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CHALLENGING TASK WHERE THE WILD THINGS ARE

PRIME NUMBER: VOLUME 33, NUMBER 1. 2018© The Mathematical Association of Victoria

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This challenging task has been developed using the much loved Maurice Sendak children’s book, Where The Wild Things Are. This text has been chosen for several reasons: it is highly engaging for students of all ages; the story lends itself to a maths task focused on time and ratios; and, importantly for us teachers, the book is readily available in school classrooms and libraries (as well as read-aloud versions online).

As suggested by Sullivan et al (2014), challenging tasks can be an eSective learning tool as they help students ‘build connections between a network of ideas’ and develop the ‘confidence... to devise solutions to problems’ by engaging with ‘mathematical tasks that are complex’ (p 124). Furthermore, children’s literature can help support the learning of mathematical concepts by helping to contextualise the maths, promote mathematical reasoning and engage students (Muir et al., 2017).

THE TASK

The idea for this task stems from my own childhood curiosity about Max, the story’s protagonist, and his trip to The Land of The Wild Things. I couldn’t fathom how, upon returning from what seemed like years in the mysterious and wondrous Land of the Wild Things, Max’s dinner was still hot! This was the first time I had considered alternative realities and the idea that time could be relative. How much more quickly did time pass in the Land of the Wild Things than it did in the real world? This key question forms the basis of this task.

Having read the book to my 5/6 class, I explained that we would be exploring a mathematical problem about Max. My students are familiar with my use of literature as a tool to launch maths tasks and one enthusiastically stated ‘I bet we will be looking at how quickly time passes when Max is away compared to at home!’

In order to explore the relative speed that time passes in the two worlds, it is first necessary to work out how long Max spends away and, then, how much time passed at home. For this investigation it is necessary to guide students to establish

the following time periods to ensure the maths is workable (although, coincidentally my class stated the exact numbers I was looking for!).

For a digital presentation to guide you through this part of the task, go to: bit.ly/wildmaths.

TIME IN THE LAND OF THE WILD THINGS

Initially, I asked my students to consider ‘How long did Max spend in the Land of the Wild Things’, by analysing the following sections of text:

And an ocean tumbled by with a private boat for Max and he sailed o7 through night and day and in and out of weeks and almost over a year to where the wild things are.

We concluded 365 days (‘almost over a year’) to get to the Land of The Wild Things.

But Max stepped into his private boat and waved goodbye and sailed back over a year and in and out of weeks and through a day

SUGGESTION: AN ALTERNATIVE TASK FOR YOUNGER STUDENTS

An exploration of how much time Max spent away and how much time passed at home (without the more complex ratio task) could be posed as a challenging task for younger students (e.g., Year 2). This would draw attention to the relationship between diSerent units measuring time (i.e., day, week, year), highlight how mathematical problems are often embedded within text, and provide students with a relatively diacult multi-digit addition and subtraction task.

Toby Russo, Bell Primary School

We concluded that this period was equal to one year + two weeks + one day, PLUS five days staying at the Land of The Wild Things or 385 days. Students calculated this total amount as the first part of the problem.

This makes a total of 750 days (365 + 385).

TIME AT HOME

I then asked my students to consider ‘How much time do you think passed at home?’, looking at this section of the text:

Where he found his supper waiting for him and it was still hot.

Student A: Well his dinner was still hot, so it couldn’t have been all that long.

Student B: I reckon about five minutes, no longer than that.

We agreed that 5 minutes had passed.

POSING THE PROBLEM: HOW MUCH TIME HAD PASSED?

• In the Land of The Wild Things: 365 days PLUS one year + two weeks + one day + five days

• At home: 5 minutes

The above information was recorded for the students and the following problems were posed:

Problem Using this information, how much time passes in the Land of the Wild Things, compared with one minute in the real world?

Extending prompt Max went to sleep, after eating his supper, at 9pm and woke up at 6.30am. He then went straight back to the Land of the Wild Things. How much time has passed there?

The next time Max returned it was four years later, at age 12. How much time had passed now?

Enabling prompt If 750 days passed in the The Land of the Wild Things compared with only five minutes in Max’s world, how can we work


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out how many days passed during one minute in Max’s world? What operation would help you solve this problem?

SOLVING THE PROBLEM

Students worked in pairs or independently to tackle the problem. Generally students found the first part of the problem straight forward: applying their knowledge of the length of a year and week, as well as addition and subtraction to determine the total time in The Land of the Wild Things (750 days). Some solved this problem mentally, while others recorded their process (see Figures 1 and 2).

The next stage of the problem was to determine the relative time that had passed at home (5 minutes) compared with The Land of the Wild Things (750 days). Most students immediately recognised the need to use division (750 ÷ 5) to solve the ratio of 1 minute = 150 days, although some required peer or teacher guidance (through

the enabling prompt).

The majority of students attempted the extension problems, making diSerent levels of progress. For example, one student used the ratio (150 days : 1 minute) to determine that one hour in Max’s world was the equivalent to 9000 days. Using a table to organise their working out, she extrapolated this amount to determine 9.5 hours (the time he was asleep) was the equivalent to 85,500 days or 234.24 years (see Figures 3 and 4). It should be noted that this student initially made an error in the total amount of days passed by a factor of 10, until this was highlighted by a peer.

