maths program proforma yr 4 t2-s tooney

Sharon Tooney

MATHS PROGRAM : STAGE TWO

YEAR FOUR

WEEKLY ROUTINE

Monday Tuesday Wednesday Thursday Friday

Whole Number 2 Terms 1-4 Number & Algebra Terms 1-4: Addition and Subtraction 2 Terms 1-4 : Multiplication & Division 2 Terms 1 & 3: Patterns and Algebra 2 Terms 2 & 4: Fractions and Decimals 2

Statistics & Probability Terms 1 & 3: Data 2 Terms 2 & 4: Chance 2

Measurement & Geometry Term 1: Length 2 / Time 2/ 2D 2 / Position 2 Term 2: Mass 2 / 3D 2 / Angles 2 Term 3: Volume and Capacity 2 / Time 2 / 2D 2 / Position 2 Term 4: Area 2 / 3D2 / Angles 2

Sharon Tooney

K-6 MATHEMATICS SCOPE AND SEQUENCE

NUMBER AND ALGEBRA MEASUREMENT AND GEOMETRY STATISTICS & PROBABILITY

TERM

Whole Number

Addition & Subtraction

Multiplication & Division

Fractions & Decimals

Patterns & Algebra

Length Area Volume & Capacity

Mass Time 3D 2D Angles Position Data Chance

K 1 2 3 4

Yr 1 1 2 3 4

Yr 2 1 2 3 4

Yr 3 1 2 3 4

Yr 4 1 2 3 4

Yr 5 1 2 3 4

Yr 6 1 2 3 4

NB: Where a content strand has a level 1 & 2, the 1 refers to the lower grade within the stage, eg. Whole Number 1 in S1 is for Yr 1, Whole Number 2 is for Yr 2.

Sharon Tooney

MATHEMATICS PROGRAM PROFORMA

STAGE: Year 4 ES1 S1 S2 S3

STRAND: NUMBER AND ALGEBRA

TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Whole Number 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: uses appropriate terminology to describe, and symbols to represent, mathematical ideas MA2-1WM checks the accuracy of a statement and explains the reasoning used MA2-3WM applies place value to order, read and represent numbers of up to five digits MA2-4NA

Background Information The convention for writing numbers of more than four digits requires that numerals have a space (and not a comma) to the left of each group of three digits when counting from the units column, eg 16 234. No space is used in a four-digit number, eg 6234. Language Students should be able to communicate using the following language: largest number, smallest number, ascending order, descending order, digit, ones, tens, hundreds, thousands, tens of thousands, place value, expanded notation, round to. Refer also to language in Whole Numbers 1.

Recognise, represent and order numbers to at least tens of thousands apply an understanding of place value to read and write numbers of up to five digits arrange numbers of up to five digits in ascending and descending order state the place value of digits in numbers of up to five digits - pose and answer questions that extend understanding of numbers, eg 'What happens if I rearrange the digits in the number 12 345?', 'How can I rearrange the digits to make the largest number?' use place value to partition numbers of up to five digits and recognise this as 'expanded notation', eg 67 012 is 60 000 + 7000 + 10 + 2 partition numbers of up to five digits in non-standard forms, eg 67 000 as 50 000 + 17 000 round numbers to the nearest ten, hundred, thousand or ten thousand

Learning Across The Curriculum Cross-curriculum priorities Aboriginal &Torres Strait Islander histories & cultures Asia & Australias engagement with Asia Sustainability General capabilities Critical & creative thinking Ethical understanding Information & communication technology capability Intercultural understanding Literacy Numeracy Personal & social capability Other learning across the curriculum areas Civics & citizenship Difference & diversity Work & enterprise

Sharon Tooney

CONTENT WEEK TEACHING, LEARNING and ASSESSMENT

ADJUSTMENTS RESOURCES REG

Recognise, represent and order numbers to at least tens of thousands

1

Doubling and Halving Write on the board a selection of whole numbers between 20 and 50:

21 24 28 32 35 38 43 46 Ask students if they can double any of the numbers straight away (e.g. 21, 32). Cross out these numbers and record on the board, for example, 21 2 = 42, 32 2 = 64. Ask students to use their books and to work in pairs to double the remaining numbers. Go through the numbers one by one, inviting students to the board to explain their method to the class. Look for these methods: using known facts, for example 19 2 is 2 less than double 20; splitting the number into tens and ones or units, for example 28 2 is double 20 + double 8; splitting the number in other ways, for example 38 2 is double 35 plus double 3. Use a diagram to show students how they can always double a two -digit number by doubling the tens and doubling the ones or units.

Ask students to use this method to double 28, then 36, doing as much as possible mentally.

Support: provide concrete materials where appropriate Extension: increase the complexity of the questions

Whiteboard, markers, paper and pencils

2

Doubling and Halving With Money Review previous lesson, show the class how the method can be extended to doubling a sum of money such as: $27.38 by splitting the dollars and the cents. Give one or two examples to practise, such as $13.09 and $36.75. Repeat the above for halving numbers, starting with some simple practice of halving

numbers to 20, including odd numbers (e.g. half of 15 is 7 ).

What do you think the answer to half of 120 will be? Why? Establish that half of 120 is the same as half of 12 multiplied by 10, so the answer is 60. Write on the board: half of 120 = half of (12 10) = (half of 12) 10 Now ask for: half of 80, half of 140, half of 320. Get students to explain their answers. Practise halving a few more multiples of 10 to 200, and multiples of 100 to 2000. Give the class some two-digit numbers under 100 to halve, inviting them to explain their strategies. Show them how they can always halve two -digit numbers by partitioning into tens and ones or units, and how to halve sums of money by partitioning into dollars and



Sharon Tooney

cents, using diagrams similar to those for doubling. Give one or two examples of amounts of money to halve, such as $8.26 and $14.50.

3

Counting In Fours or Eights Use a counting stick.

Tell students that one end is zero. Count along the stick and back again in fours. Point randomly at divisions on the stick, saying: - What is this number? How do you know? Encourage students to use multiplied by and divided by in their answers. Point out that they can use the mid-point of the stick as a reference point. For example: I know that halfway is 4 multiplied by 5, or 20, and the next point is 4 more, or 24. Say that this is a good way to remember awkward facts. To remember 10 times a number is always easy. To find 5 times a number is also easy, as it is half of 10 times the number. For example, 10 times 4 is 40, so 5 times 4 is half of 40, or 20. Repeat, this time counting in eights.

Support: multiplication tables for use as a direct reference Extension: increase the complexity of the questions

Counting stick, paper and pencils

4

Recognising Multiples of 4 or 8 (for example) Using a 100s chart. Highlight multiples of 4, for example. Ask students to discuss the patterns that they can see, and then to describe them. Cover part of the 100s chart with a square of paper and ask students to identify which multiples of 4 are hidden. For each multiple, ask one of these questions: - How many fours are in ? - What is divided by 4? - Tell me two division facts that you know for ? Move the paper square around to different positions on the grid. Repeat with other multiples, for example multiples of 8, etc.

Support: multiplication tables for use as a direct reference Extension: increase the complexity of the questions

100s Chart, paper squares, paper and pencils

5

Using Addition and Subtraction to Solve Grid Puzzles Draw an incomplete 3 by 3 grid on the board:

164 30 20 418

Ask students to complete the grid using addition down and across. Repeat with other examples. When students are confident, use this grid:

70 40

297 562 Point to the empty space at the top left and ask: - When I add 40 to this number, I get the answer 297. What is the number? How did you work it out?


3x3 grids, paper and pencils

Sharon Tooney

Repeat with the other empty spaces. Ask students to complete more examples of the second type of grid.

