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 of 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

(This is lesson 3 of a 3 part lesson)

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).

Based on the needs of all learners, any one or more of the following can be used as next steps for students and drawn out,

accordingly, over time.

Further Consolidation:

Deliberate Practice - Metacognition and Reflection:

Have students create their own meaningful note (sample template)--i.e., a note that they would write to their future selves about adding fractions. Younger students and/or those who have not experienced this type of task will need support in the early stages. These types of notes take into account existing, visible work and the process can be supported by exemplars. Templates that list the topics and put constraints on the amount of space for writing can be helpful. Teachers might also look at other media to support student expression.

• Important: Meaningful notes hold meaning for the learner--that is, they are for the student, not the teacher. It’s important that students know that their notes are not being used in an evaluative sense. When it comes to deliberate practice and meaningful notes, it’s important that teachers carefully position providing descriptive feedback to students. Teachers might choose to refer the student to their notes in support of student-teacher conferencing (Opportunity for Assessment) of other learning experiences.

  • Example:

    • Student: “Hmm...I feel like I get this, but there’s something that’s not right.”

    • Teacher: “Let’s take a look back at your meaningful notes to discuss if there’s something there that could help with this new task.”

  • Note: If there is something ‘missing’, this can be an opportunity to offer a student some guidance.

• As this lesson is drawn over 3 days (at least), students can take time at the end of each lesson to make additions to their meaningful notes. The sample template provides other areas where notes on different operations with fractions can be recorded.

Deliberate Practice - Conceptual Understanding and Skill Development:

Option 1: The following prompt can be used to further support students’ reasoning with fractions as well as teachers’ triangulation of evidence over time (e.g., Day 4 lesson prompt or at various points in time over the year).

Prompt: The sum of two, different fractions is a little less than 2. Identify pairs of fractions that have this property.

Option 2: Check Your Understanding Questions (Liljedahl, 2019)

Teachers can design a sample of questions relevant to their students to check their understanding of concepts and practice skills.

Tips:

• The questions are not intended to be checked for completion or for assigning a mark: they are for student self-assessment. Students can check their answers against a key.

• Students can work on these on their own or in groups--either on VNPS, their desks or elsewhere.

• Important: Every lesson/day does not necessarily require that students work on Check Your Understanding questions. As per teachers’ ongoing assessment, the decision to use these questions relate to student readiness and ability to practice correctly.

  • E.g., Each day, specific questions could be shared with students following the lesson or the full sample could be shared at the end of the Day 3 lesson.

  • A description of the questions for each day is provided below. Teachers can craft examples according to their students’ experiences.

• Like meaningful notes, it’s important that teachers carefully position providing descriptive feedback to students. Teachers might choose to refer the student to the work they’ve done in support of student-teacher conferencing (Opportunity for Assessment).

  • Example:

    • Student Remarks:

      • “I did a great job solving this problem. Let me show you with this example.”

      • “Hmm...I feel like I got this, but I’m not entirely sure. Can I show you what I did?”

      • “This was a difficult question. I was able to get started, but I’m not sure about next steps.”

  • In each of these cases, the focus of a teacher’s discussion with students is that of thinking and reasoning--i.e., offering support to help students to keep thinking.

Check Your Understanding Questions - Teacher Outline:

Day 1: Students will identify fractions from visual models (linear and area). The fractions they identify in this set of exercises will then be compared. For example, the first example can be structured to show equivalence; the second, showing one fraction being larger (or smaller) than the other.

Day 2: Students are presented a mixed number and must determine pairs of fractions that combine to this mixed number. This question could be complemented by the prompt in Option 1, above, which is more open.

Day 3: In this set of exercises, only numeric representations are presented. Students are asked to find sums for pairs of fractions with the following characteristics. The pairs do not necessarily need to be presented in this order. Use of models is to be encouraged.

a) Fractions with like denominators; their sum, less than one-whole (i.e., proper fraction)

b) Fractions with like denominators; their sum, more than one-whole (i.e., improper fraction)

c) Compare results in parts a) and b) for similarities and differences

d) Express the sum in b) as a mixed number

e) Fractions with unlike denominators; denominators partition easily (lowers complexity of drawing on multiplication facts for composite numbers)

f) Fractions with unlike denominators; denominators do not partition as easily (increases complexity of drawing on multiplication facts for composite numbers)

Guided Learning Groups:

If your observations and conversations are pointing to students having difficulty in either representing or representing and using their models to add fractions, then the challenge of the activities may have been greater than the skills students have already developed. Finding some time between this lesson and the next planned activity to refer to the different examples can further elicit student thinking, foster conversation, and lead towards an increased ability to reason about fraction operations (Opportunity for Assessment).

Challenge:

This particular teacher move could be used during the course of “Active Learning”, “Consolidation of Learning” and/or “Further Consolidation/Next Steps”:

• During the course of active learning, teachers might choose to have students practice justifying their choices within their group or across groups. Teachers could ask any one of the following:

  • E.g., One of the examples is incorrect. Try to find out which one it is; justify with reasons; and explain it to one another.

  • E.g., There is another pattern possible in one of the groups’ work. Try to find it, and explain to each other how you know.

  • E.g., There’s another connection possible between your groups’ work. Try to find it, and explain this connection to each other.

    • Once students have justified and explained, they can be asked to pose problems and challenge other groups to solve them.

• During “Consolidation of Learning”, teachers might choose to create opportunities for small groups to think and discuss further before continuing the math congress. Teachers might ask one or more of the questions above or one like the following:

  • If you were to explain these findings to someone else, what would you say to them? What would you have them try?

• During “Further Consolidation/Next Steps”, teachers might choose to conference with students individually or in small groups to have them justify, explain and pose problems. Teachers might choose to use a reflective prompt to engage students in these three actions.

  • E.g., If you had to create this type of activity, what improvements would you make? Why?

Extension:

During the course of the lesson activities and debrief, some students may have arrived quickly and proficiently to the learning

goal. In this case, the challenge of the activities was either below or at the level of skills already developed by students. In this

case, providing students with opportunities that lie just beyond their development can be productive in deepening their learning.

• For example, teachers might choose to open the task in a way that invites students to think about other fraction operations. In the example below, the position of the unknown invites students to think about the ratio of the sizes of “Hop A” and “Hop B” relative to the length of the closed number line (starting at zero and ending at a specified sum--i.e., a “fractional distance”--either provided by the teacher or suggested by the student).

  • In Diagram 1, the ratio of Hop A to Hop B is 1 to 2. That is,

13 of the fractional distance relates to Hop A; 23, to Hop B. Exploring division of the fractional distance by a whole number (i.e., division by 3 to determine 13 of the fractional distance) not only gives the length of Hop A, but by doubling this length, students can determine the length of Hop B. As a scaffold to meeting students where they are in their development, teachers should settle on a fractional distance that can be split into thirds. Finally, the quantities for both Hops can be verified by using one or more strategies for adding fractions.