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IntroductionThis document will provide readers with a report on a grade three students MAI

results, demonstrating his mathematical achievements and the strategies used to attain

the growth points presented from various domains. Focusing on the place value

domain, a lesson plan has been developed to accommodate Joshua in developing an

understanding of 3-digit numbers being comprised of hundreds, tens and ones. In

addition, this lesson will also expand on this learning intention, encouraging him to

demonstrate his knowledge through modelling 3-digit number representations and

justifications using the appropriate language. Through rich discussions and a range of

resources, students such as Joshua are provided with multiple opportunities to extend

on their knowledge and understanding of place value in 2-digit numbers. These

activities that are incorporated into this lesson will aid students in developing

meaningful learning of place value, focusing mainly on 3-digit numbers.

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EDMA262 Mathematics: Learning & Teaching 1Report

Preservice teacher’s name: Diana NicoliStudent ID: S00171791Student: JoshuaYear Level: 3Growth points reached:

Domain Growth point (number)

Growth point (in words)

Counting 4 Count from 0 by 2s, 5s, and 10s.Can count to any given number by 2s, 5s and 10s starting from a non-zero starting point.

Place Value 2 Reading, writing, interpreting, and ordering two-digit numbers.Can read, write, interpret and order 2-digit numbers.

Addition & subtraction strategies

3 Count back/count down to, count up from.Using strategies such as commutativity and adding on or known facts to solve problems.

Multiplication & division strategies

3 Partial modelling multiplication and division (some objects perceived)Uses skip count strategies to solve multiplication questions..

When counting forwards and backwards, Joshua uses a skip counting strategy (mostly counting by 2s) in order to reach any given target. For example, if the given target is 24, Joshua would count from 2, 4, 6… to 24. However, if a mistake does occur, such as skipping a number, Joshua would use self correction by going back to where he made the mistake and correct himself from there. In a case where Joshua is asked to calculate how many teddy bears are in four teddy vehicles, Joshua was able to subsitze and correctly guess the answer. Furthermore, in calculating what number a sign indicates on a number line, Joshua carefully compares the distance between the sign using the provided numbers as guidance. He does this through using the maximum number (such as 100) and recognises that the sign is placed directly in the middle, between 0 and 100. Joshua would then partition the number 100 into two parts (being 50 each). Through this, Joshua recognises as a known fact that half of 100 is 50, which indicates that the sign must be 50. Moreover, Joshua is able to accurately interpret, order, write and read 2-digit numbers using known facts of place value. He compared numbers through understanding the concept of more or less by using strategic skills. For instance, Joshua understands that taking a bundle of 10 from 30 would equal to 20 as it is a known fact. Lastly, Joshua uses strategies of grouping, commutativity and repeated addition to solve questions that ask him to either make an object twice as big or dividing equal teddies onto four mats. He does this through grouping teddies into 3, using repeated addition (such as 3 + 3 + 3 + 3) until he reaches the total amount of teddies such as 12.

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Mathematics lesson plan Topic: Number and Place Value Date: 5/5/2016

Year Level: Three Lesson duration: 60 minutes

Mathematical Focus: Understanding three-digit numbers as comprised of hundreds, tens and ones.

Intended learning outcome: Using a calculator or other devices to model and represent numbers. Understanding three-digit numbers as comprised of hundreds, tens and ones. Demonstrate three digit numbers using models such as icy-pole sticks, MAB blocks, unifix blocks etc. and

explaining reasoning.

Learning Intention:By the end of this lesson, students will be able to recognise that three-digit numbers consist of hundreds, tens and ones, and can can be modeled and represented in a variety of ways.Victorian Curriculum (VC): students use mathematical terminology to justify and explain their reasoning of place value. They find the total value of numbers and represent it by grouping it into sets.