Another student took an alternative approach, solving the length of time Max was asleep as 9.5 hours or 570 minutes and multiplying this by 150 days to solve the total number of days passed (85,500). They then divided this number by 365 to get 234.2 years (see Figure 5).

A number of students found the extension problem challenging and worked through it as part of a teacher-focus group. In order to elicit their thinking, I asked ‘How many hours was Max asleep for?’ Students used diSerent techniques to determine the amount of time lapsed between 9pm and 6.30am, including mental processes (‘Well there’s three hours until midnight and then six and a half more hours, which makes nine and a half hours’) while others drew a timeline.

Once we established he was asleep for 9.5 hours, we revisited what we already knew from the first problem: one minute in Max’s World is 150 days. I asked the group ‘How will we work out how much time has passed in The Land of The Wild Things’ and a student responded ‘First we need to work out how many minutes nine and half hours is. Then we times it by 150 to work out how many days all up’. Students were able to multiply 9.5 by 60 independently (either by using the distributive property, partitioning 9 x 60 and 0.5 x 60 and then adding the products, or by using an algorithm) to determine the total minutes Max was asleep as 570 minutes. I asked, ‘If one minute for Max is 150 days for The Wild Things, then 570 minutes is…?’ Students realised they needed to multiply

Figures 1 and 2: Varied approaches to solving the first part of the problem.

570 by 150 to work out how many days had passed. Some students were comfortable attempting this problem using an algorithm (one student stated ‘We can do 57 x 15 and then add two zeroes to our answer’) and others elected to use a calculator and could verify their peers answer: 85,500 days had passed for The Wild Things!

I asked my group, ‘So we know the number of days, how do we work out the number of years that have passed?’ One students was quick to suggest ‘We can divide the days by 365 because there are 365 days in a year.’ Using calculators, the group determined that it was 234.25 years! The students were amazed at how much time had passed for the Wild Things and were concerned that some of Max’s friends were still around to play with him. I reassured them they live for a very long time!

Finally, we then discussed what a quarter of a year was and worked out it was about 91 days - one student even that we should consider leap years (an extra day every 4 years), which gives a more precise answer of 234 years and 33 days!

A small group of students attempted the second extension problem: The next time Max returned it was four years later, at age 12. How much time had passed now? Another prompt was provided to support students: How many minutes passed in Max’s world over four years? Most students understood the process and attempted to work out how many minutes in Max’s life across four years. Once they established this, they returned to the ratio of 1 minute: 150 days. However due to the sheer size of the numbers and the challenge of organising their thinking in a structured way, only two students were able to solve the problem. For the record, 4 years in Max’s world is exactly 864,000 years in the Land of the Wild Things!

CURRICULUM LINKS

The curriculum links for this task are varied and are somewhat dependent on the strategies used by individual students. This task is suitable for upper primary students (grades 5-6) and lower secondary students (years 7-8), and the breadth of curriculum coverage indicates the varied entry and exit points that a rich task such as this one facilitates. Broadly this task covers the following content descriptors from the Victorian Curriculum:


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CHALLENGING TASK WHERE THE WILD THINGS ARE (CONT.)

Figures 3 and 4: One student’s approach to the first extension question.

Figure 5: An alternative approach to the extension problem.


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Level 5 Number and Algebra

• Solve problems involving division by a one digit number, including those that result in a remainder

• Use eacient mental and written strategies and apply appropriate digital technologies to solve problems


• Select and apply eacient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers

Measurement and Geometry

• Measure, calculate and compare elapsed time


• Recognise and solve problems involving simple ratios


• Solve a range of problems involving rates and ratios, including distance-

time problems for travel at a constant speed, with and without digital technologies

CONCLUDING THOUGHTS

The success of this lesson stemmed from a high level of student engagement. Although Where The Wild Things Are is more obviously suited to younger students, my upper-primary class loved being read this book; I believe their enjoyment in tackling this task was due to the way it was embedded in the narrative. It is common for children’s literature to be used in connection with mathematical learning, but often the maths is superficially linked to the text or a text is chosen for its mathematical focus. This lesson is based on a ‘Narrative-First Approach’ to lesson planning, whereby key ideas, themes, and characters from well-known children’s stories are reconstructed through a mathematical lens. For other examples of attempts to employ this approach, see Russo and Russo (2017a, b, c). If you’d like to find out more about the lesson, please feel free to email the author at [email protected].

REFERENCES

Muir, T., Livy, S., Bragg, L., Clark, J., Wells, J., & Attard, C. (2017). Engaging with Mathematics through Picture Books. Albert Park, Australia.: Teaching Solutions.

Russo, J., & Russo, T. (2017a). Harry Potter-inspired Mathematics. Teaching Children Mathematics, 24(1), 18-19.

Russo, J., & Russo, T. (2017b). One Fish, Two Fish, Red Fish, Blue Fish. Teaching Children Mathematics, 23(6), 338-339.

Russo, J., & Russo, T. (2017c). Problem solving with the Sneetches. Teaching Children Mathematics, 23(5), 282-283.

Sullivan, P., Askew, M., Cheeseman, J., Clarke, D., Mornane, A., Roche, A., Walker, N. (2014). Supporting teachers in structuring mathematics lessons involving challenging tasks. Journal Maths Teacher Education (pp. 123-140). Springer Science+Business Media Dordrecht 2014.

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