MATHEMATICS / UNIT PLANNER

EDMA310/360 Mathematics unit planner{Mary Buffon}

Unit Overview

Unit title:Decimal Fractions

Content maths area:The content area to be focused on in this unit overview is Decimal Fractions.

Grade/year level:The grade focus for this unit overview is Year 5, with the corresponding AusVELS Level 5.

Learning Focus (ideas extrapolated from AusVELS):The content strand for this unit is Number and Algebra, with the corresponding sub-strand, Fractions and decimals. The focus topic is decimal fractions. The proficiency strands for this focus of unit work are listed as Understanding, Problem Solving and Reasoning.

Rationale:The content area, decimal fractions, aligns with the AusVELS (2014) level descriptors of comparing, ordering and representing decimals, whilst recognising that the place value system can be extended beyond hundredths. Irwin (2001) explores the important role that decimals play in students everyday life knowledge, and the vital role we as teachers play in tapping into misconceptions of decimal fractions in order to assist students in correctly solving decimal fractions. Wright (2004) emphasizes the importance of understanding decimals as a vital role of number sense for students in the upper primary years, as decimals pervade everyday situations (Wright, 2004, p.606).

Assumed prior knowledge of students:It is assumed that students would have had prior experiences and therefore have prior knowledge of the development of whole number and fraction concepts including an understanding of decimal numbers. Specifically, it is assumed that students will have previously learnt, and therefore understand that our number system is based on groupings of 10. For example, 10 ones makes ten, 10 tens make one hundred, 10 hundreds make one thousand. It is also understood that students know and understand that fractions represent equal parts of a whole and that a whole, when divided into 10 equal parts, results in tenths.

Grouping strategies to support learning:Sullivan (1997) explores how differences in students learning goes beyond the rate at which people learn, extending to the style in which they learn, their orientation to learning and their motivation towards the learning. Therefore, the use of mixed ability grouping has been used in this unit planner as the teacher aims to provide a range of mathematically rich learning experiences that provide extension work for the higher achievers (Sullivan, 1997, p.20) and remedial support for those with difficulties (Sullivan, 1997, p.20). It is through the use of mixed ability grouping the higher achievers are able to extend upon their knowledge, as they are required to explain their learning through the use of a deep understanding and use of different strategies. These highly able students are challenged, as they are required to persevere with the task to help one another in scaffolding the learning. Mixed ability grouping allows students to work collaboratively, scaffolding each others learning, through peer scaffolding in the whole class environment. Differentiation= one of the most challenging but important aspects in planning

Overview of assessment:A range of assessment strategies will be used to assess students both formally and informally. Using formative assessment will be evident in order to move learning forward throughout the semester. This form of assessment is crucial in planning as teachers determine what to do with students who already know, and those who dont (DuFour, 2004). Summative assessment will be used to demonstrate what students have learnt in regards to decimal fractions at the end of the topic, using this for both formal and informal reporting. Assessment will be used to plan for differentiated learning, gauging students capabilities through out the content area, decimal fractions. Picture chats were used as a form of assessment as Ed Partnerships (n.d) explore these as a potentially rich source of insights into students context knowledge, a form of assessment that allows for mathematically rich learning conversations, exploring students justification and reasoning strategies. Gallery walks used peer assessment and were chosen as a further reflection tool, providing students with a structured time to discuss and actively listen to a range of mathematical thinking ideas and understandings, justifying and reflecting on what they have learned from one another, and from the task itself (Thinking of Teaching, 2012). References:Department of Education and Early Childhood Development. (2014, February 26). Comparing decimal numbers: Level 5. Retrieved from http://www.education.vic.gov.au/school/teachers/teachingresources/Pages/default.aspx Downton, A., Knight, R., Clarke, D., & Lewis, G. (2006). Mathematics assessment for learning: Rich tasks & work samples. Melbourne, Australia: Mathematics Teaching and Learning Centre, Australian Catholic University.DuFour, R. (2004). What is a professional learning community? Schools as Learning Communities 61(8), 6-11. Retrieved from http://www.ascd.org/publications/educational-leadership/may04/vol61/num08/What-is-a-Professional-Learning-Community Moody, B. (2011). Decipipes: helping students to get the point. Australian Primary Mathematics Classroom 16(1) 10-15. Retrieved from http://www.aamt.edu.au/ Roche, A. (2005). Longer is larger: Or is it? Australian Primary Mathematics Classroom, 10(3), 11-16. Sullivan, P. (1997). Mixed ability mathematics teaching: Characteristics of suitable tasks. Learning Matters, 2(3), 20-23. Thinking of Teaching. (2012). Math talk and two new strategies to try. Retrieved from http://thinkingofteaching.blogspot.com.au/

