Differences with two-digit whole numbers with no renaming

1. Pose difference problems with two-digit numbers in which all the digits (i.e. tens and ones) of the larger number are greater than the digits of the smaller number.
   You have 49 ice block sticks and I have 26 ice block sticks.
   How many more ice block sticks do you have than I have?
    
2. Provide time for students to work on the problem independently, without using physical models. If your students demonstrate place value misunderstandings, create models with bundled place value materials arranged on a place value board to support their thinking.
    
3. After a suitable amount of time, gather together to discuss the two main ways of solving the problem - addition and subtraction - and the efficiency of each strategy. Te reo Māori kupu such as tāpiri (add), tango (subtract), and huantango (difference) could be used throughout this learning.
   Who solved the problem by adding?
   Who solved the problem by subtracting or "taking away"?
   Which strategy, adding on or subtracting, was the easiest to do? Why?    
   If needed, model each strategy (or get students to) for the class.   
   In this case subtraction is easier because the smaller number can be subtracted from the large number without renaming.      
    

• Adding on from 26 to get to 49. This strategy cab be represented by an empty number line.
  
  
• Subtracting 26 from 49. This strategy can also be represented on an empty number line or as a vertical written algorithm. Note that the latter strategy might demonstrate procedural knowledge, rather than an understanding of place value.
  
      
    

4. Pose other similar problems in which the amount of difference is extended. Encourage students to use their number facts and place value knowledge to solve each problem. Ensure students have opportunities to express their mathematical thinking in different ways (e.g. written, verbal, drawn diagrams, acting out).
   You have 98 ice block sticks and I have 15 ice block sticks.
   How many more ice block sticks do you have than I have?
   Which strategy is easier, adding on or subtracting?
   
   If needed, support students' recognition of place value by making the quantities with bundles of materials and arranging them vertically on a place value board. Gradually mask the materials to support greater reliance on symbolic recording and mental or written strategies.
   
      
   
   Subtraction is more efficient for two reasons; the amount being subtracted is small and there is no need for renaming.
    
5. Provide examples with a focus on finding the answer, using accurate and systematic recording strategies (e.g. a number line), and on using the most efficient method. You might group students to encourage scaffolding, extension, and productive learning conversations. Ensure students have opportunities to share their understanding, ask questions, and listen to a variety of ideas in a variety of groupings.
   • Subtraction is more efficient:
     You have 75 ice block sticks and I have 23 ice block sticks.
     How many more ice block sticks do you have than I have?
   • Addition is more efficient:
     You have 82 ice block sticks and I have 79 ice block sticks.
     How many more ice block sticks do you have than I have?
   • Addition and subtraction equally efficient:
     You have 88 ice block sticks and I have 55 ice block sticks.
     How many more ice block sticks do you have than I have?
     Record the possible strategies symbolically.
     55 + [ ] = 88
     88 - 55 = [ ].

Next steps 

1. Increase the level of abstraction to the point where students can work with symbols without the need for physical models. Develop their fluency with recording strategies as addition or subtraction equations, in horizontal or vertical form. For example, the difference between 89 and 52 can be found using 89 - 52 = [ ] or 52 + [ ] = 89. or  
   
    
2. Change the questions to include the word “fewer”. For example:
   You have 88 counters and I have 35 counters. How many fewer counters do I have compared to you?

A suggested sequence for extending the difficulty of finding differences is:

1. Use smaller differences, such as between 47 and 31.
2. Use larger differences, such as between 86 and 14.