Published over 2 years ago by Ann Baker
Ideally, differentiation provides tasks that:
The following task allows for each of the above.
Notice that, while the task specifies that each of the numbers has to be used once, there is no requirement as to which operators should be used. Also notice that it does not set up competition by asking how many can be found. So if I only find one way I can feel as successful as the students who findS five ways.
The deliberate implications of this are that even the youngest or lowest attaining students can join in and have success.
A really important problem solving strategy that all students can use at some time is ‘try a simpler case’.
In this problem trying a simpler case could make this problem accessible to all because it means that I can only use two of the numbers, not all of them, or I could use only addition and none of the other operators.
The following list uses only addition and subtraction and most use only two numbers and yet each total 1 – 6 can be made. 2 – 1 = 1 3 – 1 = 2 1 + 2 = 3 1 + 3 = 4 2 + 3 = 5 1 + 2 + 3 = 6
Many other entry points are also possible and could include all numerals and the four operators, +, -, ×, ÷, or to add challenge, include the use of brackets and indices.
For students who need extra challenge use the scoring scale for each unique equation created. Weighting different criteria will push them to think well beyond just the four operators. In fact, using the scoring system as a personal best will push some students to include a division. We find that addition and multiplication are used in preference to subtraction and division by many students so ‘paying’ in points for their use also moves students along.
There are many ways to make each total 1-6 or 0-10, (you choose the range to match your students). This is useful when it comes time to share because a basic equation such as 1 + 2 + 3 =6 has a rightful place on the board along with 2 × 3 × 1= 6 or 3 × 2 ÷ 1 = 6.
Sharing the range of answers serves many purposes. In particular, it:
As discussed in the first point, try a simpler case is going to make this problem accessible to all.
The obvious problem solving strategy for this problem is to choose an operation (or more than one) and to pay attention to the results of using particular operations. This will lead into Guess, check and improve.
Guess, check and improve is a useful strategy as it will enable students to get started in any way they choose and will soon lead to them noticing that they have made the same values in many ways but have omitted some values entirely. Time for them to look at the previous examples and to adapt them to make missing values.
Some students, however, will work systematically so that eventually they will be able to prove to themselves or to others that they have found all the possibilities. Working systematically is likely to lead to a rich and extensive list of possibilities.
Note in the list below that every example uses the numbers in the order in which they were given,
–1 × 2 + 3 = 1
–1 + 2 + 3 = 4 1 + 2 + 3 = 6 1 + 2 – 3 = 0 1 – 2 + 3 = 2 1 × 2 + 3 = 5 1 × 2 × 3 = 6 1^2 + 3 = 4 1 + 2^3 = 9 and so on. The next step would be to see what happens if the list is used in the reverse order or as 2, 3, 1.
For some students this question will generate more questions, for instance:
If we tell students that this simple number problem is designed for exploration and that their job as they work on it is to notice and think about the effects of the operations used on the range and magnitude of the values they will begin to plan accordingly and to make some reasoned generalizations, e.g.,
A simple problem such as this provides so much to talk about, share and reflect on that we need to decide the most important learning to highlight and why.
The following are just a few of the thoughts for this problem:
Note: the questions above are just the tip of the iceberg but you will notice that they all have somethings in common, they:
They also encourage classrooms to become communities of learners where all ideas can be valued and built upon.
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