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Victorian Curriculum and Assessment Authority

Mathematics Version 2.0

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  1. 5

Level 5

Level 5 Description

In Level 5, learning in Mathematics builds on each student’s prior learning and experiences. Students engage in a range of approaches to the learning and doing of mathematics that develop their understanding of and fluency with concepts, procedures and processes by making connections, reasoning, problem-solving and practice. Proficiency in mathematics enables students to respond...

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Level 5 Content Descriptions

Number

  1. interpret, compare and order numbers with more than 2 decimal places, including numbers greater than one, using place value understanding; represent these on a number line (VC2M5N01)
    Elaborations
    1. making models of decimals including tenths, hundredths and thousandths by subdividing materials or grids, and explaining the multiplicative relationship between consecutive places; for example, explaining that thousandths are 10 times smaller than hundredths, or writing numbers into a place value chart to compare and order them
    2. renaming decimals to assist with mental computation; for example, when asked to solve 0.6 ÷ 10 they rename 6 tenths as 60 hundredths and say, ‘If I divide 60 hundredths by 10, I get 6 hundredths’ and write 0.6 ÷ 10 = 0.06
    3. using a number line or number track to represent and locate decimals with varying numbers of decimal places and numbers greater than one and justifying the placement; for example, 2.335 is halfway between 2.33 and 2.34, that is, 2.33 < 2.335 < 2.34, and 5.283 is between 5.28 and 5.29 but closer to 5.28
    4. interpreting and comparing the digits in decimal measures, for example, the length or mass of animals or plants, such as a baby echidna weighing 1.78 kilograms and a platypus weighing 1.708 kilograms
    5. interpreting plans or diagrams showing length measures as decimals, placing the numbers into a decimal place value chart to connect the digits to their value
  2. express natural numbers as products of their factors, recognise multiples and determine if one number is divisible by another (VC2M5N02)
    Elaborations
    1. using a certain number of blocks to form different rectangles and using these to list all possible factors for that number; for example, 12 blocks can form the following rectangles: 1 × 12, 2 × 6 and 3 × 4
    2. researching divisibility tests and explaining each rule using materials; for example, using base-10 blocks to test if numbers are divisible by 2, 5 and 10
    3. using divisibility tests to determine if larger numbers are multiples of one-digit numbers; for example, testing if 89 472 is divisible by 3 using 8 + 9 + 4 + 7 + 2 = 30, as 30 is divisible by 3 then 89 472 is a multiple of 3
    4. demonstrating and reasoning that all multiples can be formed by combining or regrouping; for example, multiples of 7 can be formed by combining a multiple of 2 with the corresponding multiple of 5: 3 × 7 = 3 × 2 + 3 × 5, and 4 × 7 = 4 × 2 + 4 × 5
  3. compare and order common unit fractions with the same and related denominators, including mixed numerals, applying knowledge of factors and multiples; represent these fractions on a number line (VC2M5N03)
    Elaborations
    1. using pattern blocks to represent equivalent fractions; selecting one block or a combination of blocks to represent one whole, and making a design with shapes; and recording the fractions to justify the total
    2. creating a fraction wall from paper tape to model and compare a range of different fractions with related denominators, and using the model to play fraction wall games
    3. connecting a fraction wall model and a number line model of fractions to say how they are the same and how they are different; for example, explaining 14 on a fraction wall represents the area of one-quarter of the whole, while on the number line 14 is identified as a point that is one-quarter of the distance between zero and one
    4. using an understanding of factors and multiples as well as equivalence to recognise efficient methods for the location of fractions with related denominators on parallel number lines; for example, explaining on parallel number lines that 210 is located at the same position on a parallel number line as 15 because 15 is equivalent to 210
    5. converting between mixed numerals and improper fractions to assist with locating them on a number line
  4. recognise that 100% represents the complete whole and use percentages to describe, represent and compare relative size; connect familiar percentages to their decimal and fraction equivalents (VC2M5N04)
    Elaborations
    1. recognising applications of percentages used in everyday contexts, for example, the bar model used for charging devices indicating the percentage of power remaining, and advertising in retail contexts relating to discounts or sales
    2. creating a model by subdividing a whole (for example, using 10 × 10 grids to represent various percentage amounts) and recognising complementary percentages (such as 30% and 70%) combine to make 100%
    3. creating a model by subdividing a collection of materials, such as blocks or money, to connect decimals and percentage equivalents of tenths and the commonly used fractions 12, 14 and 34; for example, connecting that one-tenth or 0.1 represents 10% and one-half or 0.5 represents 50%, and recognising that 60% of a whole is 10% more of the whole than 50%
    4. using physical and virtual materials to represent the relationship between decimal notation and percentages, for example, 0.3 is 3 out of every 10, which is 30 out of every 100, which is 30%
  5. solve problems involving addition and subtraction of fractions with the same or related denominators, using different strategies (VC2M5N05)
    Elaborations
