Decimals and the Four Operations

Decimals and the four operations

Must Should Could Read decimal fractions as vulgar Change denomination from tenths to fractions using place value to hundredths, or hundredths to identify denomination thousandths when reading decimal fractions Add and subtract decimals using appropriate column methods Multiply decimals involving 1 or 2 Multiply 3-digit decimals using box digits using box method method Divide decimals

Key Words: Decimal fraction, vulgar fraction, denomination, integer, manyness, muchness, tenth, tenth tenth, hundredth, tenth hundredth, thousandth, units, tens, hundreds, thousands ratio, equivalent, equivalence, same value different appearance, + meaning “get ready to get some more”, – meaning “get ready to take some away”, ÷ meaning “look and wonder how many piles of …”, × meaning “love it, do it again lots of times” , arbitrarily name cups

Starters: How many? … How much? … write lots of amounts on board (eg 2 eggs, 3 fifths etc) and ask how many (count with finger) and how much (bracket amount with hands to emphasise muchness)

Ask student to write a manyness, such as “thirty one” then use a decimal point or zeros to make the denomination, such as “tenths”; 31 tenths is written as 3•1. This process can also be used by writing amounts on the board and asking students to answer the questions; How many? Of what denomination? How much?

Eg 45 000 how many? 45 of what denomination? Thousand how much? 45 thousand 0.045 how many? 45 of what denomination? Thousandths how much? 45 thousandths

Ask “how do you make a tenth?” - get a cup and tenth it “how do you make a hundredth?” – get a cup and hundredth it “how do you make a tenth tenth?” – get a cup and tenth it, then tenth every tenth …

Activities:

Use number cards (including a decimal point card) to get the students to form amounts by moving themselves around. Eg someone shouts out how many students move their cards into correct position to show how many, then someone shouts out a denomination, student with decimal point card or students with ‘zeros’ move into correct position.

To practise applying the four operations … use the maths table and resources table to act out maths stories (making sure the denominations are the same for addition, subtraction and division.) Practice process of ‘arbitrarily naming’ cups “tenths” or “hundredths” etc for the purposes of acting out story.

Mental arithmetic to practice logical language for multiplying (eg 2 × 3 = 6 … 2 tenths × 3 tenths = 6 tenths tenths … etc) Useful to use circle template with one side showing ‘tenths’ and the other side showing ‘hundredths’ to demonstrate what a ‘tenth tenth’ is.

Plenaries: 3a: Decimals and the four operations Ensure students understand that for decimal fractions, the decimal point is used to specify the denomination. Consider 243.2 … how many? 2,432 … of what denomination? Tenths … how much? 2,432 tenths. What do +, –, ×, ÷ mean? When do we need to use logical language? (eg when multiplier is not a positive integer) Which key idea is important when adding, subtracting or dividing (denomination)

Learning Framework Questions:

 What is a fraction?  Why use cups? How do you make a tenth? How do you make a tenth tenth?  Why is it important to act out making fractions accurately?  What does + mean? What does – mean? What does × mean? What does ÷ mean?  When is denomination important?  When and why do we need to use logical language?

Resources: Circle template showing tenths and hundredths Cups See MMMS Decimals worksheets

3a: Decimals and the four operations Possible Homeworks: Multiplication, Division, Addition and Subtraction methods – practice regularly!