6

Adding and Subtracting Mentally Pairs of Two -Digit Numbers Part A Remind the class that an easy way to add or subtract 9 to or from a number is to add or subtract 10 then adjust the answer by 1. Reinforce that when adding, the answer is adjusted by subtracting 1, since an extra 1 has been added. Similarly, when subtracting, the answer is adjusted by adding 1, since 1 more than needed has been taken away. Support each explanation using an empty number line:

Ask the class to count on in nines from 75. Stop them after about ten steps, then ask them to count back in nines to 75. Discuss strategies. - What is an easy way to add or subtract 19 to or from a number? Agree it is adding or subtracting 20 then adjusting by 1. Extend to adding or subtracting 29, 39, 49, by adding or subtracting the nearest multiple of 10 and adjusting. Include crossing the 100 boundary. Ask students to record their answers. Encourage students to dispense with the support of the empty number line. Get them to count on or back for the multiple of 10, and then do the adjustments. Repeat with adding or subtracting 11, 21, 31, - What is an easy way to add or subtract 18, 28, 58?


Number lines, paper and pencils

7

Adding and Subtracting Mentally Pairs of Two -Digit Numbers Part B Establish using the nearest multiple of 10 and adjusting by 2. Provide a few practice examples as per previous lesson. For example, use an interactive whiteboard Number spinner with a spinner labelled 8, 9, 18, 19, 28, 29. Start with a score of 250. Spin the spinner. Ask students to subtract the number rolled from the score and to record their answers. The game ends when the score becomes a one-digit number. Relate the strategies to the context of money. Set a problem such as : - I bought a bag of apples for 75c and a melon for 69c. How much did they cost altogether? - How can we work this out mentally? Take feedback and jot on the board: 75c + 70c = 145c and 145c 1c = 144c. Establish that 144c is better expressed as $1.44. Repeat with a problem such as: - Melons now cost 85c. How much more do they cost? Give out copies of BLM: Shop Prices to pairs of students. Explain that list A and list B show the prices of items in two different shops. Students should select one price from each list. Working mentally, Student A should find the total of the two items, while Student B finds their difference. They then check each others answers and discuss errors. On the next turn, the students swap roles. Repeat several times.


IWB, whiteboard, markers, paper and pencils, Shop Prices BLM

Sharon Tooney

8

Review Ask students to explain how any errors in the sums and differences activity from the previous lesson were made. Write on the board: 53 + 24. Demonstrate how to do this calculation by adding the tens first. Ask students to partition the numbers. 53 + 24 = (50 + 3) + (20 + 4) = (50 + 20) + (3 + 4) = 70 + 7 = 77 Work through other examples with the class. Demonstrate an example which crosses the tens boundary: 38 + 43 = (30 + 8) + (40 + 3) = (30 + 40) + (8 + 3) = 70 + 11 = 81



9

Revision

10

Assessment

ASSESSMENT OVERVIEW

Sharon Tooney

SHOP PRICES

$2.75 86C

95C

80C

$2.50

$3.62

87C

98C

$1.43 84C

LIST A

LIST B

28C

61C

51C

49C 9C

19C

31C

78C

69C

37C

Sharon Tooney




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Addition and Subtraction 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: uses appropriate terminology to describe, and symbols to represent, mathematical ideas MA2-1WM selects and uses appropriate mental or written strategies, or technology, to solve problems MA2-2WM checks the accuracy of a statement and explains the reasoning used MA2-3WM uses mental and written strategies for addition and subtraction involving two-, three-, four and five-digit numbers MA2-5NA

Background Information Students should be encouraged to estimate answers before attempting to solve problems in concrete or symbolic form. There is still a need to emphasise mental computation, even though students can now use a formal written method. When developing a formal written algorithm, it will be necessary to sequence the examples to cover the range of possibilities, which include questions without trading, questions with trading in one or more places, and questions with one or more zeros in the first number. This example shows a suitable layout for the decomposition method:

Language Students should be able to communicate using the following language: plus, add, addition, minus, the difference between, subtract, subtraction, equals, is equal to, empty number line, strategy, digit, estimate, round to, change (noun, in transactions of money). Word problems requiring subtraction usually fall into two types either 'take away' or 'comparison'. Take away How many remain after some are removed? eg 'I have 30 apples in a box and give away 12. How many apples do I have left in the box?' Comparison How many more need to be added to a group? What is the difference between two groups? eg 'I have 18 apples. How many more apples do I need to have 30 apples in total?', 'Mary has 30 apples and I have 12 apples. How many more apples than me does Mary have?' Students need to be able to translate from these different language contexts into a subtraction calculation. The word 'difference' has a specific meaning in a subtraction context. Difficulties could arise for some students with phrasing in relation to subtraction problems, eg '10 take away 9' should give a response different from that for '10 was taken away from 9'.

Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems select, use and record a variety of mental strategies to solve addition and subtraction problems, including word problems, with numbers of up to and including five digits, eg 159 + 23: 'I added 20 to 159 to get 179, then I added 3 more to get 182', or use an empty number line:

- pose simple addition and subtraction problems and apply appropriate strategies to solve them use a formal written algorithm to record addition and subtraction calculations involving two-, three-, four- and five-digit numbers, eg

solve problems involving purchases and the calculation of change to the nearest five cents, with and without the use of digital technologies solve addition and subtraction problems involving money, with and without the use of digital technologies -use a variety of strategies to solve unfamiliar problems involving money -reflect on their chosen method of solution for a money problem, considering whether it can be improved calculate change and round to the nearest five cents use estimation to check the reasonableness of solutions to addition and subtraction problems, including those involving money

Learning Across The Curriculum

Cross-curriculum priorities Aboriginal &Torres Strait Islander histories & cultures Asia & Australias engagement with Asia Sustainability General capabilities

Critical & creative thinking Ethical understanding Information & communication technology capability Intercultural understanding Literacy Numeracy Personal & social capability Other learning across the curriculum areas Civics & citizenship Difference & diversity Work & enterprise

Sharon Tooney



Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems

1

Doubles of Multiples Ask students: - What is double 3? What is double 30? Continue with other pairs e.g. 8, 80; 6, 60 Repeat asking students to find doubles of hundreds e.g. 300, 100, 200, 400. Record doubles for reference:

5 50

500

10 100

1000 etc

- How can we use these facts to double numbers like 320? Work through double 300 and double 20 600 + 40 = 640 Repeat asking students to double other three-digit numbers up to 500. Record for reference:

120 230 180 90

240 460 360 180

Display the table below. Point to a number and ask students to halve the number. Discuss the methods the students used. Repeat.

60 190 490 180 240

460 230 90 300 470

120 70 480 30 360

380 250 500 270 150

What happens when we halve an odd multiple of ten? What is the inverse operation to halving?



2

Addition Families Write on the board: 1 + 2 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9 Ask students to add these up. Agree on finding pairs which sum to 10 and count up in 10s to get the answer. Write on board 3 + 4 + 7. Remind students of the method of finding pairs that sum to 10. Discuss responses and highlight the pair that sums to 100. Give students similar lists of three multiples of 10 to add. Discuss responses. Organise students into groups of 3 or 4 and give each group the cards from Addition Families. (see attached) The groups play a matching pairs activity. They place the cards face down. In turn they turn two cards over and keep them if they are equal e.g. 3 + 2 +7 12 When all pairs have been claimed students ask each other for pairs which will complete their family. e.g. If a student has 3 + 2 + 7 and 12 he/she could ask another player have you got 30 + 20 + 70 and 120? if the player has the cards, he/she must surrender them. At the

Support: concrete materials to support addition

Addition Families Cards, whiteboard, markers, paper and pencils

Sharon Tooney

end of game the winner is the student who has collected the most families.

3 Addition Spiders Draw on the board the first empty-box statement of a spider diagram: 140 + + 230 - What pairs of numbers could complete this number sentence? - Which pair was the easiest to find? Why? Make connections to previous lessons. Extend the spider diagram by adding more empty-box statements:

Discuss efficient methods for completing the diagram. Display the following table:

How many sets of four squares can you find that add up to 200. Have students create addition spiders to demonstrate answers.

