Comparing allows us to order quantities and determine which quantity has more and which has less.
A common unit allows us to quickly compare quantities.
Whole class
A large quantity of small items which students can take a handful of (for example: counters, pasta, dried beans). A student handful should be around 10 to 20 items.
Each student
Comparing handfuls Student sheet
Task
Pose the initial task: Take a handful and organise your collection in a way that makes it easy to see how many items are in your handful.
Allow students to count and organise their collections, then ask them to compare their organised handful with a partner.
Pose the question: Who has more and who has fewer items? It is likely that the students will have used different arrangements, which makes it more difficult to compare the two collections.
Pose the task: Take a new handful. Work with your partner to arrange your handfuls so it is easy to see how many there are in each handful, and to also see who has more and who has fewer.
Give students time to arrange their handfuls. As students are working, ask:
- How do your arrangements make it easy to see how many there are?
- How do your arrangements make it easy to see who has more and who has fewer items?
Provide students with the Comparing handfuls Student sheet. Ask students to record how they have arranged and compared their handfuls.
Both students in each pair need to arrange their items in the same way, using subitisable groups to easily see how many they have and to see who has more and who has fewer items.
- Students might make arrangements where it is easy to compare collections but not easy to see how many there are altogether. For example, making two lines of items (ensuring a common baseline and that there is one-to-one correspondence across the lines).
- Students might make arrangements where it is easy to see how many items are in each collection, but it is not easy to compare without counting. For example, using different arrangements of subitisable groups.
Allow students the chance to explore some of these options to determine if they can easily see how many and easily compare. Prompt students to think about how they might be able to arrange their collections so they can do both.
Students’ work
Look at how these different pairs of students have arranged their handfuls.
Discuss with colleagues: What do you notice? What do these students know and what have they shown that they can do? Give evidence from the pictures for your statements.
Look at how these different pairs of students have arranged their handfuls.
Discuss with colleagues: What do you notice? What do these students know and what have they shown that they can do? Give evidence from the pictures for your statements.
Review the original task (arrange your handfuls so it is easy to see how many there are in each handful, and to also see who has more and who has fewer items) and ask students to think about what they expect to see as they complete a gallery walk.
Ask students to consider the following questions as they look at others’ work: Look at how pairs of students have arranged their handfuls. Which arrangements make it easy to see how many there are and also to see who has more and who has fewer items? Why?
Conduct a class gallery walk.
Ask the students to return to their collections. Allow them to rearrange their items using the strategies they have seen, to make their collection easier to count and to see who has more and who has fewer items.
Gallery walk
The gallery walk is not a ‘show-and-tell’ activity; it is an opportunity to critically reflect on how others have solved the problem. Remind students of the task and ask them to reflect on their own experience of doing the task before they commence the gallery walk. This helps to focus their attention on the mathematics they may expect to see.
The gallery walk is not a ‘show-and-tell’ activity; it is an opportunity to critically reflect on how others have solved the problem. Remind students of the task and ask them to reflect on their own experience of doing the task before they commence the gallery walk. This helps to focus their attention on the mathematics they may expect to see.
Gallery walk
In a collaborative mathematical community, students need the opportunity to see and make sense of others’ thinking. A gallery walk provides time for all students to learn from how others approached the problem. Provide students with time at the end of the gallery walk to alter their own solutions if they would like to. Students should not think of this as copying or cheating, rather they are learning from their peers.
In a collaborative mathematical community, students need the opportunity to see and make sense of others’ thinking. A gallery walk provides time for all students to learn from how others approached the problem. Provide students with time at the end of the gallery walk to alter their own solutions if they would like to. Students should not think of this as copying or cheating, rather they are learning from their peers.
Discuss:
- Which arrangements made it easy to see how many items there were? Why?
- Subitising patterns (such as those on dice) make it easy to see at a glance how many items are in an arrangement.
- Which arrangements made it easy to see who had more and who had fewer items? Why?
- A common unit is needed for quick comparison. For example, making a line of items (ensuring a common baseline and that there is one-to-one correspondence across the lines) uses the common unit of ‘one’.
- Were there any arrangements that made it easy to see how many there are and also to see who had more? Why?
- The best way to do this is to use the same subitisable patterns. Students can quickly see how many they have and also who has more.
Ask the students to return to their collections. Again, provide time to rearrange the items if they would like to. Ask students to record their new arrangements of their handfuls on their student sheet.
Connect Phase
There are two important ideas to bring out in this Connect phase:
- Arranging items in subitisable groups makes it easy to see how many there are in a collection.
- A common unit makes it easy to compare collections to see which has more and which has fewer items.
There are two important ideas to bring out in this Connect phase:
- Arranging items in subitisable groups makes it easy to see how many there are in a collection.
- A common unit makes it easy to compare collections to see which has more and which has fewer items.