    1. using different ways to add and subtract fractional amounts by subdividing different models of measurement attributes; for example, adding half an hour and three-quarters of an hour using a clock face, adding a 34 cup of flour and a 14 cup of flour, subtracting 34 of a metre from 2 14 metres
    2. representing and solving addition and subtraction problems involving fractions by using jumps on a number line, or bar models, or making diagrams of fractions as parts of shapes
    3. using materials, diagrams, number lines or arrays to show and explain that fraction number sentences can be rewritten in equivalent forms without changing the quantity, for example, 12+14 is the same as 24+14 
  6. solve problems involving multiplication of larger numbers by one- or two-digit numbers, choosing efficient mental and written calculation strategies and using digital tools where appropriate; check the reasonableness of answers (VC2M5N06)
    Elaborations
    1. solving multiplication problems such as 253 × 4 using a doubling strategy, for example, 2 × 253 = 506 and 2 × 506 = 1012
    2. solving multiplication problems like 15 × 16 by thinking of factors of both numbers, 15 = 3 × 5, 16 = 2 × 8, and rearranging the factors to make the calculation easier, 5 × 2 = 10, 3 × 8 = 24 and 10 × 24 = 240
    3. using an array to show place value partitioning to solve multiplication, such as 324 × 8, thinking 300 × 8 = 2400, 20 × 8 = 160, 4 × 8 = 32 then adding the parts, 2400 + 160 + 32 = 2592; and connecting the parts of the array to a standard written algorithm
    4. using different strategies used to multiply numbers, and explaining how they work and if they have any limitations; for example, discussing how the Japanese visual method for multiplication is not effective for multiplying larger numbers
  7. solve problems involving division, choosing efficient mental and written strategies and using digital tools where appropriate; interpret any remainder according to the context and express results as a whole number, decimal or fraction (VC2M5N07)
    Elaborations
    1. interpreting and solving everyday division problems such as ‘How many buses are needed if there are 436 passengers and each bus carries 50 people?’, deciding whether to round up or down in order to accommodate the remainder and justifying choices
    2. solving division problems mentally, such as 72 divided by 9, 72 ÷ 9, by thinking, ‘How many nines make 72?’, □ × 9 = 72, or ‘Share 72 equally 9 ways’
    3. using the fact that equivalent division calculations result if both numbers are divided by the same factor
  8. check and explain the reasonableness of solutions to problems, including financial contexts using estimation strategies appropriate to the context (VC2M5N08)
    Elaborations
    1. interpreting a series of contextual problems to decide whether an exact answer or an approximate calculation is appropriate, and explaining their reasoning in relation to the context and the numbers involved
    2. recognising the effect of rounding addition, subtraction, multiplication and division calculations, and rounding both numbers up, both numbers down, and one number up and one number down; and explaining which estimation is the best approximation and why
    3. considering the type of rounding that is appropriate when estimating the amount of money required; for example, rounding up or rounding down when buying one item from a store using cash, compared to rounding up the cost of every item when buying groceries to estimate the total cost and not rounding when the financial transactions are digital
  9. use mathematical modelling to solve practical problems involving additive and multiplicative situations, including simple financial planning contexts; formulate the problems, choosing operations and efficient mental and written calculation strategies, and using digital tools where appropriate; interpret and communicate solutions in terms of the situation (VC2M5N09)
    Elaborations
    1. modelling an everyday situation and determining which operations can be used to solve it using materials, diagrams, arrays and/or bar models to represent the problem; formulating the situation as a number sentence; and justifying their choice of operations in relation to the situation
    2. modelling a series of contextual problems, deciding whether an exact answer or an approximate calculation is appropriate, and explaining their reasoning in relation to the context and the numbers involved
    3. modelling financial situations such as creating financial plans; for example, creating a budget for a class fundraising event, using a spreadsheet to tabulate data and perform calculations
    4. investigating how mathematical models involving combinations of operations can be used to represent songs, stories and/or dances of Aboriginal and Torres Strait Islander Peoples
  10. follow a mathematical algorithm involving branching and repetition (iteration); create and use algorithms involving a sequence of steps and decisions and digital tools to experiment with factors, multiples and divisibility; identify, interpret and describe emerging patterns (VC2M5N10)
    Elaborations
    1. simulating a simple random walk
    2. manipulating sets of numbers using a given rule, for example, if a number is even, halve it; or if a number is odd, subtract 1 then halve it
    3. creating algorithms that use multiplication and division facts to determine if a number is a multiple or factor of another number; for example, using a flow chart that determines whether numbers are factors or multiples of other numbers using branching, such as yes/no decisions
    4. identifying lowest common multiples and highest common factors of pairs or triples of natural numbers; for example, the lowest common multiple of {6, 9} is 18, and the highest common factor is 3, and the lowest common multiple of {3, 4, 5} is 60 and the highest common factor is 1
    5. using the ‘fill down’ function of a spreadsheet and a multiplication formula to generate a sequence of numbers that represent the multiples of any number you enter into the cell, and describing and explaining the emerging patterns

