1. Write a decimal number on the board (to two decimal places). 
2. Brainstorm different ways to represent it.
   For example 3.81 could be expressed as;

• Three point eight one
• 3 + 0.8 + 0.01
• (3 x 1) + (8 x 0.1) + (1 x 0.01)
• 381 / 100
• 4 – 0.19
• Three and eighty one hundredths
• 7.62 / 2

3. Give each student a decimal number and ask them to represent in as many ways as they can.  Depending on the level of individual students they could be to one, two, or three decimal places.  Less confident students could work in pairs.
4. Students could present their work on a sheet of A3 paper to be displayed on the wall.
5. As the students work encourage them to use a variety of approaches (not just a whole list of addition sums).

Getting Started

The purpose of this session is to explore the range of strategies already used by students to solve addition and subtraction problems. This lesson will enable you to evaluate which strategies need to be focused on in greater depth. In turn, you will identify students in your group as "expert" in particular strategies. There are two problems given as examples for exploration. You may want to use further examples of your own. Consider adapting the contexts used in these problems to further engage your students.

1. Problem 1: Sarah has $288 in the bank. She then deposits her pay for $127 from her part time job at a cafe. How much does she have now?
   Ask the students to work out the answer in their heads or by recording in some way. Give the students plenty of thinking and recording time. Ask the students to share their solutions and how they solved the problem with a peer. The following are possible responses:
   
   Place value (mentally, possibly with the support of equations on an empty number line):
   288 + 127 is just like 288 + 100 +20 +7. So that’s 388… 408… 415.
   
   Tidy numbers (mentally, possibly with the support of equations on an empty number line):
   If I tidy 288 to 300 it would be easier. To do that I need to add 12 to 288, which means I have to take 12 off the 127. So that’s 300 plus 115. 
   
   Algorithm (usually written):
   Students may visualise or record a written algorithm like this:
   
   Understanding is revealed by the language used to describe the strategy, such as, “8 plus 7 equals 15. I wrote 5 in the ones place and carried the extra ten into the tens place because 15 is made up of 5 and 10 and I can only record one digit in the ones place.”
   
   As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the correct strategies that are elicited, recording the strategies to reflect upon later in the unit on the class T chart (under the addition heading). You might also ask students to model how their strategies work with place value materials. Note that folding back to justify a strategy is often more difficult than performing the strategy.
    
2. Problem 2 Sarah has $466 in her bank account and spends $178 on a concert ticket. How much money does she have left in her bank account?
   Ask students to solve the problem mentally, giving them plenty of thinking and recording time. Then ask students to share their solutions and how they solved the problem with a peer. Possible responses are:
   
   Reversibility (adding on to find the difference with recording to ease memory load):
   $466 - $178 is the same as saying how much do you need to add to $178 to get $466. $178 plus $22 makes $200, plus $200 more makes $400 plus $66 makes $466. If you add up $22 plus $200 plus $66 you get $288.
   
   
   Subtracting a tidy number then compensating 
   $466 - $200 = $266. I took off $200 instead of $178 so I need to ‘pay back’ $22. $266 + $22 = $288.
   
   Equal additions:
   You round the $178 to $200 by adding $22. $466 - $200 is $266. Then you put on $22 to keep the difference the same, so it’s $288.
   
   Algorithm (usually written):
   Students may visualise or record a written algorithm like this:
   
   
   Understanding is revealed by the language used to describe the strategy, such as, “6 minus 8 doesn’t work (ignoring integers) so I changed one ten from the tens column into ones to make 16. I wrote 16 in the ones place and took one ten off in the tens place…”
   
   As different strategies arise, ask the students to explain why they chose to solve the problem in that way. Accept all the correct strategies that are elicited, and justified, recording the strategies to reflect upon later in the unit (under the ‘subtraction’ heading on the class T chart). You might also ask students to model how their strategies work with place value materials. Note that folding back to justify a strategy is often more difficult than performing the strategy.
   