Year level(s): three

Content strand(s): Number and AlgebraSub-strand: Number and place value

Content descriptors(s): Group, partition and rearrange collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting (VCMNA104)

Proficiency strand(s) and descriptor:Understanding: recognising that 3-digit number are comprised of hundreds, tens and ones, and representing these in different ways.Reasoning: analysing the place value of 3-digit numbers and explaining this through utilising various representations with the appropriate vocabulary.

Students’ prior knowledge:

Students already understand/know about this topic/mathematical focus, and the skills already used: Students demonstrate grouping numbers into various

sets. Students can use a variety of materials to represent

place value. (e.g. using bundles of icy pole sticks to represent 2 digit numbers)

students can demonstrate place value understanding of 2-digit numbers.

Assessment strategy/strategies: work samples photos observation checklist

What will you analyse, in the evidence found in the assessment? the terminology used by students

o are they using appropriate terms?o Can they effectively explain and

communicate their way of thinking through using words or diagrams?

o Can they justify and use argumentative thinking if their idea is challenged?

Students accurately displaying 3-digit numbers in at least three ways

o How many representations have they use?

Key vocabulary/terms:

Digits Ordering Units Ones Tens Hundreds grouping

Resources: interactive white board (involves students in an

engaging place value activity as a whole group, Appendix D)

MAB blocks (resource to model 3-digit numbers)

Icy-pole sticks (resource to model 3-digit numbers)

Place value Represent Number line Equations Word form

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o Do they use the same representation each time or explore through different representations?

o Are they using grouping strategies?

Place value grid (Appendix A) Place value matching cards (appendix B) Unifix blocks (resource to model 3-digit

numbers) Place value worksheet (Appendix C)

Lesson designe5: ENGAGE, EXPLORELesson introduction (Whole – TUNING IN):whole class activity: fastest finger (Sexton, 2016)Provide each students with a calculator or allow them to use their own calculators. Articulate the key points of this tuning in activity, being that students need to make a prediction of a number they believe will pop up on the calculator once they consistently press the equal sign and stop after 60 seconds.Once all students have their own number, invite two students to share it so it can be used as examples on what to do next.On the whiteboard, write the given numbers (such as 98 and 110).Probe questions:

o What value does the 9 have from the number 98 (9 tens)?o What value does the first 1 have from the number 110? (1 hundreds)o Which number is larger out of the two and why?

(draw columns of hundreds, tens and ones, with the provided numbers in the correct column)

o How can you represent these numbers using materials such as icy-pole sticks?

As students are sitting on the carpet in a circle, use icy-pole sticks to represent the given numbers. For example, use 9 bundles of 10 and 8 single icy-pole sticks.Invite students to participate through responding to questions being posed.

o How many bundles of 10 would I need for 110?

Provide students with place value matching cards (Appendix B) and invite students to work together to find each match.

o Ask students why it matches or why it does not match

Focus question/s: Which do you believe is the

larger number and why? Which do you believe is the

smaller number and why?

How many different ways can you represent your number?

What is ten more than your number? What is ten less?

What is 100 more than your number or one less than your number?

e5: EXPLORE, EXPLAIN, ELABORATEDevelopment/investigation (Part - INVESTIGATING):place value activitystudents are to take their own numbers from the whole class activity and go back to the tables. If a student has anything lower than a 3-digit number, ask to add the number 5 at the end to create a 3-digit number.Ask students to investigate their numbers and record their process down in their books. A place value grid will be provided to help students place their numbers in the correct places (Appendix A). Once students have completed this activity, they are encouraged to discuss and justify what they have done using appropriate terminology. For instance, “I have placed my 3 in the hundreds column because its 3 groups of 1 hundred”.

Walk around the classroom, observing students and asking questions to probe their thinking. During this stage, students will represent their number using a range of materials such as the MAB blocks, icy-pole sticks, unifix blocks, number lines, equations, drawings etc. In pairs, students will again work collaboratively on their numbers while explaining their own processes to each

Focus question/s: Why is place value important? How can you convince your

classmate that you have outlined the correct place value?

