Reasoning · Grade 4-2 Counting Large and Small Figures

Problem

Count figures that contain a marked cell

Figure (1) is a rectangle cut into 4 columns and 5 rows. Its star sits in the 2nd column from the left and the 4th row from the top. Figure (2) is a rhombus cut into 5 bands one way and 4 the other, with its star where the 3rd band meets the 2nd. Count the quadrilaterals containing the star in each.
Your answer
How to solve
Strategy Make a Systematic List — Hunting for four-sided figures one at a time invites double counting and misses. Instead I notice that a quadrilateral here is completely decided by two separate choices: which pair of lines forms its left and right edges, and which pair forms its top and bottom edges. That splits the job into two much easier related problems, each along a single strip of the figure, and each one is small enough to list out loud. Multiplying the two counts then gives the total, and the very same reasoning transfers to the slanted figure because slanting the picture does not change which lines cross which.
1STEP 1

See that each quadrilateral is a choice of two lines each way

A quadrilateral is two lines in each direction.

2STEP 2

Figure (1): count along the row that holds the star

In (1) the row direction gives 6.

1 + 2 + 2 + 1 = 6 (or 2 × 3 = 6)
3STEP 3

Figure (1): count along the column that holds the star

In (1) the column direction gives 8.

1 + 2 + 2 + 2 + 1 = 8 (or 4 × 2 = 8)
4STEP 4

Figure (1): pair every row choice with every column choice

Multiplying, (1) gives 48.

6 × 8 = 48
5STEP 5

Figure (2): recognise the very same grid, just tilted

(2) is just the same grid tilted.

6STEP 6

Figure (2): count along each direction and multiply

The same count gives (2) 54.

9 × 6 = 54
Answer
48, 54 quadrilaterals
6 × 8 = 48, 9 × 6 = 54
Both figures hold exactly the same number of quadrilaterals altogether: in figure (1) it is 10 ways to pick 2 of the 5 vertical lines times 15 ways to pick 2 of the 6 horizontal lines, which is 150, and in figure (2) it is 15 times 10, which is 150 again. So 48 and 54 are each about a third of everything, which is just right for a cell that is near, but not exactly at, the middle. It also makes sense that 54 beats 48: the star in figure (2) sits closer to the centre of its grid, and cells nearer the centre are trapped by more quadrilaterals than cells nearer an edge. Finally, both answers are counts, so whole numbers are the right kind of answer, and neither can be bigger than 150.
Takeaway

A rectangle is just a left-right choice and a top-bottom choice, so count each one on its own and multiply.

  • See that each quadrilateral is a choice of two lines each way
  • Figure (1): count along the row that holds the star
  • Figure (1): count along the column that holds the star
  • Figure (1): pair every row choice with every column choice
  • Figure (2): recognise the very same grid, just tilted
  • Figure (2): count along each direction and multiply