Support: provide Addition Spider BLM for students and concrete materials to support addition Extension: increase the complexity of the questions

Whiteboard, markers, paper and pencils, table of figures

4

Add or Subtract the Nearest Multiple Of 10, Then Adjust Part A Introduce quick fire questions involving multiples of 10 e.g. 80 30, 20 + 40, 50 30. Extend to adding three multiples of 10 e.g. 20 + 50 + 10 = or: 40 + 30 20 = Now consider 60 + 20. Identify answer. What if the calculation were 60 + 19? Discuss. Repeat interactively with a series of examples adding 9, 19, 29, 39 etc. Refine explanations by modelling on a number line e.g. 60 + 19 =

Now consider 57 + 20. Identify answer. What if the calculation were 57 + 19? Discuss. Repeat interactively with a series of examples starting with any two-digit number, adding 9, 19, 29, 39 etc. Refine explanations by modelling on a number line. e.g. 47 + 39

Write: 24 + 9 = 86 + 9 =

Support: provide blank number lines for students to work from Extension: increase the complexity of the questions. Encourage working mentally.

Number lines, whiteboard, markers, paper and pencils

Sharon Tooney

24 + 19 = 86 + 19 = 24 + 29 = 86 + 29 = Ask students for the answers and discuss their methods and the pattern. Ask them to extend the pattern. Discuss crossing the 100 boundary. Repeat for subtraction. Write: 34 9 = 86 9 = 34 19 = 86 19 = 34 29 = 86 29 =

5

Add or Subtract the Nearest Multiple Of 10, Then Adjust Part B Write on the board 56 30; ask students for the answer. Repeat for 56 29. Refer to previous lesson to establish prior knowledge. Give children further examples to complete e.g. 63 19, 78 39 etc. Invite children to explain their strategies. Refine explanations by modelling on an empty number line. What is 56 28? Draw on the board: 56 Invite a child to model on a number line e.g.

Establish the answer will be 28. Refine model to show the tens jumps can be replaced by one jump to the nearest 10, and then adjust with an addition. Play race to zero in pairs. Each child starts by writing 250. Take it in turns to roll the 9, 9, 19, 19, 29, 29 dice. Subtract the dice roll from their number each time. First to get down to a units number is the winner.

Support: provide blank number lines for students to work from

Number lines, whiteboard, markers, paper and pencils

10

Revision and Assessment

ASSESSMENT OVERVIEW

Sharon Tooney

ADDITION FAMILIES

9+6+1 90+60+10

9+8+4

16 160 21

6+5+4 60+50+40 90+80+40

15 150 210

4+5+3 40+50+30 9+7+2

Sharon Tooney

12 120 18

4+9+7 40+90+70 90+70+20

20 200 180

5+8+6 50+80+60 3+6+8

19 190 17

Sharon Tooney

30+60+80 5+8+4 50+80+40

170 17 190

5+7+4 50+70+20

14 140

2+4+5

11

20+40+50 110 ADDITION FAMILIES

Sharon Tooney




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Multiplication and Subtraction 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: uses appropriate terminology to describe, and symbols to represent, mathematical ideas MA2-1WM selects and uses appropriate mental or written strategies, or technology, to solve problems MA2-2WM checks the accuracy of a statement and explains the reasoning used MA2-3WM uses mental and informal written strategies for multiplication and division MA2-6NA

Background Information An inverse operation is an operation that reverses the effect of the original operation. Addition and subtraction are inverse operations; multiplication and division are inverse operations. Linking multiplication and division is an important understanding for students in Stage 2. They should come to realise that division 'undoes' multiplication and multiplication 'undoes' division. Students should be encouraged to check the answer to a division question by multiplying their answer by the divisor. To divide, students may recall division facts or transform the division into a multiplication and use multiplication facts, eg is the same as . The use of digital technologies includes the use of calculators. Language Students should be able to communicate using the following language: multiply, multiplied by, product, multiplication, multiplication facts, tens, ones, double, multiple, factor, shared between, divide, divided by, division, halve, remainder, equals, is the same as, strategy, digit. As students become more confident with recalling multiplication facts, they may use less language. For example, 'five rows (or groups) of three' becomes 'five threes' with the 'rows of' or 'groups of' implied. This then leads to 'one three is three', 'two threes are six', 'three threes are nine', and so on. The term 'product' has a meaning in mathematics that is different from its everyday usage. In mathematics, 'product' refers to the result of multiplying two or more numbers together. Students need to understand the different uses for the = sign, eg 4 3 = 12, where the = sign indicates that the right side of the number sentence contains 'the answer' and

Recall multiplication facts up to 10 10 and related division facts count by fours, sixes, sevens, eights and nines using skip counting use the term 'product' to describe the result of multiplying two or more numbers, eg 'The product of 5 and 6 is 30' use mental strategies to build multiplication facts to at least 10 10, including: using the commutative property of multiplication, eg 7 9 = 9 7 using known facts to work out unknown facts, eg 5 7 is 35, so 6 7 is 7 more, which is 42 using doubling and repeated doubling as a strategy to multiply by 2, 4 and 8, eg 7 8 is double 7, double again and then double again using the relationship between multiplication facts, eg the multiplication facts for 6 are double the multiplication facts for 3 factorising one number, eg 5 8 is the same as 5 2 4, which becomes 10 4 recall multiplication facts up to 10 10, including zero facts, with automaticity find 'multiples' for a given whole number, eg the multiples of 4 are 4, 8, 12, 16, relate multiplication facts to their inverse division facts, eg 6 4 = 24, so 24 6 = 4 and 24 4 = 6 determine 'factors' for a given whole number, eg the factors of 12 are 1, 2, 3, 4, 6, 12 use the equals sign to record equivalent number relationships involving multiplication, and to mean 'is the same as', rather than to mean to perform an operation, eg 4 3 = 6 2 - connect number relationships involving multiplication to factors of a number, eg 'Since 4 3 = 6 2, then 4, 3, 2 and 6 are factors of 12' - check number sentences to determine if they are true or false and explain why, eg 'Is 7 5 = 8 4 true? Why or why not?' Develop efficient mental and written strategies, and use appropriate digital technologies, for multiplication and for division where there is no remainder multiply three or more single-digit numbers, eg 5 3 6 model and apply the associative property of multiplication to aid mental computation, eg 2 3 5 = 2 5 3 = 10 3 = 30 - make generalisations about numbers and number relationships, eg 'It doesn't matter what order you multiply two numbers in because the answer is always the same' use mental and informal written strategies to multiply a two-digit




Sharon Tooney

should be read to mean 'equals', compared to a statement of equality such as 4 3 = 6 2, where the = sign should be read to mean 'is the same as'.

number by a one-digit number, including: using known facts, eg 10 9 = 90, so 13 9 = 90 + 9 + 9 + 9 = 90 + 27 = 117 multiplying the tens and then the units, eg 7 19: 7 tens + 7 nines is 70 + 63, which is 133 using an area model, eg 27 8

using doubling and repeated doubling to multiply by 2, 4 and 8, eg 23 4 is double 23 and then double again using the relationship between multiplication facts, eg 41 6 is 41 3, which is 123, and then double to obtain 246 factorising the larger number, eg 18 5 = 9 2 5 = 9 10 = 90 - create a table or simple spreadsheet to record multiplication facts, eg a 10 10 grid showing multiplication facts use mental strategies to divide a two-digit number by a one-digit number where there is no remainder, including: using the inverse relationship of multiplication and division, eg 63 9 = 7 because 7 9 = 63 recalling known division facts using halving and repeated halving to divide by 2, 4 and 8, eg 36 4: halve 36 and then halve again using the relationship between division facts, eg to divide by 5, first divide by 10 and then multiply by 2 - apply the inverse relationship of multiplication and division to justify answers, eg 56 8 = 7 because 7 8 = 56 record mental strategies used for multiplication and division select and use a variety of mental and informal written strategies to solve multiplication and division problems - check the answer to a word problem using digital technologies Use mental strategies and informal recording methods for division with remainders model division, including where the answer involves a remainder, using concrete materials - explain why a remainder is obtained in answers to some division problems use mental strategies to divide a two-digit number by a one-digit number in problems for which answers include a remainder, eg 27 6: if 4 6 = 24 and 5 6 = 30, the answer is 4 remainder 3 record remainders to division problems in words, eg 17 4 = 4 remainder 1 interpret the remainder in the context of a word problem, eg 'If a car can safely hold 5 people, how many cars are needed to carry 41 people?'; the answer of 8 remainder 1 means that 9 cars will be needed