How many ways can you represent your number?

Is your way of thinking different to your peers?

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other.

If students have finished, they are required to complete a place value worksheet to challenge their knowledge (Appendix C) (Base ten blocks worksheets, 2016)

When roving and observing, use the checklist provided to write notes on each student:

o Number line used to represent 3-digit numbero Equations and words used to represent 3-digit numbero Diagrams or other drawings are used to represent 3-digit numbero Use of provided concrete materials to represent 3-digit number

e5: EXPLAIN, ELABORATE, EVALUATEPlenary and conclusion (Whole – REFLECTING and GENERALISING):Go to the website listed in Appendix D, and as a whole class, answer the questions provided (Place value worksheets and resources, 2016). (An attached image will be in appendix E). Ask each student to come up and answer one of the questions using the interactive white board. During this process, another observation will take place being guided by the question:

o Who has used the appropriate terminology?

Ask students to sit in a circle on the floor in front of the whiteboard.Invite 5 students who have used at least three different representations of their given numbers. One at a time, each student will display their representations, while explaining their process and convincing their peers that it is correct using appropriate vocabulary.Ask students if they have any questions or if anything is unclear to help them understand the concepts better.

Focus question/s: What new words were used

and what do you think they mean?

What was something interesting you have learnt from these students who have presented?

Did they have similar or different ways of thinking or understanding?

Catering for diversity:Enabling prompt: ask student to create their own two-digit number. Using a place value grid as a scaffolding tool, ask the student how many tens and ones there are in the provided number. For instance, if the number is 27, ask what column does the 2 belong in and what column does the 7 belong in. Use concrete materials such as the unifix blocks to act as another scaffolding tool. Once the student is comfortable with the two-digit number, introduce a three-digit number and repeat the process.Questions to ask:

o How may ways can you bust a two-digit number?o How many tens and ones are there in the provided number?o Compare and contrast the two-digit number from the three-digit number. Is it the same process?

Extending prompt: ask the student to create their own 4 digit-number and explain their values without using the place value grid. In addition, students will then represent their number using a range of materials, including a number line being partitioned into their values. For instance, the number 145 on a number line could be represented as: +40 +5

OR (290/2 = 145)100 140 145 0 | 290

Questions to ask: How can you use a number line to represent your number? How many ways can you represent your number using only number lines?

English as additional language learner/dialect (EALL/D) learners: provide students with various cards that display the appropriate vocabulary used within the lesson, and an example of a 3-digit number. Example of vocabulary cards include hundreds, tens and ones. Students are then encouraged to use the vocabulary card as a guideline to explain what place value each digit has on the other card provided (Appendix E).

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Indigenous learners: Students will have the opportunity to utilise their own 3-digit numbers, and represent it through a drawing (an example is drawing 100 animals, 20 farms and 2 farmers) or a story.

JustificationThis lesson was designed to further develop Joshua’s skills and achievements within

the place value domain, with the hopes of increasing his growth point from 2 to 3.

This is as Joshua accurately demonstrated his knowledge of 2-digit numbers,

understanding their value and writing their numerals on a calculator. Based on this

MAI result, this lesson is designed to further extend on his achievements by

introducing 3-digit numbers. This lesson will also help Joshua develop argumentative

thinking through explaining and justifying the place values a 3-digit number holds.

This is through discussing their process in pairs, and then later demonstrating it to the

whole class. Teaching place value can help learners think about larger quantities in

addition to enhancing their abilities to invent computation strategies (Van de Walle,

Karp and Bay-Williams, 2015). In utilising the place value grid, Joshua will be able to

recognise and understand that 140 equals 7 hundreds, 2 tens and 4 ones, rather than

700 hundreds, 2 tens and 4 ones (this was a misconception outlined in the MAI

interview). This grid will also help Joshua develop a relational understanding of the

concept, knowing why, what and how to do it (Skemp, 1978).

