MathTrailCounting and enumerationGrades 1–6
A flat net of six squares folds into a cube; a folded sheet, punched once, opens with several holes; a die rolled along a row shows a number on top that can be worked out. These tasks are solved by picturing the folding, the rolling or the unfolding, one step at a time.
In a net, two squares in a row with one square between them become opposite faces of the cube.
A sheet folded once has 2 layers where the fold lays it, and folded again 4. A punch makes a hole in every layer.
On a die the numbers on opposite faces add up to 7: 1 and 6, 2 and 5, 3 and 4.
Fold, roll or unfold in the mind one step at a time, and note what each step does.
| What is done | What changes |
|---|---|
| a square of the net folds up | it becomes the next face of the cube |
| a sheet is folded once | its layers double where it lies |
| a die rolls over one edge | a side face comes up on top |
| a die rolls twice straight on | the bottom comes up on top |
Name the square, the fold or the roll that comes first, say what it does, then go on to the next. Where you can, cut out a paper net or use a real die to check. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 1–2
A net of a cube is a cross: a column of 4 squares numbered 1, 2, 3 and 4 from the top, and squares 5 and 6 on either side of square 2. Which number is opposite 2 when it is folded?
Squares 1 and 3 are next to 2 in the column, so they touch it.
Squares 5 and 6 are beside 2, so they touch it too.
Square 4 is two squares below 2, with 3 between them: it lands opposite 2.
Answer The number 4.
Example 2 · Grades 1–2
A square sheet is folded in half, then in half again. One hole is punched through the folded sheet. How many holes are there when it is opened?
The first fold makes 2 layers.
The second fold doubles them: 4 layers.
The punch goes through all 4 layers, so the opened sheet has 4 holes.
Answer 4 holes.
Example 3 · Grades 3–4
A die stands with 2 on top. It rolls over its edge straight ahead twice. Opposite faces of a die add up to 7. Which number is on top now?
The first roll brings the face from the back up to the top.
The second roll brings up the face that was at the bottom at the start.
The bottom was opposite the 2, so it is 7 − 2 = 5.
Answer The number 5.
MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.
A face, a layer or a case is missed: a square that folds round to touch another, or a layer the punch also goes through.
What to say
“Is there a face, or a layer, not counted yet?”
A rule of the task is lost: squares side by side are taken for opposite faces, or the 7 that opposite faces of a die make is forgotten.
What to say
“Does this fit the rule the task gives?”
The working stops one step short: after the first fold, or after one roll of the die.
What to say
“Is there one more step to follow?”
Draw a cross of six squares, cut it out and fold it into a cube. Which squares end up opposite?
Fold a sheet twice, make one hole with a pencil, and guess the holes before opening it.
Look at a die: which numbers are opposite? Roll it along the table and say the top number before looking.
Give me a task about cubes and paper