One of our preferred forms of differentiation is to provide tasks that:
- have multiple entry and success for all
- have more than one right answer
- require use of at least one problem-solving strategy
- allow for what if and what else questions
- involve reasoning and generalizing
- lend themselves to whole class sharing and reflection
The following task allows for each of the above.
Multiple entry points and success for all
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 enter 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 use only 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 for more challenge the use of brackets and indices.
For students who need extra challenge scoring each equation as follows 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 moves students along.
More than one right answer
There are many ways to make each totals 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:
- immerses students in methods that they did not try or were not quite ready for
- suggests to some students that they were no pushing themselves as far as they could have to move outside their immediate comfort zone
Require the use of more than one problem solving strategy
As discussed in the earlier paragraph, 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 now 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 used the order 2, 3, 1.
Allow for what if’s and what else’s
For some students this question will generate more questions, for instance:
- What if we use 4 numbers, e.g., 1, 2 3, 4?
- What if we use 3 different numbers, e.g., 2, 3, 4?
- What if we use a division in each example, what values will we be able to make then?
- Will using only odd numbers mean we can only make odd values?
Involve reasoning and generalizing
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.,
- “The value after my first operation will be unchanged if I divide by 1.”
- “I can only divide an even number by 2 to get a whole number value.”
- “The largest values are made by using multiplication or indices.”
- “I can make values less than 1 when I want to make a small value altogether.”
Whole class sharing and reflection
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:
- What different ways did we find for making the value 4?
- How could we organize the list for making the value 4 so that we can see the range and whether new possibilities can be found?
- Was dividing by 1 a useful strategy? Let’s put up some examples and check it out.
Note: the questions above are just the tip of the iceberg but you will notice that they all have something’s in common, they:
- Promote sharing, comparing and connection mathematical ideas
- Focus on process and mathematical thinking not just on answers
- Are accessible to all students and actual examples can be selected from students at any level
They also encourage classrooms to become communities of learners where al ideas can be valued and built upon