   Subtraction strategies tend to be more difficult to control than addition strategies, given comparable numbers. Look out for students compensating the wrong way (taking more off) in the tidy number strategy and making errors when using the algorithm.
   
   Use your observations to plan for your subsequent teaching from the exploring section outlined below.

Exploring

Over the next two to three days, explore different strategies for addition and subtraction of whole numbers. Give the strategies a name so students can tell others which strategy they are preferencing for a given problem. Highlight when certain strategies are most efficient, for example, In the problem 357 + 189 tidy numbers would be a useful strategy because 189 is close to 200.

Follow a similar lesson structure each day to the introductory session, with students sharing their solutions to the initial questions and discussing why these questions lend themselves to the strategy being explicitly taught. Conclude each session by asking students to make statements about when the strategy would be most useful and why the certain problem is appropriate, e.g., tidy numbers when one number is close to 100 or 1000, standard place value (hundreds, tens, and ones) when no renaming is needed and reversibility when neither of the other two numbers are easy for subtraction. It is important to record examples of strategies as they will be used for reflection at the end of the unit. Some strategies may require more teaching time, greater use of materials, and more scaffolded and individualised teaching. Ensure that students who demonstrate proficiency with the strategies early on in each session have adequate opportunities for practice, extension, and supporting their peers (if appropriate).

The questions provided are intended as examples for the promotion of the identified strategies. If the students are not secure with a strategy you may need to make up some of your own questions to address student needs. Consider adapting the contexts reflected in these problems to further engage your students.

Tidy numbers then compensating

1. Room 9 are selling muesli bars at lunchtime to raise money for their camp. They had 434 at the beginning of lunchtime and sold 179, how many did they have left to sell?
   The tidy numbers strategy involves rounding a number in a question to make the question easier to solve. In the above question, 179 can be rounded to either 180 (by adding 1), or 200 (by adding 21). If 200 is subtracted (434 – 200 = 234) then the result is 21 less than the answer. 234 + 21 = 255. An empty number line shows this strategy clearly:
   
   For addition questions, one addend can be tidied by taking from the other addend. Alternatively, both addends might be tidied, and compensation used to adjust for the tidying.
2. The following questions can be used to elicit discussion about the strategy.
   • What tidy number could you use that is close to one of the numbers in the problem?
   • What do you need to do to the other number if you tidy up this number? Why?
   • Why is this strategy useful for this problem?
   • What knowledge helps you to solve a problem like this?
3. If the students do not understand the tidy numbers concept, use place value equipment (place value blocks, place value people, beaNZ) and/or an empty number line to model the problems physically. Students should be encouraged to record equations and number lines to track their thinking and reduce load on working memory.
4. Use the following questions for further practice if required:
   • Ariana has scored 739 runs for her cricket club this season. Last season she scored 294, how many did she score in total in the last two seasons?
     Shifting six between the addends gives 739 + 294 = 733 + 300 = 1033.
   • Nigel sold his guitar for $587 and his amp for $395. How much money did he make from both sales?
     Rounding 587 to 600 and 395 to 400, then compensating gives 587 + 395 = 600 + 400 – 13 – 5 = $982.
   • Farmer Samsoni has 1623 sheep, and he sells 898 sheep at the local sale. How many sheep does he have left?
     Rounding 898 to 900 gives 1623 – 898 = 1623 – 900 + 2 = 725
   • Other examples might be:
     568 + 392
     661 - 393
     1287 + 589
     1432 - 596
5. Note that the problems posed here are using a tidying up strategy rather than tidying down i.e. 103 down to 100 as in these situations place value tends to be a more useful strategy.

