Rich tasks
- These are problem solving open-ended questions that require students to investigate special numbers. They 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.
- Investigating primes
- True or false
- End digits
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1. Investigating primes
- There are many interesting facts and theories to investigate about prime numbers. Some are complex and it could be a good idea to check what students are planning before they start.
- Some sites to start with are:
www.ducksters.com/kidsmath/prime_numbers.php
www.hugin.com.au/prime/index.htm
www.en.wikipedia.org/wiki/Prime_number
- Students could present their work in an interesting way to the rest of the class.
2. True or false
- This has been proved true up to very large numbers. It has never been proved true for ALL even numbers greater than 4.
- True.
3. End digits
- This is a good chance for students to use a spreadsheet to confirm their predictions.
- Encourage students to work out why the patterns are as they are.
Squares and square roots
- Know the squares of numbers up to 10 and their corresponding square roots.
- Be able to find the square or square root of a number using the calculator.
- Find areas and lengths of squares using squares and square roots.
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Teaching ideas
Displays the Getting started on page 71.
The second dot relates to square numbers.
Displays pictures of squares and asks students to find the area of the square. Displays other squares where the area is given and asks them to find the length of one side.
Discuss how 7 x 7 is written as 72 and why this might be.
Ask students about the number 1 and why it is a square number.
It is important to make the connection between squaring numbers and finding the area of a square.
In a large space, ask the class to think of an imaginary grid drawn on the ground. Get the students to stand on this imaginary grid to form a solid square of students using as many people in the class as possible.
Example 16 students could stand on the grid to make a square with 4 students along each side of the square.
How did they work out how to do it? If people are left over ask why this might be. To make sure students understand the connection between the number of people along each side, and the total number of people forming the square, ask them if there are any other size squares that they could make if they used different numbers of students.
You could extend this by asking the class to find the largest square that could be made using all the students in the school.
Teacher notes
- Point out that squaring and finding the square root undo each other (are inverse operations like multiplying and dividing).
- Check that the students’ calculators have a square root button. Students can solve problems without the square root button, but it is helpful for checking answers.
- Students should recognise the first 10 or 12 square numbers.
- Different calculators may have different buttons for finding squares and square roots. Make sure students are familiar with these.
Cubes and cube roots
- Use the calculator to find the cube and cube root of a number.
- Find volumes and lengths of cuboids using cubes and cube roots.
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Teaching ideas
This link displays pictures of cubes and asks students to find the volume of the cube.
It then displays other cubes where the volume is given and asks them to find the length of one side.
It is important to make the link between cubes and cube roots and the volume of a cube and finding the length of the side given the volume of a cube.
Discuss how 7 x 7 x 7 is written as 73.
You could use these as class discussions to introduce cubes and cube roots.
Teacher notes
- Point out that cubing and finding the cube root are inverse operations.
- Check that students’ calculators have a cube root button.
- Students should recognise the first 4 or 5 cube numbers.
- Different calculators may have different buttons for finding cubes and cube roots.
Prime numbers
- Understand and use prime numbers.
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Teaching ideas
If you have them, give peg boards and pegs out to groups of students.
You could use dotty arrays or counters.
Give each student 30 pegs.
Ask students to find out for which numbers between 1 and 30 they can make a rectangle (or square) where each side is at least 2. For example, you can make 10 with a rectangle of 2 by 5 or 5 by 2. Give a time limit or give different numbers to different groups of students to investigate.
Ask students to list the numbers that they couldn’t use to make a rectangle. They should end up with a list of the prime numbers from 0 to 30.
Ask students what is special about the number 1.
It isn’t a prime number but it is both a square and cube number.
It is the building block of our number system.
Ask students what is special about the number 2.
It is the only even prime number.
Discuss the reason why the multiples of any number are not prime numbers is that they have more than 2 factors.
You could use this as a class/group discussion.
https://nrich.maths.org/1150
http://www.hbmeyer.de/eratosiv.htm (Seive of Eratosthenes)
Teacher notes
- Students should learn the prime numbers below 30.
- Emphasise that prime numbers only have two factors, themselves and 1. They have no other factors.
- Remind students that 1 is not a prime number because it only has one factor, itself.
- 2 is a special number because it is the only even prime number. All other even numbers have the factor 2 plus themselves and 1 and so are not prime.
- No multiples of numbers are prime numbers because a multiple must have at least three factors – itself, 1 and the number of which it is a multiple.
Highest common factor and lowest common multiple
- Understand and use highest common multiple and lowest common factors.
- Understand factorials.
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Teaching ideas
This link displays a set of number cards to print and the instructions below.
Give one set of cards to each group.
Instructions
1. Deal all of the cards out to your group.
2. Write all of the factors of the number that is on each of your cards.
3. Make sets of cards that have a common factor.
4. Write down the cards in each set.
5. What is the highest common factor for each set?
The group that finds the most sets wins.
This link displays a set of number cards to print and the instructions below.
Give one set of cards to each group.
Instructions
1. Deal all of the cards out to your group.
2. Write the first ten multiples of the number that is on each of your cards. Make sets of cards that have a common multiple.
3. Write down the cards in each set.
4. What is the lowest common multiple for each set?
The group that finds the most sets is the winner.
You could work through this with the class/group.
It is a quick way of finding the HCF or LCM of a number.
Teacher notes
- Encourage students to have a systematic method for recording the factors of numbers to ensure that they do not miss any.
For example, 1 times the number, then try 2 times something, then try 3 times something etc.
(Getting started 4)