Missing numbers II

This task is about finding missing numbers that complete number patterns.

For each number pattern:

Put in the correct numbers to complete the pattern.
Write down how you worked out what numbers to put down.

Diagnostic and formative information:

	Common error	Likely calculation	Likely misconception
b) i) c) i)	46, 58, 70, 82 36, 30, 24, 18	Adds 12 Subtracts 6	Consistent application of rule which is incorrect by 1.
d) i)	4, 8, 12, 30	Adds 4	Makes first four terms differ by 4 to give the sequence 4, 8, 12, 16, then adds 4 to 26.
a) c)	1, 12, 23, 34 63, 56, 49, 42		Repeats the pattern.
a)-d)	Various		Application of a rule with an incorrect difference or with varying differences between the terms.

Student strategies
The strategies that students use can determine their level of development within the Numeracy framework. Strategies mentioned were multiplicative or place-value based, additive (including subtraction), or generic descriptions such as "I used my fingers", "I guessed", or "I did it in my head".

Usage of strategies

Multiplicative or place value strategies, such as referring to times tables, or place-value strategies accounted for 7% (42 out of 604) of the strategies mentioned in parts a) to d).
Additive strategies, such as "add on 7" or "take away 5", accounted for 58% of the strategies mentioned in parts a) to d).
Generic descriptions accounted for around 6% of the strategy statements in each of the parts. Another 10% of the strategies were of an unspecific nature.

Success rates using the strategies

Multiplicative or place value strategies led to the highest success rates at continuing the number pattern correctly, with an average of 93% (39 out of 42) of the students who described these strategies getting all 4 members of the sequence correct, and 100% got at least 2 correct in all sequences.
Additive strategies, stating how much more to add on or take away from each term, were reasonably successful, with an average of 81% of the students who described these strategies getting all 4 members of the sequence correct, and 97% getting at least 2 of the 4 numbers in a sequence correct.
Generic descriptions led to far lower success rates, with an average of only 23% of students getting all numbers in the pattern correct, and just 49% getting at least 2 correct.
Around 7% of students who stated a wrong rule got the number patterns completely correct.
42% of the time unspecific strategies were given, students got all 4 correct with 63% getting at least 2 correct.

Attrition rates
Only 8% of students gave no strategy for part b), but this steadily increased to reach 27% for part d). Students who could not give any strategy were far less likely to be successful on the latter two number patterns which were harder for students to answer. In parts a) and b), close to 40% of these students got a completely correct answer, but this dropped to about 5% for parts c) and d).
For more details on the success rates of student strategies, click on the link Appendix of student strategies.

Next steps:

Students should be encouraged to develop a repertoire of strategies, and move towards using a multiplicative strategy to solve these problems.

Pattern repeating
A small number of students merely repeat the pattern. While this is not incorrect, it is more productive to encourage students to develop patterns that are steadily increasing or decreasing, as this better prepares them for relational and functional thinking. Get these students to work on physical patterns, especially spatial patterns (click on spatial patterns) or number patterns which have real-life stimulus which indicates the patterns are increasing in a resource such as Exercise programme.

Wrong rule used
Several students came up with rules that did not fit all the numbers in a sequence. Encourage these students to state a rule by which they come up with numbers, and get them to check if it works for all given numbers. For spatial or "real-life" patterns, get them to see if their rule reflects the stimulus material which led to the pattern.

Number Framework
Sequences and patterns are addressed in the Number Framework (strategies) at stages 3 and above (see Book 9, p31 for a table of the relationship between number strategy stages and sequences and patterns). Most students in this resource used strategies from the advanced counting or early additive stages. Students using multiplicative strategies are indicating stage 6 (Early multiplicative).

Appendix of student strategies for Missing Numbers II
Table 1. Frequencies and percentages of strategies in each part of the question.

Strategy used	Part a) Freq, %	Part b) Freq, %	Part c) Freq, %	Part d) Freq, %	TOTAL Freq, %
Multiplicative / Place value	15, 9%	20, 12%	5, 3%	2, 1%	42, 7%
Adding on / subtracting	124, 78%	89, 54%	68, 48%	72, 54%	353, 58%
Generic - use fingers	0, 0%	5, 3%	0, 0%	1, 5%	6, 1%
Generic - guessed	1, 1%	4, 2%	7, 5%	6, 1%	18, 3%
Generic - mentally	1, 1%	4, 2%	4, 3%	2, 1%	11, 2%
Adds on incorrect amount	4, 3%	22, 13%	39, 27%	21, 16%	86, 16%
Adds on 4 to make 4, 8, 12, 16 (relevant only to part d)	- -	- -	- -	15, 11%	15, 2%
Repeated pattern	0, 0%	2, 1%	1, 1%	1, 1%	4, 1%
Unspecific pattern statement	13, 8%	18, 11%	16, 11%	13, 10%	60, 10%
Other statement	1, 1%	4, 2%	3, 2%	1, 1%	9, 1%
TOTAL strategies¹	159, 100%	168, 100%	143, 100%	134, 100%	604, 100%
No strategy given²	24, 13%	15, 8%	40, 22%	49, 27%	128, 17%
TOTAL students	183	183	183	183	732

Based on a representative sample of 183 students.

