Mathematicians are fascinating people – they notice patterns, make connections and create solutions based on research. Without mathematicians, where would we be today? We certainly wouldn’t have computers, skyscrapers or even basic knowledge of the natural world.
Thinking like a mathematician is truly a powerful skill that we can all learn, which is why Cambridge International added an element to its Cambridge Primary and Lower Secondary Mathematics curriculum frameworks to help students do just that. It is called Thinking and Working Mathematically (TWM).
If you are not familiar with TWM, then you can check out our previous blog on the 8 TWM characteristics. For those of you who would like to jump in and see what these activities look like in practice, you’re in luck! In this blog, you’ll find activities from our primary mathematics series Stages 1 and 2. Each one helps students practise and develop TWM skills. Download them below and try them with your class.
Specialising
Download specialising worksheet
Learner’s book 2, project 5
This task develops learners’ awareness of the structure of the 100 square and challenges them to both add and subtract tens and ones from the numbers within it. The final part of the activity gives learners the opportunity to specialise (TWM.01) as they find examples that satisfy the criteria. This project relates to learning objective 2Ni.04 (Estimate, add, subtract whole numbers with up to two digits (no regrouping of ones or tens).
Possible approach
For these activities, first make sure that there are no 100 squares displayed around the classroom. Introduce the task by showing the whole class a small section of a 100 square with all of the numbers filled in.
Ask learners to talk to their partner about the patterns that they notice. When learners share their ideas with the class, prompt them to ask any questions that they might have about the structure of a 100 square. Display a small section of a 100 square with missing numbers, and provide time for learners to discuss this with their partners. Can anyone explain how they know which numbers are missing? What patterns in the 100 square did you use to find the missing numbers?
Give learners some time to work together in pairs. Encourage them to discuss their ideas with each other, explaining how they know what the missing numbers will be. Bring the class back together for a mini plenary and prompt learners to explain how the missing numbers can be found in different ways.
Finally, provide time for learners to work together on the last square. If any learners do not notice the special property, work with them to help them see that the number in the top corner has reversed digits from the number in the bottom corner. For the plenary, ask learners to share the grids they found with this property. What do all of these grids have in common? Some learners might notice that the digit reversal happens in grids where the diagonal is the same diagonal as the 100 square (e.g. with 11, 22, 33, 44, 55 as the diagonal). Why does this happen?
Key questions
- How do you know which number will go in that square?
- What patterns do you notice?
- Why are there lots of numbers ending with the same digit or starting with the same digit?
Possible support
Some learners will benefit from having a 100 square in front of them and you asking them to find particular numbers in the 100 square. If learners need extra support, prompt them to count on or back from 1 or 10 and then ask if they can see a quicker way of working that out. For example, learners can count on ten to work out 27 + 10 = 37 and then they might notice that the tens digit has increased by one, and the new number is one row down in the 100 square.
Possible extension
Encourage learners to draw a new ‘square’ from 1 to 99 with only nine numbers going across in each row instead of ten. What patterns do they notice now?
Characterising and classifying
Download the characterising and classifying worksheet
Learner’s book 1, exercise 15.1
Ask learners to work out their age in years and months. Learners usually know how many years old they are. They will need to work out how many months they have been alive.
Learners will be characterising their age, putting it into the same format as Zara’s so that they can compare. We then ask learners to compare with Zara, who is 6 years and 3 months old, classifying themselves as older or younger than her or perhaps even the same age as her. Some learners may be able to extend the classifying to draw up a list of those who are younger than Zara and those who are older. They will be generalising the repeating pattern of the months of the year and specialising to find out their own age.
Learners will also be characterising their age, putting it into the same format as Zara’s so that they can compare. We then ask learners to compare with Zara, who is 6 years and 3 months old, classifying themselves as older or younger than her or perhaps even the same age as her. Some learners may be able to extend the classifying to draw up a list of those who are younger than Zara and those who are older.
Conjecturing and convincing
Download the conjecturing and convincing worksheet
Learner’s book 1, project 1
This task provides opportunities for learners to develop language to describe and compare (in particular) length and thickness, and to find different ways of measuring these two attributes. Prompt learners to make conjectures before the activity begins. They can ten develop their reasoning skills in order to convince others.
Possible approach
Begin by inviting learners to create their own snake using their choice of materials (within reason!). You may wish to gather a selection of paper, card, ribbon, dough, pipe cleaners, cubes, glue, tape, etc. for this purpose.
Try not to say too much by way of introduction so that learners have the freedom to decide how their snake will look and to create it in whatever way they choose. You may wish to give them some time to consider how they will make it before you give them the materials.
It might be worth asking a few learners to share their plans with the whole class before everyone begins the practical aspect of the task, as this may support those who are struggling for ideas.
As the learners complete their snakes, ask them to sit in pairs or small groups. Invite each learner to tell the others about their snake, then pose the questions in the learner’s book:
- What is the same about your snakes?
- What is different?
- Who has made the longest snake?
- Who has made the shortest snake?
- How do you know?
- Whose snake is widest?
- Whose snake is thinnest?
- How do you know?
- What else could you say about your snakes?
As you circulate round the room, look out for those learners who are justifying their conjectures, for example, by carefully lining up snakes to compare lengths, or by using cubes to measure lengths of individual snakes.
In the plenary, you could ask some learners to share their ways of convincing others. You could write up their justifications and display them in the classroom along with the snakes.
Key questions
- Tell me about your snake
- How do you know which one is longest or shortest?
- How would you measure the length/thickness?
Possible support
Some learners may need help deciding how to determine whether one snake is longer/thicker than another, particularly if the two are quite similar. If this is the case, you could invite another learner or group to show them what they have found.
Possible extension
You could challenge learners to line up the snakes in order of length/thickness. You could also invite them to make a longer/shorter/thinner/thicker snake than any in the set so far.
Classifying
Download the classifying worksheet
Learner’s book 1: exercise 14, question 5
This task provides opportunities for learners to practise using common words to describe data, including ‘more’, ‘less’, ‘most’ or ‘least’. Learners will be characterising and classifying as they identify and sort animals to find out how many different pets there are.
Lists and tables can be confusing and some learners may not remember which is which. However, if you write a long list with the class and link the beginning sound of list and long, they sound the same.
This will be remembered, especially if you make a long list with fun items that the class can relate to. Block graphs could be made using actual blocks before the class see ones printed. This will reinforce the word ‘block’ when being asked to complete a block graph. Always make connections between the words and real-life examples that learners can relate to.
Block graph: a graph that is made using blocks, with each block represents the same amount of something
List: a record of short pieces of information usually arranged one below the other so that they can be read easily or counted
Table: an arrangement of facts and numbers in rows or blocks
Critiquing and improving
Download the critiquing and improving worksheet
Learner’s book 2, exercise 1.3
Zara asks learners whether it is always, sometimes or never true that they only need to look at tens of numbers. Zara is really offering learners three slightly different conjectures.
Learners will need to investigate comparing and ordering numbers to decide which, if any, of the conjectures they agree with. By trying to convince themselves that one of the conjectures is true, learners may discover that the conjecture is incorrect. This is critiquing and improving.
When learners have reached a conclusion, ask them to explain how they know they are correct. This will give you the opportunity to assess what the learners think is enough to convince themselves and you.
More Thinking and Working Mathematically activities
If you have enjoyed these activities, you can find the more in our Cambridge Primary and Lower Secondary Mathematics resources.
To find out how you can help your students think like mathematicians, go to our hub page and check out our learner’s books, workbooks and teacher’s resources.
