Introduction
This Webquest is designed for Year 5 students to learn about fractions. The key objective will include identifying and generating equivalent fractions, adding or subtracting fractions with related denominators, converting between mixed numbers and improper fractions. The students will learn to multiply proper fractions by whole number and also to relate decimals to fractions, for example : 0.71 = 71/100.
According to (Siemon, et al, 2020), fractions can be thought about in two related but different ways that loosely reflect the distinctions between the counting and splitting based approaches to multiplicative thinking. Hence two different ways of thinking about fractions are counting based interpretations in which numerator counts, denominator indicates what is being counted. The second thought is splitting based interpretations in which the numerator is a multiplier, the denominator is a divisor (Siemon, et al, 2020).
The notion of hypothetical learning trajectories is a unique and substantive contribution to this field as the construct differs from other models in that it involves self-reflective constructivism and includes the simultaneous consideration of mathematical goals, models of children's thinking, teachers and researchers models of children's thinking, sequences of instructional tasks and the interaction of these at a detailed level of analysis of processes ( Clements & Sarama, 2004). This learning trajectory will focus on significant expansion of fraction concepts, moving from basic understanding to more complex operations and the integration of decimals.
Task
Lesson 1 : Introduction of Fractions
This lesson is about giving students the idea of how fractions operation. The main objective is on building strong foundation, real world understanding with formal mathematical notation. According to the Australian Curriculum (Version 9.0 ) students should achieve the compare and order milestone which is about comparing unit fractions and represent them beyond one whole. The students should also have an understanding of how to add and subtract problems involving fractions with the same or related denominators.
The teacher will introduce the trajectory of learning which supports the shift from using words unlikely or likely to assign fractions. According to (Clements & Samara, 2004), hypothetical learning trajectory includes three aspects which are:
- the learning goal
- developmental progressions of thinking and learning
- sequence of instructional tasks
Australian Curriculum Descriptions:
- (AC9M5N04) - This descriptor focuses on Year 5 students understanding the relationship between percentages, fractions and decimals but specifically regarding to fractions, this descriptors emphasizes on linking familiar fractions (for example halves, quarters and tenths) to their percentages and decimal counterparts. For example the children can connect equivalents through recognising that:
- 1/2= 50% =0.5
The descriptor also focuses on representing the relative size as well as on benchmark relationships which helps students to understand fractions of quantity and relative proportions. For example :
- The students can use decimals to describe and compare the size of fractional parts.
2. (AC9M5N05) focuses on developing proficiency in solving problems involving the addition and subtraction of fractions with the same or related denominators in Year 5. This is where one denominator is a multiple of other, for example :
- 1/2 + 1/4
- 1/3 + 2/6
The students will use various strategies to solve problems such as:
- Number lines - they will use jumps to add or to subtract fractions.
- Bar Models or diagrams- visualise fractions as parts of shapes.
- Equivalent fractions - Converting fractions to a common denominator.
- Mixed numerals : this involves adding and subtracting numbers for example 2 1/4 - 3/4 by partitioning or regrouping whole numbers.
- Real - World Contexts - the students will apply these skills to practical situations such as measurements for example hours on a clock.
- Decomposing fractions - the students will break down fractions into smaller unit fractions to assist with calculation
This moves students from basic fraction representation to flexible thinking with fraction operations and procedural fluency.
Lesson 2 : Learning Fractions through Number line.
This is designed to develop a strong sense of fraction magnitude and relationships. The number line helps position fractions as actual numbers that can be compared, ordered and used in calculations. The students will learn to place proper fractions (between 0 and 1), improper fractions and mixed numbers on a line by partitioning the space into equal intervals based on the denominator. They will understand that fraction is a single number located at a specific point on the number line, it is not just a part of a whole shape.
The students will understand the magnitude and ordering of fractions in a number line by placing various proper fractions which are between 0 and 1 and improper fractions which are beyond 1 on a number line. Students will learn to compare and order them based on their positions.
Students will use benchmark fractions for example 0, 1/2 , and 1 to estimate the position of other fractions. For example
- 3/7 is less than 1/2
Proficiency Strands : This lesson falls under Number Strand in mathematics. It falls mainly on the sub-strands of Number Sense and Numeration or Number and Operations (Fractions). It is the foundation for developing a conceptual understanding of fractions as numbers with specific locations.