Algebra

  1. recognise and explain the connection between multiplication and division as inverse operations and use this to develop families of number facts (VC2M5A01)
    Elaborations
    1. using materials or diagrams to develop and explain division strategies, such as halving, using the inverse relationship to turn division into a multiplication
    2. using arrays, multiplication tables, and physical and virtual materials to develop families of facts, for example, 3 × 4 = 12, 4 × 3 = 12, 12 ÷ 3 = 4 and 12 ÷ 4 = 3
    3. demonstrating multiplicative partitioning using materials, diagrams or arrays and recording 2 multiplication and 2 division facts for each grouping (for example, 4 × 6 = 24, 6 × 4 = 24, 24 ÷ 4 = 6 and 24 ÷ 6 = 4), explaining how each grouping is different from and connected to other groupings in the materials, diagrams or arrays
    4. using materials, diagrams or arrays to recognise and explain the inverse relationship between multiplication and division (for example, solving 240 ÷ 20 = □ by thinking 20 × □ = 240) and using the inverse to make calculations easier (for example, solving 17 × □ = 221 using division, □ = 221 ÷ 17)
  2. find unknown values in numerical equations involving multiplication and division using the properties of numbers and operations (VC2M5A02)
    Elaborations
    1. using knowledge of equivalent number sentences to form and find unknown values in numerical equations; for example, given that 3 × 5 = 15 and 30 ÷ 2 = 15, then 3 × 5 = 30 ÷ 2, and therefore the solution to 3 × 5 = 30 ÷ □ is 2
    2. using relational thinking, and an understanding of equivalence and number properties to determine and reason about numerical equations; for example, explaining whether an equation involving equivalent multiplication number sentences is true, such as 15 ÷ 3 = 30 ÷ 6
    3. using materials, diagrams and arrays to demonstrate that multiplication is associative and commutative but division is not – for example, using arrays to demonstrate that 2 × 3 = 3 × 2 but 6 ÷ 3 does not equal 3 ÷ 6; demonstrating that 2 × 2 × 3 = 12 and 2 × 3 × 2 = 12 and 3 × 2 × 2 = 12; and understanding that 8 ÷ 2 ÷ 2 = (8 ÷ 2) ÷ 2 = 2 but 8 ÷ (2 ÷ 2) = 8 ÷ 1 = 8
    4. using materials, diagrams or arrays to recognise and explain the distributive property, for example, where 4 × 13 = 4 × 10 + 4 × 3
    5. constructing equivalent number sentences involving multiplication to form a numerical equation, and applying knowledge of factors, multiples and the associative property to find unknown values in numerical equations; for example, considering 3 × 4 = 12 and knowing 2 × 2 = 4, then 3 × 4 can be written as 3 × (2 × 2) and, using the associative property, (3 × 2) × 2 so 3 × 4 = 6 × 2 and so 6 is the solution to 3 × 4 = □ × 2

