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MTP501 - Teaching Number, Algebra and Probability in The Primary Years
Assessment: (Two)
Date: 03/01/2022
1 | P a g e
Introduction
The proficiency strands of problem-solving, reasoning, fluency, and understanding are fundamentally included within the mathematics curriculum within the three content strands of Algebra and Number, Geometry and Measurement, and Probability and Statistics (ACARA, 2016). The skills support the importance of working mathematically within the taught content and assist in defining how the content is developed. They provide language to build in the developmental aspects of learning mathematics. The achievement standards mirror the content whilst encompassing the skills (ACARA, 2016).
Number sense is referred to the individual's knowledge and understanding of operations and numbers and flexibly applying knowledge and understanding when answering mathematical problems (Siemon, 2007). Furthermore, studies confirm that students struggle to understand the concept of decimals and fractions (Reys et al., 2020). Studies further suggest that evidence-based instructional learning such as visual and concrete representation, systematic instruction, and explicit teaching strategies often increases student achievement regarding skills and fraction concepts (Shin & Bryasnt, 2015). Reys et al., 2017 confirm that developing an understanding of fractions is vital as it is the fundamentals of numbers as it allows students to understand various mathematical concepts, including subtraction, addition, division, and multiplication (ACARA, 2016; Reys et al., 2020). Subsequently, students will then be taught how to utilise algebraic thinking and apply it to solve problems (ACARA, 2016). The following teaching plan focuses on the Australian Curriculum for Mathematics, years five and six. It explores the three content strands to identify how students solve problems by utilising various strategies that address understanding, fluency, problem-solving, and reasoning whilst understanding a sense of numbers. Furthermore, the plan demonstrates a concept map linking big ideas of algebra, numbers, and probability. Based on the Australian curriculum and specific activities that enhance students' conceptual knowledge and understanding of the taught content (ACARA, 2016). 2 | P a g e
Identify the possibilities Of day-to-day events that occur
Solving problems modelling jump skip counting
Explore and describe
number patterns
resulting from
performing
multiplication
Inverse relationships
Terms used to express
probability
(Fractions/Decimals/
Percentage)
Utilising geometry to create patterns
Create patterns with fractions, decimals and whole numbers resulting from addition and subtraction
Representing and
connecting numbers in
various ways
Presenting fractions on a number line Various ways a whole number is presented
Measuring duration and time by through
the incorporation of measurement
Concept map 3 | P a g e
Chance
Number lines
Estimating
Estimation
Multiplication
and Division Whole numbers
Number
Statistics and
Probability
Probability
Algebra
The four
operations Measurement
and geometry Statistics and
data Patterns Percentages Time
Fractions Decimals
Math Links
Content Descriptor
Analysis
Fraction
s
and Decimal
s
ACMNA077 – Investigate equivalent fractions used in contexts.
ACMNA102 - Compare and order common unit fractions and locate and represent them on a
number line (ACARA, 2016)
ACMNA127 - Find a simple
fraction
of a quantity where the result is a
whole number
, with and without digital technologies (ACARA, 2016)
ACMNA131 – Make connections between equivalent fractions, decimals,
and percentages (ACARA, 2016)
Decimals, fractions, and percentages share comparable aspects as they convey
part numbers (ACARA, 2016). Students tend to convert fraction-based questions to percentages or decimals to gain a sound understanding (Reys et al., 2020). Subsequently, students can express fractions in the form of decimals
by simply dividing the ratio. An example of this includes ¾ is equivalent to 0.75
(Kwarteng & Agia 2022). Percentages, fractions, and decimals are different ways one can express a proportion of a value (ACARA, 2016). Depending on the context, the student will need to know and understand to determine which concept to apply (ACARA, 2016; Reys et al., 2020). For example, when shopping, one will often see discounts written as 40% off, however never see 0.40 off the original cost. Comparably, when one converts currency, the exchange rate will not be quoted
as a percentage; instead, it will be quoted as a decimal (Kwarteng & Agia 2022).
Subsequently, all three concepts share the same mathematical equivalence; however, by default, use the appropriate one to the problem/context (Kwarteng
& Agia 2022; Reys et al., 2020).
Patterns
and Algebra
ACMNA081 - Explore and describe number patterns resulting from performing multiplication (ACARA, 2016)
ACMNA107 - Describe, continue, and create patterns with fractions, decimals and whole numbers resulting from
addition and subtraction (ACARA, 2016)
Patterns and Algebra assist with supporting student reasoning, thinking, and working mathematically (ACARA, 2016). Teachers assist students to extend their knowledge and understanding beyond what they see to generalise about situations involving the unknown (Reys et al., 2020). Patterns and algebra draw
together fundamental relationships and properties that guide arithmetic to algebraic thinking (
Blanton & Kaput, 2015; Reys et al., 2020).
Teachers must efficiently convey fractions as numbers and implement a number line to represent decimals and fractions (Reys et al., 2020). In turn, this
allows students to visually understand the relation between fractions, decimals,
and whole numbers (Baker & Ward, 2015). For example, 8/4 is the same as 2
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Part 2 Year Five Lesson One – Number line and fractions.
