Big Idea: Addition of Fractions

Expectations:

Number

• B2 use knowledge of numbers and operations to solve mathematical problems encountered in everyday life

• B2.1 use the properties of operations, and the relationships between operations, to solve problems involving whole numbers, decimal numbers, fractions, ratios, rates, and whole number percents, including those requiring multiple steps or multiple operations

• B2.5 add and subtract fractions with like and unlike denominators, using appropriate tools, in various contexts

Social Emotional Learning Skills in Mathematics and the Mathematical Processes

• A1. Throughout this grade, in order to promote a positive identity as a math learner, to foster well-being and the ability to learn, build resilience, and thrive, students will apply, to the best of their ability, a variety of social-emotional learning skills to support their use of the mathematical processes and their learning in connection with the expectations in the other five strands of the mathematics curriculum.

• In this lesson, to the best of their ability, students will learn to think critically and creatively and learn to build relationships and communicate effectively, as they apply the mathematical processes reasoning and proving (develop and apply reasoning skills (e.g., classification, recognition of relationships, use of counter-examples) to justify thinking, make and investigate conjectures, and construct and defend arguments) and communicating (express and understand mathematical thinking, and engage in mathematical arguments using everyday language, language resources as necessary, appropriate mathematical terminology, a variety of representations, and mathematical conventions) so they can make connections between math and everyday contexts to help them make informed judgments and decisions and work collaboratively on math problems - expressing their thinking, listening to the thinking of others, and practising inclusivity - and in that way fostering healthy relationships.

teacher background

Prior to this lesson, students may have had the opportunity

to...

Represent composite numbers as a product of their prime factors, including through the use of factor trees (Gr 6, B2.6)

Represent equivalent fractions from halves to twelfths, including improper fractions and mixed numbers, using appropriate tools, in various contexts (Gr 5, B1.3)

Compare and order fractions from halves to twelfths, including improper fractions and mixed numbers, in various contexts (Gr 5, B1.4)

Describe relationships and show and show equivalencies among fractions, decimal numbers up to hundredths, and whole number percents, using appropriate tools and drawings, in various contexts (Gr 5, B1.7)

Add and subtract fractions with like denominators, in various contexts (Gr 5, B2.5)

Use virtual manipulatives (beneficial, but not a prerequisite)

Have created their own meaningful notes or other meaningful demonstrations of their learning

• Meaningful Notes - Liljedahl (2019) writes: “Notes should consist of thoughtful notes written by students to their future selves. The students should have autonomy of what goes in these notes and how they are formatted and should be based on the work that is already existing on the boards from their own work, another group's work, or a combination of work from many groups” (p1 - 12).

To begin, open the lesson with a Think-Pair-Share strategy to help activate student thinking about representations and models of fractions.

Think (Prompt 1 of 2): Today, we’ll be representing our thinking about fractions in a variety of ways. Show how you could represent a fraction.

Notes:

• Prompt 2 follows later in “Starting Learning” and focuses on “relationships”.

• Teachers might find that student exploration with prompts 1 and 2 can take up much of one, 60-minute class period (Day 1). “Active Learning and Consolidation” and “Further Consolidation and Next Steps” will extend into Day 2 and Day 3.

Tips:

• Distribute a mini dry-erase board, dry-erase marker and wipe to each student. Ensure that students have access to manipulatives.

• Provide students with ample time to prepare one or more representations of their thinking.

• By splitting their board in half and/or using the reverse side, students will be able to record more than one example.

Pair: Have students share their thinking with a partner. Encourage opposite partners to explain how their partner’s thinking represents a fraction.

Tips:

• What are you looking and listening for? What mathematical ideas are students representing through their physical models, pictures, and/or numbers?

  • Relationships: part-whole, part-part, fraction as quotient

  • Models: set (collection of items), linear (number line), area (rectangular)

  • Numerical/Types of fractions:

    • Proper, unit

    • Improper, mixed numbers

• If you notice that the majority of students’ fractions are one type (proper, improper, mixed number), and/or models of one type (set, linear, area), consider prompting students to share another example on a different part of their dry-erase board. For example:

  • “I've noticed and heard lots of discussion about numerical representations of fractions where the numerator is smaller than the denominator. What other examples can you come up with? And what do they mean?”

  • “I’ve noticed and heard lots of discussion about fractions used to represent parts of a region (or shaded parts of an area). How might you also model the use of fractions when measuring length (or travelling a distance)?”

Share: Invite students to share their thinking during a whole-class math conversation.

Tips:

• Record students’ contributions on a vertical surface for reference. With subsequent lessons, you might choose to prepare an anchor chart.

• Ensure that you have a good representation of all students’ mathematical ideas

  • With respect to the goal of students using a linear model, include more examples of this type during the course of the math conversation.

• Invite students to share their representations and how it relates to a fraction.

  • Ask the class how many represented similarly; follow this with those that represented differently. Repeat this process--this time, prompt students to compare the current with the previous representation.

  • Ask which representation they prefer and why. Pay attention to any contextual clues students may provide in their explanations. For example, are they noting a particular constructive use of fractions? Counting unit fractions, measurement (length or hops on a number line; area or how the whole has been divided into congruent, fractional regions), or fraction as a quotient?