Sharon Tooney



Recall multiplication facts up to 10 10 and related division facts Develop efficient mental and written strategies, and use appropriate digital technologies, for multiplication and for division where there is no remainder Use mental strategies and informal recording methods for division with remainders

5

Remainders Students explore division problems involving remainders, using counters eg We have to put the class into four even teams but we have 29 students. What can we do? Students make an array to model the solution and record their answer to show the connection with multiplication eg 29 = 4 7 + 1. Students could interpret the remainder in the context of a word problem eg Each team would have 7 students and one student could umpire. Students could record the answer showing the remainder eg 29 4 = 7 remainder 1. The teacher could model recording the students solutions, using both forms of recording division number sentences. The teacher sets further problems that involve remainders eg A school wins 125 computers. If there are seven classes, how many computers would each class receive? Since only whole objects are involved, students discuss possible alternatives for sharing remainders. Students write their own division problems, with answers involving remainders.

Support/Extension: adjust complexity of questions accordingly

Counters, paper and pencils

6

Ancient Egyptian Long Multiplication The teacher explains to the students that the Ancient Egyptians had a different number system and did calculations in a different way. They used doubling to solve long multiplication problems eg for 11 23 they would double, and double again, 1 23 = 23 2 23 = 46 4 23 = 92 8 23 = 184 1+ 2 + 8 = 11, so they added the answers to 1 23, 2 23 and 8 23 to find 11 23. 23 46 184 + 253. Students are encouraged to make up their own two-digit multiplication problems and use the Egyptian method to solve them.

Support: concrete materials, multiplication tables to support answering questions

Paper and pencils

7

Factors Game The teacher prepares two dice, one with faces numbered 1 to 6 and the other with faces numbered 5 to 10. Each student is given a blank 6 6 grid on which to record factors from 1 to 60. Students work in groups and take turns to roll the two dice and multiply the numbers obtained. For example, if a student rolls 5 and 8, they multiply the numbers together to obtain 40 and each student in the group places counters on all of the factors of 40 on their individual grid ie 1 and 40, 2 and 20, 4 and 10, 5 and 8. The winner is the first student to put three counters in a straight line, horizontally or vertically.

Support: concrete materials to aide answering questions

Dice, 6x6 grids, counters, paper and pencils

8

Tag Students find a space to stand in the classroom. The teacher asks students in turn to answer questions eg What are the factors of 16? If the student is incorrect they sit down.

Support: multiplication tables as direct reference

Sharon Tooney

The teacher continues to ask the same question until a correct answer is given. When a student gives a correct answer, they take a step closer to another student and may tip them if within reach. The tipped student sits down. The question is then changed. Play continues until one student remains, who then becomes the questioner. This game is designed for quick responses and repeated games.

9 New From Old Students are asked to write a multiplication and a division number fact. Each student uses these facts to build new number facts eg Starting with12 3 = 4 Starting with 3 2 = 6 24 3 = 8 6 2 = 12 48 3 = 16 12 2 = 24 96 3 = 32 24 2 = 48 Possible questions include: - what strategy did you use? - what other strategies could you use? - what strategy did you use? - did you use the relationship between multiplication and division facts?

Support: concrete materials where needed

paper and pencils

10


ASSESSMENT OVERVIEW

Sharon Tooney




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Fractions and Decimals 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: uses appropriate terminology to describe, and symbols to represent, mathematical ideas MA2-1WM checks the accuracy of a statement and explains the reasoning used MA2-3WM represents, models and compares commonly used fractions and decimals MA2-7NA

Background Information In Stage 2 Fractions and Decimals 2, fractions with denominators of 2, 3, 4, 5, 6, 8, 10 and 100 are studied. Denominators of 2, 3, 4, 5 and 8 were introduced in Stage 2 Fractions and Decimals 1. Fractions are used in different ways: to describe equal parts of a whole; to describe equal parts of a collection of objects;

to denote numbers (eg is midway between 0 and 1 on the

number line); and as operators related to division (eg dividing a number in half). Money is an application of decimals to two decimal places. Refer also to background information in Fractions and

Decimals 1. Language Students should be able to communicate using the following language: whole, part, equal parts, half, quarter, eighth, third, sixth, fifth, tenth, hundredth, one-sixth, one-tenth, one hundredth, fraction, numerator, denominator, whole number, number line, is equal to, equivalent fractions, decimal, decimal point, digit, place value, round to, decimal places, dollars, cents. The decimal 1.12 is read as 'one point one two' and not 'one point twelve'.

Refer also to language in Fractions and Decimals 1.

Investigate equivalent fractions used in contexts (ACMNA077) model, compare and represent fractions with denominators of 2, 4 and 8; 3 and 6; and 5, 10 and 100 model, compare and represent the equivalence of fractions with related denominators by redividing the whole, using concrete materials, diagrams and number lines record equivalent fractions using diagrams and numerals Recognise that the place value system can be extended to tenths and hundredths, and make connections between fractions and decimal notation (ACMNA079) recognise and apply decimal notation to express whole numbers, tenths and hundredths as decimals investigate equivalences using various methods identify and interpret the everyday use of fractions and

decimals, such as those in advertisements state the place value of digits in decimal numbers of up to two decimal places use place value to partition decimals of up to 2 decimal places partition decimals of up to two decimal places in non-standard forms apply knowledge of hundredths to represent amounts of

money in decimal form model, compare and represent decimals of up to two decimal places apply knowledge of decimals to record measurements, interpret zero digit(s) at the end of a decimal recognise that amounts of money are written with two

decimal places use one of the symbols for dollars ($) and cents (c)

correctly when expressing amounts of money use a calculator to create patterns involving decimal

numbers place decimals of up to 2 decimal places on a number line round a number with one or two decimal places to the nearest whole number




Sharon Tooney



Investigate equivalent fractions used in contexts Recognise that the place value system can be extended to tenths and hundredths, and make connections between fractions and decimal notation

4

A Pikelet Recipe Students explore dividing wholes into equal parts and use sharing diagrams to divide by fractions. The activity aims to promote partwhole conceptual understanding and to assist students perform fraction computations based on using a sound understanding of the fraction concept. 1. Place 4 identical empty cylindrical clear plastic tumblers near each other on a table. - I want to pour half a glass of drink. Who can show me where about on the glass I would need to fill it to? Provide the student with a thin piece of masking tape to record his or her answer. A marking pen can be used to identify the exact level. - Who thinks that this is the place we should fill the tumbler to get half a glass? Allow an opportunity for class discussion and if the student wishes, he or she can move the tape. - How can we know if we are right? 2. Put out another transparent tumbler with vertical sides. - Can you show me where I would have to fill this glass to get one-quarter of a glass? Attach a small piece of thin black tape at the indicated location. - Does this look correct? (Adjust as directed.) Draw a sketch of the tumbler on the board. Ask one student to add a line to your diagram on the board to show one-quarter of a glass. 3. Put out three empty transparent tumblers with vertical sides and one tumbler full of water. - By pouring, and using any of these other glasses, show me exactly a third of a glass of water? What fraction remains in the glass? Draw a sketch of the three tumblers on the board. Ask one student to add a line to your diagram on the board to show one-third of a glass. - Who can show me two-thirds of a glass by drawing a line on the glass I have drawn on the board?

