Saving for a pet
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.
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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.
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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.
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| Y4 (11/2007) | ||
| a) |
137 Working that involves addition of 9, 56 and 72:
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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.
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.
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 | |
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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. |
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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. |
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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%).
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).
- 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.
- Buying groceries
- Buying books II
- School swimming sports
- Numbers at an art show
- Bank account
- Temperature changes
- City populations
- Sharing fruit and vegetables
- Addition wheel
- Writing word problems
- Estimate these II
- Keep fit programme
- Sharing Jelly beans
- Knock the can
- At the canteen
- Number line addition II
- Estimating with addition III
- Estimating addition IV
- Student answers
- Number line addition and subtraction III
- Estimating with addition
- Estimation or not?
- Eating fractions of pie, pizza and cake
- Fractions of cake
- Solving maths problems
- Some maths problems
- Collecting beads
- Making a cake
- Fruit and vegetables