QUESTION IMAGE
Question
ms. squillace’s sorting.
ms. squillace grouped the sequences as shown in the picture.
why do you think she did this?
4 cards
8, 16, 32, 64, ……
6, 11, 16, 21, 26, ……
2, 5, 8, 11, 14, ……
1, 5, 25, 125, ……
4 cards
40, 43, 46, 49, ……
100, 50, 25, 12.5, ……
30, 26, 22, 18, ……
99, 33, 11, \\(\frac{11}{3}\\)……
Looking at the first group of 4 cards:
- The first sequence (8, 16, 32, 64,...) has a common ratio of \( \frac{16}{8}=2 \), \( \frac{32}{16}=2 \), so it's a geometric sequence with ratio 2.
- The second sequence (6, 11, 16, 21, 26,...) has a common difference of \( 11 - 6 = 5 \), \( 16 - 11 = 5 \), so it's an arithmetic sequence with difference 5.
- The third sequence (2, 5, 8, 11, 14,...) has a common difference of \( 5 - 2 = 3 \), \( 8 - 5 = 3 \), arithmetic sequence with difference 3.
- The fourth sequence (1, 5, 25, 125,...) has a common ratio of \( \frac{5}{1}=5 \), \( \frac{25}{5}=5 \), geometric sequence with ratio 5. Wait, maybe I mis - grouped. Wait, looking at the second group of 4 cards:
- First sequence (40, 43, 46, 49,...): common difference \( 43 - 40 = 3 \), arithmetic.
- Second sequence (100, 50, 25, 12.5,...): common ratio \( \frac{50}{100}=0.5 \), geometric.
- Third sequence (30, 26, 22, 18,...): common difference \( 26 - 30=- 4 \), arithmetic.
- Fourth sequence (99, 33, 11, \( \frac{11}{3}\),...): common ratio \( \frac{33}{99}=\frac{1}{3} \), geometric.
Wait, maybe the grouping is based on the type of sequence: arithmetic or geometric. Let's re - check:
First 4 - card group:
- 8,16,32,64,...: Geometric (ratio 2)
- 6,11,16,21,26,...: Arithmetic (difference 5)
- 2,5,8,11,14,...: Arithmetic (difference 3)
- 1,5,25,125,...: Geometric (ratio 5)
Second 4 - card group:
- 40,43,46,49,...: Arithmetic (difference 3)
- 100,50,25,12.5,...: Geometric (ratio 0.5)
- 30,26,22,18,...: Arithmetic (difference - 4)
- 99,33,11, \( \frac{11}{3}\),...: Geometric (ratio \( \frac{1}{3}\))
Wait, no, maybe the first group (top 4 cards) has two arithmetic and two geometric? Wait, no, let's check the differences/ratios again.
Wait, the first row of 4 cards (top):
- 8,16,32,64,...: ratio \( r = 2 \) (geometric)
- 6,11,16,21,26,...: difference \( d = 5 \) (arithmetic)
- 2,5,8,11,14,...: difference \( d = 3 \) (arithmetic)
- 1,5,25,125,...: ratio \( r = 5 \) (geometric)
Second row of 4 cards (bottom):
- 40,43,46,49,...: difference \( d = 3 \) (arithmetic)
- 100,50,25,12.5,...: ratio \( r = 0.5 \) (geometric)
- 30,26,22,18,...: difference \( d=-4 \) (arithmetic)
- 99,33,11, \( \frac{11}{3}\),...: ratio \( r=\frac{1}{3} \) (geometric)
Ah! Now I see. The top 4 - card group has two geometric sequences (first and fourth) and two arithmetic sequences (second and third). The bottom 4 - card group has two arithmetic sequences (first and third) and two geometric sequences (second and fourth). Wait, no, maybe the grouping is based on the type of sequence: arithmetic or geometric. Let's check the number of arithmetic and geometric in each group.
Top group:
- Geometric: 8,16,32,64 (ratio 2); 1,5,25,125 (ratio 5) → 2 geometric.
- Arithmetic: 6,11,16,21,26 (diff 5); 2,5,8,11,14 (diff 3) → 2 arithmetic.
Bottom group:
- Arithmetic: 40,43,46,49 (diff 3); 30,26,22,18 (diff - 4) → 2 arithmetic.
- Geometric: 100,50,25,12.5 (ratio 0.5); 99,33,11, \( \frac{11}{3}\) (ratio \( \frac{1}{3}\)) → 2 geometric.
So Ms. Squillace grouped the sequences by the type of sequence: each group (top 4 and bottom 4) has 2 arithmetic and 2 geometric sequences. Alternatively, looking at the common difference or ratio:
In the top group:
- The first and fourth are geometric (multiplicative pattern), second and third are arithmetic (additive pattern).
In the bottom group:
- The first and third are arithmetic (additive pattern: 40 + 3, 30-4), second and fourth are geometric (multiplicative pattern: 100×0.5, 99×\( \frac{1}{3}\)).
So the reason for grouping is that within…
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Ms. Squillace grouped the sequences by their pattern type (arithmetic or geometric). The top 4 - card group and bottom 4 - card group each contain 2 arithmetic sequences (with a common difference) and 2 geometric sequences (with a common ratio), allowing for comparison of additive (arithmetic) and multiplicative (geometric) sequence patterns.