Number: Counting your odds and evens
View Sequence overviewWe can use properties of odd and even numbers to reason about sums without calculating.
We can justify general statements about numbers using diagrams, counters, or other representations.
We can identify which aspects of a problem matter for a result, and which do not.
Whole class
Counting your odds and evens Slides
Task
Show Slide 8 of the Counting your odds and evens Slides. This slide displays two 3 × 3 number mazes.
Ask students to move through each maze and find a path that gives an odd total. They can move right and down only. Challenge students not to calculate every total precisely—the focus is on noticing patterns across paths rather than finding exact sums.
Discuss their strategies and any patterns they may have noticed.
- What do you notice about the first maze? Does the result change depending on which path you take?
- Every path gives an odd total.
- The result never changes, no matter which path is chosen.
- Why might that be? What do you notice about the numbers in the first maze?
- There is only one odd number in the maze, and every path must pass through it.
- All the other numbers are even.
- What about the second maze? Do all paths give the same result?
- Some paths give an odd total and some don’t.
- The second maze is less predictable than the first maze.
- What is different about the second maze?
- There are two odd numbers. Some paths pass through one of them, some pass through both, and some pass through neither.
- Can you find a path in the second maze that gives an odd total? What do you notice about which odd numbers it passes through?
- Paths that pass through exactly one odd number give an odd total.
- Paths that pass through both odd numbers, or neither, give an even total.
Pose the challenge: Can we design a 3 $\times$ 3 maze so that we can predict whether the total will be odd or even, without adding all the numbers?
Briefly invite a few initial ideas, without evaluating or resolving them. Let students know they will be testing and refining their ideas through their own designs.
Give students time to design a 3 $\times$ 3 number maze. You might let students choose their own design goal, or, depending on your class, suggest one of the challenges below.
| Design challenge | A possible approach | A deeper approach |
|---|---|---|
| Design a maze that always gives an even total, no matter which path is taken. | Fill the maze with even numbers, check that several paths all give even totals, and conclude it works. | Reason that any sum of even numbers must be even, so the path chosen cannot matter. Then ask whether a maze containing some odd numbers could also always give an even total, and under what conditions. |
| Design a maze that always gives an odd total. | Try different combinations of odd and even numbers until several paths all give odd totals. | Reason that every path through a 3 $\times$ 3 maze passes through 5 squares, which is an odd number of steps. If every square contains an odd number, every path will add 5 odd numbers, which always gives an odd total. |
| Design a maze where only one path results in an odd total. | Try different number arrangements until only one path gives an odd total, checking each path by calculating. | Reason about which squares are shared by multiple paths and place odd numbers so that only one path collects an odd number of odd squares. |
| Design a maze where changing just one number switches all paths from odd totals to even totals. | Try changing different numbers one at a time until finding one that switches all paths. | Reason that only a square that appears on every possible path could affect all totals at once, and identify which square that is before deciding what to change. |
Once a maze is designed, students swap mazes with another group and test each other’s designs. Students should check more than one path and be prepared to explain why the maze behaves the way it does.
Students now explore whether the patterns they noticed in 3 × 3 mazes still hold when the maze changes size.
Ask students to design a new maze using a different grid size, for example 4 × 4 or 2 × 4, and explore paths through it. Examples are provided on Slide 9.
Pose the question: Does changing the size of the maze change anything about which paths give an odd total?
As students work, encourage them to test several paths and record whether each gives an odd or even total. Push students to explain their results rather than just report them. Ask:
- How many odd numbers are in your path?
- Does it matter how many squares are in the path altogether?
- Can you design a maze of this size that always gives an odd total? What do you need to control?
Draw out the key idea: what matters is not how many squares are in a path, but how many of those squares contain odd numbers. A longer path is not harder to predict. It follows exactly the same rule.
The challenges below vary in how much structure they provide. The first two are good starting points for most classes. The later challenges are better suited to students who have already articulated the core rule clearly.
You might have the whole class work on the same challenge, assign different challenges to different groups, or let students choose their own. Each approach has merit: a shared challenge makes whole-class discussion easier, while different challenges can produce a richer variety of designs to compare.
- Use no more than three odd numbers in the entire maze.
- Place odd numbers so that every path passes through at least one.
- Design a maze that is not a rectangular grid, for example an L-shape or a grid with missing squares.
- Design a maze where changing just one number switches every path from odd to even.
- Design a maze where paths have different lengths from start to finish. What does this tell you about whether the result is predictable?
Using the mazes students have just created, pose the question: What happens if we multiply the numbers along a path instead of adding them? Can we still predict whether the result will be odd or even?
Students should be encouraged to predict first, before calculating.
Spotlight: Bring the class back together. Ask students to place their maze designs where others can see them.
Select one or two student mazes to spotlight. Good candidates are:
- a maze where every path gives the same result, because all odd numbers are unavoidable.
- a maze where the student has successfully controlled the outcome by placing odd numbers deliberately.
- a maze that surprised its designer, where the outcome was not what they expected.
Make the selected maze visible to the whole class. Ask the student to explain their thinking:
- What were you trying to design?
- How did you decide where to put the odd numbers?
- Did it work the way you expected?
- Is there anything you noticed that surprised you?
Invite the class to ask questions and respond:
- Does anyone have a maze that behaves differently? Why do you think that is?
- What would happen if we moved just one odd number in this maze?
- Can you see a pattern in which paths give odd totals?
Use the discussion to draw out the key ideas without stating them directly:
- Even numbers in a path never affect whether the total is odd or even.
- What matters is how many odd numbers a path passes through, not how many squares it contains.
- We can design a maze with a predictable outcome by controlling where the odd numbers are placed.
Once these ideas have surfaced through student voice, ask: So if even numbers don't matter for the outcome, what role do they play? And what does that tell us about odd numbers?
Allow a brief moment for students to turn and talk before taking responses. This sets up the idea that odd and even numbers behave in fundamentally different ways when added, which connects to the generalisation in the next task.
Spotlight
A spotlight pauses the class to draw attention to a piece of student work worth examining together. Rather than telling students what to think, the teacher uses a selected student’s work as a starting point for discussion, allowing mathematical ideas to emerge from the class itself.
In this lesson, a spotlight works particularly well because students will have generated genuinely varied maze designs. Some will have controlled the outcome deliberately, some will have been surprised by their results, and some may have revealed a misconception worth exploring. All of these make rich material for a whole-class discussion.
When selecting a student to spotlight, look for someone who:
- has noticed something about where the odd numbers need to go.
- has a maze that behaved differently from what they expected.
- has made their thinking visible in a way others can engage with.
The goal is not to showcase a “correct” answer, but to use one student’s thinking as a lens through which the whole class can examine the mathematics together. Students should feel invited to question, agree, or push back on what they see.
A spotlight pauses the class to draw attention to a piece of student work worth examining together. Rather than telling students what to think, the teacher uses a selected student’s work as a starting point for discussion, allowing mathematical ideas to emerge from the class itself.
In this lesson, a spotlight works particularly well because students will have generated genuinely varied maze designs. Some will have controlled the outcome deliberately, some will have been surprised by their results, and some may have revealed a misconception worth exploring. All of these make rich material for a whole-class discussion.
When selecting a student to spotlight, look for someone who:
- has noticed something about where the odd numbers need to go.
- has a maze that behaved differently from what they expected.
- has made their thinking visible in a way others can engage with.
The goal is not to showcase a “correct” answer, but to use one student’s thinking as a lens through which the whole class can examine the mathematics together. Students should feel invited to question, agree, or push back on what they see.