Measurement

  1. choose appropriate metric units when measuring the length, mass and capacity of objects; use smaller units or a combination of units to obtain a more accurate measure (VC2M5M01)
    Elaborations
    1. ordering metric units from the largest unit to the smallest, for example, kilometre, metre, centimetre, millimetre
    2. recognising that some units of measurement are better suited to some tasks than others; for example, kilometres are more appropriate than metres to measure the distance between 2 towns
    3. deciding on the unit required to estimate the amount of paint or carpet for a room or a whole building, and justifying the choice of unit in relation to the context and the degree of accuracy required
    4. measuring and comparing distances (for example, measuring and comparing jumps or throws using a metre length of string and then measuring the part metre with centimetres and/or millimetres) and explaining which unit of measure is most accurate
    5. researching how the base units are derived for the International System of Units (SI), commonly known as the metric system of units, recognising that the metric unit names for the attributes of length and mass are international standards for measurement
  2. solve practical problems involving the perimeter and area of regular and irregular shapes using appropriate metric units (VC2M5M02)
    Elaborations
    1. investigating problem situations involving perimeter, for example, ‘How many metres of fencing are required around a paddock, or around a festival event?’
    2. using efficient ways to calculate the perimeters of rectangles, such as adding the length and width together and doubling the result
    3. solving measurement problems such as ‘How much carpet would be needed to cover the entire floor of the classroom?’, using square metre templates to directly measure the floor space
    4. creating a model of a permaculture garden, dividing the area up to provide the most efficient use of space for gardens and walkways, labelling the measure of each area, and calculating the amount of resources needed, for example, compost to cover the vegetable garden
    5. using a physical geoboard or a virtual geoboard app to recognise the relationship between area and perimeter and solve problems; for example, investigating what is the largest and what is the smallest area that has the same perimeter
    6. exploring the designs of fishing nets and dwellings of Aboriginal and Torres Strait Islander Peoples, investigating the perimeter, area and purpose of the shapes within the designs
  3. compare 12- and 24-hour time systems and solve practical problems involving the conversion between them (VC2M5M03)
    Elaborations
    1. using timetables written in 24-hour time, such as flight schedules, to plan an overseas or interstate trip, converting between 24- and 12-hour time
    2. converting between the digital and analog representation of 24-hour time, matching the same times represented in both systems; for example, setting the time on an analog watch from a digital alarm clock
  4. estimate, construct and measure angles in degrees, using appropriate tools, including a protractor, and relate these measures to angle names (VC2M5M04)
    Elaborations
    1. using a protractor to measure angles in degrees and classifying these angles using angle names; for example, an acute angle is less than 90°, an obtuse angle is more than 90° and less than 180°, a right angle is equal to 90° and a reflex angle is more than 180° and less than 360°
    2. estimating the size of angles in the environment using a clinometer and describing the angles using angle names
    3. using a ruler and protractor to construct triangles, given the angle measures and side lengths
    4. using a protractor to measure angles when creating a pattern or string design within a circle
    5. recognising the size of angles within shapes that do and do not tessellate, measuring the angles and using the sum of angles to explain why some shapes will tessellate and other shapes do not

