Number: Taking handfuls
View Sequence overview10 is a useful benchmark to quantify and compare larger collections.
Each group
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.
Ten-frames
Each student
Using 10 Student sheet
Blank A5 or A6 card
Task
Introduce the task: Today we are going to take another handful. We are going to take a handful of [whatever small item you are using].
Take a handful, making sure you have more than 10 items.
Ask: How many items do you think are in my handful? Take suggestions from students. The point is not to get the right answer but to show that there are a lot more than 10.
In the last task, we used 5 to help us work out how many we had in our handful. But this time there are a lot more than 5 items in my handful! I wonder what we might use this time. Take suggestions from students. They may suggest 10.
Let’s put two 5s together and use 10 as our benchmark.
Introduce the ten-frame as a tool to organise items. Look at the fact that the ten-frame is made up of 2 rows of 5, which is like 2 towers of 5 cubes (the towers made in the previous task), and that there are 10 spaces in all. Allow students time to explore the ten-frame.
Pose the task: Take a handful. Use the ten-frames to see how many items are in your handful.
Allow students time to organise their collections using the ten-frames.
Ask students to show their collection (organised onto ten-frames) to a partner and share how many items they have and how 10 helps them work out the total. For example:
- I have 14 items. There are 4 more than 10.
- I have 23 items. There are 10, 20 and then 3 more.
Ask students to compare their collections, using 10 as a benchmark to determine who has more and who has fewer.
Provide each student with a blank card and ask them to write down the number of items in their collection. Provide them with Using 10 Student sheet and ask them to use the ten-frames on the sheet to record their collection.
Students can repeat the activity if there is time.
Ask students: How many items are in your collection? Watch how they determine the total.
- Count multiple times and then count all: When the collection is composed of multiple parts, students may count all in each smaller part and then count the collection as a whole.
- For example, a collection of 14 items might be made up of a full ten-frame of ten and a ten-frame of four items. Students might first count the 10, next count the 4, and then finally count the two ten-frames together to find the total.
- The more we count, the more room there is for error. Prompt students to consider how they can determine the total without doing too much counting.
- Count all at once: Students count the whole collection; they don’t hold the 10 and count on. Ask students if they need to count from one, or if they might be able to count on from the 10; you may even cover/hide the 10.
- Count on from 10: Students hold the 10 and count on from 10 or use 10 as a benchmark.
10 as a benchmark
Using 10 as a benchmark allows us to see the “10-ness” of a number.
In the previous task, students learnt that 5 is a powerful benchmark number. 10 is even more powerful as a benchmark number. Indeed, the power of 5 as a benchmark is based on the fact that it is half of 10. This task introduces 10 as a benchmark.
10 is the first two-digit number in our number system. This is because our number system is structured on 10. All two-digit numbers can be thought of as a count of 10s and some more (Siemon et al., 2021).
Through this task students begin to develop a sense of the “10-ness” of numbers. Students take a handful greater than 10 and describe it in terms of 10 and some more. The ten-frame is a very useful tool to think about the 10-ness of a number, as it is possible to see quickly whether a number is larger or smaller than 10.
Using 10 as a benchmark allows us to see the “10-ness” of a number.
In the previous task, students learnt that 5 is a powerful benchmark number. 10 is even more powerful as a benchmark number. Indeed, the power of 5 as a benchmark is based on the fact that it is half of 10. This task introduces 10 as a benchmark.
10 is the first two-digit number in our number system. This is because our number system is structured on 10. All two-digit numbers can be thought of as a count of 10s and some more (Siemon et al., 2021).
Through this task students begin to develop a sense of the “10-ness” of numbers. Students take a handful greater than 10 and describe it in terms of 10 and some more. The ten-frame is a very useful tool to think about the 10-ness of a number, as it is possible to see quickly whether a number is larger or smaller than 10.
Review the task that was posed (Use the ten-frames to see how many items are in your handful) 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:
- How are other students’ handfuls different? How are they the same?
- Who has the most items? Who has the fewest? How do you know?
Conduct a class gallery walk.
After the gallery walk, come together for a whole class discussion.
Focus this Connect phase on the power of 10 as a benchmark number. 10 is a very important number in our number system, which also makes it an easy number to work with. Students may notice the place value patterns that emerge through activity which further emphasises the power of 10, however, these patterns are not the primary focus of the investigation.
Discuss:
- What did you notice that was the same?
- Everyone used ten to arrange their items. The use of ten makes it easy to see how many and to compare the collections.
- What did you notice that was different?
- Different students would have different numbers of items in their handfuls.
- Who has the most items? Who has the least? How do you know?
- Using 10 as a benchmark makes it easy to compare numbers.
Select some students to come to the front, with the cards from Step 2 where they recorded their total number of items. Ask them to hold up their cards and invite the other students in the class to arrange these students in ascending order. Keep using 10 as benchmark to determine the order of numbers.
What is the same and what is different?
Asking “what is the same and what is different?” is a powerful prompt to guide students’ inquiry. In mathematics, when we find same-ness we can make generalisations. Difference shows us how generalisations can be applied to specific examples.
In this task, we see same-ness in the use of 10 to organise and compare collections. All collections are the same in that they can all be described as "10 and some more" or "10 and some fewer". We see difference in how many more than 10 and how many fewer than 10 each collection is.
Asking “what is the same and what is different?” is a powerful prompt to guide students’ inquiry. In mathematics, when we find same-ness we can make generalisations. Difference shows us how generalisations can be applied to specific examples.
In this task, we see same-ness in the use of 10 to organise and compare collections. All collections are the same in that they can all be described as "10 and some more" or "10 and some fewer". We see difference in how many more than 10 and how many fewer than 10 each collection is.
Discuss with the students the way that they used 10 as a benchmark number. For example, 15 can be thought of as 10 and 5 more, and 26 can be thought of as 2 tens and 6 more.
Explain: 10 is an important number in our number system. It is a very useful number to compare larger collections.