Task 1 • Where will you stand?

Explain that you are going to place some lamingtons on three tables. Ten students will then take turns to choose which table to stand at. The students at each table will divide the lamingtons at their table fairly between themselves. The aim is to be at the table which gets the largest share of lamingtons.

Select ten students to go outside the room, or to the back of the room, so that they cannot see the tables.

Place three tables at the front of the room and place a different number of lamingtons on each table:

• one lamington on the first table.
• two lamingtons on the second table.
• three lamingtons on the third table.

One at a time, invite each of the selected students to the front of the class to choose a table to stand at. Remind them that once they have chosen a table they must remain at that table.

When eight students have chosen a table to stand at, ask the rest of the class to consider where they think the last two students would choose to stand and why. Student suggestions might include:

• the table with the most lamingtons, as they assume that students will get more because there are more lamingtons to share.
  • This works at the start, but the share gets smaller as more students join.
• the table with the fewest students, as they will get a larger portion of each lamington because the lamingtons are being shared with fewer students.
  • This works at the start, but the share gets smaller if more students join.
• the last two students should choose their tables to make sure that they balance students already at the tables.
  • A similar number of students at each table ensures that the students at the table with the most lamingtons will get the larger share.

Invite the remaining two students to choose where to stand. Make a note of:

• the number of students at each table.
• the number of lamingtons.

Record this somewhere, such as a classroom whiteboard, for it to be clearly seen.

Pose the task: Find out how much lamington each student at each different table gets. On which table did students get the most lamington? Explain why they got the most.

Organise students into small groups. Provide each student with a sheet of A3 paper, a Lamingtons Sheet, a pair of scissors and a glue stick.

Explain that students are going to use the information recorded on the board to find how much lamington each of the students at each table get and what the largest share is.

Students use the pictures on Lamingtons Sheet to support their thinking about the problem. They may represent their reasoning in any way they choose to create a poster of their mathematical thinking, which includes:

• the number of students and the number of lamingtons.
• their strategy for sharing/dividing the lamingtons on each table between the students.
• the fraction of lamington that each table of students will receive.
• which table of students gets the most lamington each and why.

You may choose to Spotlight student strategies that may benefit the whole class.

Ask students to display their posters in preparation for the gallery walk.

Re-Launch: If this phase happens in the subsequent lesson, briefly review last lesson’s mathematical task that was posed and ask students to think about what they expect to see as they complete the gallery walk. If you are continuing the current Explore phase, move straight into the gallery walk.

Provide each student with some sticky notes to write feedback and questions about what they see.

Ask students to consider the following questions as they look at others’ work:

• What do you notice that is the same about how students have shared the lamingtons? What do you notice that is different?
• Which strategy/strategies do you find most helpful for working out the share that each student gets? Why?
• Which strategy do you find most helpful for finding which group of students each get the largest share?

Conduct the class gallery walk.

At the end of the class gallery walk, allow students time to read and reflect on any sticky notes left on their work.

Select student posters which use different strategies to discuss during the Connect phase.

Invite selected students to share their work and explain their strategy and thinking.

Reflect on the different strategies students noticed during the gallery walk.

Discuss:

• Did you see a strategy that was helpful for working out the share that each student got? Why do you think this?
  • Some examples of student strategies might include:
    • cutting each lamington up to match the number of students so each piece can be counted out and shared.
    • cutting each lamington into half and sharing the halves equally, then cutting the remaining lamington halves into enough smaller parts for each student.
    • noticing a relationship between the number of lamingtons and the number of shares and connecting this with fraction notation.
• Which strategy made it easier for you to compare who got a bigger fraction? Why do you think it made it easier to compare the fractions?
  • Students may think:
    • it is hard to compare different sized shares.
    • the student with most parts gets most.
• Can you find the fraction each student will get without working it out?
  • Some students may have noticed the connection between:
    • the number of lamingtons.
    • the number of students who get a share.
    • the fraction notation.

Show slides 3-5 of Lamington slice Slides, which are animated with each click, to guide class discussion. There are operating notes for each slide which can be seen when using the slideshow in presenter mode.

Slide 4 shows fractions as division:

• 1 shared between 2 is equal to $\frac{1}{2}$, so $1 \div 2 = \frac{1}{2}$.
• 2 shared between 3 is equal to $\frac{2}{3}$, so $2 \div 3 = \frac{2}{3}$.
• 3 shared between 5 is equal to $\frac{3}{5}$, so $3 \div 5 = \frac{3}{5}$.

Slide 5 breaks the shares down into unit fractions, that is, a fraction with a numerator of 1. Unit fractions can make it easy to compare who got the most lamingtons:

• 1 shared between 2 students equals $\frac{1}{2}$ for each student.
• 2 shared between 3 students equals a share of $\frac{1}{2}$ and $\frac{1}{6}$ for each student.
• 3 shared between 5 students equals a share of $\frac{1}{2}$ and $\frac{1}{10}$ for each student.

Discuss:

• Do students get different shares by using the different strategies? Why do you think that?
  • Students may not see that the fractions are equivalent, and it does not matter which strategy is used.
  • Emphasise that when we divide the lamingtons between the students, they each get an equal share or fraction of the amount of lamington.
  • Encourage students to justify their thinking. How do they know that each student in the group has an equal share?

Explain: When we share a quantity, this is the process of division. The portion each student receives can be written as a fraction.

You may finish up this task by sharing the lamingtons from the Launch with the whole class.