Possible misconceptions
- Students may not understand the question and pick two notes or two coins.
- Students may not understand the value of notes and coins (eg. select $50 and $1 as the change).
- Students may not understand place value for decimals (eg. how many cents in a dollar).
- The size of coins can be problematic as they “are not proportional to their value” (Siemon et al., 2015, p. 667).
- Students may invent denominations of coins which in reality are restricted to 5c, 10c, 20c, 50c, $1, $2 (Siemon et al., 2015).
Solution 1 |
Solution 2 |
Solution 3 |
From the problem, I know I had $50 and now I have 1 note and 1 coin. The problem does not tell me which note or coin so I select one of each. The note has to be less than $50 so it could be $5, $10 or $20 and it could be any coin (as all coins are less than $50).
I select a $20 note and $1 coin. This is equivalent to $21. I need to add notes and coins to reach $50.
I can add $5, $10, $20 or coins. I start with the largest value to get as close as possible to $50. If I add $5 it is $26, $10 it is $31 or $20 it is $41. $20 gets me the closest to $50. If I add $5 I will make $46 but if I add $10 I will go over $50. I add $5 to get $46.
Adding $5 will go over $50 so I need to look at coins. I can add $2 to make $48 and another $2 to make $50.
I combine the notes and coins added to make up $50 and calculate their total.
The present cost $29. |
I have a $10 note and 0.50 cents in change. This is equivalent to the monetary value of $10.50.
I subtract $10.50 from $50 to find the cost of the present Therefore, the present cost $39.50.
This can also be shown using money as per below. If I add $10.50 to $39.50 I can confirm they are equal to $50.
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Misconception regarding value of coins
I have a $10 note and 5 cents in change. This is equivalent to $10.05. I need to add notes and coins to reach $50. I add $20, $10 and $5 to reach $45.05.
I need to add coins to reach $50. I add $2, then $2 to reach $49.05.
I add 5c to reach $50.
I combine the notes and coins added to make up $50 and calculate their total.
The cost of the present was $39.05. In this solution the student was mistaken in thinking that the value of 2 five cent coins is $1 instead of 10 cents. They would require additional support to understand equivalent combinations of coins. |
There are 15 combinations possible amounts for the change using $20, $10, $5, $2, $1, 50c, 20c, 10c and 5c.