Base 2 Notation: 4.5
Supporting Materials
- Related Progression Points
- Developmental Overview of Numbers and Operations (PDF - 2.8Mb)
- Developmental Overview of Numeration: Base Ten and Place Value Properties (PDF - 28Kb)
Indicator of Progress
At this level, students express a base ten numeral as a base 2 numeral and vice versa, and are able to explain the meaning of the digits in a numeral expressed in base 2. For example, they know that the number which is written in base ten as 13 is written as 1101 in base 2 (or 11012) and they can explain that this is because 13 = 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20.
Before this, students will have good understanding of the way in which numbers are conventionally written in our base ten place value system. They will understand how powers of ten give the place values, why the largest digit that can be in any place value column is 9, and be able to use expanded notation (e.g. 4507 = 4 × 103 + 5 × 102 + 0 × 102 + 7 × 100).
Learning about the base 2 system is important because this is the system used by computers.
Notes:
- Base 2 notation is also called binary notation. We use the term base 2 here to emphasise the parallels with base ten.
- Teachers must be clear in both writing and speech about which system is being used. Here we are writing the word ‘ten’ instead of using the numeral 10 for clarity (because the numeral 10 is ‘two’ in the base 2 system). We can use the numeral 2 because it is not ambiguous in either system. The word ‘two’ can also be used.
Illustration 1: Confusing base 2 with base ten
Students who are successful will know that 110012 means 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 and that this is equivalent to 16 + 8 + 1 = 25ten. Students who do not understand base 2 notation may think that 11001 2 means eleven thousand and one (base ten) or that it means
1 + 1 + 0 + 0 + 1 = 3. A common error is to see 102 and think that it means ten, rather than two.
Illustration 2: Understanding base 2 numeration
Students who understand base 2 numeration will:
- know that 112 must be read as ‘one one’ and not as ‘eleven’
- know that the place values increase from right to left, just as for base ten numbers, and that the smallest place value is 20. Students with only a partial understanding of base 2 notation may think that the smallest place value is 21
- be able to relate the place-value concepts encountered in base ten numbers to base 2 numbers
- recognise that a zero in a particular place-value column means an absence of that power of 2.
Successful students will have no difficulty understanding the subscript 10 (ten) or 2 as referring to the base. However it is likely that some students will at first confuse these with indices. For this reason it may be preferable to indicate the base by writing base 2 or base ten in brackets after the number, rather than using subscript, e.g. write 13 (base ten) or 1101 (base 2) rather than 1310 (or 13 ten) and 11012.
Important note: There is a difference between the number of objects we have, and the numeral we use to represent that number. Consider the set of objects below:
In base ten, which is what we are most used to, we would say that there are 13 objects, in base 2 we would say there are 1101 objects, in base 5 there are 23 objects, and in Roman numerals there are XIII objects. The actual number of objects does not change, but the numeral describing that number depends on what numeration system we have. Students need to appreciate that different numerals may mean the same number. It should also be noted that sometimes the word ‘number’ is used when ‘numeral’ is more correct.
Teaching Strategies
Learning to work with numbers represented in base 2 notation builds on students’ understanding of place value and in turn strengthens it. These activities focus on developing understanding of powers of 2 as the place values, rather than powers of ten, and on highlighting the significance of zero in a column.
Activity 1: Building numbers from powers of 2 is designed to focus students’ attention on powers of 2 as building blocks.
Activity 2: From base ten to base 2 builds on Activity 1, with students writing base ten numerals as base 2 numerals using a place value table.
Activity 3: From base 2 to base ten is designed to give students practice in writing base 2 numerals as base ten numerals by writing the value in base ten of each base 2 digit.
Activity 4: Writing coded messages gives students further practice in moving between base ten and base 2 representations, linking base 2 numbers with computer bytes.
Activity 1: Building numbers from powers of 2
For this activity, students will need a set of cards depicting counters in powers of 2 and a student resource sheet (containing the table below) to be used for recording work, if desired. Students use the cards as building blocks to form the base ten numbers in the left hand column of the table.
The resources are found here:
- Counter cards (PDF - 19Kb)
- Student resource sheet - Building numbers from powers of two (PDF - 27Kb)
In building a number, students can use each card either once or not at all. For example, in building the number 13, they will need to use the cards with 8, 4 and 1 counters. It is important that students recognise that 1 is a power of 2 (20 = 1).
This set of cards will allow students to build all the whole numbers from 1 to 63. The emphasis should be on how many (either one or zero) of each card is needed. Students write a 1 in the table if one of that card is needed and a zero if that card is not used. Many students will soon become sufficiently confident to fill in the table without needing to use the cards.
In the base 2 system, each card can be used only once or not at all (i.e. less than 2 times). Discuss the reason for this with students and make the link to the base ten system (with cards showing 1, 10, 100, 1000, and so on), where a ‘card’ would be able to be used up to 9 times, but not ten.