Place Value (written algorithm)

1. For the community hangi, 356 potatoes are peeled and there are 233 left to be peeled. How many potatoes will there be altogether?
   The place value strategy involves adding the ones, tens, hundreds, and so on. In the above problem:
   300 + 200 is added
   Then 50 + 30
   And finally 6 + 3
   As an algorithm the calculation is represented as:
   
2. The following questions can be used to elicit discussion about the strategy:
   • How can you use your knowledge of place value to solve this problem?
   • Why is this strategy useful for this problem?
3. If the students do not understand the strategy, use place value equipment to show the problems physically. Students will find it useful to record and keep track of their thinking .
4. Use the following questions for further practice if required:
   • Zac has $498 available on his eftpos card and spends $243 on a new BMX bike, how much money does he have left?
   • Other problems might be:
     3221 + 348
     4886 - 1654
     613 + 372
     784 – 473

Reversibility (adding one rather than subtracting to find the difference)

1. Faloa is helping his Mum build a path. There were 438 bricks in the pile and they used 169 of them yesterday. How many bricks have they got left to lay today?
   The reversibility strategy involves turning a subtraction problem into an addition one so the problem above becomes:
   169 + ? = 438
   Using tidy numbers to solve the problem makes calculation easier:
    
   Or
   
2. The following questions can be used to elicit discussion about the strategy:
   • How could we think of this as an addition equation?
   • What do you need to add to make it easier to solve?
   • How can you keep track of how much you have added altogether?
   • Why is this strategy useful for this problem?
   • What knowledge helps you to solve a problem like this?
3. If the students do not understand the concept, use a number line to show the problems graphically. Students will find it useful to record and keep track of their thinking using a number line.
4. Use the following questions for further practice if required:
   • At the mail sorting office there were 547 letters to be sorted, 268 of these were distributed to private boxes, how many were left to be delivered?
   • The school library has a total collection of 1034 books and 459 are issued at the moment, how many are on the shelves?
   • Other problems might be:
     628 - 342
     537 - 261
     742 - 353
     1521 - 754
     1762 - 968
     1656 - 867

Reflecting

As a conclusion to the week’s work, give the students the following five problems to solve. Pose the problems in contexts that are relevant and engaging for your students. Ask students to discuss, with a peer or in small groups, which strategy they think will be most useful for each problem and justify their view. For some problems many strategies may be equally efficient. After students have solved the problems, engage in discussion about the effectiveness of their selected strategies.

Some students may have a favourite strategy that they use, sometimes to the exclusion of all others. The best approach is to pose problems where the preferred strategy may not be the most efficient. For example, 289 + 748 is most suited to using tidy numbers and compensation.

Problems for discussion
1318 - 747
763 - 194
433 + 452
1993 + 639
4729 - 1318

You might also like to also try some problems with more than two numbers in them, such as:
721 – 373 - 89
663 - 61 - 88
63 + 422 + 49
42 + 781 + 121
84 + 343 - 89

1. Discuss the different strategies explored during the week and ask students to explain in their own words what types of problem each strategy would be useful for solving, and what types of problem each strategy would not be useful for solving. Ask the students to draw a strategy ‘from a hat’ and write questions specific to that strategy for a partner.
2. Conclude the unit by showing the students the questions asked in the initial session again. Discuss whether they would solve the problems in a different way now, and how their thinking has evolved. With a partner, students could create a short presentation demonstrating how they would solve one addition and one subtraction problem, naming the strategies used, and justifying the use of their chosen strategies.

This unit includes five sessions, each with a number of learning activities. These sessions could be taught in succession over a number of days to provide a concentrated focus on building place value knowledge. Alternatively, selected sessions or activities could be used to support place value understanding, as part of small group teaching sessions, while other students work on solving number problems. You could use the activities as an introduction to a particular number concept, and then use the remaining time to revisit and consolidate knowledge of these concepts, with specific students, in response to their demonstrated knowledge.

Session 1 

SLOs:

• Appreciate and understand the size of one million.
• Understand and describe the multiplicative nature of our base ten number system.