MATHEMATICS UNIT PLANNERTopic: Fractions and Decimals Year Level: 5Term: Week: Date:

Key mathematical understandings (2-4 understandings only; written as statements believed to be true about the mathematical idea/topic): Decimals are numbers that can be represented on a number line. Understanding decimal numbers requires a deeper understanding of place value. The place value of each number is important when comparing, ordering and representing decimals and can be extended beyond thousandths.

Key AusVELS Focus / Standard (taken directly from AusVELS documents):Content strand(s):Number and AlgebraMeasurement and GeometryStatistics and Probability Sub-strand(s):Fractions and Decimals

Level descriptions: Level 5 Compare, order and represent decimals (ACMNA105) Recognise that the place value system can be extended beyond hundredths (ACMNA104) Proficiency strand(s):UnderstandingProblem SolvingReasoningUnderstanding: Understanding involves making connections between representations of numbers and decimals and being able to represent these in a variety of ways. This also includes correctly comparing and ordering a range of decimals, recognizing that the place value system can be extended beyond tenths and hundreds to thousandths. Problem Solving: Problem solving includes being able to formulate and solve authentic problems using whole numbers to create, order, compare and represent a range of decimal fractions. Reasoning: Reasoning involves investigating strategies to efficiently perform calculations, represent patterns involving decimals whilst also using the correct mathematical language to compare, order and represent decimal fractions.

Key skills to develop and practise (including strategies, ways of working mathematically, language goals, etc.) (4-5 key skills only): Locating decimals on a number line attending to scale. Comparing and ordering decimals using benchmarking, equivalence and partitioning. Finding a decimal fraction of a quantity using multiplicative thinking. Using correct language and terminology in regards to decimal notion e.g. hundredths, thousandths rather than zero point one.

Key equipment/resources: Six-sided die Ten-sided die Three in a row decimal number line worksheet Decipipes Mathematics Assessment for Learning: Rich Tasks & Work Samples (2nd ed.). Mini-whiteboards Post-it-notes String/wool Camera Interactive whiteboard

Key vocabulary (be specific and include definitions of key words appropriate to use with students) Decimal- relating to or denoting a system of numbers based on the number ten, tenth parts, and powers of ten Decimal fraction- A fraction written as a decimal. Thousandths- the ordinal number of one thousand in counting order. Hundredths- the ordinal number of one hundred in counting order. Place value- The value of each digit in a number depends on its place or position in that number. Number line- A line on which equally spaced points are marked. The points correspond, in order, to the numbers shown. Position- Describes the place where something is.

Possible misconceptions (list of misconceptions related to the mathematical idea/topic that students might develop): Decimal notion, leading to the whole number thinking misconception. This misconception indicates that many students treat decimals as another whole number to the right of the decimal point. (Roche, 2005) The longer is larger misconception. One of the most prevalent misconceptions as students believe that the more digits, the larger it is. E.g. 0.45 is larger than 0.8 because 0.45 has more digits. (Roche, 2005). The shorter is larger misconception. Many children lack the ability to coordinate the numerator and denominator of a fraction and therefore conclude that a shorter decimal, such as 0.3, is greater than a larger decimal, such as 0.92 because they think only about the size of the parts and therefore cannot simultaneously consider how many parts there are (Moody, 2011). Key probing questions (focus questions that will be used to develop understanding to be used during the sequence of lessons; 3 5 probing questions): Can you show me what youre thinking? Can you write down what you are thinking? How could you check this? How can you do this another way and still get the same answer/result? Is this the most efficient way? How do you know that? Can you convince me?Links to other contexts (if applicable, e.g., inquiry unit focus, current events, literature, etc.): Literacy will be incorporated as students use the correct language and terminology in regards to decimal notation. E.g. hundredths, thousandths rather than saying 0.01 or 0.001. Students will incorporate the general capability of personal and social learning as they work together to participate in mathematical investigations and learning experiences. The students are currently undertaking an inquiry unit in which they are actively taking particular interest in the kinds of foods they consume, the nutrition information on the boxes and labels of foods they eat. As students begin to understand, compare and order decimal numbers they will be able to use this to compare items in the supermarket to recognise the nutritional content of the items they are consuming.

