Rich tasks
- These are problem solving open-ended questions that require students to investigate number properties. The tasks can be given at the start of a topic and done as the students gain enough knowledge to complete them, inserted into a topic at an appropriate place, or given as homework, or extension.
- Digit reversal
- True or false
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1. Digit reversal
- This is a challenging investigation.
- Students might be able to see that if you multiply the tens digit, 3 with the tens digit 2 you get 6. If you multiply the units digits together 6 x 1 you also get 6. The same thing happens once the digits are reversed. It is possible to show this algebraically but this would be beyond most students at this level.
- Other examples are 48 x 42 = 2016, 32 x 46 = 1472.
2. True or false
- Encourage students to find exceptions to the rules if they can.
- The answers are:
a is always true. It is not true if you multiply 6 by an odd number.
b is always true.
c is only sometimes true. It is not true when the digits of the number being multiplied add to more than 9. The steps could be adapted so that it was always true – they are just more complicated.
Counter example 64 x 11 = 704 6 + 4 = 10 (1 hundred and 0 tens) You can’t put 10 in the middle but if you put 0 in the middle and add the 1 to the hundreds e.g add 100 to 600 to get 700 then it is true.
d is true except for the one case of (1 ÷ 1) ÷ 1 = 1 ÷ (1 ÷ 1).
You could show how this works using materials.
Multiples and factors
- Understand the terms multiple and factor and when it is useful to know how to find these.
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Teaching ideas
- Circle multiples
Ask the class to stand in a circle.
Choose someone to start and give them the number 4.
Ask them to go around the circle saying all the multiples of 4 (counting in 4s). Discuss the connections between counting in 4s, the 4 times table and multiples of 4.
- Factor sharing
Pegs and a pegboard or Multilink cubes or counters.
Give groups of students 30 pegs, Multilink cubes or counters.
Ask them to find as many ways as possible to share them into equal sized groups (1, 2, 3, 5, 6, 10, 15, 30).
Discuss that all of these numbers divide exactly into 30 and are called factors.
Teacher notes
- When finding factors students often forget about 1 and the number itself.
Multiplication strategies
- Understand and practise the use of each of these multiplication strategies.
Using basic facts, including estimating answers
Place value partitioning
Rounding and compensating
Using factors
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Teaching ideas
This link displays the Getting started on page 15.
This link displays a set of card dominoes for you to print out and cut up.
The dominoes have a selection of partitioned numbers and their equivalent unpartitioned number and rounded numbers and their equivalent whole numbers. Students play dominoes by matching the numbers one way or the other.
This link displays a sheet with some multiplications for students to estimate, which you can print to make cards. Students are asked to match up cards with the same answer.
Teacher notes
- All of the strategies rely on students understanding place value well.
You may need to revise place value. - Emphasise that the main strategies are the three given. Students may know other strategies that work in particular cases such as multiplying by 25 or 5 but it is important that their main understanding is of using basic facts, rounding and compensating, place value partitioning and factors.
- Emphasise the importance of estimating all answers first.
- Students need a lot of practice at recognising which strategy is most efficient for particular numbers. Encourage sharing of ways of doing this. Rounding and compensating is best used with numbers that can be easily rounded to the nearest 10 or 100. Place value partitioning can be used with any numbers but sometimes using rounding and compensating leads to fewer or easier mental calculations so might be more efficient.
- Sometimes there is no “best” strategy. Encourage pupils to develop their own strategies that work for them in these cases. What is efficient for one pupil might not be for another because of their number knowledge differences.
- Encourage students to understand and use a range of strategies as some students use the same strategy for all calculations.
Division strategies
- Understand and practise the use of each of the division strategies
Using basic facts, including to estimate answers
Place value partitioning
Rounding and compensating
Changing both numbers
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Teaching ideas
This link displays a sheet with some divisions for students to estimate, which you can print to make the cards. Students are asked to match up cards with the same answer.
This link displays a sheet with a set of 12 numbers to halve as quickly as possible, and a set of 12 numbers to third as quickly as possible. You could divide the class/group into halves and get them to race each other to halve all the numbers. Repeat this with thirding the second set of numbers.
Teacher notes
- Students often have trouble estimating the answers to divisions because they round the one or both numbers to the nearest 10 or 100 rather than looking for multiples of the divisor (“nice numbers”).
For example, 358 ÷ 7 is better rounded to 350 ÷ 7 than 360 ÷ 7 or 360 ÷ 10.
Choosing a strategy for multiplication and division
- A selection of problems for which students must choose an appropriate strategy to solve.
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Teaching ideas
This link displays a multiplication problem and a division problem.
Ask students to work them out the most efficient way they can. Discuss as a class/group which way is the most efficient and why.
Students could argue their point and lobby for their way. You could take a vote at the end. The problems lead to the calculations 18 x 8 and 108 ÷ 6.
Ask questions about which strategies are generally used with which numbers. Rounding and compensating is best used with numbers that can be easily rounded to the nearest 10 or 100 for example. Place value partitioning can be used with any numbers but sometimes using rounding and compensating leads to fewer or easier mental calculations so might be more efficient in some cases etc.
Teacher notes
Using all four operations
- A selection of problems for which students must use a selection of the four operations to solve.
- This section ends with some problem solving questions.
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Teaching ideas
This link displays several words that are common in problems, such as altogether, total, product, sum, times, shared,
in groups, difference.
Discuss which operation these words usually belong to but point out that there are exceptions.
Teacher notes
Interesting website
This website has a lot of potential maths that can be explored.
You could give it to students and ask them to summarise it or do some of their own calculations using the data given at the site.
https://www.science20.com/science_20/strange_climate_math_driving_better_pollution_walking-2888
