MathTrailCounting and enumerationGrades 1–6
A drawing of a few lines holds more triangles and squares than first meet the eye: small ones, and big ones made of several pieces. These tasks ask for all of them, or for the matches to take away to change a figure. Counting by size, smallest first, is the whole skill.
A square of 2 by 2 cells holds 4 small squares and 1 big square round them. A figure made of several pieces counts as well.
A triangle counts only when each of its sides lies on a line of the drawing. Two lines from corner to corner make 8 triangles in a square.
Where two squares of matches touch, they share one match. A row of 3 squares takes 10 matches, not 12.
List the figures by their size: the smallest first, then those of two pieces, then the bigger ones, and add the counts up.
| The drawing | The smallest | The bigger ones | In all |
|---|---|---|---|
| a square of 2 by 2 cells | 4 | 1 | 5 |
| a square of 3 by 3 cells | 9 | 4 + 1 | 14 |
| a rectangle of 2 by 3 cells | 6 | 2 | 8 |
Name each size of figure the drawing can hold, count each size on its own, then add. For matches, ask which squares each match is a side of. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 1–2
Two lines go from the top corner of a triangle down to its base. How many triangles are there in the drawing?
The two lines cut the triangle into 3 small triangles.
Two small triangles side by side make a triangle: there are 2 such pairs.
The whole triangle is 1 more. In all, 3 + 2 + 1 = 6.
Answer 6 triangles.
Example 2 · Grades 3–4
A square of 3 by 3 cells is drawn with all its lines. How many squares of any size are there?
Squares of one cell: 3 rows of 3, that is 9.
Squares of 2 by 2 fit in 2 rows of 2 places, that is 4.
The whole square of 3 by 3 is 1 more. In all, 9 + 4 + 1 = 14.
Answer 14 squares.
Example 3 · Grades 5–6
12 matches make a square of 2 by 2 cells. What is the least number of matches to take away so that no square of any size is left?
There are 5 squares: 4 small ones and the big one.
A match is a side of 2 squares at most, so 2 matches break at most 4 squares, and one is left.
The 2 inside matches that stand up and down break all 4 small squares, and 1 match of the top breaks the big one: 3 matches.
Answer 3 matches.
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 size of figure is missed: the big square round the small ones, or the triangles made of two pieces.
What to say
“Is there a bigger figure made of several pieces?”
One figure is counted twice, from two of its pieces, or a shared match is counted for both of its squares.
What to say
“Has this one been counted already?”
The count stops at the smallest figures, or at the first answer that looks right.
What to say
“Are there figures of another size?”
Draw a square of 2 by 2 cells and count its squares together. Then try 3 by 3.
Lay a row of squares with matches or sticks. How many are needed for 4 squares? For 5?
When counting, say each size out loud: small ones, then two pieces, then bigger. Write the numbers down and add them.
Give me a task about counting figures