Learning strategies/ skillsAnalysingCheckingClassifyingCo-operatingConsidering optionsDesigningElaboratingEstimatingExplainingGeneralisingHypothesisingInferringInterpretingJustifyingListeningLocating informationMaking choicesNote takingObservingOrdering eventsOrganisingPerformingPersuadingPlanningPredictingPresentingProviding feedbackQuestioningReadingRecognising biasReflectingReportingRespondingRestatingRevisingSeeing patternsSelecting informationSelf-assessingSharing ideasSummarisingSynthesisingTestingViewingVisually representingWorking independentlyWorking to a timetable

MATHEMATICALFOCUS

(what you want the students to come to understand as a result of this lesson short, succinct statement)TUNING IN(WHOLE CLASS FOCUS)(a short, sharp task relating to the focus of the lesson; sets the scene/ context for what students do in the independent aspect. e.g., It may be a problem posed, spider diagram, an open-ended question, game, or reading a story)INVESTIGATIONS SESSION(INDEPENDENT LEARNING)(extended opportunity for students to work in pairs, small groups or individually. Time for teacher to probe childrens thinking or work with a small group for part of the time and to also conduct roving conferences)REFLECTION & MAKING CONNECTIONS SESSION(WHOLE CLASS FOCUS)(focused teacher questions and summary to draw out the mathematics and assist children to make links. NB. This may occur at particular points during a lesson. Use of spotlight, strategy, gallery walk, etc.)ADAPTATIONS

- Enabling prompt(to allow those experiencing difficulty to engage in active experiences related to the initial goal task)- Extending prompt(questions that extend students thinking on the initial task)ASSESSMENTSTRATEGIES

(should relate to objective. Includes what the teacher will listen for, observe, note or analyse; what evidence of learning will be collected and what criteria will be used to analyse the evidence)

Session 1 Developing students understanding and prior knowledge on decimals to be able to explain and relate common decimal fractions.

Reviewing what children already know

Mini WhiteboardsWhiteboards are powerful tools to elicit evidence of students prior knowledge. The teacher can quickly frame a question e.g. Which decimal is larger? Students record their answer on the whiteboard and when asked, hold up. The teacher quickly scans the students responses, making observations on prior knowledge. Have students work individually to participate in the game everything about my decimal. Have students write down everything they know about the decimal, represent their decimal on a number line, write down the fraction equivalence of the decimal, represent the decimal as part of a metre and cut a piece of wool or string the appropriate length. Encourage students to represent their decimal in a range of different ways. Teacher roams during the game to observe, record and gauge students level of decimal knowledge and understandings. Questions to probe thinking: -How could you check this?- Can you use materials to help?-Can you explain your thinking to me?- How do you know its correct?- Can you do it another way?(Downton, Knight, Clarke & Lewis, 2006). Students participate in a gallery walk to explore other students knowledge, understanding and strategies such as benchmarking and the use of equivalence to represent their knowledge of decimal fractions. What do you notice about the way Anna has represented her decimal value using wool? Is it bigger/larger or smaller than Victorias decimal? How can you tell? Ensure students can provide a convincing argument to demonstrate their knowledge. Strengthen the gallery walk by having children post post-it-notes writing a strength, question or wondering they have after viewing their peers work. Discuss these notes in pairs or small groups, discuss the need for the feedback they provide to assist their peers in moving learning forward.

Enabling prompts: Provide a benchmark for the string/wool e.g. , 50cm. Less questions e.g. just focus on number line.Extending prompts: What is the most efficient strategy to compare these decimals? How do you know this? Which two decimals were the easiest to compare? Why? Which two decimals were the hardest to compare? Why?

