Christmas turkey

This task is about working out how long to cook turkeys of different sizes and completing a rule to describe this relationship.

Tanu’s Turkeys provides this chart showing how long to cook turkeys of different sizes. The larger the turkey, the longer it takes to cook.

Size of turkey	No. 2	No. 3	No. 4
Cooking time (in hours)	1\(3 \over 4\)	2\(1 \over 2\)	3\(1 \over 4\)

a) Tanu’s Turkeys raised some extra large turkeys. Show how long a No. 6 turkey should cook for.

Cooking time = __________ hour(s)

b) If a turkey needs to be cooked for 8\(1 \over 2\) hours, show how to work out what size it would be.

Size __________

c) Describe how to work out the cooking time for a turkey of any size.

d) If you know the size of a turkey (x), complete this equation so that it gives the cooking time (y) the turkey needs.

y = ______________________________

Diagnostic and formative information:

	Common error	Likely misconception
a) b)	6 ³/₄ and 7	Uses 1 ³/₄ (the cooking time) for the increment rather than ³/₄
a) b)	5 ¹/₂ and 9	Increments the whole hours by 1 (i.e. 1, 2, 3, 4, 5..) and cycles the fractions (³/₄, ¹/₂, ¹/₄, ³/₄, ¹/₂...)
a)	5 ¹/₄	Does not use a constant increment.
a)	5	Adds the time for a No. 2 and a No. 4 turkey, or doubles the time for a No. 3 turkey. This assumes the function has a zero intercept.
d) c)	³/₄ x (or 45x) Equivalent verbal description	Ignores the intercept or just attempts to write the recursive rule in a functional form.
d) c)	⁴/₄ x + 1 Equivalent verbal description	Uses an incorrect intercept, possibly where x=1 rather than 0.
d)	x + ³/₄, x + 0.75, x + 1 ¹/₄, x + ¹/₄, ³/₄ , 0.75 etc.	Attempts to complete the functional form, but uses the increment additively, indicating recursive thinking of the difference between successive terms.
d)	y + ¹/₄	Attempts to write a recursive formula, but uses the wrong increment.

Student strategies
Two major strategies were used by students, with a third minor option used by some. These were:

Recursive. This means looking at the pattern "horizontally", by seeing what needs to be added to one term to get the next.
Functional (or explicit relationship). This is where an explicit relationship is given between the size of the turkey (x), and the cooking time (y).
Additive. This assumes there is a functional form, but that the function passes through the origin (i.e. has a zero intercept) and so any two terms can be added to get a third, e.g., the cooking time for a No. 6 turkey is the cooking time for a No. 2 plus that for a No. 4.

The functional form of the relationship (y = ³/₄ x + ¹/₄) is seen as more sophisticated at this level. However, students who can correctly express the recursive form (yn = yn-1 + ³/₄) are rare, but probably highly able. This recursive form has powerful applications in later mathematics and should not be seen as inferior.

For parts a) and b), the recursive strategy predominated, with about three quarters of those who answered using this strategy, about 10% using the additive strategy, and about 5% using the functional strategy.
For part c) about two thirds of those answering used recursive strategies and nearly a fifth used functional ones.
For part d), about a half of those answering were still using recursive strategies, even though these were often disguised as functional forms. Just under a half of those answering wrote functional forms, but only about a third of these were correct.
The number of students not answering rose from 45 for part a), to 55 for part b), 67 for part c), and 98 for part d) indicating students were having increasing difficulty answering as the questions demanded a functional approach.

Next steps:

At this level, students need to move from recursive strategies to functional ones. One simple approach is to introduce or revisit the concepts of intercept and slope.

The intercept in this situation is the cooking "set up time" that each turkey must have, or alternatively the cooking time for a No. 0 Turkey. This can be obtained from "working backwards" to get the cooking times for the No. 1 and then the No. 0 turkey. It is imperative to give attention to the fact that the intercept is not zero.

For other resources about intercepts click on this link.

The slope is just the extra in cooking time for the next size up of turkey (i.e. ³/₄ hour). Exploring how this increment can be expressed as part of the functional form rather than recursively is useful.
The functional form is then just: y = intercept + slope × x where x is the size of the turkey.

For students who get different functional forms of the equation, exploring the equivalence of them is worthwhile. In many ways the form y = 1 + ³/₄ (x – 1) is preferable to y = ³/₄ x + ¹/₄ , as there is no useful interpretation of a No. 0 turkey, so the sequence should start at 1.

		Y10 (05/2007)
a)	4 ³/₄ States the general rule used or shows further extrapolation of the table or other working.	moderate difficult
b)	11 States the general rule used or shows further extrapolation of the table or other working.	difficult difficult
c)	Any rules that are: recursive (works horizontally), e.g., last cooking time plus ³/₄; or functional (works vertically), e.g., size of turkey times ³/₄ plus ¹/₄.	difficult
d)	Any 1 of: ^{(3x + 1)} / ₄ ³/₄x + ¹/₄ 1 + ³/₄(x-1) Also accept other algebraically equivalent expressions.	very difficult

You are here

Resource user links

Christmas turkey

Christmas turkey