4. I have 6 cups of milk. A recipe needs of a cup of milk. How many times can I make the

recipe before I run out of milk? Can you draw your answer?

5. I have 6 cups of milk. A recipe needs one-quarter ( ) of a cup of milk. How many times

can I make the recipe before I run out of milk? Can you draw your answer? 6. Draw what would happen if I have 6 cups of milk and a recipe needs three-quarters

( ) of a cup of milk. How many times can I make the recipe before I run out of milk?

7. Who can draw what would happen if I have 6 cups of milk and a recipe needs one-third

( ) of a cup of milk? How many times can I make the recipe before I run out of milk?

8. I have 6 cups of milk. A recipe needs two-thirds ( ) of a cup of milk. How many times can I

Support: representations of fractions as a reference

A pouring jug full of water (food colouring or cordial, optional), 4 cylindrical clear plastic tumblers, thin strips of masking tape or similar.

Sharon Tooney

make the recipe before I run out of milk? Can you draw your answer?

5 Lamington Bars : Forming Equivalent Fractions Students encounter partitioning a rectangle in two directions. The activity aims to promote partwhole conceptual understanding leading to simple fraction multiplication. 1. Lamingtons are pieces of sponge cake covered in chocolate icing and dipped in shredded coconut. Mrs Packer makes excellent lamingtons and she likes to put a layer of whipped cream in the middle of her lamingtons. Mrs Packer starts with a large rectangular sponge cake. 2. Distribute rectangular sheets of brown paper. Show by folding the piece of paper how Mrs Packer could make four lamington bars.

Check to see which way the paper has been divided. If your students use different methods to form quarters ask them if each person would still get the same cut of cake. If all students create quarters by folding in the same direction take your piece of paper and fold it a different way to the direction the class has chosen. Compare the different ways of forming quarters shown above. Ask your students to show how the pieces of cake are equal. 3. I am going to make eight smaller lamington bars. Fold the rectangle into eighths as below.

If I wanted to eat this much (show three-quarters of the horizontally divided rectangle) how many of the smaller lamington bars would this be equal to? Remember that you have to explain your answer.


Brown paper, paper and pencils

6

Mrs Packers Visitors : Comparing Fractions Students encounter partitioning a rectangle into different amounts and comparing the resulting fractions. 1. Mrs Packer was expecting guests. She made five lamington bars and put them on two tables ready for the guests. As each guest arrived Mrs Packer asked the guest to choose a table. Once seated, the guests cannot change tables but must equally share the lamington bars with all the guests at the table. 2. Mrs Packer has placed one lamington bar on one table and four lamington bars on the other table.

Place one rectangular sheet of brown paper on one table and four rectangles of brown paper on another table. 3. Mrs Packer is expecting eight guests. I want eight of you to play the part of the guests. The aim is to get as much of the lamington bars as you can but you cannot change tables after you sit down and everyone must wait until the last person sits down to share the


Tables, brown paper, paper and pencils

Sharon Tooney

lamington bars at their table. Send eight students out of the class and give each one a number to represent the order in which they should return. As each student comes in and sits down, ask the class to record how much each person at that table will receive. Remember that as you sit down you will have to explain why you chose the table you sat at. 4. Show by folding the piece of paper how much each person on your table receives. What would be the best solution? Record your answer. 5. Repeat the activity with two lamington bars on one able and three on the other.

7

Related Fractions 1 : One-Half, One-Quarter and One-Eighth

Students explore the relationships between the unit fractions , and through dividing a

continuous unit. They then express the equivalence between various units, as well as the relationship between the unit fraction and the whole. The activity aims to promote an understanding of the relationship between unit fractions with related denominators.

1. Write the fractions one-half ( ), one-quarter ( ) and one-eighth ( ) on the board. Hold

up a paper streamer approximately 90 cm long. Using this paper streamer, how could you make one of these fractions? Allow the students some time to think about the question. Which of these fractions will be the easiest to make? Why? Focus the questions on: How do you know that you have one-half (or one-quarter or one-eighth)? 2. Fold the paper streamer in half and then fold one half in half. Unfold the streamer and display it to the class. Point to each part in turn and ask: - What fraction of the streamer is this part? How do you know? 3. If I fold one-quarter in half, what will I have? Fold the quarter in half and, as before, point to each part in turn and ask: - What fraction of the streamer is this part? How do you know? Emphasise reversibility: If I fold the quarter in half I get two-eighths and two eighths is the same as one quarter. 4. Which is the biggest part? Which is the smallest part? Can anyone see two fractions that would be the same as another fraction? 5. Show me two-eighths. Show me two-quarters. Show me two-halves. 6. Draw the streamer and show how halves, quarters and eighths are related to each other.


Whiteboard, markers, paper and pencils, paper streamers

10


ASSESSMENT OVERVIEW

Sharon Tooney



STRAND: MEASUREMENT AND GEOMETRY

TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Mass 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: uses appropriate terminology to describe, and symbols to represent, mathematical ideas MA2-1WM selects and uses appropriate mental or written strategies, or technology, to solve problems MA2-2WM measures, records, compares and estimates the masses of objects using kilograms and grams MA2-12MG

Background Information In Stage 2, students should appreciate that formal units allow for easier and more accurate communication of measures. Students are introduced to the kilogram and gram. They should develop an understanding of the size of these units, and use them to measure and estimate. Language Students should be able to communicate using the following language: mass, measure, scales, kilogram, gram. The term 'scales', as in a set of scales, may be confusing for some students who associate it with other uses of the word 'scales', eg fish scales, scales on a map, or musical scales.

These other meanings should be discussed with students.

Use scaled instruments to measure and compare masses (ACMMG084) recognise the need for a formal unit smaller than the kilogram recognise that there are 1000 grams in one kilogram, ie 1000 grams = 1 kilogram use the gram as a unit to measure mass, using a scaled instrument associate gram measures with familiar objects, eg a

standard egg has a mass of about 60 grams (Reasoning) record masses using the abbreviation for grams (g) compare two or more objects by mass measured in kilograms and grams, using a set of scales interpret statements, and discuss the use of kilograms

and grams, on commercial packaging (Communicating, Problem Solving)

interpret commonly used fractions of a kilogram, including

, , , and relate these to the number of grams

solve problems, including those involving commonly used fractions of a kilogram (Problem Solving)

record masses using kilograms and grams, eg 1 kg 200 g




Sharon Tooney



Use scaled instruments to measure and compare masses

1

Calibrated Elastic Band Students work in a small group to hang known masses (in grams) from a large, thick elastic band. Masses will need to be large enough to stretch the elastic band or use thinner elastic bands for smaller masses. Students calibrate the stretch by marking and labelling the levels on a backboard of cardboard or paper. Students use the scale to measure and record the mass of other objects from the classroom. It may be necessary to have a supply of elastic bands available as the elastic band may not return to its original length if used repeatedly or with objects of a large mass. Check the mass of the objects by measuring with a set of scales.

Support: peer tutor grouping strategies

Large elastic bands, nail or hook, cardboard, paper clips, objects to compare, paper and pencils

2

How Heavy Are My Books? Students work individually or in pairs to select six books and estimate the mass of the books in kilograms and grams. Students select appropriate scales to weigh the books. Students find and record the mass of each individual book and then calculate the mass of the six books by adding the six results. Students check their calculation by weighing the six books and commenting on any variation from their calculation. The final report should include the reasons for the selection of the measuring device.


Assorted scales, books, pencils and paper

3

Which Scales? Students work in small groups to trial and record the smallest and largest masses that can be accurately measured on various measuring devices. The devices may include bathroom scales, kitchen scales, balances, etc. Ensure that the students are conversant in how to read the scale on each measuring device and that the scales are set at zero. Students may need reminding to handle the equipment carefully and to check that scales have been placed on a firm, flat surface.