- Collect students work samples to indicate level of understanding e.g. equivalence, benchmarking, and comparisons. Use these to scaffold/extend future lessons. Roving conferences- - Use these to hear students mathematical language and evidence of learning in the moment e.g. understandings and knowledge of decimals. These roving conferences give the teacher the opportunity to articulate students learning, providing them with the opportunity to justify and reason why they have done what they have done. It also assists the teacher to identify and correct misconceptions as they arise through authentic learning conversations, not possible by only collecting work samples. - Use both forms of assessment to document and record students understanding.

Session 2 Relating tenths, hundredths and thousandths and using place value to record numbers involving thousandths

Teacher models the previous session everything about my decimal, writing down a number and modelling how to place this upon a number line, taking the investigation a little deeper and building on what was previously explored. Add/remove 1 or 2 digits from the number to make connections between tenths, hundredths and thousandths, representing each one of these using place value to record the numbers. Have a student model the activity, using their number from the previous day this time using decipipes with teacher scaffolding to introduce the following activity.

Have students explore the decipipes on the table. Discuss as a class after exploration what these may be, what they may represent such as 0.1, 0.32 and 0.875 and have a student model the activity with teacher scaffolding. Have students work in pairs, using the think, pair, share thinking routine of mixed ability grouping to explore and represent a range of decimals, using reasoning strategies to convince their partner of their representation and the corresponding decimal. The teachers role during activity/lesson is to observe for strategies using place value to record numbers involving thousandths with students extending the place value system beyond hundredths. Teacher also observes for the correct use of mathematical language. Questions to probe student thinking may include: How could you write that? Could you model that in a different way? What do these remind you of?Teacher may provide benchmarking such as if this represents one, what will one half (0.5) look like?At the end of the lesson students will participate in a gallery walk around the classroom to look at other students representations using the decipipes. Students ask questions and share their strategies of using place value to record numbers involving thousandths such as using strategies of benchmarking. Again, strengthen the pedagogical use of sticky notes, students can write new strengths or observations, different to the previous lesson e.g. something new I learnt, noticed or that I am still wondering. Encourage students to share ideas and use reasoning strategies to provide a convincing argument to acknowledge the place value system can be extended beyond hundredths.

Enabling prompts: What do they remind you of? How could you write this? Use benchmarks such as this (tenths) represents 1 whole. Initially have only tenths, using fewer materials. Extending prompts: Add an extra number e.g. 227 tenths Challenge students to persevere by encouraging them to orally share their justifications and reasonings with their peers. Using picture chats have mathematically rich conversations to challenge students thinking strategies and reasoning to understand their thinking and further knowledge. Have students take a few photos and choose one to explain what was happening in the photo, what strategy they were using and how this strategy assisted them to solve the problem.- Listening for mathematical language of tenths, hundredths and thousandths. - Looking for correct place value representations to record numbers (tenths, hundredths and thousandths).

Session 3 Relating tenths, hundreds and thousandths and using place value to record numbers involving thousandths

Have decipipes on table and have children revise these and make a number with as many decimal points as they like. Teacher poses question what does your number look like in standard form? Write this down. What does your number look like in base ten form? Draw this. Have children use the think, pair, share activity and work in pairs with different numbers e.g. tenths, hundredths, thousandths discuss/relate their numbers. Emphasize that students will continue with place value for today, deepening their understandings on what they previously learnt. Introduce the game Make a Number (Downton et al., 2006). Participate in the game using cards, numbered 6-9 and one (thus, extending beyond hundredths, into thousandths) with a decimal point (see appendix 2). In pairs, initially have students come up approximately 5-7 questions e.g. using your cards, what is the smallest number you can make? Students use the instruction sheet (see appendix 2) and then are encouraged to see how many questions/answers they can come up with relating decimal place value, decimal density and beginning to compare decimals. Selected students will be asked to share strategies to emphasize partitioning of decimals (tenths, hundredths and thousandths) when relating the three. What do you notice about the way Anna uses place value to record her decimals for example, when she uses her cards to make the smallest decimal? Have students record some new strategies and ideas to be able to justify and reason the answer they have arrived at.

