Differences with two-digit whole numbers using renaming
The purpose of this activity is to support students using their knowledge of place value to solve difference problems with two-digit numbers, with renaming.
About this resource
New Zealand Curriculum: Level 2
Learning Progression Frameworks: Additive thinking, Signpost 4 to Signpost 5
These activities are intended for students who understand addition to be the joining of sets and subtraction to be the removal of objects from a set. Difference problems require the student to compare the numbers of objects in two different sets and can be solved using either addition or subtraction.
Target students should have already developed a degree of part-whole understanding in addition and subtraction contexts (joining and separating). They should have some number facts to call on, particularly number bonds to ten. Understanding of two-digit place value, including the structure of ‘teen’ and ‘ty’ numbers, will support many of the activities described in this intervention.
Differences with two-digit whole numbers using renaming
Achievement objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
Required materials
- place value materials: individual items grouped into tens, such as BeaNZ in film canisters, iceblock sticks bundled with rubber bands (hundreds with hair ties), or a paper form, such as Place value people 1. Bundled materials are important as they allow partitioning and combining without the need for “trading” tens blocks for ones.
- A place value board (Place value with 2 digits 1) can be used to organise the materials in columns and support calculation strategies. Three-column and four-column place value boards are available in Place value to 4 digits 2.
See Materials that come with this resource to download:
- Place value people 1 (.pdf)
- Place value with 2 digits 1 (.pdf)
- Place value to 4 digits 2 (.pdf)
1.
Pose difference problems with two-digit whole numbers in which the ones digit of the larger number is less than the ones digit of the smaller number. For example:
- You have 32 ice block sticks and I have 18 ice block sticks.
- How many more ice block sticks do you have than I have?
2.
Provide time for students to work out the difference 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, addition and subtraction are similar because both involve acts of renaming.
Adding on from 18 to get to 32. This strategy can be represented by an empty number line.
Subtracting 18 from 32. 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. For example:
- You have 74 ice block sticks, and I have 29 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.
Addition might be considered more efficient because the renaming is easier. Students might also find easier subtraction strategies, such as 74 – 30 = 44; 44 + 1 = 45.
5.
Provide examples with a focus on finding the answer, using accurate and systematic recording strategies (e.g., a number line), and 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 93 ice block sticks, and I have 17 ice block sticks.
- How many more ice block sticks do you have than I have?
- 93 – 20 = 73, 73 + 3 = 76 is very efficient.
Addition and subtraction are equally efficient.
- You have 84 ice block sticks, and I have 78 ice block sticks.
- How many more ice block sticks do you have than I have?
- Record the possible strategies symbolically.
- 78 + [ ] = 84
- 84 – 78 = [ ]
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 61 and 18 can be found using
2.
Change the questions to include the word “fewer”.
For example,
- You have 76 counters, and I have 29 counters.
- How many fewer dinosaurs do I have compared to you?
A suggested sequence for extending the difficulty of finding differences is:
- Use smaller differences, such as between 41 and 27.
- Use larger differences, such as between 96 and 28.
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