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# Base 2 and complementary arithmetic

Keywords:
Achievement Objectives:

Achievement Objective: NA5-3: Understand operations on fractions, decimals, percentages, and integers.
AO elaboration and other teaching resources
Achievement Objective: NA5-2: Use prime numbers, common factors and multiples, and powers (including square roots).
AO elaboration and other teaching resources

Purpose:

These exercises, activities and games are designed for students to use independently or in small groups to practise number properties. Some involve investigation (see Related Resources) and may become longer and more involved tasks with subsequent recording/reporting. Typically an exercise is a 10 to 15 minute independent activity. While the involve operating with number, the contexts for the number operations in these exercises are not formally in the curriculum, and are therefore most suitable as ‘broadening’ or extension work with more able students.

Specific Learning Outcomes:

solve addition problems using complementary numbers
solve subtraction problems using complementary numbers
explore complementary numbers in base 2
explore how computers solve mathematical operations

Description of mathematics:

Addition and subtraction, AM (Stage 7)

Required Resource Materials:
sample electronic circuits, old computer CPU board
calculator
Activity:

### Prior Knowledge.

demonstrate strong understanding of place value
demonstrate an understanding of how traditional vertical algorithms work for addition and subtraction
understand powers and power notation
demonstrate a knowledge of an algorithm as a sequence of instructions

### Background:

These activities involve students in learning about using mathematics in base 2. This presupposes that they are already fluent (and understand) mathematics in base ten.

While base 2 is not formally in the curriculum, the activities provide an example of powerful extension work that can be used with more able, and gifted, students. The focus on how computers "do" maths in these exercises should be of interest to good mathematics students.

Exercise 1: Making Up

Asks students to choose from a selection of numbers those that add to 9, 99, 999 etc. The skill of making to 9 on a digit by digit basis is needed to explain how computers do arithmetic. This exercise aims to develop this skill and emphasises an "I notice…" assessment to help form new ideas.

Exercise 2: Make a Complement

Asks students to again practise ‘making to 9’, only this time the phrase ‘making the complement’ has been introduced. Later problems include writing numbers in expanded form, and looking at the generalisation of this, where different shapes are used for each digit; for example, represents a two digit number where both digits are different. Shapes rather than letters are used here as in another context, bc means b x c, and the confusion between this and the use of symbols to represent a two digit number, where both digits are different, is worth avoiding when students are only starting to learn algebra. The sections with symbols are extension work for students who are at stage 7.

The final problems get students to look for the complements of some unknown multi-digit numbers. These are again shown with shapes. Here students need to have come to the realisation that if a number and its complement add up to nine, then the missing number can be found by subtracting the written digit from 9; for example, the digit 3 can be described as having the complement 9 – 3. Shapes have been used to encourage students to develop this generalisation.

Showing a multi-digit number using this generalisation is also hard, as if regular brackets are used this can cause confusion with existing conventions that students are learning about; for example, (a + d)(b + c) meaning (a + d) x (b + c). Discussing this possible confusion between conventions is important, and getting the students to invent their own notation to show the complement of a 4 digit number like is a valuable exercise, as it inducts the students into the use of the language of mathematics and the way that mathematical notation and conventions develop.

Teaching lessons preceding this exercise could include:

• modelling with similar problems as suggested
• modelling to suggest generalisation of the procedure.

Understanding how the generalisation works is essential before progressing onto exercise 3.

### Exercise 3: Complementary Subtraction

Asks students to use the complement in subtraction problem. This exercise introduces students to the use of the complement to change a subtraction into an addition. This is introduced by example, though later students are asked to provide a conjecture about why the method works. For them to be successful with this, some students may need the clue that adding the complement can be shown in a number of ways; for example: 51 – 27 becomes 51 + 72, which can also be written as 51 + 99 – 27 (hence the answer is always 99 more than the original problem).

Teaching lessons preceding this exercise could include:

• modelling this procedure carefully

This is further skill building and promotes the recording of student views to assist forming learning.

Exercise 4: More Complementary Subtraction

Asks students to practice using the complementary addition strategy. It draws attention to the fact that the numbers needs to have the same columns (same number of columns and the same column values) for the method to work.

Teaching lessons preceding this exercise could include:

• modelling this procedure carefully

This is further skill building using base 10.

Exercise 5: Martian Maths 12 + 12  = 102

Asks students to explore base 2. This exercise introduces students to working base 2, and provides a pattern for students to follow when converting our base ten numbers to base two. A good activity to assist students develop an understanding of different bases is to introduce them to how our place value system uses powers of ten, where each additional column is one power of ten bigger (or smaller) than the last (so if the Martians with only 2 fingers invented a similar number system, what powers would be used to work out the value of each column?) Exploring the conversion of base ten to base two numbers, and vice versa, is also worthwhile.

The last questions introduce student to the idea that numbers in different bases needs to be signalled in some way to prevent confusion. Here the term convention may need to be explained before students attempt these. Different suggestions that students have come up with can be discussed before the formal mathematical notation is introduced.

Teaching lessons preceding this exercise could include:

• the concept of number systems in different bases

This activity can be formed by patterns but it should lead to the development of being able to read small binary numbers meaningfully.

Exercise 6: Complements in Base 2

Asks students to explore complements in Base 2. This is a simple but very important concept to be developed and mastered. The language of Logic Chart, Input, and Output is also important.

Teaching lessons preceding this exercise could include:

• an electronic demonstration of an inverter

Asks students to explore the concept of adding in base 2. A teaching session introducing students to this concept before they start the exercise would assist those students who have only previously worked in base ten. Having a good understanding of the process of adding in another base is important for the following exercises.

A demonstration of the AND gate would be meaningful to students but is more difficult because only the NAND gate is manufacturered because it is a much more useful device. See the Forrest Nim Engineer’s handbook for straightforward instruction.

Exercise 8: Computers, Multiplying

Asks students to explore multiplication wiht a computer loop. Before starting this activity, students may need to be led through the process of reading and understanding a simple computer program, and the process of constructing a table to check the logic process the computer is following.

Some features in the program to note:

• The line numbers at the start of each line
• The use of letters as the name of a memory store
• The use of a decision box in line 90 to create a repeated loop

Showing such a program as a flow diagram could also be useful to students.

Teaching lessons preceding this exercise could include:

• revision of the meaning of multiplication as repeated addition

Exercise 9: Computers, Dividing

Asks students to explore division in a computer loop. This exercise gets complicated in places, and may require support, as division is processed as a repeated subtraction, which in turn is completed as repeated additions of the complement.

Teaching lessons preceding this exercise could include:

• revision of the meaning of division as repeated subtraction

Exercise 10: Computers, Powers and other Functions

Asks students to investigate how computers do mathematical operations.  This exercise is a summary of the unit and suggests a project for the student to have an opportunity to publish learning in a chosen topic. This is an opportunity to assess the communication key competency.

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