# The numbers get larger

Purpose

In this unit students, are encouraged to extend their use of part/whole methods from working with small numbers to working with multi-digit numbers. This work follows naturally from the Smart Doubling and Make a Ten units.

Achievement Objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
Specific Learning Outcomes
• Use part/whole strategies to solve multi-digit addition and subtraction problems.
• Use the most efficient mental strategy for a given problem.
Description of Mathematics

In this unit students, are encouraged to extend their use of part/whole methods from working with small numbers to working with multi-digit numbers.

Students need to be able to understand a range of simpler part/whole strategies before working on this unit. These strategies are described in the units Smart Doubling and Make a Ten, and include:

• The students work out 38 + 39 by removing 1 from the 38 to leave 37, add this 1 to the 39 to make 40 then add 37 and 40 to give 77.
• The students work out 38 + 39 by 30 + 30 = 60, 8 + 9 = 17, 60 + 17 = 77.
• The students work out 38 + 39 by 40 plus 40 = 80 and then remove 2 (to get to 38) and remove a further 1 (to get to 39). So 80 - 2 - 1 = 87 gives the answer.
• The students work out 8 + 5 by removing 2 from the 5 to leave 3, add this 2 to the 8 to make 10 then add the 3 to the 10 to give 13.

• The students work out 38 + 8 by removing 2 from the 8 to leave 6, add the 2 to the 38 to give 40, then add 6 to the 40 to give 46.

Required Resource Materials

Tens frames (Material Master 4-6)

Activity

#### Getting Started

To start with, students solve addition and subtraction problems involving one single-digit number and one multi-digit number. Once they can confidently solve these, progress to problems that involve two multi-digit numbers.

1. Pose the problem:
Minnie has \$8 and her grandmother gives her \$19 for her birthday. How much money does Minnie have now?
2. Students model the problem on pre-printed tens frames.
3. Encourage students to identify dots that could be moved to make the problem quicker, or easier, to solve.  Note that because pre-printed tens frames are used the dots don’t actually move. For example: Mentally becomes 4. Repeat with similar problems, sharing the Make to Ten strategies used.

#### Exploring

Over the next 3-4 days pose addition and subtraction problems for the students to work on, individually, in pairs or in a small teaching group.  Encourage the progression from using  pre-printed tens frames to imagining the tens frames and then to completing the problem mentally.

1. Pose the problem:
Malcolm has \$27 and he spends \$9 on lunch. How much does he have left?  Students model this on pre-printed tens frames.
2. Share solutions. Students are likely to imagine the 7 is removed leaving 20, then 2 is removed from one of the tens to leave 18. Other methods are possible. 3. Pose the following problems for the students to work on.  Give the students pre-printed tens frames to help in their solutions:
4. 28 + 7, 33 - 6, 24 - 5, 38 - 6, 24 + 9, 33 + 5, 40 - 6
5. Pose the following problems.  For these problems encourage the students to imagine the tens frames:
38 + 6, 23 – 8, 44 – 5, 39 – 9, 9 + 34, 3 + 45, 30 – 9
6. When the students are able to solve problems involving a single and double-digit numbers extend the problems to involve two 2-digits numbers. Have pre-printed tens frames available.

Pose the problem:
Kiri has 37 Jaffas and 26 wine gums. How many lollies does Kiri have altogether?  Discuss the students’ ideas for solving the problem.

• One strategy involves putting the tens together to get 50 sweets and then combining the 7 + 6 to get 13.  The 13 is then added to the 50 to get 63.
• Alternatively they may combine the tens to give 50, then put the 9 with the 50 to give 59, then use previous part/whole thinking to add 3 to the 57 to give 60 then add the remaining 3 to give 63.
7. Pose problems that involve the use of doubles.  As with single-digit problems doubling strategies are useful for certain problems. For example:
24 + 25   (25 + 25 = 50, then subtract 1)
15 + 16,  35 + 36,  24 + 26,  97 + 103, 402 + 398
8. Pose the double-digit subtractions problems, for example:
Eva has \$45. She buys a top for \$29.  How much does she have left? Discuss the ideas that the students have for solving this problem.  Strategies may include:
• take away 20 from the 45 to leave 25. Another 9 needs to be removed. Remove the 5 to leave 20 then remove 4 more to give an answer of 16.
• take away 20 from the 45 to leave 25. Another 9 needs to be removed. Remove 9 from a 10 frame to leave 1. There is another 10 and a 5 to add to 1 to give 16.
• take away 30 from the 45 to leave 15. But taking away 30 is 1 too many. So add 1 to 15 to give 16.
1. Pose problems for the students to work on.
47 – 28, 50 – 27, 100 – 68,  91 – 12,  63 – 23,  42 – 38,  103 – 6,  103 – 98,  81 – 34,
200 - 188
2. Extras for experts.

In these problems the students are encouraged to add and subtract more than 2 numbers in which there are smart ways to do them. Begin by discussing the following problem:

Harry adds up the number of lunches his class orders in the week. By pairing up some of the numbers Harry quickly noticed the total lunches sold for the week was 60. How did Harry work this answer out so quickly? (Pairing the 28 with the 2 gives 30. Pairing the 17 with the 3 gives 20: 30 + 20 + 10 = 50.)

1. Molly has \$56 when she goes shopping. She buys a CD for \$28 and a pair of shoes for \$26. To work out how much money she has left over she writes down 56 - 28 - 26. (Write this expression on the board.
Her friend Kate sees this and almost immediately says Molly has \$2 left. How did Kate do this so quickly?

Discuss the students’ ideas:

• 56 - 26 = 30, 30 - 28 = 2
• 28 + 26 = 54 (from 27 + 27) and 56-54 =2
1. Ask the students to find "clever" ways to work out the following problems:
198 + 65 + 2                        345 - 99 - 245                88 + 45 + 12 + 55
100 - 34 - 66                        9 + 456 + 191                7 + 25 + 33 + 25

#### Reflecting

At the end of each session gather the students together to share strategies.