Activity 1

1. Begin the lesson by writing a 4, 5 or 6 digit number on the place value houses, using an erasable whiteboard marker. Include numbers that have one or more zero as a placeholder. Have students read the number to themselves then read it aloud to a partner. Address any gaps in knowledge that arise around correctly reading out the names of these numbers.
2. Together, select a digit to remove and erase this from the chart. Ask students to tell a partner how much was removed (e.g. five hundred, three thousand). 
   What was the value of the digit that was removed?
   Leave the empty space and see whether the students tell you that a zero must be written there. If not, do this and ask the students why the zero is needed.
3. Introduce the game Zap (Book 4 Teaching Number Knowledge, p26) to the class.
4. Distribute calculators and have students play Zap in pairs. Have them record the numbers made and the amounts removed as they take their turns.

Activity 2

1. Show the dot image of 10 000 from Copymaster 1.
   Ask:
   How many dots are here and how do you know? 
   Repeat with the image of 100 000 dots. Pose the question:
   What would 1 million dots look like?
   Have students discuss this in pairs and allow time for them to write down their ideas.
2. Read How much is a million? by David M. Schwartz and discuss. You may wish to do this activity linked to the book.
3. Have students share their ideas of how much one million is. Display 10 pages of 100 000 dots, asking "Is this one million? How do you know?”
4. Draw one dot on the class chart and write beside it 1 x 10 = 10. Draw ten dots beside the expression.
5. Begin listing this pattern, having the students complete each equation. 1 x 10 = 10 10 x 10 = (100) 100 x 10 = (1000) 1000 x 10 = (10 000) 10 000 x 10 = (100 000) 100 000 x 10 = (1 000 000)
6. Ask students what they notice about the notation 1 000 000 and record their ideas about “How much is a million?”. Ensure students notice that one million is one thousand lots of one thousand. You could make a ‘One Million Book’ in which children record their ideas in an ongoing way and in which are pasted copies of the 10 pages of 100 000 dots. This could also be done by small groups or pairs, with the final ‘One Million Book’ being collated by the teacher.
7. Introduce exponent notation, 10², 10³, 10⁴, 10⁵, 10⁶, explaining that this is another way (a shorthand way) of showing how many times ten is multiplied by itself. For example, write and show 10 x 10 x 10 = 1000, ten multiplied together three times is one thousand.

Activity 3

1. Have students plan a “One Million Investigation”. For example:
   • How long would my favourite book be if it had one million words?
   • How old would you be if you lived one million seconds?
   • How long would you have lived if you’d had one million breakfasts?
   • How long would it take you to take one million breaths?
   • How long would the line be if you lined up 1 million one centimetre cubes side by side? Which two real-world locations are this distance apart?

Ensure you set clear parameters for this investigation - such as method of presentation (e.g. act it out, created with a relevant digital tool, written, verbal, video, poster, PowerPoint). These investigations could also include the most interesting facts students have learnt about one million.  Consider organising students into groups with mixed levels of mathematical knowledge and confidence in order to encourage greater peer learning and scaffolding, and collaboration.

1. Have students present their findings to the class. Include these findings in the class ‘One Million book’ (developed in the previous session). Have students discuss how they could share what they have learned with their parents and whānau.

Session 2

SLOs:

• Recognise the repeating pattern of hundreds, tens, and ones in numbers up to, and beyond, one million.
• Demonstrate understanding of the powers of ten.

Activity 1 

1. Ask a student to write one million in numerals on the class chart, or ask students to do this independently (e.g. on a mini whiteboard, on working paper). Note the spaces between 3 digits (1 000 000) and ask for an explanation of this. Look for students to recognise that the hundreds, tens and ones naming structure is being repeated. 
2. Have students predict what the Millions place value house will look like, then display this.
   
   Write a range of 7, 8 and 9 digit numbers. Emphasise the hundreds, tens and ones naming in each house, and highlight how to correctly read numbers including zeros (e.g 703 206).
3. Write a 7-digit number and have students give some examples of things that are counted in millions in the real-world. For example, money ($3,000,000 might buy an expensive house), populations (there are over 5 million people in NZ), distances in space (it is 152 million km from earth the sun), years dating to the past (dinosaurs became extinct about 65 million years ago), and stars (there are 10 million stars in the giant star cluster Omega Centauri).
4. Highlight for your students that we use commas to separate the groups of three digits when we are writing amounts of money. However, spaces are used in other contexts (except when writing 1000).