Different measuring devices, different objects to weigh, paper and pencils

4

Toy Story Students bring a toy from home to be weighed. In small groups, students weigh and record the mass of each toy. Students find the total mass of all toys in the group by weighing, by also by adding the masses.

Extension: students graph the mass of each toy in the group.

Toys, scales, paper and pencils

5

Fruit Salad Students work in pairs or small groups to select a measuring device and then measure the mass of individual pieces of fruit, or vegetables. Students estimate then calculate how many pieces would be needed to make a kilogram. Students check their calculations by working with other groups to weigh and count 1 kilogram of the fruit or vegetables.


Fruit, vegetables, kitchen scales, pencils and paper

6

Aussies Abroad Students work in small groups to investigate the gross and net weights of small plastic jars and large glass containers of vegemite. If several different examples are used, each container can be examined by a small group and then rotated to the next group. Students determine which containers would hold the greater volume of vegemite and find how many of each container would fit into a 10kilogram carton

Support: use of calculators Extension: compare the vegemite containers by finding the best value for money.

Different size jars of vegemite, scales, calculators, paper and pencils

Sharon Tooney

7

Mass Measurement Story Problem Provide students with a variety of problems involving mass, in which they need to determine the operation required to solve the problem. Examples include: 1. Selmas body weight is 22 kilograms, while Kiaras body weight is 3 kilograms heavier than Selma. How heavy is Kiara? 2. Aquilas mum wants to make a cake. She bought 585 grams of flour, 250 grams of eggs, and 150 grams of sugar. What is the total weight of the things that Aquila bought? 3. Andi had 1 kilogram of candy. After she gave some to Nadia, she still has 290 grams left. How heavy was the candy that Andi gave to Nadia? 4. The limit of the baggage that each person can bring in the aeroplane is 20 kilograms. Mitchells baggage weighs 24000 gram. How much over the limit is this? 5. Zandas mum bought 17 kg of rice, while Wendy and Cassies mum bought 15 kg and 22 kg. What is the total weight of rice that was bought? Discuss how the students solved each problem and their results.

8

Making Chocolate Cake Present the following recipe to the class: Recipe for Chocolate Cake: 4 eggs (1 egg is about 75 gram) 150 gram of sugar 100 gram of hazelnut, finely ground 5 tablespoons cocoa powder (1 table spoon is about 10 gram) 300 g dark chocolate 100ml whipping cream (50 ml is about 50 gram) Possible questions: - What is the total mass of this chocolate cake? - What is the total mass of 5 chocolate cakes? - If I ate half of the cake, what would be the mass of the part I ate?

Support: concrete materials to answer questions

Chocolate cake recipe, paper and pencils

9

Light Challenges Students use the "feel" of 10 grams to make some guesses about light objects. They are not allowed to use any measuring scales to help with Their guesses. Students put each of Their guesses on named pieces of paper in the challenge containers to be checked at the end of the lesson. Possible challenge stations include: Challenge 1: How many paper clips in 10 grams? Challenge 2: How many drawing pins 20 grams? Challenge 3: How many cm cubes in 50 grams? Challenge 4: How many marbles in 40 grams? Challenge 5: How many teaspoons of rice in 30 grams? etc After students have moved through the challenges and posted their prediction, determine the answer to each as a class, using scales to measure. Check student predictions and have students make generalisations about their prediction

10 gram weights, variety of materials, paper and pencils

Sharon Tooney

verses the correct answer for each challenge.

10 Revision and Assessment

ASSESSMENT OVERVIEW

Sharon Tooney




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Angles 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: uses appropriate terminology to describe, and symbols to represent, mathematical ideas MA2-1WM checks the accuracy of a statement and explains the reasoning used MA2-3WM identifies, describes, compares and classifies angles MA2-16MG

Background Information A simple 'angle tester' can be made by placing a pipe-cleaner inside a straw and bending the straw to form two arms. Another angle tester can be made by joining two narrow straight pieces of card with a split-pin to form the rotatable arms of an angle. Language Students should be able to communicate using the following language: angle, arm, vertex, right angle, acute angle, obtuse angle, straight angle, reflex angle, angle of revolution. The use of the terms 'sharp' and 'blunt' to describe acute and obtuse angles, respectively, is counter-productive in identifying the nature of angles. Such terms should not be used with students as they focus attention on the external points of an angle, rather than on the amount of turning between the arms of the angle.

Compare angles and classify them as equal to, greater than or less than a right angle (ACMMG089) compare angles using informal means, such as by using an 'angle tester' recognise and describe angles as 'less than', 'equal to', 'about the same as' or 'greater than' a right angle classify angles as acute, right, obtuse, straight, reflex or a revolution describe the size of different types of angles in relation

to a right angle, eg acute angles are less than a right angle (Communicating)

relate the turn of the hour hand on a clock through a right angle or straight angle to the number of hours elapsed, eg a turn through a right angle represents the passing of three hours (Reasoning)

identify the arms and vertex of the angle in an opening, a slope and/or a turn where one arm is visible and the other arm is invisible, eg the bottom of an open door is the visible arm and the imaginary line on the floor across the doorway is the other arm create, draw and classify angles of various sizes, eg by tracing along the adjacent sides of shapes draw and classify the angle through which the minute

hand of a clock turns from various starting points (Communicating, Reasoning)




Sharon Tooney



Compare angles and classify them as equal to, greater than or less than a right angle

1

Measuring Angles in the Classroom Students use the windmill pattern as an angle tester to measure and record at least three different angles found in the classroom. Students record an acute, an obtuse and a right angle. Discuss how the angles on the windmill sheet can be used as informal units to measure other angles. - What have we learnt about angles? Discuss strategies that students might use to copy and measure angles in the classroom in terms of windmill units. - How would I measure an angle? - How have we used the windmill angle tester to measure other angles? Students measure, draw and label at least three corners (including an acute, an obtuse and a right angle) in the classroom. Discuss students responses and the range of angle sizes found. - Who had the smallest angle? - Who had the largest angle? - How big are these angles? - How many windmill angles fit into an acute angle? - How many windmill angles fit into an obtuse angle?

Support: individual assistance as required

Bent straws, Windmill BLM or transparencies, pencils and paper

2

Measuring Body Angles Students investigate and record angles made by parts of their body, using the windmill angle tester to measure the angles. Discuss different angles that can be made with the human body. Stand with one arm straight out to the side, then bend your arm at the elbow. Have students do the same, and discuss the angles they can make. Revise the use of terminology arms and vertex, before discussing the arms and vertex in the body angles. Ask a student to hold her arm out straight and ask what the angle is at the elbow. Introduce the term straight angle. - Can you show us how to make an angle with a part of your body? - What angles can you see when I bend my arm like this? - How would you describe these angles? - What angles can you make with your elbow? - When I make an angle with my elbow, where are the arms and the vertex of the angle? - How big is the angle at the elbow when an arm is held straight out from the body? - What could we call this angle? Have your students work in pairs to: make different body angles and discuss these with their partner complete the body angles sheet. Discuss students answers to the body angles questions. Focus on the largest and smallest

Extension: Point out that some angles go beyond a straight angle, e.g. most people can bend their wrist more than six windmill angles. Such angles are called reflex angles. Find some more examples.

Windmill BLM, Body Angles BLM, pencils and paper

Sharon Tooney

angles which students can make by bending their wrists. What were the easiest angles to find or make? Can anybody tell us about body angles which we havent already discussed? What are the largest and smallest angles you can make with your wrist? Can you estimate the size of these in windmill angles?