Enabling prompts: - Use only tenths, hundredths or thousandths rather than a combination of all three at one time. Use prompting strategies to encourage students to check the cards matches how she has placed her cards, with questioning techniques such as, Is there another way to work that out? Can you do it another way? Teacher provides a list of questions they may answer rather than students creating their own. Extending prompts: Represent decimals that go beyond thousandths. E.g. tens of thousandths. Use more variables e.g. add a fifth or sixth card. Students create their own questions that go beyond smallest and largest e.g. decimal density, creating a rubric if they finish early. - Use the learning dispositions triangle to gather evidence for learning including skills & capabilities, learning dispositions and students understandings. This way the learners voice is heard, valuing student discussions to identify future learning needs, experiences and contexts. Teacher roves the classroom, listening to learning conversations. Add these to teacher journal for students mathematical content. Observation of the use of the correct mathematical language e.g. saying one and three tenths rather than 1.3 (one point three). Observation of the use of correct decimal place value e.g. one, two and three decimal places. Observe the use of decimal density e.g. recognising how many decimals they can make between two numbers on their cards.

Session 4 Comparing and ordering decimals

Display a 10 x 10 grid on the interactive whiteboard. Have students come up and shade parts, asking questions such as what does the shaded part represent? Revise previous lesson by emphasizing the first place to the right of the decimal point is the tenths place, the second hundredths and so on. Have one student model their number from the previous lesson using the 10 x 10 grid board, moving into comparisons for this lesson with the teacher scaffolding and student modelling where their number would be placed upon a number line, in comparison with the start and endpoints.

Introduce the game Three in a row decimal number line. Use two students to model the game with the teacher scaffolding. Have the students play the game three in a row in pairs (see appendix 2). The teachers role will be to observe for reasoning strategies and explanations of ordering decimals and comparisons using mathematical language. Questions to probe students thinking may include: What did you visualise when you were working on this problem? Why is this greater than this? Why is this less than this? Why did you place this number here? How did you work out where the number is placed upon the number line?Emphasise that when students are playing the game they are to record their reasoning using numbers sentences of why they placed the decimal where they did.Choose students to share their strategies in small groups, allowing for more discussion opportunities for all students to explore the use of benchmarking and equivalence. Ask the class questions such as: What are some different ways to compare 0.1 and 1.2? Look for language such as less then, greater than in comparisons.

Enabling prompts: Use one die instead of two to only make tenths. Use a six sided die Have the number line just 0-1. Provide benchmarks Extending prompts: Use a ten-sided die Add an extra die Therefore, number line will be extended from 0-21.

Picture chats: Take photos of students working. Ask students what was happening in this picture? Use to articulate students knowledge and reasoning skills. Have students take a range of photos, choosing one to share something they are proud of and something they find challenging, identifying areas for further improvement and collaborating on how you can address this challenge together. During this time observe students reasoning strategies of why they ordered decimals the way they did, how they compared decimals using appropriate mathematical language in comparisons such as less than, greater than, equal to.

Session 5 Comparing and ordering decimals

Teacher poses open-ended question create a decimal number using tenths, hundredths, thousandths. Teacher instructs students to order themselves from smallest to largest. Use probing questions to help students order themselves such as: How could you check that your decimal is smaller? Can you use place value to explore whose decimal is larger?Use of comparing and ordering task to deepen learning from the previous day, linking this to continue the mathematical focus in this lesson. The teacher introduces the game number between by writing a pair of numbers far apart on the board (smallest on the left) and calls on a student to write a number in between the pair. When a correct answer is given, call on another student to write a number between the new number and one of the earlier endpoints. Divide the students into teams to play the number between game, challenging the correctness of the answers of the other teams. The teachers role during the game is to observe for strategies of comparisons as students compare decimal numbers with different numbers of digits after the decimal point. (Department of Education and Early Childhood Development, 2014). Have selected students share their strategies with the class to explore the use of benchmarking. Ask questions such as: If you know this is 1 and this is 2, where would 1.5 be? Where would 0.9 be? Locate decimals in order upon the number line. How do you know where to place these numbers? Can you show your thinking?

Enabling prompts: Use only tenths Provide benchmarks e.g. 0, 0.5 and 1Extending prompts: Extend beyond tenths to hundredths and thousandths

Use roving conferences to: Observe the use of appropriate skills such as equivalence and benchmarking in justifying their strategy and answer. Observation of the use of correct and appropriate mathematical language, e.g. two and eight tenths rather than 2.8 (two point eight). Add these anecdotal notes to students checklists.

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