Teaching Methods/Points: Place Value – taken from MMMS Paper 06

Place Value is NOT a ‘Big Idea in maths. Let me state why Place Value is not a Big Idea in maths and then explain it carefully. The Big Ideas in maths are theoretical constructs Kth that are judged to be sufficient to provide a framework for learning maths. They clarify the manner in which, for instance, 1/5 × 1/3 is the ‘same’ as 2 × 4. Once learnt, the Big Ideas serve to help pupils make sense of subsequent work. Place Value is not like that. It is a post hoc summary, a rough and ready description, of the possession of certain knowledge and skills; but it is not very useful. It is not widely applicable. It is certainly the case that in every number we can make reference to the ‘place value’ of its digits. But reference to ‘place value’ certainly does not help the novice to do anything worthwhile. It is a mistake to think that Place Value is a Big Idea in learning maths. And that mistaken belief has had a seriously damaging influence on the way maths is presented to novices. Let me clarify this with an example. I do not in some abstract way teach you ‘how to play hockey’ so that you learn the rules, learn how to hit the ball in a legal manner, how to dribble it, how to support the player with the ball and so on. What I do, in fact, is to teach you the rules, teach you to hit the ball in a legal manner, teach you to dribble it, teach you how to support the player with the ball and so on. When you can do these things in the context of a game I make a judgement, in consideration of your level of experience (possibly related to your age) that “You can play hockey”. My statement is a widely accepted shorthand that conveys, roughly, that you know the rules, can hit the ball in a legal manner, can dribble it, can support the player with the ball and so on. The fact that you can “play hockey” makes no impact on the quality of your game. It is useful only to the extent that I can convey something of your achievement without spending too much time conveying its nature. Place Value is like that. When you can write 2371 and 462 accurately in vertical format (I train you to do that) and you can look at 263 and, pointing appropriately, utter the words “there is two hundred here”; “there is six-ty here”; there is three here” (your ability to do that is based on the fact that ‘the logic of the language tells you the answer’: that is a Big Idea), then I say “You know Place Value”. The fact that you “know Place Value” makes no impact on the quality of your maths. It is useful only to the extent that I can convey something of your achievement without spending too much time conveying its nature.

Reading decimal fractions and decimal integers The decimal point serves the purpose of describing denomination. The figures tell us how many. So decimals can be read, for example, as follows; 21.3 how many? Two hundred and thirteen. Of what denomination? Tenths. How much? 213 tenths. Students learn to read the decimal point as “start thinking about tenths”; so the first column is tenths, the second column is hundredths, the third column is thousandths etc. The logic of the language also tells us that if the first column is tenths, then the second column is tenths tenths and the third column is tenths tenths tenths etc [to make a tenth tenth: get a cup and tenth it by cutting it into ten equal sized pieces, then tenth every tenth by cutting every tenth into ten equal sized pieces – this will make a hundred equal sized pieces called hundredths]. The decimal system [base 10] includes integers, of course. For the purposes of teaching the four operations with decimal integers, it is useful to identify 4 anomalies in the language of the base 10 system: Consider; 1 2 3 4 5 6 7 8 9 3a: Decimals and the four operations One Two Three Four Five Six Seven Eight Nine 10 20 30 40 50 60 70 80 90 Ten Twenty Thirty Four * Fifty Six Seven Eight * Nine ty ty ty ty ty 100 200 300 400 500 600 700 800 900 One Two Three Four Five Six Seven Eight Nine hundred hundred hundred hundred hundred hundred hundred hundred hundred 1000 2000 3000 4000 5000 6000 7000 8000 9000 One Two Three Four Five Six Seven Eight Nine thousand thousand thousand thousand thousand thousand thousand thousand thousand 1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000 9000000 One Two Three Four Five Six Seven Eight Nine million million million million million million million million million * Forty and eighty work phonetically although the spellings are slightly different Reading decimal integers follows the pattern of how many (eg 1, 2, 3, 4 etc) and of what denomination (hundred, thousand, million etc) at all times except with the values 10 [ten], 20 [twenty], 30 [thirty] and 50 [fifty]. For the purposes of adding, subtracting, multiplying and dividing, it is useful to refer to these values as 1ty, 2ty, 3ty and 5ty. So ty us written in symbolic form as ‘0’, hundred as ‘00’, ty ty as ‘0 0’ etc 4 ty ≡ 4 0 2 hundred ≡ 2 00 31 ty ≡ 31 0 465 hundred ≡ 465 00 12 thousand ≡ 12 000 16 ty ty ≡ 16 0 0 3 hundred ty ≡ 2 00 0 92 ty ≡ 92 0 etc Reading decimal fractions, and remembering that the decimal point is used to specify the denomination of the decimal fraction, the logic of the language works as follows;

4 ty tenths ≡ 4 . 0 31 hundred hundredths ≡ 31. 00 416 ty hundredths ≡ 41.6 0 37 ty ty tenths ≡ 37 0. 0 4 thousand tenths tenths ≡ 4 0.00 68 ty hundred thousandths ≡ 68. 0 00

