Weighing for the post
| a) |
2.96 kg + 3.57 kg + 1.04 kg
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b) |
0.16 kg + 0.4 kg + 0.306 kg |
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c) |
1.21 kg + 2.046 kg + 0.9 kg |
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- Correctly adding decimals with the same number of decimal places (1 or 2) without recombining
- Correctly adding decimals with the same number of decimal places with recombining as required (not including tenths to ones).
- Correctly adding decimals of different significant figures without recombining.
- Correctly adding decimals of different significant figures and recombining as required (including tenths to ones).
| Y8 (10/2010) | ||
| a) |
7.57 Working that involved any of the following:
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moderate easy |
| b) |
0.866 Working that involved any of the following:
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difficult moderate |
| c) |
4.156 Working that involved any of the following:
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difficult moderate |
Based on a representative sample of 191 Y8 students.
This resource is about adding decimal values with varying number of decimal places. This is a better measure of whether students can "add decimals" as they must understand how place value is maintained and something of the nature of decimals.
Prior knowledge
Students should have experienced learning situations where they have split and recombined decimal numbers into their component parts (e.g., place value partitioning). They should also have some awareness of the nature of decimals and have an understanding of how to maintain place value of all the parts of the decimal numbers under addition.
Many students could successfully add these decimal numbers together. The most common addition strategy (used by students in this trial) for this problem was vertical form, followed by place value partitioning and then adding the numbers in the order they were written cumulatively (i.e., add the 2nd number to the 1st, then the 3rd number to that, etc). The vertical form and place value partitioning strategies had a similar level of accuracy (success for getting the correct answer). However students who used cumulative addition of the numbers had a higher accuracy for the correct answer.
Students made a significant number of errors based on place value: maintaining place value: adding tenths and hundredths together as if they were of the same place value; and recombination: some students don't know how to cross the threshold from tenths to ones.
See Student work samples [pdf] for a range of strategies that students' used.
| Common response | Likely calculation | Likely misconception | |
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a) b) c) |
757 326 or 0.326* 2.176* |
16+4+306 121+9(=130)+2046 |
Working with the decimals as whole numbers Students add the decimal fractions as if they were whole numbers without adding the digits with their correct place value. *Then compensate to maintain the decimal value. |
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b) c) |
0.506 3.346, 3.76 or 3.076 |
0.16+0.4(=0.2)+0.306 1.21+0.9(=1.3)+2.046 1.21+2.046=2. 57 |
Place value error: maintaining place value Students add the decimal fractions but make a place value error where digits are added without maintaining their place value - this could also be combined with dropping place holder zeros. |
| a) |
6.157 or 6.57 |
2.96+1.04 (=3.100)+3.57 2.96+1.04 (=3)+3.57 |
Place value error: recombining decimals Students add the decimal fractions but make an error recombining the tenths: either carried over to a newly created decimal place before the tenths or did not carry at all. |
Working with the decimals as whole numbers and maintaining place value error
Students who added the decimal numbers as whole numbers are very likely to make place value errors - especially if the addends have varying numbers of significant figures [particularly questions b) and c)]. Students should be seeing decimals (tenths, hundredths and thousandths, etc.) as an extension of the whole place value system with the same base 10 rules (ten of these makes one of those) and using these rules to solve decimal addition problems.
Ensure that students have strategies to successfully add whole numbers involving recombination. Students need to develop their understanding of what decimal values are. They should explore the base 10 relationship of the decimal values in a similar way to how they may have explored the relationship between ones, tens, hundreds, etc. This can be done using place value blocks that represent tenths, hundredths and thousandths, an abacus that represents ones, tens, hundreds and tenths, hundredths and thousandths, or arrow cards (NZ Maths). The resource Using place value blocks looks at using place blocks to represent a selection of decimal values. Students could also explore the base 10 nature of decimals using decimal pipes. The above learning experiences should also support students who added the digits to other digits of different place value (e.g., tenths added to hundredths as if they were the same). These students indicated that they didn't recognise how to add decimal numbers of different place values.
Place value error: recombining decimals
These students who added the decimal numbers mostly maintaining correct place value, except they had difficulty recombining the decimal components: especially recombining the tenths where they either carried over to a newly created decimal place before the tenths or failed to "carry" anything over to the ones.
Before moving on to adding decimal values students should have an understanding about the size of decimals. The resource Gymnastic competition involves ordering decimals up to 3 decimal places. This would indicate their understanding of decimal size which underlies the understanding about decimal place value.
The use of a 6-pronged abacus to represent whole number and decimal values to add decimals can be used much in the same way a 3-pronged abacus would be used to learn about adding whole numbers and what happens to the smaller unit when there are ten (or more of them). For example, using to add 2.67 + 1.50 (in particular, how 11 tenths is shown).
Students can also explore how to recombine with simpler decimal values. For example the resources Temperature changes, Buying groceries and Checking the bank account should support students to add the tenths and hundredths involving recombination.