Providing students with hands-on materials such as the place-value cards can help

them visualise and construct patterns of number systems (Cotter, 2000). For example,

students will be able to comprehend known mathematical facts of 3-digit numbers

being comprised of hundreds, tens and ones. Encouraging students to model 3-digit

representations using a range of resources such as the MAB blocks can facilitate a

deeper understanding of place value. Furthermore, this will also help student’s “think,

learn and discuss” (Boaler, 2014, p.2), through reflecting on their mistakes and self

S00171791Diana Nicolicorrecting themselves when using concrete models (e.g. icy-pole sticks) and rich

discussions with their peers. The facilitation of new thinking is reinforced through the

invited students sharing their work to the whole class, discussing how, what and why

they labelled their numbers into certain place values.

ConclusionThe central learning need this lesson focuses on is place value, emphasising the

concept of 3-digit numbering being comprised of hundreds, tens and ones. To support

Joshua’s development and comprehension of place value was informed through

having achieved a growth point 2 due to successfully reading, writing, and ordering

two-digit numbers. This lesson was developed to further extend on his understanding

and achievements of this through introducing 3-digit numbers. Developing activities

that incorporate a place value grid and interactive games requiring students to guess a

number value from a 3-digit number can help facilitate his understandings further.

This is as these activities give a visual of how 3-digit numbers are comprised of

hundreds, tens and ones. In addition, rich discussions with peers explaining their

thinking and reasoning aid in Joshua gaining a deeper insight of place value.

Appendix A (base ten blocks worksheet, 2016)

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Appendix B (Place value matching cards)

384 3 hundreds8 tens4 ones

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Appendix C

384

102400274109

3 hundreds8 tens4 ones

1 hundred0 tens2 ones

4 hundreds0 tens0 ones

2 hundreds7 tens4 ones

1 hundred0 tens9 ones

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Appendix D

S00171791Diana NicoliWebsite: http://www.snappymaths.com/counting/placevalue/interactive/to1000imm/to1000imm.htm

Appendix E

102 Hundreds Digits

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Appendix F (observation checklist)

102

300625999734

Hundreds

Tens

Ones

Groups

Place value

Digits

Order

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Observation: checklist Name:Satisfactory Unsatisfactory

Number line used to represent 3-digit numbers

Equations and words used to represent 3-digit numbers

Diagrams or other drawings are used to represent 3-digit numbers

Creativity

Appropriate vocabulary

References:

Base ten blocks worksheets. (2016). Mathworksheets4kids. Retrieved 9 May 2016,

from http://www.mathworksheets4kids.com/blocks.php

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Boaler, J. (2014). The mathematics of hope. Retrieved from

http://www.youcubed.org/wp-content/uploads/Mathematics-of-Hope-

Paper2.pdf

Cotter, J. A. (2000). Using language and visualization to teach place

value. Teaching Children Mathematics, 7(2), 108. Retrieved from

http://search.proquest.com.ezproxy2.acu.edu.au/docview/214137984?

accountid=8194

Place value worksheets and resources. (2016). Snappymaths. Retrieved 9 May 2016,

from http://www.snappymaths.com/counting/placevalue/placevalue.htm

Skemp, R. R. (1978). Relational understanding and instrumental understanding. The

Arithmetic Teacher, 26(3), 9-15.

Sexton, M. (2016). EDMA262 tutorial 8: Role of invented and standard written

algorithms [PowerPoint slides]. Retrieved from

http://leo.acu.edu.au/pluginfile.php/1565655/mod_resource/content/8/

EDMA262%20Week%208%20tutorial%202016.pdf

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2015). Elementary and

middle school mathematics: Teaching developmentally (9th ed., global ed.).

Boston, MA: Pearson

Victorian Curriculum and Assessment Authority (2015). Victorian Curriculum:

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Mathematics. Melbourne, Australia: VCAA.