3. Do We Have Enough Space?

• Measuring tapes and rulers
• Grid paper: docx PDF

Explain to students that it is important for engineers and town planners to know how many people can fit safely in a space and that internal spaces general have less capacity than external spaces. 
Introduce to students the following scenario: 

"Each time tickets to the Olympics or Commonwealth Games are released, they quickly sell out. The Games Committee would really like to increase the number of spectators and therefore they have decided to add standing room only areas in each stadium. 

You and your fellow students are responsible for deciding how many spectators can fit in each standing room space at the next Olympics."

Implementation options:

• Provide students with written dimensions of the standing room space; or
• Use a classroom as the model space 

Ask students to determine how many spectators can fit in the space safely and comfortably. Encourage students to work in small groups to come up with a solution.  

When choosing the dimension of the standing room only spaces take into consideration the current knowledge and skills of your students. If students are beginning to understand area, then you may decide to use whole numbers which can be easily measured out, for example 4 m x 5 m. For more experienced students you may decide to create an irregular standing room only shape. 

Regardless of the space chosen initially, ask students to estimate how many spectators they believe can fit in the space. Have students record this estimate in their workbooks or on the board and inform them they will compare their estimates, with the actual measurement at the end of the activity. 

Estimation is a valuable skill - it helps students to develop the capacity to determine the reasonableness of their answer. Where possible, have students estimate a likely answer prior to commencing the formal solving of a problem. 

Provide students with adequate time to explore and test their thinking, and to come up with a method of their own. At the conclusion of the activity, ask the students to suggest ways in which they could improve their methodology next time, and lead them to suggest ways they could tackle the problem more efficiently. Record the strategies used, and the suggested improvements on an anchor chart for future problem-solving activities.

Enable students by modelling the activity and placing classmates within a defined space in the room to determine how much room each participant will need. Demonstrate how to draw a scale model of the standing room space on grid paper. Next have them partition the space on their scale model into user-friendly sizes, based on their modelling in the classroom. How many spectators (students) can they fit in this space? 

Extend students by informing them that the organisers of the upcoming Olympics want to increase the spectator capacity of the Mathematics Sports Arena from 18,200 people (all seated) to 30,000, with a mix of seated and standing only room. The seating area of the stadium currently occupies a space of 12,397 square metres and each of the new tickets (to get them to the 30,000 limit) will be standing room tickets. How much more space do they need to add so that they can safely accommodate these additional spectators?  

At the conclusion of the activity, have students engage with the following questions using the Think, Pair, Share collaboration strategy:

• I notice your answer says that we can fit X people into this space. How practical is that?
• Are people more likely to put up with tight spaces if the event is quick? Is it feasible if the sporting event goes for a long time i.e., a baseball match (3+ hours) will people be okay with the tight spaces?
• Are people more likely to put up with a small space if the event is hard to get tickets to i.e., an NBA basketball match starring the season's top scorer?
• How much room to people need around them to be comfortable or to feel safe? Would you be more comfortable with your family or friends closer to you than a stranger?
• Can you think of situations you have experienced where we need to allow for access ways so people can go to the toilet or to get food?
• What do you need to consider to ensure you meet public health measures, for example as dictated by COVID-19 restrictions?

These prompting questions provide students with additional learning opportunities, specifically to reinforce the difference between theoretical solutions and practical solutions.

Prior to commencing this activity, ensure that students understand the difference between perimeter and area. Introduce the following scenario to students:

"Alina is building a pen for her new pet, a goat called Ginger. During the day Ginger can roam freely however, in the evening she must be fenced in to keep her safe. Alina only has 28 metres of fencing wire and will build a rectangular enclosure. Describe the different goat pens Alina can build for Ginger. Which pen will give Ginger the most area to run around?" 

Encourage students to work in pairs to come up with all the possible solutions to this  scenario. 

The number 28 provides seven different combinations using lengths that are integers. Ensure concrete material and grid paper are available for all students to support their problem-solving.

Enable students by using a number that provides fewer combinations, such as 16 m of fencing wire. Demonstrate how to use concrete materials such as coloured tiles or grid paper to model and draw the problem when finding solutions.

Students will approach this problem in different ways. Encourage students to brainstorm and develop problem-solving strategies before tackling the solution and share their ideas with peers for feedback. This will encourage students to become creative thinkers; students need time for trial and error and to develop their capacity to experiment and make connections. They need to see that they will not always be right the first time they try something. 

Do not tell students to tabulate their information; instead, support them to come up with this idea on their own. Encourage them to discuss with their partner how they might display this information in a more ordered manner. Use prompting questions, such as:

• I am having trouble seeing all your information clearly, is there a way you can make it easier for me to see?
• Do you have all possible outcomes? How do you know?
• What do you notice about the relationship between the area and the perimeter? 

Once students have tabulated their information ask them which block of land they think Ginger would prefer to live in. Ask them to explain their reasoning. Prompt students to see that most animals prefer large open spaces where they can roam freely. 

The seven combinations for a perimeter length of 28 m are tabulated below. When set out in graduating widths, students are supported to see they have recorded all combinations. It also helps them to approach their thinking in a more methodical manner. Resist the urge to show students this format in the beginning. If at the conclusion of the activity you notice that a group has their information tabulated in this manner, encourage them to share this strategy with the class. Demonstrate how to tabulate their data if none of the groups used this strategy.

Enable students by demonstrating how to model the problem using grid paper or square tiles and to draw a diagram. 

Extend students by asking them ‘Now that you know how many possible rectangular blocks of land can be made with a perimeter of 28 metres can you come up with a rule to describe this relationship?’, ‘If you were given a new measurement could you tell me how many combinations are possible?’, ‘Could you tell me the smallest land space possible or the largest?’, "what combinations could you make if you used half-meter measures, such as 2.5 m wide and 11.5 m long?'

Introduce students to a new scenario:

"Alina’s friend Thao is also building a goat pen. She has 34 metres of fencing wire. Describe the different pens Thao can build. Which pen of Thao’s would give Ginger the most area to run around? Which will give Ginger the least?"

Areas for further exploration

This task can be explored by building a three-dimensional structure on the chosen block of land. Students select the block of land which has the largest area. Encourage students to draw out a scale drawing of this block of land. From here students design a three-dimensional home for the goat. 

Ask students to determine the appropriate size the shelter should be to house the goat. Provide the following information to students:

The approximate dimensions of a grown goat are 80cm tall, 60cm wide and 1.1 metres long. Can you design a shelter which will comfortably house Ginger?

Enable students by supporting them to use cubes to create a model their goat shelter, and then create nets of their 3-dimensional structure using grid paper. 

This stage supports students to extend their knowledge and understanding of area. At the conclusion of this stage, you may choose to set a variety of problem-solving tasks for students to consolidate their new knowledge and skills. 

You might decide to set area problems with a different unit so that students must first convert before solving the area equation. When writing area-based questions, keep in mind that questions which ask students to calculate the area of a given shape are less challenging than those which ask students to prove a solution is correct or which require students to ‘work backwards’ to come up with the possible dimensions.

Australian Association of Mathematics Teachers, n.d. How many people can stand. [Online]
Available at: https://maths300.com/lessons/129
[Accessed 15 March 2022; Note: access to this resource requires a maths300 subscription].

State Government of Victoria (Department of Education and Training), n.d. Diverse Dimensions. [Online]
Available at: https://www.education.vic.gov.au/Documents/school/teachers/teachingresources/disciplicrowssquarene/maths/assessment/lafzone5intro.pdf
[Accessed 15 March 2022].