It is important that students make the link between base ten, where numbers are expressed in terms of ones, tens, hundreds, thousands, etc. (that is, powers of ten), and base 2, where numbers are expressed in terms of ones, twos, fours, eights, etc. (that is, powers of 2). The ones and zeros they have written in their table are in fact an expression of each base ten numeral in base 2, for example, 57ten is 1110012.
Just as we do not write unnecessary zeros to the left of a whole number in base ten (for example, we do not write 0014 for 14), we do not normally write unnecessary zeros in base 2 numerals. For example, 52 is written as 101, not as 000101. However, we must write the 0 between the two ones because this shows us that there are no twos and tells us that the first 1 is in the fours (two squared) place value column.
Base 2 cards
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Number |
Thirty-twos |
Sixteens |
Eights |
Fours |
Twos |
Ones |
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2×2×2×2×2 |
2×2×2×2 |
2×2×2 |
2×2 |
2 |
1 |
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25 |
24 |
23 |
22 |
21 |
20 |
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5 |
0 |
0 |
0 |
1 |
0 |
1 |
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9 |
0 |
0 |
1 |
0 |
0 |
1 |
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15 |
0 |
0 |
1 |
1 |
1 |
1 |
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57 |
1 |
1 |
1 |
0 |
0 |
1 |
Activity 2: From base ten to base 2
The next activity follows on directly from Activity 1. Again it is important that students recognise that base 2 numerals are using powers of 2 as the place values. Students should attempt this activity without using the cards, but the cards should be available for those students who still need them. They record ones or zeros in each place value column as before, then write the base two numeral without the leading zeros in the right-hand column. A student resource sheet is provided here: From base 10 to base 2 (PDF - 33Kb).
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Base ten Number |
Sixty-fours |
Thirty-twos |
Sixteens |
Eights |
Fours |
Twos |
Ones |
Base 2 |
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2×2×2×2×2×2 |
2×2×2×2×2 |
2×2×2×2 |
2×2×2 |
2×2 |
2 |
1 |
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26 |
25 |
24 |
23 |
22 |
21 |
20 |
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1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
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2 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
10 |
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3 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
11 |
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4 |
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5 |
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etc |
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20 |
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31 |
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35 |
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69 |
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100 |
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My age |
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My age in ten years’ time |
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My house number |
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Variation
Completing a table such as that above with the numbers in sequence 1 – 63 highlights the patterns in the base 2 system. Discuss with students how the pattern corresponds to the patterns in base ten counting.
Activity 3: From base 2 to base ten
This activity is designed to give students practice in renaming numbers from base 2 into base ten. It is important that they place the digits of the base 2 numeral into columns, starting at the ones column. This replicates what happens with base ten numerals.
Base 2 to base 10 (PDF - 32Kb).
|
Base 2 Number |
64 |
32 |
16 |
8 |
4 |
2 |
1 |
Calculation |
Base ten number |
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26 |
25 |
24 |
23 |
22 |
21 |
20 |
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100 |
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1 |
0 |
0 |
1 × 4 + 0 × 2 +0 × 1 = 4 |
4 |
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1101 |
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1 |
1 |
0 |
1 |
1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 13 |
13 |
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10111 |
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111000 |
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10001 |
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1001001 |
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1111111 |
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1000011 |
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1000000 |
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110011 |
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1110011 |
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1111001 |
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Activity 4: Writing coded messages
This activity has two main purposes:
- to make a link between the use of base 2 (binary) numbers in computers and
- to give students practice in converting between base 2 and base ten numerals.
Every keyboard character on a computer is represented by a number code, with a number from 0 to 127 (base ten), that is, from 0 to 1111111 (base 2). These codes are referred to as ASCII codes (American Standard Code for Information Interchange). List of the codes found here: ASCII codes (PDF - 42Kb).
Students can write a message in words, translate it into the ASCII code number, then in turn translate this into base 2 numbers. For example, the ASCII code for Hello Kim would be
72 101 108 108 111 32 75 105 109.
Turning this into base 2 numbers, the coded message becomes:
01001000 01100101 01101100 01101100 01101111 00100000 01001011 01101001 01101101. Notice that here the zeros are always written in front so that each base 2 number has exactly eight digits. This 8-digit base 2 number is called a byte. Students could count how many bytes are in their message, remembering that spaces, full stops, etc. all count as bytes. The message Hello Kim has 9 bytes. This gives meaning to the size of computer files, such as 150 kB (kilobytes) or 1.4 MB (megabytes). Students may also be interested to know that a kilobyte is not 1000 bytes, but 210 bytes, that is 1024 bytes.
Students enjoy this activity, provided the message is short and the activity is limited to just part of a lesson so that it does not become tedious.