3

Drawing Two-Line Angles Students draw diagrams that can represent angles in any situation. They investigate the similarity between two-line angles in different locations. Revise and discuss situations in which the size of an angle may change. These may include body angles, the hands of a clock, or scissors. Discuss how angles on objects or in different situations can be fixed or changeable. - We have discussed how the angles on some objects are fixed or dont change, and angles on other objects can change by opening or turning. - Tell us about some angles in this room that are fixed. - Tell us about objects in this room that have changeable angles Discuss how to draw an angle diagram that could represent any of these situations and ask students to demonstrate on the board. - How can you draw an angle so that it can look like either a fixed angle or one that can be changed? Ask students to suggest the angles on objects or shapes that could be represented by the angle diagrams on the chalkboard. Introduce and discuss the drawing two-line angles sheet. Have your students complete the drawing two-line angles sheet. Discuss students answers to the worksheet questions. Review the different types of angles students have identified. Review the different parts of angles on a variety of objects. - What is the same about all the angles you have found? - What can you tell us about the parts of these angles? - What have you learnt about angles?

Support: have students work in pairs to complete the drawing two-line angles sheet. extension: Discuss what it means to say that angle is an abstract concept (Angle is an abstract concept because it represents the same idea occurring in different situations; it is abstracted from all those contexts. Similarly, the angle diagrams above are called abstract diagrams because they do not represent any particular angle but what is common to all angles of that size, in different situations.)

Objects with movable arms, Drawing Two-Line Angles BLM, pencils and paper, access to angle testers and pattern blocks

4

Measuring The Angle Of Opening Of Doors Students are introduced to the concept of a one-line angle by measuring the angle of opening of a door. Students measure the angle of opening of a door using the house activity sheet and a floating door, using pattern block corners. Open and close the classroom door slowly. Discuss how the door turns or pivots on the hinges. Discuss the angle of opening of the door by looking at the top edge and then the bottom edge. Discuss how to visualise the arm formed by the doorway at the bottom edge. Demonstrate opening the classroom door to about 45 and the door on the house worksheet or the model house to about 45 and use a bent straw to check that the angles are equal. Discuss how the angle could be measured with pattern block corners. - Describe what is happening when this door opens and closes? - What allows the door to swing this way? - How could we describe this in mathematical terms? - How could I measure the angle of opening?

Support: individual assistance as required, peer tutor grouping strategies

House BLM, A5 card, pattern blocks, scissors and bent straw; optional model house for teachers demonstration

Sharon Tooney

- How could I make the same angle of opening with the model door or house worksheet door? How could I measure this angle? Activity A Have pairs of students prepare their house worksheets and lay the sheets on their desks. Explain how Student A will select a pattern block angle and open the house door to match the angle without their partner seeing. Student B will estimate which pattern block angle was chosen. The players measure the angle and then reverse the roles. Activity B Demonstrate to the students how to fold the A5 card to make a floating door. Hold the floating door upright on a desk. Discuss how one arm of the angle must be imagined when the door is opened. Ask your students to make a floating door and repeat the activity of measuring the opening with a pattern block. Discuss the different angles that can be made when the door is opened. Ensure students understand that part of the angle when a door is opened needs to be imagined or remembered, as it cannot be seen. - What are the largest and smallest angles you can make when you open the door? - In an angle of opening, where is the vertex? Where are the arms of the angle?

10


ASSESSMENT OVERVIEW

Sharon Tooney

Windmill

Sharon Tooney

Body Angles

Raise one arm at your side, like this:

What angle sizes can you make?

Draw the smallest angle and the largest angle.

Make your hand flat and then make an angle

At your wrist, like this:

What angle sizes can you make?

Draw the smallest angle and the largest angle.

Complete the drawing of a school student to

make the following angles:

angle right arm raised = 3 windmill angles

angle at right elbow = 2 windmill angles

angle left arm raised = 5 windmill angles

angle at left elbow = 4 windmill angles

Sharon Tooney

Drawing Two-Line Angles

Part 1. Each of these objects makes an angle. Draw the angles on each object.

Part 2. Draw the three angles separately here:

Part 3. Find a way to check that the angles you drew in Part 2 are the same size as the angles you found in Part 1. Write how you measured the angles.

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

Sharon Tooney

Part 4. Here is another angle:

Draw and label three different objects that make an angle this size:

Part 5. What is an angle?

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

______________________________________________________________________________________

Sharon Tooney

House

Sharon Tooney




TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: 3D 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: uses appropriate terminology to describe, and symbols to represent, mathematical ideas MA2-1WM checks the accuracy of a statement and explains the reasoning used MA2-3WM makes, compares, sketches and names three-dimensional objects, including prisms, pyramids, cylinders, cones and spheres, and describes their features MA2-14MG

Background Information When using examples of Aboriginal rock carvings and other Aboriginal art, it is recommended that local examples be used wherever possible. Consult with local Aboriginal communities and education consultants for such examples. Refer also to background information in Three-Dimensional Space 1. Language Students should be able to communicate using the following language: object, two-dimensional shape (2D shape), three-dimensional object (3D object), cone, cube, cylinder, prism, pyramid, sphere, top view, front view, side view, isometric grid paper, isometric drawing, depth. Refer also to language in Three-Dimensional Space 1.

Investigate and represent three-dimensional objects using drawings identify prisms (including cubes), pyramids, cylinders, cones and spheres in the environment and from drawings, photographs and descriptions investigate types of three-dimensional objects used in

commercial packaging and give reasons for some being more commonly used (Communicating, Reasoning)

sketch prisms (including cubes), pyramids, cylinders and cones, attempting to show depth compare their own drawings of three-dimensional

objects with other drawings and photographs of three-dimensional objects (Reasoning)

draw three-dimensional objects using a computer drawing tool, attempting to show depth (Communicating)

sketch three-dimensional objects from different views, including top, front and side views investigate different two-dimensional representations of

three-dimensional objects in the environment, eg in Aboriginal art (Communicating)

draw different views of an object constructed from connecting cubes on isometric grid paper interpret given isometric drawings to make models of three-dimensional objects using connecting cubes




Sharon Tooney



Investigate and represent three-dimensional objects using drawings

2

Identify Prisms Identify and name 3D objects, using concrete models, including; cubes, pyramids, cylinders, cones and spheres. Discuss the properties of each shape. Using the school play equipment as an example of 3D shapes in the environment, have the students make a detailed sketch of the equipment. Students should then be encouraged to identify the different 3D shapes used in the make-up of the school play equipment and label these on their diagram. Discuss the different 3D shapes used in the play equipment. Possible questions: - What 3D shapes did you identify within the play equipment? - What was the predominant shape used? Why do you think this was the case? - What purpose did each shape play in the functionality of the play equipment? - Is there any piece of the equipment that you think could have better utilised a different 3D shape? Why?


3D models, paper and pencils

3

3D Shapes in Construction Provide students with a variety of pictures of buildings from around the world. Have them identify and draw the 3D shapes that make up the buildings construction. Explain to students that a photograph is a 2D representation of a 3D object and so we are unable to see the entire object. Keeping this in mind, have the students predict, how many of the 3D shapes they believe went into the construction of the building (obviously this does not involve predicting the number of bricks, for example, but rather columns etc)

Students should be encouraged to compare their drawings to exact drawings of 3D shapes to check for accuracy and to determine how to improve on their attempts.


3D models, pictures of buildings and/or landmarks, paper and pencils

4

Different Views Provide students with concrete examples of 3D shapes. Students work in small groups to share a selection of shapes. Students need to select one shape at a time and draw and label the shape. They then need to draw the shape from different views including; top, front and side views. Students should present their sketches in a table using the headings; 3D Shape, Top View, Front View and Side View at the top of each column.