Space

  1. connect objects to their nets and build objects from their nets using spatial and geometric reasoning (VC2M5SP01)
    Elaborations
    1. designing and constructing exact nets for packaging particular-shaped items or collections of interest, taking into consideration how the faces will be joined and how the package will be opened
    2. visualising folding some possible nets for a range of prisms and pyramids, predicting which will work and which cannot work, and justifying their choices, based on the number, size and position of particular shapes in each diagram
    3. sketching nets for a range of prisms and pyramids considering the number, shape and placement of the faces, and testing by cutting and folding
    4. investigating objects designed and developed by Aboriginal and/or Torres Strait Islander Peoples, such as those used in fish traps and instructive toys, identifying the shape and relative position of each face to determine the net of the object
  2. construct a grid coordinate system that uses coordinates to locate positions within a space; use coordinates and directional language to describe position and movement (VC2M5SP02)
    Elaborations
    1. understanding how the numbers on the axes on a grid coordinate system are numbers on a number line and are used to pinpoint locations
    2. discussing the conventions of indicating a point in a grid coordinate system; for example, writing the horizontal axis number first and the vertical axis number second, and using brackets and commas
    3. comparing a grid reference system to a grid coordinate system (first quadrant only) by using both to play strategy games involving location; for example, in playing the game Quadrant Commander, deducing that in a grid coordinate system the lines are numbered (starting from zero), not the spaces
    4. placing a coordinate grid over a contour line, drawing and listing the coordinates of each point in the picture, asking a peer to re-create the drawing using only the list of coordinates, and discussing the reasons for the potential similarities and differences between the 2 drawings
  3. describe and perform translations, reflections and rotations of shapes, using dynamic geometry software where appropriate; recognise what changes and what remains the same, and identify any symmetries (VC2M5SP03)
    Elaborations
    1. understanding and explaining that translations, rotations and reflections can change the position and orientation of a shape but not its shape or size
    2. using pattern blocks and paper, tracing around a shape and then conducting a series of one-step transformations and tracing each resulting image, and then finally copying the original position and end position on a new sheet of paper
    3. demonstrating how different combinations of transformations can produce the same resulting image
    4. challenging classmates to select a combination of transformations to move from an original image to the final image, noting the different combinations by using different colours to trace images
    5. investigating how animal tracks can be interpreted by Aboriginal and Torres Strait Islander Peoples using the transformation of their shapes, to help determine and understand animal behaviour

Statistics

  1. acquire, validate and represent data for nominal and ordinal categorical and discrete numerical variables to address a question of interest or purpose using software including spreadsheets; discuss and report on data distributions in terms of highest frequency (mode) and shape, in the context of the data (VC2M5ST01)
    Elaborations
    1. recognising that ordinal data is a form of categorical data even though the data being collected might be numbers, for example, a rating scale using numbers 1–5 to represent the categories people can choose from when asked, ‘What rating would you give this film out of 5?’
    2. determining the mode for a set of data and discussing that there may be more than one mode
    3. identifying the best methods of presenting data to illustrate the results of investigations and justifying the choice of representations
    4. acquiring data through chance experiments, discussing and reporting on the distribution of outcomes and how this relates to equal and unequal outcomes
    5. using digital systems to validate data; for example, recognising the difference between numerical, text and date formats in spreadsheets, and setting data types in a spreadsheet to make sure a date is input correctly
    6. investigating data relating to the reconciliation process between Aboriginal and Torres Strait Islander Peoples and non-Indigenous Australians, posing questions, discussing and reporting on findings
  2. interpret line graphs representing change over time; discuss the relationships that are represented and conclusions that can be made (VC2M5ST02)
    Elaborations
    1. reading and interpreting different line graphs, discussing how the horizontal axis represents measures of time such as days of the week or times of the day, and the vertical axis represents numerical quantities or ordinal categorical variables such as percentages, money, measurements or ratings such as fire hazard ratings
    2. interpreting real-life data represented as a line graph showing how measurements change over a period of time, and make simple inferences
    3. matching unlabelled line graphs to the context they represent based on the stories of the different contexts
    4. interpreting the data represented in a line graph, making inferences; for example, reading line graphs that show the varying temperatures or ultraviolet (UV) rates over a period of a day and discussing when would be the best time to hold an outdoor assembly
  3. plan and conduct statistical investigations by posing questions or identifying a problem and collecting relevant data; choose appropriate displays and interpret the data; communicate findings within the context of the investigation (VC2M5ST03)
    Elaborations
    1. posing questions about insect diversity in the playground, and collecting data by taping a one-metre-square piece of paper to the playground and observing the type and number of insects on it over time
    2. posing a question or identifying a problem of interest; collecting, interpreting and analysing the data; and discussing if the data generated provides the information necessary to answer the question
    3. developing survey questions that are objective, without opinion, and have a balanced set of answer choices without bias
    4. exploring Aboriginal and/or Torres Strait Islander ranger groups’ and other groups’ biodiversity detection techniques to care for Country/Place, posing investigative questions, and collecting and interpreting related data to represent and communicate findings

