Saving for a pet

Saving for a pet

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about showing how to solve some addition problems with money.
 
a) Maraea is saving to buy a cat. In 3 weeks she has saved $9, $56 and $72. Show how to work out how much she has saved altogether.

 
 
 
 
Total money saved: $__________  
 
b) Tom is saving to buy a dog.  In 4 months he has saved $36, $78, $203, and $123.  Show how to work out how much he has saved altogether.

 
 
 
 
Total money saved: $__________  
 
c) Te Ao is saving to buy a horse.  So far she has saved $17, $160, $72, and $148.  Show how to work out how much she has saved altogether.

 
 
 
 
Total money saved: $__________  
 
 
Task administration: 
This task is completed with pencil and paper only.
Level:
3
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
addition, place value
Description of task: 
Students solve addition story problems involving numbers of different magnitude to explore their place value understanding.
Curriculum Links: 
This resource can help to identify students' ability to apply additive and simple multiplicative ideas flexibly to combine or partition whole numbers to solve addition problems.
 
Key competencies

This resource involves recording the strategies students used to solve addition problems. This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Additive thinking, sets 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y4 (11/2007)
a) 137
Working that involves addition of 9, 56 and 72:

  • Place value partitioning: 50 + 70 + 9 + 6 + 2 = 120 + 17
  • Derived place value partitioning: 56 + 70 + 9 + 2 = 126 + 11
  • Vertical algorithm (working form)
  • Accumulated addition: 9 + 56 = 65, 65 + 72 = 130 + 7
  • any other working that involves the correct operation (addition), all the numbers, and is complete.
 difficult
moderate
b) 440
Working that involves any of the above strategies and the addition of 36, 78, 203, and 123.		 difficult
moderate
c) 397
Working that involves any of the above strategies and the addition of 17, 160, 72, and 148.		 very difficult
moderate

Based on a representative sample of 283 Y4 students.

Teaching and learning: 

This resource uses numbers of different magnitude to explore how students solve addition problems with 3 or more numbers. Using these numbers ensures that students have to deal with the place value aspect of addition in the tens and hundreds. This resource is also concerned with the range of strategies that students use. In the context of the classroom these strategies can be shared and students would critique the best, fastest, easiest and/or most effective strategies.

Prior knowledge
Students should be able to add two numbers before they are asked to add three or more. They should also be able to add three simple or similar numbers before adding three numbers of differing magnitudes.

Diagnostic and formative information: 

There was a wide range of computational and conceptual errors involving a combination of place value and simple calculation errors. About half of all the answers given across all three questions were "close" to the answer.  "Close" was deemed to be plus or minus 10 for question a) and plus or minus 20 for questions b) and c).  This margin of error was purely used as a broad guideline rather than any measure of standard error.  This broad accuracy fell as the numbers became more difficult to add (questions a, b, and c: 63%, 46%, and 35%, respectively).

  Common error Likely misconception
a)
b)
c)		 1037
10034
30010		 Compacting hundreds, tens and ones (compact numerals)
Some students when combining the hundreds and tens and ones to make a number did not collapse the place holding zeros to create a compact number.  For example 100 + 34 becomes 10034. These errors can generally be recognised by "large" numbers with lots of zeros.
a)
b)
c)		 136, 138
438, 439
396, 398		 Computational error
Students did not add digits together correctly and the solution is off by 1 or 2.
a)
b)
c)		 127
430 or 340
297		 Computational error: not maintaining place value
Students added digits together but did not take into account place value issues moving from ones to tens (lost the tens that they needed to account for).  For vertical algorithms this is referred to as not "carrying the ten"

Strategies
The most common strategy that students used to solve the problems was place value and derived place value partitioning (together about 35% of students across all questions).  A small number of students used the vertical algorithm (8%). Over half of these used the vertical algorithm to add all the numbers, while others added the numbers in pairs, using a number of vertical procedures to complete the problem. Another strategy that 15% of students used was accumulating addition – adding two numbers together then adding the next number on to that, effectively getting a cumulative total until all the numbers were added.  A small number of students used compatible numbers to make tidy numbers, and horizontal jumps (like number lines).
Across all three questions the most successful (deriving a correct answer) strategies that students used were place value partitioning (46%) followed by the vertical algorithm (45%), and followed by derived place value (36%) and accumulating addition or adding in pairs (25%).

Next steps: 

Adding zero place holders
A very small number of students were still keeping the zero place holders in when they added into the hundreds.  These students could explore numbers using an abacus (Abacus numbers), place value blocks (Place value blocks), arrow cards or tens frames to see what happens with the place holding zeros when numbers are constructed.  Students could also look at compacting and expanding numbers (Compact and expand).

Computational error
Students who made simple calculation errors could be encouraged to check over their additions and look at how they are developing their working to find the answer. Basic facts are an important part of developing the strategies to solve problems. Students may need to explore the strategies they use to add two numbers together, explain what happens when the digits add to more than ten, and apply this understanding to adding 3 numbers together.

For those students who successfully add the three numbers, have them check the reasonableness of their work, by estimating the size of the answer (Estimation as a Check, Book 5: Teaching Addition, Subtraction, and Place Value).

Links to the Number Framework
  • Adding numbers of different magnitudes into the hundreds is early-advanced additive part whole (stages 5-6), adding three numbers is more advanced additive part whole (stage 6)
  • Saving Hundreds (Early additive part-whole) and Three or more at a time (Advanced additive part-whole), Book 5: Teaching Addition, Subtraction, and Place Value, 2007.