Support: individual assistance as required, peer tutor grouping strategies, provide ready-made tables

3D models, paper and pencils

5

Drawing Shapes on Isometric Paper Teach the students how to draw a cube on isometric paper:

Extension: make and draw shapes, such as, the following:

Isometric paper, pencils. Whiteboard, IWB isometric paper

Sharon Tooney

Provide students with examples of isometric drawings of interlocking cubes. Have the students use the images to create the shapes themselves using interlocking cubes and then draw the shapes on isometric paper using the diagrams as a guide. Possible examples:

6

Creating and Drawing Shapes from Interlocking Cubes Have students create a series of shapes using 3,4, 5............ interlocking cubes. Have students draw their shapes on isometric paper or alternatively, have the students work in pairs and draw their partners shapes. Provide students with a series of interlocking cube shape diagrams and have them determine the number of cubes within each one. Possible examples:

Support: Allow students who are unable to visualise the number of cubes used in the 2nd part of the lesson, make a model using interlocking cubes Extension: Use more complex diagrams, such as, the following:

Isometric paper, pencils. Whiteboard, IWB isometric paper

7

Drawing 3D Shapes With Computers Have students experiment with drawing individual 3D shapes using computer software and/or online tools. When students have become proficient in drawing individual shapes, see if they can create a 3D image of a building, for example. Possible online tools/downloadable programs include: http://www.sketchup.com/ Alternatively 3D shapes can be created using Word: Step 1: Launch Microsoft Word, and click the Insert tab at the top of the screen, then click the Shapes button. Step 2: Click one of the shapes, such as a circle, from the drop-down selection menu. None of the shapes are 3D; youll add that look in a later step. The cursor turns into a plus sign. Step 3: Drag the cursor on the Word page to form the shape. Click the shape to open the new orange Drawing Tools tab at the top of the screen and the related ribbon below the

Support: individual support as needed Extension: encourage students that are capable of creating shapes without assistance to incorporate shapes into 3D designs and or constructions

Computers, paper and pencils

Sharon Tooney

tab. Step 4: Click the 3-D Effects button on the ribbon. Without clicking, hover the cursor over the options available, moving from button to button for options such as turning a flat circle into a 3D cone. Step 5: Experiment with hovering over the options in the drop-downs fly-out menus as well, with 3D shape lighting and direction choices. Step 6: Click an actual 3D effect to see it instantly take shape on the Word page.

10


ASSESSMENT OVERVIEW

Sharon Tooney



STRAND: STATISTICS AND PROBABILITY

TERM: 1 2 3 3

WEEK: 1 2 3 4 5 6 7 8 9 10

SUBSTRAND: Chance 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student: uses appropriate terminology to describe, and symbols to represent, mathematical ideas MA2-1WM describes and compares chance events in social and experimental contexts MA2-19SP

Background Information Theoretically, when a fair coin is tossed, there is an equal chance of obtaining a head or a tail. If the coin is tossed and five heads in a row are obtained, there is still an equal chance of a head or a tail on the next toss, since each toss is an independent event. Language Students should be able to communicate using the following language: chance, event, possible, impossible, likely, unlikely, less likely, more likely, most likely, least likely, equally likely, experiment, outcome.

Describe possible everyday events and order their chances of occurring (ACMSP092) use the terms 'equally likely', 'likely' and 'unlikely' to describe the chance of everyday events occurring, eg 'It is equally likely that you will get an odd or an even number when you roll a die' compare the chance of familiar events occurring and describe the events as being 'more likely' or 'less likely' to occur than each other order events from least likely to most likely to occur, eg 'Having 10 children away sick on the same day is less likely than having one or two away' compare the likelihood of obtaining particular outcomes in a simple chance experiment, eg for a collection of 7 red, 13 blue and 10 yellow marbles, name blue as being the colour most likely to be drawn out and recognise that it is impossible to draw out a green marble Identify everyday events where one occurring cannot happen if the other happens (ACMSP093) identify and discuss everyday events occurring that cannot occur at the same time, eg the sun rising and the sun setting Identify events where the chance of one occurring will not be affected by the occurrence of the other (ACMSP094) identify and discuss events where the chance of one event occurring will not be affected by the occurrence of the other, eg obtaining a 'head' when tossing a coin does not affect the chance of obtaining a 'head' on the next toss explain why the chance of each of the outcomes of a second

toss of a coin occurring does not depend on the result of the first toss, whereas drawing a card from a pack of playing cards and not returning it to the pack changes the chance of obtaining a particular card or cards in future draws

compare events where the chance of one event occurring is not affected by the occurrence of the other, with events where the chance of one event occurring is affected by the occurrence of the other, eg decide whether taking five red lollies out of a packet containing 10 red and 10 green lollies affects the chance of the next lolly taken out being red, and compare this to what happens if the first five lollies taken out are put back in the jar before the sixth lolly is selected




Sharon Tooney



Describe possible everyday events and order their chances of occurring Identify everyday events where one occurring cannot happen if the other happens Identify events where the chance of one occurring will not be affected by the occurrence of the other

1

Take-away Dice In pairs, students play the following game to investigate the concept of fairness. In turns, they throw two dice and subtract the smaller number from the larger number eg if 4 and 6 is thrown, they calculate 6 4 = 2. Player A scores a point if the answer is 0, 1, or 2. Player B scores a point if the answer is 3, 4, or 5. Students play the game and are asked to comment on whether the game is fair and why. Students are asked how the rules of the game could be changed to make the game fairer and how they could be changed so it is impossible for one student to lose.

Support: concrete materials to solve subtraction problems, peer tutoring strategies for grouping

Dice, paper and pencils

2

Sample Bags Students place four counters or blocks (eg three blue and one white) into a bag. The teacher discusses with the students the chance of drawing out a blue block. Possible questions include: - would you have a good chance or a poor chance of drawing out a blue block? Why? - what colour block is most likely to be drawn out? Why? Students could trial their predictions by drawing a block out of the bag a number of times, recording the colour and replacing the block each time. Students discuss their findings. The colours are then swapped to three white blocks and one blue block. The teacher discusses with the students the chance of drawing out a blue block from this new group. Possible questions include: - would you have a good chance or a poor chance of drawing out a blue block? Why? - what colour block is most likely to be drawn out? Why? Students complete a number of trials and discuss the results. Students are encouraged to make summary statements eg If there are lots of blue blocks you have a good chance of getting a blue block.

Support: summary statement that only require insertion of word explaining likelihood

Counters or blocks, bag, paper and pencils

3

Is It Fair? The class is organised into four teams. Each team is allocated a colour name: red, blue, green or yellow. The teacher has a bag of counters composed of 10 red, 5 blue, 4 green and 1 yellow. The students are told that there are twenty counters and that each colour is represented in the bag. The composition of counters is not revealed to the students. The teacher draws a counter from the bag and a point is given to the team with the corresponding colour. The counter is then returned to the bag and the process is repeated for twenty draws. Individually, the students are then asked to predict the composition of coloured counters in the bag, explain their prediction and state whether the game is fair. Possible questions include:

Support: peer tutoring strategies for grouping

Bag, counters, paper and pencils

Sharon Tooney

- what happens if one colour is not included? - have you tried using a diagram to help you with your predictions? - what are some possible explanations? - how will you know if your generalisations are reasonable? Students are then told the composition of colours in the bag and are asked to name the colours most and least likely to be drawn out.

4

Musical Chairs Students play the game Musical Chairs removing one chair each time. The chance of each student getting a chair is discussed. The game is repeated with three or more chairs removed at a time and students are asked to comment on whether there is more or less chance of getting out compared to the original game. Variation: Other games could be played where an aspect of the game is changed to affect the chance of various outcomes occurring.

Chairs, music CD player

5

Combination Dressing Students are told that they will be given three t-shirts and two pairs of trousers and are asked to predict how many different combinations of clothes they could make from them. They work out a strategy and follow it to calculate the number of combinations and compare the results to their predictions.

Support: provide cardboard cut-outs of clothes to make combinations with

Paper and pencils

10


ASSESSMENT OVERVIEW

Light Challenges

maths program proforma yr 4 t2-s tooney

Documents

number algebra terms

order numbers

digit number

largest number

understanding of numbers

smallest number

sequence number

place value of digits