Activity 2

Show the students Copymaster 2 and have them work in pairs to write down five interesting things they notice about this chart. Have them write some of the numbers themselves, making the connection as they do so between the exponent and the number of zeros. Share the discussion points with them.

Activity 3

Have the students play the game Powers or Bust (Copymaster 3).

Activity 4

Challenge students to write, in less than 200 words, a response to the question, “Our base ten numeration system is useful because….” Encourage students to consider relevant contexts in which they use the base ten system. Have the students share their responses. This should generate some valuable discussion. Use this opportunity to identify and address any place value misconceptions.

Session 3

SLOs:

• Understand what a decimal fraction is and recognise that they arise out of division.
• Correctly read and name whole numbers and decimals up to, and beyond, one million.

Activity 1

1. Begin the lesson reviewing the place value structure of whole numbers, using place value houses. Emphasise the repeated hundreds, tens and ones naming structure.  
2. Have students discuss what happens when you divide a big number by ten. (You may like to refer to their experiences playing Powers or Bust.)
   Write 1 000 000 on the PV house and discuss dividing this number by 10. It may be helpful to use the ten images of 100 000 dots (1 million) and demonstrate that one tenth of the ten images is one of them (100 000). Model repeated division by ten until one is reached. This can be powerfully demonstrated using the dot images from Copymaster 1.
3. Ask the students to discuss in pairs which number is the start of our whole number system (responses may include zero, one, ten). Establish that one is the number with which we begin. We either increase 1 by powers of ten (e.g. 10, 100, 1000, and so on) or decrease 1 by powers of ten (e.g. 10ths, 100ths, 1000ths, and so on).
   
    
4. Write 1 in the ones place on the place value houses. Have students discuss what happens when we move to the right of one, in other words if we divide, and keep on dividing. Make links to students’ knowledge of multiplying (e.g activity two in session one looked the following expressions: 1 x 10 = 10 10 x 10 = (100) 100 x 10 = (1000) 1000 x 10 = (10 000) 10 000 x 10 = (100 000) 100 000 x 10 = (1 000 000).
5. Record the students’ ideas. Listen for important misconceptions. For example “there is a decimal point next then it goes ones, tens, hundreds the other way” (an understanding of symmetry around the decimal point) or this may be expressed as “oneths, tenths, hundredths”. Alternatively no symmetry may be assumed. Some may suggest that the naming pattern continues unchanged hundreds, tens, ones.

Activity 2

1. Distribute a small ball of playdough, a plastic knife, chart paper and a pen to student pairs. Explain that the playdough represents "one" (like the one dot shown on Copymaster 1).
2. Challenge the students to work with the materials to show and record what happens when one is divided. This should reflect the place value pattern of numbers.
   Do not mention tenths, hundredths etc. at this point. Also explain that they may have to reshape their play dough ‘one’ to work with it more easily.
3. Observe the students' actions. Have them pair share their charts and explanations. Summarise what the students’ work has revealed and discuss this.
4. Model the activity for the class to see. Take the one ball and divide it into ten balls, then take one of those and repeat. Emphasise how quickly the size of the balls changes and highlight the magnitude of division by ten each time.
5. Comment that students may have made a ‘sausage’ with their ball and worked that way. The representation is the same.
6. Use a brightly coloured, round sticker and place this over the decimal point to emphasise its importance. Have students discuss in pairs and write an explanation to complete this statement:
   This very important dot is...
7. Have students share what they have written. If key ideas are not been highlighted by the students, emphasise:

• the decimal point is a convention that separates a whole unit (one) from parts of a unit (tenths)
• the importance of one as the start of our number system (the tenths place is tenths of ONE, and the tens place tens of ONES)
• that the numbers that follow the decimal point are called decimal fractions, which are a special kind of fraction with a tens base
• decimal fractions are the result of division.