Students will develop the knowledge and understanding of evaluating fractions when displayed number lines.
Describe, continue, and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction (ACMNA107) (ACARA, 2016).
Activity one - Fractional trail game (Appendix A)
The activity necessitates students to collaboratively work in groups.
The activity incorporates concrete materials to develop the student's knowledge and understanding of concepts regarding fractions.
Each group will be provided with the following:
-
Fraction cards
-
Empty fraction board
-
Post-it stickers
Students are prompted to put two post-it stickers on the board game before they
begin to play (Appendix B).
-
Students are divided into groups of two so that there are two teams.
-
Teacher to prompt students with a fraction and teams to collectively agree on
where they should lay the post-it note on the fractional track board.
-
Students are provided with opportunities to ask questions on misconceptions they require clarification on.
-
Students can move more than one post-it note, provided that the total of the fraction has not been surpassed.
Students are required to think about the following prompts when the activity is carried out
Have you noticed any variances amongst the groups? Are there any patterns that you have taken notice of?
Conclusion
-
Teacher to ask students that they are required to represent numbers using decimals; this is done by incorporating their skill in numbers to convert the number into a decimal. -
Teacher to ask students to expand the fractions; an example of this is ½ is equivalent to 50/100.
-
Extension students can label with chance terms.
-
The team that completes all assigned tasks first will be declared the winner.
4 | P a g e
Year Five Lesson two – Make connections amongst percentages, decimals, and fractions.
Students will be required to work out the percentages and confirm their relationship with decimals. Students will demonstrate fractions in various ways and incorporate mental strategies to identify the percentage (ACARA, 2016).
Make connections between equivalent decimals, percentages, and fractions (ACMNA131).
Teacher to prompt students with the following questions to engage in a class discussion and stimulate pre-requisite knowledge collaboratively.
What are percentages? When would you use percentages in your day-to-day life?
When have you used percentages? How can one benefit from using percentages?
Teacher scaffolds, discuss, and prompts students with the following:
-
Percentages are numbers presented in hundredths; for example, the fraction 50/100 is equal to 0.5 as a decimal and 50% as a percentage.
-
Students will be asked to write the number 65 into their books and convert it into a percentage, fraction, and decimal.
-
4/5 is equivalent to 80%. Do you disagree or agree? In your books, justify why
you agree or disagree. Once completed, students will share answers with peers and justify their responses.
Fractions, decimals, and percentages bingo (Appendix C)
-
Students will engage in an interactive activity to enhance their learning experience of fractions, decimals, and fractions.
-
Teacher to explain the rules of the game.
-
Students will be provided with a scorecard made up of 25 squares.
-
Teacher will prompt students with questions; students who answer five questions correctly in a row will win. Once completed, students are prompted with the following questions to check
for understanding:
Confirm the equivalence of the following numbers:
-
Identify the decimal and fraction of 15% (0.15 or 15/100)
-
Identify the decimal and percentage of ½ (0.50 or 50%)
-
Identify the percentage and fraction of 0.22 (22% or 22/100)
Diagnostic assessment (Appendix D)
5 | P a g e
Students will be required to complete the following worksheet to allow the teacher to check and confirm student understanding. Teacher to scaffold first question to allow students to gain a better understanding. Once completed teacher to ask students to Swap paper with the person next to them and mark together as a class. Teacher to go through answers and ask students to correct wrong answers through a class discussion.
Year Five Lesson Three – Division and Multiplication task cards
Students will be required to demonstrate their knowledge and understanding of inverse relationships and multiplication to solve equational problems.
Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and division where there is no remainder (ACMNA076) (ACARA, 2016).
-
Teacher to designate five workstations within the classroom and place a different task-card consisting of multiplication and divisional questions on each station.
-
Students will be placed in groups of five and advised that they will take turns visiting each station and answering the task card collaboratively.
First station: Convert the multiplication question into a division question
Students will be required to write these addition sentences as multiplication sentences into their workbooks:
-
5+5+5+5+5+5= 30
-
10+10+10+10+10=50
-
10+10+10+5+5=40
Second station: Critical multiplication and division thinking
Students will be required to answer the following questions.
Would you rather have three boxes with ten biscuits in or five boxes with five boxes in? Explain your reasoning. And would you rather have four trays with five apples in? Explain your reasoning.
Third station: Missing numbers
Insert the missing numbers to make these number sentences correct.
20 =? TIMES? ? DIVIDED BY? = 10
Fourth station: Number sentences (Appendix E)
Can you write some number sentences to link the following numbers together?
For example, 2 TIMES 4 = 8 Or 8 DIVIDED BY 4 = 2 Or 8 DIVIDED BY 2 = 4
Fifth station: Mathematical problem (Appendix F)
6 | P a g e
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Two friends would like to buy some sweets from the shop but want to share them
equally with non-left over. Which bag of sweets should they buy? How do you know?
-
Each group will collaboratively present answers with the rest of the class, and
students will have the opportunity to correct one another.
