Seeing dots

The students will need photocopies of square dot paper (Seeing dots CM) to draw the figures involved and to work out the number of pins.

Activity 1

The students can use a numeric approach to find the dots in a windmill. The windmill has 12 sides, three for each of its four arms. For each successive windmill, a dot is added to each side, so the difference between the number of dots in successive windmills is a constant of 12.

The students can find the number of dots for any windmill by using either addition or multiplication. (See the notes for page 1 of the student booklet.)

Addition:

Multiplication:

The students can use spatial reasoning to find the area of the windmills. Each windmill can be enclosed within a square. The windmill takes up half the area of the square.

The areas for windmills of other sizes can be found in the same way.

Activity 2

The table of values for the house contains many interesting patterns and relationships:

Ask the students to explain why they think these patterns occur. Explanations might include: “The outline of the house has five sides, including the base. The base number of dots increases by two and the other sides gain another dot each time.”

“Each larger houses surrounds the previous house, so the total number of dots on the previous house becomes the total for the dots inside the next house.”

Investigation

The students may wish to investigate both the dots and the area of the submarine pattern. Here are some results:

The students should be able to explain the increases with reference to the spatial pattern.

For example, “The hull of the submarine is increased in length by two boxes* each time. This means the number of dots increases by four (two at the top of the hull and two at the bottom), and the area increases by two boxes*.

(* Where a box is a square with a dot on each corner).

Activity 1

1.

a. Windmill 4: 45, Windmill 5: 57

b. Three possible rules are:

i. Each windmill has 12 more pins on the outline than the previous windmill.
ii. Number of pins for windmill 1 = (4 x 2) + 1, Number of pins for windmill 2 = (4 x 5) + 1

• Number of pins for windmill 2 = (4 x 5) + 1
• The number (5) in the above increases by 3 for each successive windmill

iii. Number of pins = (12 x windmill number) – 3

c. 117

2.

a. Windmill 4: 32 boxes*. Windmill 5: 50 boxes*.

b. Area = (windmill number)² x 2. For example, for windmill 2, the area would be 2² x 2 = 8 s.

c. 200

Activity 2

1.

Answers will vary. Patterns seen could include: The dots on the outline increase by 6 for each house. The dots inside increase by the next multiple of 6 (that is, + 6, + 12, …). The total number of dots also increases in multiples of 6 (+ 12, + 18, …).

2.

Discussion will vary (some ideas are in the teachers’ notes).

3.

Investigation

You may notice that each submarine increases by four dots (9, 13, 17). The base and the top deck increase by two dots each time, although the periscope stays the same. The area also increases by two boxes* each time.

(* Where a box is a square with a dot on each corner).