This unit begins with Freudenthal’s (1983) annihilation model for demonstrating the addition and subtraction of integers then goes on to introduce other representations. It is designed for students working at Stage 7 of the Number Framework who are able to choose appropriately from a broad range of mental strategies to estimate answers and solve addition and subtraction.
This unit of work is useful for students working at stage 7 of the Number Framework, Advanced Multiplicative. Students at this stage select from a broad range of strategies to solve addition problems with decimals, subdividing and recombining numbers to simplify problems.
Although students at this stage will be very familiar with the number line as a model for addition and subtraction this unit does not promote the use of the number line as it is considered problematic as a model to show subtraction of negative numbers.
The key teaching points are:
Getting started
Begin with a true/false conjecture on board for students to discuss. Ask students:
Do you think the statement is true or false and why? 1 – ^{-}4 = 5Ask students to discuss their ideas with a learning partner. Listen for student explanations. Discuss the idea of positive and negative numbers; introduce the term integer if it does not arise in the discussion.
Ask students to explain what is happening in the equation. Ask how this equation could be demonstrated with equipment.
3 – 4 =
Discuss the answer and how they used the equipment to model it.
Repeat this process with the following:
^{-}3 + 2 =
4 –^{ -}1 =
^{-} 5 + ^{-}2 =
^{-}3 – 2 = ^{-}1
Exploring
Over the next few days introduce the students to a selection of other models and contexts for addition and subtraction of integers from the list below, students could if they wished use the double coloured counters to model some of the calculations. As they work through tasks, add any new ideas to the modelling book.
Reflecting
3 –^{ -}12 =^{ -}15
Ask the students to decide whether this conjecture is true or false and to explain their thinking to their learning partner.
Revisit the summary of ideas where students have recorded what they have noticed about adding and subtracting integers. Ask students to work with their learning partner to develop some guidelines related to adding and subtracting integers for another classroom, for example adding a negative number is just like…, subtracting a negative number is like… or introduce Alistair McIntosh’s thinkboard, show an equation in the middle and model completing the sections.
Give students an equation to solve on their own thinkboard or with their learning partner. These could be differentiated in relation to student abilities.
Family and Whanau,
This week we have been investigating positive and negative integers. Your child will be bringing home a game to share with you as a way to practice adding and subtracting integers. If they get stuck they will be able to use a diagram of canceling out to solve the problem. Have fun!