Option: Continue the lesson with another Think-Pair-Share strategy (Prompt 2, below).

Tip:

• Use this prompt if you’re interested to gain a better understanding of student thinking about fractions as they relate to relationships or key concepts--i.e., these ideas may not have come about during the first Think-Pair-Share. For example: the whole, equivalency, and comparing and ordering.

Think (Prompt 2 of 2):

• List a pair of fractions and think about how they relate to one another. How would you describe or show this relationship?

• Now, list a second pair of fractions that relate to each other but in a way that is different to how your first pair of fractions relate. How would you describe or show this relationship?

Tips:

• Provide students with time to work with their first pair of fractions before asking them to work with their next example.

• If students are finding it difficult to find relationships, teachers can use some additional and scaffolded prompts to help students move forward:

  • “I wonder how you might compare the fractions you’ve chosen?”

  • “I wonder how you might tell which fraction would come first--say, on a number line?”

  • “I wonder how a physical model of a fraction (area model) might help you compare and order fractions?”

  • “What do the denominators tell you about each fraction and how they relate?”

Opportunities for differentiation:

Earlier, teachers may have chosen to supply their students with any one or more of the following: virtual manipulatives (e.g., number lines, fraction towers/strips, pattern blocks).

At this juncture, teachers might choose to hone in on the use of one or two manipulatives to support students’ thinking and representation in relation to the learning goal.

Recall: The virtual nature of these manipulatives can also serve students in hybrid/online learning models.

Pair: Have students share their thinking with a partner. Encourage opposite partners to explain how their partner’s thinking represents a fraction.

Share: Invite students to share their thinking during another whole-class math conversation.

Similarly, use the sharing tips provided from the first Think-Pair-Share.

Tips:

• Record students’ contributions on a vertical surface for reference.

• Ensure that you have a good representation of all students’ mathematical ideas

  • With respect to the goal of students using a linear model, include more examples of this type during the course of the math conversation.

• Invite students to share an example and describe any relationships.

  • Teachers might consider re-purposing the tips (additional, scaffolded prompts), above to support students’ descriptions, to bring clarity to explanations, and/or to present as questions for other students to respond to (i.e., in relation to the current example being shared).

  • Ask the class how many had identified a similar relationship, and invite another student to share their pair of fractions.

    • Pay particular attention to representations and explanations that allow for fractions to be compared.

      • E.g., using one or more of the following: benchmarks, estimation through reasoning about the size of the whole (i.e., the denominator), common numerators, common denominators, equivalent fractions (involves either the merging or splitting of partitions, which can be modelled using a number line)

    • Through their representations, students will be better supported when it comes time to add pairs of fractions.

    • Encourage students to compare the newest example with the previous.

  • Follow this with those that reported different relationships between their fractions. Repeat this process--this time, prompt students to compare the current with the previous example.

  • Of all the relationships presented, ask what relationships, as well as representations, might be helpful when combining fractions--say, if they needed to be added?

    • Encourage students to provide a rationale: Why? And can we show that with one of our examples?

    • Note: See differentiation note, right.

Wrap up this part of the lesson with a summary (see below). Several aspects of this summary--various representations and connections--may have already been recorded by the teacher on a surface visible to students.

Opportunity for Differentiation:

At this juncture, it’s important not to push for a formal means to add fractions, nor to expect that all students will be ready to add fractions. The “Active Learning (Day 2)” portion of the lesson intends to offer students an opportunity to explore further their understanding of fractions, their representations, and time to explore informally the addition of fractions.

For the time being, teacher-facilitated sharing (i.e., making use of students’ contributions) can offer support for those who may not yet be making connections between fractions and the representations used to model comparisons.

Summary:

So far, you’ve had an opportunity to share your understanding of fractions--what they mean to you, how you can represent them, and how you can describe relationships with pairs of fractions.

Together, we’ve captured many of these ideas here (i.e., the record on the surface visible to students) and we’ll come back to them later--after the next task--to work on creating success criteria together.

In the next part of our lesson, you’ll be thinking about how these representations, including linear models, can help you to combine fractions.

Opportunity for Differentiation

To “lower the floor” (or level of difficulty) while using this strategy, as well as increasing the number of representations and models that students share, teachers might choose to provide students with access to any one or more of the following: physical or virtual manipulatives (e.g., number lines, fraction towers/strips, pattern blocks).

The virtual nature of these manipulatives can also serve students in hybrid/online learning models.

Opportunity for Assessment

Listen for key terms, gestures, explanations, as well as any questions that students are asking of their partner. Your observations may become important later on when providing descriptive feedback and co-creating success criteria with students.

By encouraging partners’ to listen to and review one another’s explanations, teachers are encouraging peer assessment through active listening and giving and receiving feedback.

Listen for key terms, gestures, explanations, as well as any questions that students are asking of their partner. Your observations may become important later on when providing descriptive feedback and co-creating success criteria with students.

By encouraging partners’ to listen to and review one another’s explanations, teachers are encouraging peer assessment through active listening and giving and receiving feedback.

SEL Self-Assessments (English) and Teacher Rubric