Different subtraction number sentences
- Numbers of students become uncomfortable when they meet questions d)-f). These have the equals signs near the beginning of the equation rather than just before the "answer" (a = b – c). They may very well ask about these questions or state that they do not make sense. Encourage them to give the best answer they can.
- Have a class (or group) discussion afterwards on what the equals sign means. What is equal? gives a comprehensive guide on how a class discussion can be managed on mathematical classroom discourse.
- The location of both the equals sign and the box for the missing number (called the unknown) is in corresponding positions. For example a) has it in the position generally used in number problems (a – b = ?), while b) is of similar form (a – ? = c).
| Y4 (06/2009) | ||
|
a) b) c) d) e) f) |
2 7 13 18 6 11 |
very easy very easy moderate difficult moderate moderate |
Based on a representative sample of 164 Y4 students.
| Common error | Likely misconception | |
|
a) b) c) d) e) f) |
1 or 3 6 or 8 12 or 14 17 or 19 5 or 7 10 or 12 |
Counting on or counting back incorrectly Solves by using counting strategies, but miscounts by 1. Bolded numbers indicate 5 or more students in the sample responded this way, indicating that for subtraction questions they are equally likely to be short by 1 or over by 1. |
|
c) d) e) f) f) |
5 6 24 21 0 |
Adding instead of subtracting: Incorrect use of the inverse operation
Students use the incorrect operation on the two given numbers to get their answer, adding instead of subtracting. Nearly 40% of students made this error in d), and over 10% of students in c), e), and f).
Attempts 5 – 16 and says 0 because it is less than zero.
|
| d)-f) | No answer given |
Cannot understand the position of the equals sign Students do not know how to respond to the equals sign coming early in the number sentence or equation (? = b – c). |
These students are likely to correctly understand the number sentence, and should be given credit for this. However, they are probably not solving the problems using the inverse operation, which is the more algebraic approach, and may need to either work on:
- Part-whole strategies (e.g., in b) saying ? = 15 – 9 , 15 – 10 = 5 and adjusting), or
- improving their basic facts in addition or subtraction so the don't have to count on or back.
Click on the Basic facts concept map for more information.
Adding instead of subtracting – Incorrect use of the inverse operation
Students may have some realisation that they need to subtract the two numbers to get the answer, and may well be beginning to think algebraically. Other students may simply be performing addition or subtraction because they can't think of what else to do.
- Discuss with the students what they did and why.
- Get them to check if the number sentence they get is true.
- Discuss what they think the equals sign means. They may think it means "and the answers is" rather than "is the same as" (or more correctly "is the same amount as").
Cannot understand the position of the equals sign
Students are used to the equals sign coming near the end of a number sentence. Question a) is of the form students usually meet,
i.e., 9 – 7 = ?
Students need to realise that the equals sign can come elsewhere in a number sentence or equation, and that equality is about the balance of numbers on both sides of the equals sign.
i.e., ? = 9 – 7 is a different way of asking the same question.
Location of the missing number
This affects how the student responds to the question.
-
a) is of the usual number form (9 – 7 = "what"?)
e) is closely related to this, but is harder for students. -
The other questions place the box in a position where the number to be found is part of an expression, e.g., 13 –
- Get students to say the problem out loud, and see if this makes sense,
e.g.,13 minus "what" equals 6;
"what" minus 4 equals 9;
5 equals 16 minus "what".
-
Discuss the different strategies students use, both number types or algebraic types:
Examples: 13 – = 6
|
Numeric
|
"12, 11, 10, 9, 8, 7, 6 so its 7" "I know 13 – 7 = 6" "10 + 3 = 13 and 10 – 4 = 6 so it is 3 + 4 =7" The answer must be 13 – 6 |
Counting back Basic facts Part-whole thinking Additive inverse |
Strategies
Students who used counting on or counting back incorrectly or added instead of subtracted had a significantly lower mean ability that students who got a correct answer. These students typically had a higher mean ability than those who got other wrong answers or did not answer the question, indicating that many of them had some level of understanding of the problem.
- Good as Gold (Number Sense and Algebraic thinking, L2-3, book 2, pages 12-13).
- The Fish Hooks of Ngake (Number, L3-4, book 2, pages 20-21).
- Crunch Machine (Algebra, L2-3, page 17 Activities 2 and 3).
Equality as balance: refer to
- A Balancing Act (Book 5: Teaching Addition, Subtraction, and Place Value) and
- The Equals Sign Again (Book 8: Teaching Number Sense and Algebraic Thinking).