Activity 3

1. Using the class/group chart, write decimal numbers to one or two places (in words). Have students come forward and write the numeral representation.
2. Reverse the process in step 1 above. Write a decimal number and have students correctly read the decimal number and write it in words. Listen for, and highlight correct reading, for example 0.43 is read as "zero point four three" NOT "zero point forty three". Address any misconceptions about reading a decimal number that arise. 

Activity 4

Conclude the lesson by having the students individually write a reflection on what they have learned.
Encourage students to illustrate their reflections with pictures/diagrams.
Display some of the students’ work from this session.

Session 4

SLOs:

• Recognise the importance of zero as a place holder in whole and decimal numbers.
• Recognise that our base ten place value system extends indefinitely in two directions.

Activity 1

1. Draw a decimal place value house on the class chart and write several decimal numbers. Distribute decimal arrow cards to the students.
2. Have students take turns making the numbers using the arrow cards and reading them to each other in at least two ways.
   For example:

4.953 is read as "four point nine five three"; "four and nine tenths, five hundredths and three thousandths"; "four and nine hundred and fifty three thousandths"; or as "four and ninety five hundredths and three thousandths". 

3.05 is read as "three point zero five" or as "three and five hundredths".
7.029 is read as "seven point zero two nine"; "seven and two hundredths and nine thousandths"; or as "seven and twenty nine thousandths".

3. Discuss that thousandths are 1 ÷ 10 ÷ 10 ÷ 10 and that that pattern of dividing by 10 can continue.

Activity 2

1. Have students work in pairs to brainstorm contexts they can think of in which ONE might be divided into tiny parts. Examples might include time (Usain Bolt can run 100m in 9.58 seconds, measurement of tiny things that can only be seen under a powerful microscope (a bacterial cell is 5 millionths of a metre long), weight (a small apple might weigh one tenth of a kilogram) and volume (one teaspoon is 5 thousandths of a litre). 
2. As a class share ideas and record these on a class chart. Demonstrate how the decimal place value chart continues to the right (go to six places: tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths). Ask, “Does it keep going?’ and discuss.
3. Challenge students to investigate ‘infinity’ and present their own statement or diagram to describe ‘how it works’.

Activity 3

1. Present students with this statement:
   Zero is worth nothing and it is therefore of no use in decimal numbers.
   In pairs or small groups have students consider and investigate this, be prepared to agree or disagree with the statement, and give evidence to support their position (justify).
2. When the students are ready, have them move to one side of the room or the other, showing their agreement or disagreement.
   Have them present their rationale using illustrations to make their point.
   NB. Zero is an important placeholder, just as in whole numbers. However, unlike with whole numbers, if a zero is added to a decimal number it does not change its value. For example 4.5 = 4.50

Session 5

• Read and write decimal numbers to three places.
• Recognise the importance of zero as a place holder in whole and decimal numbers.
• Recognise that the grouping pattern extends to the right of the decimal point.
• Recognise that our base ten place value system extends indefinitely in two directions.

Activity 1

1. Show students the digital learning object Modeling Numbers: decimals. Explain that it provides a model for representing numbers using place value equipment. Clicking on the arrow above a place adds one unit in that place to the model of the number, and clicking on the arrow below a place removes one unit from that place in the model.
   Note that clicking the right arrow at the bottom of the screen will show different representations of the number: using place value houses, in standard form or represented on a three bar abacus.
    
2. Show students how the learning object works by making the numbers 6, 7, 8, 9 and 10 (by clicking on the arrow above the ones place). Ask the students what they think will happen when you make the number 11, then make the number and watch what happens. Ask the students what they think will happen if you count backwards 11, 10, 9. Then watch the place value equipment change as you remove one at a time using the arrow below the ones place. 
    
