This is a level 4 algebra strand activity from the Figure It Out series.
A PDF of the student activity is included.
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use a table to find a rule for a geometric pattern
write a rule for a relationship as a linear equation
FIO, Level 4+, Algebra, Book Four, Patterns and Designs, pages 10 -11
square dot paper (see copymaster
In this activity, students devise and use rules for studying patterns in different designs.
In question 1, the students draw diagrams to show how two different short cuts work (see the Answers). The short cut 2 x 2 + 3 leads to a rule for the total number of coloured squares: double the number of green squares and add 1 more than the number of green squares. This means that if there are x green squares, there are x x 2 + (x + 1) or 2x + x + 1 coloured squares altogether. So a section of Greer’s design that has 73 green squares has a total of 73 x 2 + (73 + 1) = 220 coloured squares.
The short cut 2 x 3 + 1 leads to the rule 3x + 1 for the number of coloured squares when there are x green squares. The first rule, 2x + (x + 1), can be simplified as follows: 2x + (x + 1) = 2x + x + 1= 3x + 1
So, although the two rules are expressed differently, they produce the same results for designs with particular numbers of green squares.
In question 1d, the students will be able to use either of the two rules from the short cuts in questions 1b and 1c to find the total number of coloured squares in the first table. The second table requires the students to calculate the number of green squares, given the total number of coloured squares. The students who attempt this using the first rule will find that they are having to guess and check the results. Those who use the second rule, 3x + 1, will find that the calculations are much more straightforward.
To reverse this rule, the students must first subtract 1 from the value for the total number of coloured squares and then divide the result by 3. The following backtracking flow chart shows how this works:
In question 2b, the students try to devise their own short cut for poutama designs. Two of the possible short cuts are shown in the Answers. In question 2c, the students examine Hine’s short cut.
The following table shows how the short cuts lead to algebraic rules that all simplify to 5x + 1.
As with the questions above, there are several ways to visualise short cuts and rules for question 3. Some are shown in the Answers. A further short cut, using additional blocks that must be subtracted, is used by Tracey in question 4.
The following table shows how the three short cuts used for questions 3 and 4 lead to algebraic rules.
Answers to Activity
1. a.
b. i. A possible diagram is:
ii. 28 x 2 + 29 = 85 coloured squares
c. The diagram below shows 2 sets of 3 coloured squares forming an L shape, and 1 additional square.
So the number of coloured squares is 2 x 3 + 1 = 7.
d.
2a.
b. Answers may vary. One possible short cut is 7 x 5 + 1 = 36. It works like this for 3 steps:
3 x 5 + 1 = 16
So the pattern with 7 steps has 7 x 5 + 1 = 36 crosses.
Another short cut for 3 steps is:
1 set of 6 + 2 sets of 5 = 6 + (2 x 5) = 16
So the pattern with 7 steps has 6 + (6 x 5) = 36 crosses.
c. Yes, it is correct. Hine included an additional cross (white) in all but the final step, so each step has 6 crosses. The 4 white crosses must be subtracted, so there are 5 x 6 – 4 = 26 crosses altogether.
d. Short cuts for the Your rule column are based on the short cuts suggested in 2b.
3. a.
b. Answers may vary. One possible short cut works like this:
2 plus signs
16 + 2 x 13 = 42 cubes
3 plus signs
16 + 1 x 13 = 29 cubes
A possible rule is: 16 for the first plus sign + the number of extra plus signs x 13. So the short cut for 5 plus signs is 16 + 4 x 13 = 68.
Another short cut is:
3 sets of 13 cubes + 3 cubes
3 x 13 + 3 = 42 cubes
A possible rule is: 13 for each plus sign plus an extra 3. So the short cut for 5 plus signs is 5 x 13 + 3 = 68.
c.
The 5 plus signs use 68 multilink cubes as predicted by the short cuts 16 + 4 x 13 and 5 x 13 + 3.
d. The calculations are based on the two rules given in 3b.
4. a. When there are 4 plus signs, the first short cut from question 3 gives 16 + 3 x 13 = 55 cubes. Tracey’s rule, which allows for extra cubes to be subtracted, gives 4 x 16 – 3 x 3 = 55. Both rules give the answer as 55, so Tracey’s rule for 4 plus signs is correct.
b. Tracey’s short cut is 8 x 16 – 7 x 3 = 107. Other possible short cuts, such as 16 + 7 x 13 = 107 and 8 x 13 + 3 = 107, should give the same result.
c. 1 000 x 16 – 999 x 3 = 16 000 – 2 997
= 13 003