In this unit five-based bead strings and number lines are used to solve addition and subtraction problems. The aim is to get students that use an early additive strategy to solve problems using a tidy number strategy with 10.
There are several things happening in this unit. All of them are aimed at enabling students to become more fluent in number.
The students need to realise that making a 10 is a good strategy for solving addition problems. This strategy is reinforced by the use of bead strings and the number line, so students need to understand how these representations work, and see their relevance for addition and subtraction work.
It is important that the students gradually learn to work without the bead strings and number line, so they are encouraged to ‘image’ these objects. Instead of actually using the devices they should start to think about what is happening in their heads. The next stage is for these number facts to become quickly recalled. This will take a reasonable amount of practice for most students. In the process, students are exposed to problems in context and finally they are given examples of their own to work on.
This unit is important in at least two ways for later work in mathematics in school, tertiary studies and even life itself. First, number is at the base of many ideas in mathematics so it is important to be fluent in addition and subtraction and to have strategies for carrying out these processes. Second, devices like the number line are not just useful to understand about addition and subtraction. Number lines are used extensively in co-ordinate geometry where two perpendicular number lines are used as axes. In this situation they enable us to visualise quite complicated functions. So even at this early stage in school, students are developing skills that will be useful throughout their school life as well as ideas that will grow into powerful and deep mathematics.
Note the following useful prior knowledge:
Over the next three days the aim is to slowly remove the number lines and bead strings and encourage students to visualise what would happen on the bead string or bead frame. This is called imaging.
Begin by using a bead string 1-20 coloured in 5’s like this.
This session is to use the number line (Copymaster 3) and bead string to solve problems and then the number lines and bead strings are taken away to encourage students to start imaging.
Freda the flea starts on 9 and hops forward 7 more spaces. Where does she end up?
Ask a couple of students to take the number line and pegs away and work out the answer. Ask the students remaining to visulaise what the others will be doing on the number line. The following questions may prompt the students to image the number line.
Where did the flea start?
How far does the flea have to go to get to 10?
Ask the students who took the number line away to share what they did to solve the problem.
Repeat with other problems. The following characters could be used to create similar story problems: Kev the kangaroo, Gala the grasshopper or Freckles the frog.
Encourage the students to visualise what they would do on either the number line or bead string. Extend some of the problems to numbers beyond 20.
The following types of problems will continue to challenge the students further.
Greg the grasshopper jumps 4 more spaces and ends up on 10. What number did he start on?
Frances the frog starts on 3 and jumps along the number line and ends up on 8. How many spaces did she go?
In this session, problems are placed on the top of a large sheet of paper. Students move around each bus stop, solving the problem. They record their working on each sheet.
Warm up with some whole class problems like the ones that have been shared in the previous sessions. Get students to talk to their neighbour and share how they worked out the answer. Record the different ways students solved the problem by writing it on the board.
Place each of the problems from Copymaster 4 on to a large piece of paper. Place the sheets around the room. Students can either rotate around the bus stops in pairs randomly or in a sequence to solve each problem. They are to show their thinking on the large sheet of paper.
Use this session to share the solutions students came up with for each of the bus stop problems. Encourage students to act out the problems where appropriate and to remodel their answers on the number lines or bead strings.