This lesson will help the students to move from viewing fractions as parts of a whole to viewing them as a distance from zero. The students will be asked the following questions during the lesson:
- What is a denominator?
- What is a numerator?
- What does a number line density demonstrate?
By the end of the lesson, students should be able to understand the full definition of a denominator being the total number of equal parts between two whole numbers such as 0 and 1 and the numerator being the number of equal parts counted from zero. Hence, the number line density demonstrates that fractions are numbers and that here are infinitely many fractions between any two points.
The students will engage on the following steps:
- They will locate and plot the correct position of proper fractions, improper fractions and mixed numbers on the line.
- They will build fluency in fractional steps by moving forward and backward along the number line.
- Identify that different fractions can occupy the same spot on the number line. For instance 1/2 and 2/4.
- They will use the line to model fraction operations, particularly when adding and subtracting fractions with the same denominator.
The Australian Curriculum considers using number lines for fractions as an evidence- based strategy that helps students to avoid incorrectly applying whole number rules to fractions.
Lesson 3 : Visualise fractions as part of shapes (using Bar Model)
This lesson will focus on visualising fractions as parts of a whole fraction through the bar models method. The students will compare fractions using bar models which provide a firm visual transition from basic shape partitioning to complex fraction operations. The students will learn how to identify numerators or denominators, comparing fraction sizes and finding equivalent fractions. The students will use coloured pencils, whiteboards and Bar Model Templates.
The key learning objectives for this lesson is to represent fractions, improper fractions, proper fractions and mixed fractions by using bar models. The students will understand the numerator and the denominator by using bars to represent a whole divided into equal parts. They will identify equivalent fractions by comparing different bars. The other key learning objectives for this lesson is to demonstrate adding and subtracting fractions with the same denominators.
The students will engage on the following steps :
- They will draw a rectangle on the board and hence explain that the rectangle represents 1 whole.
- The students will demonstrate how the denominator shows the number of equal parts to divide the bar into,
- They will use the numerator to show how many parts to shade,
- They will draw two identical bars, one above the other,
- Then divide the top bar into 2 parts and shade 1 (1/2) and divide the bottom bar into 10 parts The students should be able to tell how many must be shaded to match the top bar 5/10.
Australian Curriculum Descriptors:
- (AC9M5N05) - Expresses that using bar models focuses on visualising whole relationships, addition or subtraction and equivalence of fractions with related denominators. Bar models help students represent fractions, solve word problems by breaking down problems visually and find fractions of amounts.
Lesson 4 : Converting Fractions to common denominators using independent practice and games.
The main focus for this lesson plan is to equip students with skills to find a lowest common denominator to compare and order fractions with unrelated denominators.
Australian Curriculum Descriptor;
(AC9M5N05) - Focuses on adding, subtracting and comparing fractions with unrelated and related denominators.
The students will engage on the following practice ideas:
- The students will find the correct common denominator moving from start to finish on a worksheet created by the teacher.
- They will draw fraction bars or area models for two fractions, for example 1/2 and 1/3 and then use horizontal lines to partition them into the same number of equal parts.
- The students will find the common denominators to compare or combine values using a set of fraction word problems task cards.
The students will engage on the following games:
- The teacher will hand out cards.
- Each student will flip two cards to create a fraction. The students in the game must convert both fractions to a common denominator to find out who's larger. The winner will take the cards.
- The students will write the equivalent fractions quickly by pulling out a bundle of 3 to 5 random fraction cards from a bag identifying a common denominator for all of the cards.
- We will use online challenges as well that are interactive such as Math Playground or Splash learn for digital common denominator puzzles.
Process
Teaching fractions in Year 5 students involves focusing on mixed numbers, equivalent fractions and unlike denominators. The students transition from basic visual models to more complex abstract operations. The students will be given the definition of fraction which according to (Siemon, Warren, & Beswick, 2020), he states that fractions tend to be regarded as positive ,possibly because they are most commonly described as a part of a whole or as the results of physical dividing a collection or quantity into a given number of equal shares.
The students will need to be taught in a way that will make them proficient in their level before moving on to the next level. The Year 5 students will gain the knowledge of Math learning trajectories which have three parts. The parts are explained below as according to (Sarama & Clements 2009):
- Goal -These will include the big ideas of Math. The students will aim on achieving their desired outcomes that will also represent their endpoint developmental path.
