What happens when these new operations come into play? We also address how to solve operations at the same level: from left to right, just like in reading. This way, students understand that, if two subtractions or two consecutive divisions appear, they should solve them following that order.
Furthermore, since we know it is important to offer various representations, we suggest teachers show a video where students discover two ways to represent the resolution of the same combined operation:
- In columns: the classic one, one below the other, repeating numbers when necessary.
- In tree form: where they can see at a glance the hierarchy of operations. This is a very interesting representation because it is easier to visualize that, in a combined operation, we can start with two operations in parallel without them being at the same level. For example, in this case, although they wanted to start with the parentheses, they have seen that 8 × 9 and (9 – 6) can be solved simultaneously.
Productive practice sessions
In the third and fourth sessions, students focus on practicing order of operations through productive practices. That is, situations in which, from an open question, students are presented with a context and an objective that requires practice to produce a response.
In the third session, for example, we propose a very powerful activity: a roulette that generates random digits from which they must construct order of operations that result in 24. Without realizing it, and guided by the challenge, they are practicing order of operations meaningfully.
In the fourth session, we continue deepening, doing so with equally rich proposals, such as the date activity. The process is simple: we write the date of the day in one of the possible formats (dd/mm/yy or dd/mm/yyyy) and, without changing the order of the digits, they must establish an equation using order of operations.
For example, for the date 17/11/2025, we write on the board: 1 7 1 1 2 0 2 5. From here, proposals emerge, such as:
- 1 × 7 − 1 − 1 + 20 = 25
- 1 × (7 − 1 − 1) × 2 ÷ 02 = 5 (where they reinterpret 02 as 2)
- Or even versions using only the last two digits of the year, such as 1 × 7 + 1 − (1 + 2) = 5.
Thanks to this activity, they discover the importance of parentheses, visualize the hierarchy of operations, and, when we solve some expressions in tree form, they understand that sometimes it is more efficient to start with a specific part of the operation.
The broken calculator: the grand finale
And so we arrive at the last session of this sequence, focused on the well-known broken calculator activity.
The session is structured into different moments. The first, as a whole class, consists of finding three different combinations that result in 37 using only the digits 2, 5, 6, 8, 9, and 0, and all available symbols.
The second moment consists of finding the results from 1 to 10 with the keys 2, 5, ×, and −. How can we achieve the results using the available keys?
And finally, the session ends with individual work in the logbook. And it is at this point, where teachers walk around the classroom to accompany them, that very different and interesting moments arise. Look at everything we can observe in the video!
A very rich moment is the discussion that two students have about how to solve a problem. The problem statement is as follows:
For the first solution, one student rewrites 495 as 496 − 1 and 856 as 846 + 10. That is, he has maintained the same “base” of numbers and has reached the result by adjusting them slightly.
In contrast, his classmate uses a compensation strategy: changes 856 to 867 and, so that the final result does not vary, has adjusted 495 to 484.
Both have solved the activity correctly, but following different strategies. Furthermore, something important is that, while the student explains her procedure, the teacher refines her language so that she becomes aware of the place value of the digits: we don’t talk about “8”, but about “80”.
But these moments are also opportunities to detect errors and learning opportunities. One of the students confuses place value and treats the digits of a number as independent units. When trying to represent 885 without using 5, he writes 8 + 3 + 2 + 26, ignoring that 8 represents 800. This misunderstanding leads to a powerful conversation with the teacher, in which the importance of understanding numbers as quantities rather than just symbols is made explicit.
Another very rich moment is when the teacher guides a student with the jumping strategy, reminding her that she can “go too far and then subtract”, helping her verbalize an idea she was already approaching.
After going through these five sessions, the answer to the initial question (How many order of operations problems can a student do in five sessions?) becomes evident: very many. More than fit in a simple list of exercises. But, above all, order of operations that make sense and that are built on understanding. Because it’s not just about how many they do, but about how they think while doing them.