-
Teacher to discuss correct answers with the class.
Reflection
Students will complete the following sentence in their groups,
7 | P a g e
Conclusion
Overall, it is evident that students come across various concepts that incorporate
the big idea of numbers. This includes percentages and fractions are commonly incorporated, knowledge regarding decimals as they are portrayed daily. Students who confidently understand the big idea will effectively demonstrate the ability to distinguish and apply suitable mathematical strategies to solve problems (ACARA, 2016; Van De Walle & Lovin, 2006; Reys et al., 2020). Consequently, it has been emphasised that fractions uncover the development of
proportional reasoning, and they act as a foundation for understanding different mathematical concepts such as similarity and coordinate geometry (Van De Walle & Lovin, 2006; Reys et al., 2020). It is recommended that teachers explicitly model concepts such as percentages, decimals, and fractions. Physical and visual stimuli such as MAB blocks can be incorporated to allow students to visualise and enhance their knowledge and understanding (Downton, 2013; Reys
et al., 2020).
8 | P a g e
References
Australian Curriculum Assessment and Reporting Authority. (2016). The Australian Curriculum, All curriculum elements.
http://www.australiancurriculum.edu.au/download/f10
.
Australian Curriculum and Assessment Reporting Authority. (2016). Year 5, Mathematics. Australian Curriculum and Assessment Reporting Authority. (2016). Year 6, Mathematics.
Australian Institute for Teaching and School Leadership. (2017).
Australian Professional Standards for Teachers
.
https://www.aitsl.edu.au/teach/standards.
Baker, C., & Ward, J. (2015).
Base Ten and Place Value
. National Council of Teachers of Mathematics.
Beckman, S. (2006).
Mathematics for Elementary Teachers
(2nd ed., pp. 7-12). Sydney: Pearson. Blanton, M., & Kaput, J. (2015). Characterising a Classroom Practice That Promotes Algebra in Reasoning.
National Council of Teaching Mathematics
,
37
(1), 34-42.
Carr, K. (2017). What Are The Fundamental Differences & Similarities Between Fractions & Decimals,
36 (5).
Clarke, D., Roche, A., & Mitchell, A. (2008).
Ten practical tips for making fractions
come alive and make sense. Teaching Children Mathematics
(13th ed., pp.
372-380).
Department of Education and Training in Western Australia [DETWA], (2004). First Steps in Mathematics
: Book number 1. Perth, Western Australia: Roberts, J. Downton, A. (2013). Making Connections Between Multiplication and Division.
The Arithmetic Teacher
.
Hurrel, D., & Hurst, C. (2016).
Multiplicative thinking quiz (v.11).
[Lecture notes]. https://lms.curtin.edu.au/webapps/blackboard/content/listContentEditable.j
sp?content_id=_9441894_1&course_id=_112507_1&mode=reset
Jong, C., & Fisher, M. [n.d] Decimal Dilemmas: Interpreting and Addressing Misconceptions
. Ohio Journal of School Mathematics, 14-18.
9 | P a g e
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Kwarteng, F., & Agia, F. (2022).
Understanding Fractions in Relation with Division
| Fractions in Maths
. https://byjus.com/maths/understand-fractions-and-division/
.
Mathematics Mastery., Common Errors and Misconceptions in Column Addition and Subtraction.
Toolkit.mathematicsmastery.org. (2014). http://toolkit.mathematicsmastery.org/app/webroot/js/tiny_mce/plugins/
moxiemanager/data/files/Common%20errors%20and%20misconceptions
%20in%20column%20addition%20and%20subtraction.pdf
.
Mix, K., & Prather, R. (2015). Young Children's Interpretation of Multi-digit Number Names:
Emerging Competence to Mastery. Child Development, 85(3).
Numbers and Operations in Base 10 and Numbers and operations – Fractions
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Pfister, M., & Opitz, E. (2015). Scaffolding for Mathematics teaching in inclusive primary classrooms.
ZDM Mathematics Edition, 47(7). Popsicle Stick Regrouping Fun – Lesson This. Lessonthis.com. http://www.lessonthis.com/math/popsicle-stick-regrouping-fun/
Reys, R., Lindquist, M., Lambdin, D., Smith, M., Rogers, A., & Cooke, A. et al. (2020).
Helping children learn mathematics
(3rd ed.). Milton, QLD: Wiley.
Roche, A. (2010). Decimats: Helping Students To Make Sense of Decimal Place Value. https://files.eric.ed.gov/fulltext/EJ891799.pdf
.
Ross, S. (2016). Research, Reflection, Practice: Place Value: Problem Solving and Written Assessment.
Teaching Children Mathematics
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Steinle, V. (2004). Changes with Age in Students Misconceptions of Decimal Numbers.
Department of Science and Mathematics Education – The University of Melbourne.
Van De Walle, J., & Lovin, L. (2006).
Teaching Student-Centred Mathematics
(2nd
ed., pp. 192-196). Boston: Pearson.
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Appendix A
Appendix B
Appendix C
Appendix D
11 | P a g e
Appendix E
Appendix F
12 | P a g e
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