The numbers get larger

Purpose

This unit follows naturally from the Smart Doubling and Make a Ten units.
In this unit students are encouraged to develop part/whole addition and subtraction mental methods with multi-digit numbers.

Specific Learning Outcomes
  • Use mental strategies for addition and subtraction problem that use part/whole strategies for multi-digit problems.
  • Use the most efficient mental strategy for a given problem.
Description of Mathematics

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

Examples of more advanced "part/whole" methods using make a ten:

  • 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.

Students will need to have successfully understood the part/whole methods in the Make a Ten unit before attempting this unit.

Required Resource Materials

Tens Frames (Material Master 4-6)

Activity

Getting Started

In this addition and subtraction unit initially only one of the numbers is 2 digits or more, while the other number is normally a single digit. This is because the processing load for the students in having both numbers with multi-digit is much higher.  When the students have successfully coped with these problems 2 digit plus or minus 2 digit problems can be introduced. 

  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 this on pre-printed tens frames.
  3. Discuss the dots that could be moved to make the addition "easy".  One example is:
    diagram of tens frames.  Mentally becomes  diagram of tens frames.
    (Because pre-printed tens frames are used the dots don’t actually move.)
  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.
    diagram of tens frames.
  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?

    diagram of tens frames.diagram of tens frames.

    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?
    diagram of tens frames.
    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?

diagram of problem.

(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.