- Developmental path - This consists of levels of thinking leading to achieving the mathematical goal. The developmental path describes a typical learning route children should follow in developing understanding of and skill in particular mathematics topics. Sarama & Clements (2009 ) also states that learning trajectories are important because young children's ideas and their interpretations of situations are different from those of adults, hence teachers must interpret what the child is doing and thinking and attempt to see the situation from the child's view.
- Instructional tasks- These are designed to help children learn the ideas and practice the skills needed to master that level. The teacher will use the instructional task to promote students'growth from one level to the next level (Sarama & Clements 2009).
The teacher will use area models such as rectangles and circles to show 5 halves as two circles an one half. The students will be taught how to convert a mixed number for example 2 1/4 to an improper fraction which is 9/4, this will be done by multiplying the whole number by the denominator and adding the numerator.
Evaluation
According to the Australian Curriculum, the strand for teaching fractions is number. The children are expected to develop a deeper understanding of improper fractions, mixed numbers, ordering, equivalence and adding and subtracting fractions with related denominators.
The following evaluation will be done on the Year 5 students:
- The teacher will check for conceptual understanding to see if the students understand fractions as numbers even when using number lines.
- Can students generate and identify equivalent fractions and order fractions with related denominators?
- Evaluate if the students are able to add and subtract fractions with the same or related denominators this includes mixed numbers.
- Evaluate if the students apply fraction knowledge to real world problems for example calculating total amounts.
Conclusion
To conclude this lesson, the teacher will have a reflective discussion with the students. The teacher will ask students about the strategies that made it easier to compare or simplify the fractions given? The teacher will use think boards, this is where students Make it, Draw it, and Write it.
The teacher will give out small activities for the students to do, to gauge their understanding. For example, the students can compare two fractions with the same denominator or they can identify an equivalent fraction. The teacher will solidify the lesson by showing how fractions work in the real world.
The teacher together with the students will recap the lesson. They will have the students brainstorm scenarios that happen in the real world that would result in a fraction. The students can identify objects in the classroom and describe them using fractions. For example the bookshelf is 1/3 full of books.
Credits
- Australian Curriculum, Assessment and Reporting Authority [ACARA]. (n.d.). The Australian Curriculum: Mathematics
https://www.australiancurriculum.edu.au
- Siemon, D., Warren E., & Beswick K. (2020). Teaching Mathematics Foundations to Middle Years. Oxford University Press Australia & New Zealand. Retrieved from
https://CDU.leganto.exlibrisgroup.com/leganto/public/61CDU_INST/citatio…
- Clements, D. H., & Samara, J. (2004). Learning Trajectories in mathematics education. Mathematical thinking and learning. Taylor & Francis Journals Auto- Holdings Collection.
https://CDU.leganto.exlibrisgroup.com/leganto/public/61CDU_INST/citatio…
- Sarama, J.,& Clements, D. (2009). Teaching Math in the Primary Grades: The Learning Trajectories Approach. YC young children, 64 (2). Retrieved from
https://CDU.leganto.exlibrisgroup.com/leganto/public/61CDU_INST/citatio…
Teacher Page
This Year 5 Learning Fractions lesson is focused on deepening understanding through converting between improper fractions and mixed numbers, identifying equivalent fractions, adding and subtracting fractions with different denominators. Clements & Samara, (2004) describes the three research findings of teaching challenges and interests in maths. They state that the importance of maths learning in the Primary Grades, all children's potential to learn maths, teachers need to understand children's learning development ( Clements & Samara, 2004). The following approaches can be used to implicate teaching in primary school:
- Know and use learning Trajectories,
- include a variety of instructional activities, so that the children can invent new strategies and build new knowledge.
- Use a combination of teaching strategies such as to discuss a problem with a group, follow up by having children work in pairs, and then have children to share solution strategies.
Year 5 Key Learning Objectives:
- Name, identify and write equivalent fractions using multiplication/ division and diagrams.
- Order, recognise and convert improper fractions and mixed numbers.
- Divide by the denominator and multiply by the numerator to calculate fractions of amounts.
- Use denominators that are multiples of the same number to add and subtract fractions.
The teacher will activate prior knowledge by modelling teaching strategies such as finding a common denominator for addition and subtraction.