3. Repeat this, starting with the number 1 and asking what will happen when you remove one-tenth (by clicking the arrow below the tenths place). Zoom in using the magnifying icon and watch what happens.
    
4. Repeat twice more, removing one hundredth from one tenth, and one thousandth from one hundredth.

Activity 2

Have students work in pairs to model numbers on the learning object. They can click the die at the bottom left of the screen and a number will appear in words for the students to build using the place value equipment. They can click on the question mark symbol to check to see whether their model is correct.

Activity 3

Have students work in pairs to explore making their own number, saying it aloud and then checking whether they are reading the number correctly using the speaker icon.

Using Materials

Problem: Hina has 4.3 metres of fabric and uses 1.7 metres to make a skirt. How much fabric does Hina have left over?

Students model 4.3 as four wrapped bars (wholes) and three small pieces (tenths). Students may do 4.3 – 1 = 3.3 at first. Then they may unwrap one bar, add the ten tenths to the three tenths, and remove seven tenths to leave six tenths: 4.3 – 1.7 = 2.6. Alternatively, they may unwrap one bar first to take away the seven tenths and then take away the one whole.

Notice whether students understand that the number after the decimal point always represents tenths, that the maximum digit in any place is 9, and that to perform subtraction calculations, it is usually necessary to break one whole into ten tenths.

Examples:

4.6 – 1.7          3.3 – 1.6         5 – 3.6                        4.2 – 0.4

3.6 – 2.9          2.3 – 1.8          3 – 2.9                        3.3 – 0.6

7.2 – 3.5          9.2 – 7.3          11 – 1.6                       12.8 – 0.9

15.8 – 8.9        15.1 – 1.8       16 – 3.9                       10.3 – 0.8

Using imaging

Problem: Work out 1.5 – 0.8.

Place one wrapped bar and five small pieces where the students can see but not touch them. Listen to see who talks the language of place value as they break the one whole into ten tenths and then subtract the eight tenths from the fifteen tenths. Students who are unable to solve the problem by imaging may need to fold back to “Using materials”.

Examples:

3 – 2.9           3.3 – 0.6           7.2 – 3.5           9.2 – 7.3           11 – 1.6           12.8 – 0.9 …

Using number properties

Students need to repeat this kind of problem until they can predict the answer and explain their reasoning without using or referring to materials.

Examples:

15.8 – 8.9           15.1 – 1.8           16 – 3.9           10.3 – 0.8 …

When teachers make up their own examples, the digit in the tenths column of the number being subtracted needs to be greater than the digit in the tenths column of the number it is being subtracted from. Thus 3.3 – 1.7 is suitable, but 5.6 – 3.4 is not.

Prior knowledge.

• Explain place value with decimals
• know combinations of tenths that add to one
• Use the idea of compatible numbers to add whole numbers

Background

This activity uses the strategy of compatible numbers to add decimal numbers.  Students at this stage should be familiar with using the strategy with whole numbers.

Comments on the Exercises

Exercise 1
Asks students to identify tenths that add to one.

Exercise 2
Asks students to use <, > in sentences to show they can identify tenths that add to one, are less than or more than one.

Exercises 3 – 5
Asks students to solve problems requiring addition of decimal fractions (tenths) using compatible numbers strategy.
Exercise 3: All numbers are less than one and students make numbers up to one.
Exercise 4: Compatible numbers include one number less than one and one number greater than one.
Exercise 5: Compatible numbers could both be greater than one.

Exercise 6
Asks students to use compatible numbers in subtraction (decimal fractions).

“Mrs Hau collects $10 from each of her students for a visit to the zoo. She has a total of $180.

How many students are going to the zoo?”

Give the students $180 as a $100 note and eight $10 notes and have them break the $100 note down into 10 $10 notes. Discuss the answer. Record $180 = 18 x $10 on the board or modelling book.

