Partitions

In this session, students investigate the fact that whole numbers can be partitioned in a number of ways.

1.

Show the students a bag of 8 plastic farm animals and a piece of paper with 2 rectangles drawn to represent 2 paddocks. Tell the students you are going to find all the ways of splitting the animals between the 2 paddocks.

2.

Ask the students:

• How many animals should I put in the first paddock? (Put that number (for example, 3) in that paddock and the rest in the second paddock.)
• How many animals are in the second paddock? Make sure the students understand there are still 8 animals.

3.

Record for the students: 3 + 5 = 8.

4.

Continue working with the students to find all the pairs of numbers that add to 8. For example, 6 + 2 and 2 + 6.

5.

Show the students a length of 7 unifix cubes. Work with students to find different ways of partitioning the 7 cubes. Record the partitions, for example, 4 + 3.

6.

Students can continue to explore partitioning numbers for numbers up to 10. Students can work in pairs or as individuals to partition the number and record the results. Students can use unifix cubes, sets of counters, and strips of squared paper as materials. Students who know the basic addition and subtraction facts for 10 will be able to partition numbers without materials.

In this session, students investigate partitioning teen numbers using the number ten. It is easiest to start by making teens numbers like 10 + x.

1.

Show the students a packet of pens (or an item that comes in packs of 10) and 5 single pens.

2.

Ask the students, "How many pens do I have?" (Students may count on 11, 12, 13, etc.)

3.

Record the answer as 10 + 5 = 15. Do several more examples.

3.

Show the students two tens frames, one complete with 10 dots and another with 6 dots. Ask the students how many dots there are. Again, record the answer as 10 + 6 = 16. Use the tens frame to work with the students to solve more examples.

4.

Students also need to be able to understand that teens numbers can be partitioned. Tell the students that in today’s session, the numbers are going to be split into 10 and something.

5.

Show the students two blank tens frame and a pile of 16 counters. Tell the students you know there are more than 10 counters in the pile, but how many more? Put the counters on the tens frames.

6.

Ask the students: "16 can be split into 10 plus what?"

7.

Record the answer as 16 = 10 + 6.

8.

Name some objects in the classroom that are between 11 and 20 cm. Ask the students to measure the length using their ruler and record the answer on the table.

9.

Other measurement contexts can be used to provide practice activities. For example, capacity. Pour enough water into a bottle to make about 18 ice cubes. Ask the students to pour the water into the ice cube tray and count how many ice cubes it would make. Ask the students to write their answer as 10 + ? Bottles with different amounts of water can be used.

In this session, students solve addition problems by partitioning numbers. They use the strategy “make a ten”.

1.

Show the students a number strip (see Resources) from 0 – 20 and colour in the 10 square. Place 7 counters on the strip. Tell the students that you are going to use the number strip to help add 7 + 5. Show the students a group of 5 counters.

2.

Ask the students, "How many counters will it take to get to 10? (3)"

3.

Put on the 3 counters, then ask the students:

• How many counters are there to add on? (2)
• What do 10 and 2 make?
• What two numbers did we split the 5 into? (3 +2)
• Why did we choose 3, then 2? (To make it up to 10)

4.

Write the problem 8 + 6 for the students to see.
Ask the students:

• How many counters would be needed to get to 10? (2)
• How many counters are there still to add on? (4)
• What is 10 plus 4? (14)

Check the answer using counters and the number strip.

5.

Draw a number line and pose the question 7 + 4.
Start at 7, and ask the students:

• How many jumps do we need to add on? (4)
• How many jumps is it to get to 10? (3)
• How many are left over from the 4? (1)
• What is 10 and 1? (11)

6.

Students can practice using the make a ten strategy on number lines. Pose questions using the context of measurement and encourage students to write the correct units beside the answer. Possible questions are:

• The bucket had 9 litres in it, and Kitiona poured in another 6 litres. How many litres are now in the bucket?
• Anna put 7 cups of juice on the tray, and Kiri added another 5 cups to the tray. How many cups were there altogether?
• The temperature was 8^oC and it rose 3 degrees during the morning. What is the temperature now?
• Rangi left George’s house and ran for 8 minutes, then walked for 5 minutes before he got home. How many minutes did it take him to get home?
• Mum bought 7 kilograms of kumara and 4 kilograms of carrots. How much did the vegetables weigh?

In this session, students solve addition problems by partitioning numbers. They will solve problems that involve adding a 1-digit number to a 2-digit number. The “make a ten” strategy is applied to decade numbers; for example, 28 + 7 is solved by making it up to 30 and then adding the remaining 5.

1.

Pose the problem: If the plant was 37cm tall and it grew 8cm, how tall is it now? Show the students a number line that ranges from 0 – 100, with the 10s numbers coloured in.

2.

Ask the students, "Using the make-a-ten strategy we used yesterday, how could we solve this problem?"

3.

Work with students to jump 3 to get to 40, then jump the remaining 5 to get to 45.

4.

Pose the question: The suitcase weighed 17 kilograms and the backpack weighed 5 kilograms. How much did the luggage weigh altogether?
Show the students how they can draw a number line to suit the question.
Ask the students: 

• What numbers are we adding together? (17 and 5).

5.

Draw a line and the number 17 underneath.
Ask the students: 

• What number can we jump to from here? (20)
• How many jumps is that? (3)
• How many of the 5 are left? (2)
• What is the answer? (22)6.

6.

Students can practice using the make-a-decade strategy on their own number lines. Pose questions in the context of measurement and encourage students to write the correct units beside the answer. Possible questions are:

• Jane was going home on the bus. The bus took 26 minutes, then she walked for 8 minutes. How long did it take for Jane to get home?
• Peni’s plant was 48cm tall. It grew another 7cm. How tall is Peni’s plant now?
• Dad went to the garden shop and bought 16kg of compost and 7kg of fertilizer. How much did it all weigh?
• The painter had 65 litres of paint, and he bought another 8 litres. How much paint does he now have?
• In the first three weeks of October, Wellington had 88mm of rainfall; in the rest of the month, another 7mm fell. How much rain is that altogether?

In this session, you may wish to continue to give the students opportunities to practice addition partitioning with the make-a-ten or make-a-decade strategy. Problems could focus around one measurement theme, for example, length. Or problems could focus around a theme such as camping and involve more than one measurement context, for example, weight of packs, time spent on activities, capacity of shower water, length of washing lines, etc.

Alternatively, you may wish to use the formats of sessions 3 and 4 to show students how this partitioning strategy can be used to solve subtraction problems. For example, from 44 – 7, it takes 4 jumps to get back to 40, and then the remaining 3 jumps take us back to 37.