NOTES:
1. All percentages listed above this line in the table are based on the Total number of strategies for that part.
Examples:

The percent of strategies for part a) that were multiplicative is 15/159=9%.
The percent of strategies overall that were multiplicative is 42/604=7%.

These percentages looks at the relative frequency of different strategies, compared with the total number of strategies given.

2. All percentages in this line are based on the Total number of students for that part.
Examples:

The percent of students who gave no strategy for part a) is 24/183 =13%.
The total percent of students who gave no strategy is 128/732 =17%.

Interpretation of the relative number of students giving each strategy can be made from the raw frequencies, as the denominator is 183 for each part (or 4 times 183 = 732 in the Total column).

Table 2. Frequencies and percentages of successful strategies in each part

Strategy used	Part a) Freq (%)	Part b) Freq (%)	Part c) Freq (%)	Part d) Freq (%)	TOTAL Freq (%)
Multiplicative or Place value	all 13 (87) 2+ 15(100)	all 19 (95) 2+ 20(100)	all 5(100) 2+ 5(100)	all 2(100) 2+ 2(100)	all 39 (93) 2+ 42 (100)
Adding or subtracting	all 115 (93) 2+ 122 (98)	all 64 (72) 2+ 85 (96)	all 45 (66) 2+ 64 (94)	all 61 (85) 2+ 71 (99)	all 285 (81) 2+ 342 (97)
Generic - fingers	- -	all 1 (20) 2+ 4 (80)	- -	all 0 (0) 2+ 0 (0)	all 1 (17) 2+ 4 (67)
Generic - guessed	all 0 (0) 2+ 0 (0)	all 1 (25) 2+ 1 (25)	all 0 (0) 2+ 3 (43)	all 0 (0) 2+ 1 (17)	all 1 (6) 2+ 5 (28)
Generic - mental	all 0 (0) 2+ 0 (0)	all 2 (50) 2+ 3 (75)	all 2 (50) 2+ 2 (50)	all 2(100) 2+ 2(100)	all 6 (55) 2+ 7 (64)
Adds incorrect amount	all 1 (25) 2+ 2 (50)	all 2 (9) 2+ 5 (23)	all 0 (0) 2+ 8 (21)	all 2 (10) 2+ 4 (19)	all 5 (6) 2+ 19 (22)
Adds on 4	- -	- -	- -	all 0 (0) 2+ 0 (0)	all 0 (0) 2+ 0 (0)
Repeated	- -	all 0 (0) 2+ 2(100)	all 0 (0) 2+ 1(100)	all 0 (0) 2+ 0 (0)	all 0 (0) 2+ 3 (75)
Unspecific	all 9 (69) 2+ 12 (92)	all 13 (72) 2+ 17 (94)	all 1 (6) 2+ 5 (31)	all 2 (15) 2+ 4 (31)	all 25 (42) 2+ 38 (63)
Other	all 0 (0) 2+ 1(100)	all 2 (50) 2+ 2 (50)	all 0 (0) 2+ 0 (0)	all 1(100) 2+ 1(100)	all 3 (33) 2+ 4 (44)
TOTAL strategies	all 138 (87) 2+ 152 (96)	all 104 (62) 2+ 139 (83)	all 53 (38) 2+ 88 (62)	all 70 (52) 2+ 85 (63)	all 365 (61) 2+ 464 (77)
No strategy	all 8 (33) 2+ 9 (38)	all 6 (40) 2+ 7 (47)	all 2 (5) 2+ 6 (15)	all 3 (6) 2+ 5 (11)	all 19 (15) 2+ 27 (21)
TOTAL students	all 146 (80) 2+ 161 (88)	all 110 (60) 2+ 146 (80)	all 55 (30) 2+ 88 (62)	all 73 (40) 2+ 90 (49)	all 384 (52) 2+ 491 (67)

Based on a representative sample of 183 students.

All percentages are based on denominators from the frequencies in Table 1.
Examples:

In part a), 115 out of 124 (93%) of students giving additive strategies got all 4 parts of the pattern correct, and 122/124 (98%) got at least 2 correct.
In total [i.e. averaged over parts a) to d)] 285 out of 353 (81%) additive strategies led to answers with all 4 parts of the pattern correct, and 342/353 (97%) got at least 2 correct.
104 of the 168 (62%) students giving any strategies in part b) got the entire pattern correct, and 139/168 (83%) got at least 2 correct.
Overall, 365 of the 604 (61%) of strategies mentioned in parts a) to d) led to a completely correct pattern, and 464/604 (77%) led to at least 2 correct.

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