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 Day 2 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).

Instructions for students

In this part of the lesson, you and the members in your small group will be extending what we’ve experienced so far for combining fractions.

You’ll be working in visibly random groups on a task. Let’s create those groups and you’ll move into them once we’ve had a chance to introduce the task.

Let’s all gather around one of the dry-erase boards in the classroom.

At the end of the last lesson, you were working on describing relationships between pairs of fractions.

(Prompt 1): So, let’s use the knowledge and skills you gained. Let’s say that I want to combine two, or more, fractions to a sum of 1.

What could some of these fractions be? And how do you know?

Teacher Moves

Using either a deck of cards or random group generator, create visibly random groups (VRGs) of 3. You can read more about VRGs here.

Instead of providing the task entirely in text and/or immediately, try establishing a narrative or story to develop the problem with students.

Gather students together in one, central location for this. In subsequent lessons, intentionally choose different locations in the classroom to hold whole-class meetings so that you’re de-fronting (p1-11) your classroom practice.

Hold off on recording students' suggestions at this point. Try to keep this part of the lesson delivery short with instructions being given orally.

Tip: If students are unsure how to respond to “And how do you know?” provide an opportunity for Turn-and-Talk or to redress the question as “How could you show that your fractions add to 1?” (This might invite students to describe tools and representations they would use.)

Continue to encourage students to visualize, think aloud with one another (Turn-and-Talk), and share with the group for Prompts 2 and 3.

(Prompt 2): What if the sum you were trying to get is less than 1? What might you say about the fractions you’re combining?

Alright, let’s get crazy now! What if we wanted a sum that was close to but not exactly equal to 2? What might you say about the fractions you’re combining?

Great sharing and discussion, everyone!

Alright, before you move into your groups, remember that it’s one marker per group and remember our protocols for collaborating in small groups.

Here’s your task:

• (Prompt 1) Start by creating a model for 2 4/10.

Teachers Moves:

When working on vertical non-permanent surfaces (VNPS), it’s typical for groups to have but one marker. And the person speaking is not doing the writing. This protocol helps to ensure that one person is not doing all of the thinking. Where you notice that much of the writing is being done by one student, walk over to the group and direct them to pass it onto another group member.

Space the delivery for each part (of 3) of the task.

Encourage students to draw and include any representations on their vertical, dry-erase boards. Drawing in a vertical space will allow for greater mobility of knowledge. In addition to using VNPS, students can continue to access both physical and/or virtual manipulatives to support their thinking.

Tips:

• When students are experiencing difficulty, remind them that they can use their autonomy to check in with other groups to find inspiration for a next step.

• This prompt will serve to layer your observations with students’ conversations. I.e., There is an opportunity to develop correspondence between what has been shown and what students are explaining. This will also allow you to reconsider where you might make some changes in the student work you select, sequence and share during the math congress (see “Consolidation of Learning”).

• This prompt can also serve as a key prompt to ask students during whole-class consolidation through a math congress.

• What might students be saying?

  • E.g, I can make equivalent fractions using 12, 15.

  • E.g., I can see there are 2 tenths inside each fifth.

  • E.g., What I noticed is one fraction started at 110 and grew by 110 until it got to 1210. The other fraction started at2310 and grew less by 110 until it got to 1210. I figured out 1210 is half of 2410 so any more tries and I would just find the same pairs only reversed (shown below, mathies notepad file - download. Help using downloaded file.).

Opportunity for Differentiation:

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.

To use visual and concrete models to further develop our reasoning about how fractions can be combined through addition with a focus on the use of linear models.

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

A virtual number line can be accessed here.

Taking stock of the information students have brought forward (i.e., through “Starting Learning” and “Active Learning”), it’s now time to connect their learning back to the learning goal and draw out some success criteria (anticipated and incidental).

Learning Goal / Success Criteria

To help students make connections and draw out success criteria, consider leading students through a math congress.

Math Congress:

Using a small sample of student work collected (e.g., pictures taken), these examples could be--one at a time and sequenced--projected onto a central display. Alternatively, since students were writing on VNPS, teachers could number particular dry-erase boards to represent the sequence in which the congress unfolds. This would honour a teacher’s efforts to de-front their classroom practice.

The teacher and students would then shift their attention to the various surfaces, one after the other. In the case where students were working with virtual manipulatives or using physical manipulatives in other spaces, teachers would also sequence this into the gallery that comprises the congress.

Throughout the congress, focus on encouraging student sharing--explaining their thinking, listening to others, and asking questions of their peers.

The following discussion prompts can be used throughout to help start and guide student sharing and conversation (the first prompt is the same used towards the end of “Active Learning”).

Discussion Prompts:

1. Take a look back at the fraction pairs you’ve created. What were some of the patterns or connections you saw between the different pairs of fractions?

2. How did these patterns or connections help you to add fractions together?

To start, focus on an example that is more representative of the place to which all students were able to get to. This could involve an example like Example 1.

Next, teachers should move to examples that are more complex, yet continue to align with any hints or extensions they offered to students during the active learning component of the lesson. That is, draw on two or more examples that focus on adding fractions with unlike denominators and/or examples that demonstrate how students were able to model and add fractions flexibly (and evidenced by the representations they used). These examples could involve those in Example 2 and Example 3.

Lastly, as the sharing comes to a close, remind students that across all the examples and through conversations, they were focused on using visual and concrete models to further develop their reasoning about how fractions can be added together (Learning Goal). Along the way, they noticed that they could use their models to help them add fractions--be they cases where the denominators were the same, denominators were unlike, or they were working with mixed numbers.

If teachers sense that their students are prepared to move a bit further--i.e., towards a formal means for adding fractions--the consolidation might end differently.

Note: Again, it’s important not to push for a formal means to add fractions--especially when your assessment points to further consolidation with informal means of combining fractions is necessary.

To close out this portion of the lesson differently, see what students will say or do when asked how they would add fractions together.

Discussion Prompts:

3. How do you add fractions together with like denominators?

Students might offer up their own example or teachers can choose to provide an example different from what was included in the “Active Learning” portion of the lesson.

As students share their methods, if reference is not given to a visual model (linear or area model), represent students’ explanations and numeric examples, accordingly, so that students who are not yet ready for a purely numeric argument are supported.

4. How do you add fractions together with unlike denominators?

The key to addressing fraction addition with unlike denominators is to further partition parts of a whole so that each whole has the same number of partitions. In the case of the “Active Learning” example of 2410, 1, 2, 4 and 5 multiplication facts are important. There are some cases where partitions can be merged, thus reducing a fraction to having the same number of larger-sized partitions.

During the Mind's On prompt, 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.

While students are working in their groups, teachers can circulate to observe and monitor conversations for providing timely, descriptive feedback; flexibly using direct instruction to highlight key ideas and address any misconceptions; and preparing to consolidate learning by relating students’ thinking and understanding to anticipated success criteria and those observed incidentally.

Throughout consolidation, return to further develop and refine success criteria. These shared ideas and language will empower students, as they continue to assess their own learning and can be used as descriptive feedback by both teachers and students in subsequent lessons.

SEL Self-Assessments (English) and Teacher Rubric