Repeat for $230, $140, $200, $160 ... Encourage the students to get the answer before modelling the problem with the money if they can.

Repeat by selecting numbers that are too large to model on materials. For example $980. Discuss the pattern that has emerged. Record $980 = 98 x $10 on the board or modelling book.

Repeat for $860, $900, $340, $660 ...

Activity

Reverse the process. Record 19 x $10 = on the board or modelling book and discuss what it equals. Model with $10 notes if needed. Record 19 _ $10 = $190.

Repeat for 13 x $10, 15 x $10, 78 x $10, 34 x $10, 77 x $10 ...

Repeat for 273 x $10, 615 x $10, 798 x $10, 304 x $10, 730 x $10 ...

Activity

“Mrs Wallace’s class is going on a trip that costs $100 per student. Mrs Wallace collects $1,600.

How many $100 notes does Mrs Wallace have?” The students model $1,600 with play money and swap $1,000 for 10 $100 notes. Discuss the pattern that emerged from the material. Record $1,600 = 16 x $100 on the board or modelling book.

Repeat for $2,000; $1,300; $2,100; $1,100 ... Encourage the students to get the answer before modelling if they can.

Repeat, without money, for $6,000; $7,800; $2,400; $6,600; $5,670 ...

Activity

Record 130 x $10 = on the board or modelling book and discuss what it equals.

Record 130 x $10 = $1300 on the board or modelling book.

Repeat for 120 x $10; 490 x $10; 720 x $10; 840 x $10; 470 x $10 ...

Reverse: Discuss how many $10 notes make $1,000.

Find the number of $10 notes in: $4,000; $7,000; $2,000; $13,300; $2,600; $4,560; $2,380 ...

Extension Activity

How many $1,000 notes to make $1,000,000; $3,500,000; $4,567,000 ... ?

How many $10,000 notes to make $670,000; $560,000 ...?

How many $100,000 notes to make $2.5 million; $5 million ...?

1. Write a decimal number on the board (to two decimal places). 
2. Brainstorm different ways to represent it.
   For example 3.81 could be expressed as;

• Three point eight one
• 3 + 0.8 + 0.01
• (3 x 1) + (8 x 0.1) + (1 x 0.01)
• 381 / 100
• 4 – 0.19
• Three and eighty one hundredths
• 7.62 / 2

3. Give each student a decimal number and ask them to represent in as many ways as they can.  Depending on the level of individual students they could be to one, two, or three decimal places.  Less confident students could work in pairs.
4. Students could present their work on a sheet of A3 paper to be displayed on the wall.
5. As the students work encourage them to use a variety of approaches (not just a whole list of addition sums).

The Problem

Gill is playing with her name and with numbers. She lets all her consonants equal 1.3 and all her vowels equal 0.5. So the value of Gill’s name is 1.3 + 0.5 + 1.3 + 1.3 = 4.4

What is the value of your name?

Change the rules so that the value of your name is 4.253

Teaching Sequence

1. Tell the students Gill’s story
2. Have each student find the value of their own first name. As the students solve the problem circulate asking questions that focus on their understanding of the addition of decimal numbers.
   How are you adding these numbers?
   What is the value of the … digit in the number?
   Is your answer reasonable? How do you know? (estimation)
   How do you know that you are correct?
3. With a partner, check that they have both found the correct value for their name.
4. Have students locate themselves in order by the total value of their name, around the side of the classroom. 
5. Challenge the students to work in groups to find a way of ending up with the same number value, 4.253, for their own name.
6. As solutions emerge, have students share these and explain how they have changed the rules.
7. Discuss and have students solve the extension problem.

Extension

Using Gill’s original substitution, what is the biggest and smallest value that you can find using names in your class?

Solution

The answers that you get for the first part of the question will depend upon the names of the students in the class.

To get a name with a value of 4.253 will require some imagination. Students may abandon the consonant/vowel value rule and use arbitrary values for individual letters in their name, adjusting the final letter to ensure that the total value is 4.253.