Probability

  1. list the possible outcomes of chance experiments involving equally likely outcomes and compare to those that are not equally likely (VC2M5P01)
    Elaborations
    1. discussing what it means for outcomes to be equally likely and comparing the number of possible and equally likely outcomes of chance events; for example, when drawing a card from a standard deck of cards there are 4 possible outcomes if you are interested in the suit, 2 possible outcomes if you are interested in the colour or 52 outcomes if you are interested in the exact card
    2. discussing how chance experiments that have equally likely outcomes can be referred to as random chance events; for example, if all the names of students in a class are placed in a hat and one is drawn at random, each person has an equally likely chance of being drawn
    3. commenting on the chance of winning games by considering the number of possible outcomes and the consequent chance of winning
    4. investigating why some games are fair and others are not; for example, drawing a track game to resemble a running race and taking it in turns to roll 2 dice, where the first runner moves a square if the difference between the 2 dice is zero, one or 2 and the second runner moves a square if the difference is 3, 4 or 5, and responding to the questions ‘Is this game fair?’, ‘Are some differences more likely to come up than others?’ and ‘How can you work that out?’
    5. comparing the chance of a head or a tail when a coin is tossed, whether some numbers on a dice are more likely to be facing up when the dice is rolled, or the chance of getting a 1, 2 or 3 on a spinner with uneven regions for the numbers
    6. discussing supermarket promotions such as collecting stickers or objects and whether there is an equal chance of getting each of them
  2. conduct repeated chance experiments, including those with and without equally likely outcomes, and observe and record the results; use frequency to compare outcomes and estimate their likelihoods (VC2M5P02)
    Elaborations
    1. discussing and listing all the possible outcomes of an activity and conducting experiments to estimate the probabilities (for example, using coloured cards in a card game and experimenting with shuffling the deck and turning over one card at a time) and recording and discussing the results
    2. conducting experiments, recording the outcomes and the number of times the outcomes occur, and describing the relative frequency of each outcome; for example, using ‘I threw the coin 10 times, and the results were 3 times for a head, so that is 3 out of 10, and 7 times for a tail, so that is 7 out of 10’
    3. experimenting with and comparing the outcomes of spinners with equal coloured regions compared to unequal coloured regions; and responding to questions such as ‘How does this spinner differ to one where each of the colours has an equal chance of occurring?’, giving reasons
    4. comparing the results of experiments using a fair dice and one that has numbers represented on faces more than once, explaining how this affects the likelihood of outcomes
    5. using spreadsheets to record the outcomes of an activity and calculate the total frequencies of different outcomes, representing these as a fraction; for example, using coloured balls in a bag, drawing one out at a time and recording the colour, and replacing them in the bag after each draw
    6. investigating Aboriginal and/or Torres Strait Islander children’s instructive games (for example, Diyari koolchee from the Diyari Peoples near Lake Eyre in South Australia), to conduct repeated trials and explore predictable patterns, using digital tools where appropriate

Level 5 Achievement Standard

By the end of Level 5, students use place value to write and order decimals including decimals greater than one. They express natural numbers as products of factors and identify multiples and divisors. Students order and represent, add and subtract fractions with the same or related denominators. They represent common percentages and connect them to their fraction and decimal equivalents. Students use their proficiency with multiplication facts and efficient mental and written calculation strategies to multiply large numbers by one- and two-digit numbers and divide by one-digit numbers. They check the reasonableness of their calculations using estimation. Students use mathematical modelling to solve financial and other practical problems, formulating and solving problems, choosing arithmetic operations and interpreting results in terms of the situation.

Students apply properties of numbers and operations to find unknown values in numerical equations involving multiplication and division. They design and use algorithms to identify and explain patterns in the factors and multiples of numbers.

Students choose and use appropriate metric units to measure the attributes of length, mass and capacity, and to solve problems involving perimeter and area. Students convert between 12- and 24-hour time. They estimate, construct and measure angles in degrees. Students use grid coordinates to locate and move positions.

Students connect objects to their two-dimensional nets. They perform and describe the results of transformations and identify any symmetries.

Students plan and conduct statistical investigations that collect nominal and ordinal categorical and discrete numerical data with and without digital tools. Students identify the mode and interpret the shape of distributions of data in context. They interpret and compare data represented in line graphs.

Students conduct repeated chance experiments, list the possible outcomes, estimate likelihoods and make comparisons between those with and without equally likely outcomes.

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