Column methods for adding and subtracting decimals Remember that + means “get ready to get some more” and – means “get ready to take some away”. You can only add and subtract decimals if the denominations are the same. Changing the denomination of a decimal value is easy; for example, 2.3 ≡ 2.30 ≡ 2.300 etc (23 tenths ≡ 230 hundredths ≡ 2300 thousandths) Use the column method (and ensure the denominations are the same) as follows; 2.3 + 3.04 14.52 – 3.4 5 – 2.7

2 . 30 14 . 52 5 . 0 + 3 . 04 - 3 . 4 0 - 2 . 7 5 . 34 11 . 12 2 . 3 Division methods for decimals Remember that ÷ means “look and wonder how many piles”. You can only divide decimals if the denominations are the same. Use division methods as follows; 2.4 ÷ 0.2 3.54 ÷ 0.4 0.3 ÷ 32

200 120 34 5 3

3a: Decimals and the four operations Multiplication (love it, do it again lots of times) and Logical Language with decimals Remember that × means “love it, do it again lots of times” and can be demonstrated with students by asking them to perform actions a certain number of times; Eg “Blink three times … now blink four times … now blink 2 thirds times … now blink 2x times … etc” Multiplication makes sense (in real life) when the multiplier is a positive whole number, ie each repetition can be counted with the finger as “once, twice, three times, four times …” If the multiplier is a muchness, then the maths can no longer be acted out. So, this is when the students use the key concept of “the logic of the language” to derive the solution. [refer to notes on multiplying fractions Arithmetic 2b] Consider;  2 thirds × 4 thirds = 8 thirds thirds [get a cup and third it then third every third ≡ ninths]  2 a × 5 b = 10 ab  3 ty × 6 ty = 18 ty ty (3 0 × 6 0 = 18 00)  2 tenths × 3 tenths = 6 tenths tenths (0.2 × 0.3 = 0.06)  2 hundred × 3 thousand = 6 hundred thousand (2 00 × 3 000 = 6 00 000)  7 ty × 4 tenths = 28 ty tenths (7 0 × 0.4 = 28. 0) To multiply 2 digit decimals together, use the box method to separate the denominations; 23 × 32 4.7 × 3.4 4.08 × 8.5

Decimals and the four operations Help Sheet + means “get ready to get some more” × means “love doing it, do it again lots of times” – means “get ready to remove some” ÷ means “look and wonder how many piles” Denomination can be described as, “the thing that you are dealing in” Decimal fractions use decimal points to tell you denomination. The figures tell you how many.

Eg 12.3 How many? One hundred and twenty three. Of what denomination? Tenths. How much? 123 tenths Decimal integers can be read as follows; 1 ≡ ‘one’ 10 ≡ ‘one ty’ 100 ≡ ‘one hundred’ 1000 ≡ ‘one thousand’ etc 2 ≡ ‘two’ 20 ≡ ‘two ty’ 200 ≡ ‘two hundred’ 2000 ≡ ‘two thousand’ etc etc (where ty refers to a pile of 10 cups, hundred refers to a pile of 100 cups and so on …) Addition, subtraction, multiplication and division methods with decimals You can only add and subtract decimals if the denominations are the same. Use the column method (and ensure the denominations are the same) as follows; 2.3 + 3.04 14.52 – 3.4 5 – 2.7

2 . 30 14 . 52 5 . 0 + 3 . 04 - 3 . 4 0 - 2 . 7 5 . 34 11 . 12 2 . 3 You can only divide decimals if the denominations are the same! Use division methods as follows; 2.4 ÷ 0.2 3.54 ÷ 0.4 0.3 ÷ 32

200 120 34 3a: Decimals and the four operations 5 3 You can only multiply in real life if the multiplier is a whole number (eg ×2 ×3 ×4 ×5 etc). If the multiplier is a decimal, then we use logical language. Examples of logical language: 2 tenths × 3 tenths = 6 tenths tenths [0.2 × 0.3 = 0.06] 2 ty × 9 hundredths = 18 ty hundredths [20 × 0.09 = 1.8 0] To multiply 2 digit decimals together, use the box method to separate the denominations;

23 × 32 4.7 × 3.4 4.08 × 8.5

3a: Decimals and the four operations