Seeing Dots

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Purpose

This is a level 4 algebra strand activity from the Figure It Out series.
A PDF of the student activity is included.

Achievement Objectives
NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
Student Activity

Click on the image to enlarge it. Click again to close. Download PDF (274 KB)

Specific Learning Outcomes

find a rule to describe a geometric pattern

Required Resource Materials
geoboard or square dot paper (see Copymaster)

FIO, Level 3-4, Algebra, Seeing Dots, page 10

classmate

Activity

The students will need photocopies of square dot paper (see the copymaster at the end of these notes) to draw the figures involved and to work out the number of pins.


Activity One

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:

table.
Multiplication:

equation.

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.

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


Activity Two

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

diagram.

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.”

house.
“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:

table.
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 box.  s 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 box.  s.

Answers to Activities

Activity One
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

diagram.

This number (5) increases by 3 for each successive windmill
iii. Number of pins = (12 x windmill number) – 3
c. 117
2. a. Windmill 4: 32 box. s.
Windmill 5: 50 box.  s.
b. Area = (windmill number)2 x 2.
For example, for windmill 2, the area
would be 22 x 2 = 8 s.
c. 200
Activity Two
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.

table.
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 box. s each time.

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Level Four