Unit Representing Fractions: Day 1: Exploring Representations Junior

MO 25 min

A 20 min

C/D 15 min

60 min

Math Learning Goals

Students will:

  • represent fractions as parts of a whole using a variety of models
  • reason about meaning of a fraction and the relationship between numerator and denominator
  • communicate strengths of different representations for different students and in certain contexts e.g., use of benchmarks to support and refine the meaning of fractions

Materials

  • a selection of manipulatives and tools (rulers, markers)

Minds On...

Pairs Activity

Ask students to show 25 in as many different ways as possible. Insist that students use a variety of manipulatives, tools, materials. Have them record their representations on a piece of chart paper. Circulate, observe, question.

Action!

Whole Group Gallery Walk

Students circulate around the room and review the different representations. Students consider which representation they think most clearly shows the fraction, and indicate their preference by placing a sticky note with their name by their first choice. Ask students to be prepared to discuss any similarities or differences they notice between the representations of 410 and 25.

Whole Group Discussion:

Ask students the following key questions:

  • What did you see that made sense for you? What was it about this model that you found particularly meaningful?
  • What is the role of the numerator in this representation? Where is that shown in the representation?
  • What is the role of the denominator in this representation? Where is that shown in the representation?
  • Explain why this representation is helpful.

Consider the following options for the discussion:

  • highlight those representations that students found most helpful with annotation
  • organize the representations by type (e.g., area models, linear models, symbolic representations, set/discreet models) and label accordingly

Consolidate/ Debrief

Whole Class Anchor Chart

Have students generate an anchor chart based on the types of representations used. They can include key information such as how to identify which type of representation is being used, critical components of the representation, and appropriate uses.

Representations include:

  • Area models
  • Set/Discreet models
  • Linear models

Home Activity or Further Classroom Consolidation

Unit Representing Fractions: Day 2: Introducing and Comparing Models Junior

MO 20 min

A 30 min

C/D 10 min

60 min

Math Learning Goals

Students will:

  • represent fractions as parts of a whole using a variety of models
  • reason about meaning of a fraction and the relationship between numerator and denominator
  • communicate strengths of different representations for different students and in certain contexts e.g., use of benchmarks to support and refine the meaning of fractions

Materials

  • A variety of manipulative materials, including square tiles, 2 colour counters, glass beads, pattern blocks
  • Elmo/document camera

Minds On...

Independent Math Journal

Students respond to the prompt: 38 shows the number of red lollipops Johnny collected on Halloween. How would you represent this fraction? Use materials in the classroom to show the story in two ways. Record your representations in your math log.

Whole Class Discussion

Students share their representations (using Elmo if available). Students explain why they selected that representation and how they understand it to connect to the problem posed.

Highlight the key points of set/discreet models from the student responses (that the number of pieces forms the whole; that the numerator identifies the number of parts that meet one specific criterion; that the parts do not need to be the same size). Discuss why the set/discreet model is an appropriate representation for this problem.

Action!

Individual Task

Students complete BLM 2.1

Pair/Share Discussion

Students share their responses and reasoning to the first question. Encourage students to extend their partner’s thinking by asking clarifying questions and connecting to their own thinking.

Whole Group Discussion

Ask one pair who agreed upon their responses to share their reasoning. Ask if other students had different reasoning but the same answer. Have them share. Allow other students to ask questions and to make connections to their own responses. If some students disagree then ask them to share their reasoning.

Adapted from Comparing and Ordering Number Lines

Consolidate/ Debrief

Whole group Discussion

Ask students to share their stories for the area model and the set/discrete model. Discuss the stories to ensure that they are appropriate and that students understand the connections between the story and the models (i.e., What is the whole? What are the parts?).

Ask the following key questions and record student responses using a Venn diagram. How are the stories of the set model (c) different from the stories of the area model (b)? How are they similar?

Home Activity or Further Classroom Consolidation

BLM 2.1

  1. Suzie says all 3 representations here show the same fraction. Is that true?
    a) linear model
    b) area model
    c) discrete model
  2. Write a short math story for each of these three representations. For example, if I were to use a coloured counter story for c), I could say There are four counters in total. 3 are yellow. I can describe this using a fraction by saying three-fourths of the counters are yellow.

Unit Representing Fractions: Day 3: Focus on Linear Models Junior

MO 5 min

A 45 min

C/D 15 min

60 min

Math Learning Goals

Students will:

  • represent fractions as a number as well as parts of a whole using a number line
  • reason about meaning of a fraction and the relationship between numerator and denominator
  • communicate strengths of different representations for different students and in certain contexts e.g. , use of benchmarks to support and refine the meaning of fractions

Materials

Minds On...

Whole Class Discussion

Refer students back to BLM 2.1 and ask them to look at the linear model (a) again. Prompt students to share connections that they made between the linear model and their understanding of fractions. Students may also wish to share what connections they make between the linear model and other items they are familiar with.

Action!

Whole Class Discussion

Students share their stories for a) from BLM 2.1.

Discuss the stories to ensure all students understand the connection between the model and fraction. Extend their thinking by asking: How are the stories of the area model (b) different from the linear model (number line) in a)?

Whole Class Discussion

Review the critical components of a number line if necessary (see note in sidebar).

Show students a number line with 0 on one end and 1 on the other end. Ask them to consider the following questions:

  • Where would 12 go?
  • Where would 14 and 34 go?
  • Can you show another fraction on this number line?





A number is a linear model that shows distance between points (intervals). It can be open without end intervals or closed with end intervals. The number line is helpful to order and compare fractions.

Consolidate/ Debrief

Individual Math Logs

Ask students to revisit their stories from yesterday. Have them respond to the following:

  • What changes would you make with your stories to better match the representations?
  • Write a new story for the linear model (number line) representation.

Home Activity or Further Classroom Consolidation

Unit Representing Fractions: Day 4: Connecting Number Lines Junior

MO 15 min

A 25 min

C/D 20 min

60 min

Math Learning Goals

Students will:

  • order fractions using a number line
  • reason about meaning of a fraction, including as a number, as a part-whole relationship and as a quotient
  • connect fractions to other number systems, such as decimals and percents

Materials

Minds On...

Whole Class Acting

Each student selects a number (fraction, percent or decimal) and writes it on a sticky note. Students then organize themselves from smallest to largest along the front of the classroom, displaying their sticky note as they discuss the order with their classmates. Once they feel they are in order students read out their numbers from smallest to largest to see if everyone agrees.

Action!

Small Groups Activity

Provide each group with the chart paper with the number line constructed on it and the set of fraction cards (BLM 4.1). Tell them that there has been a mishap and some fractions fell off the number line. Ask them to place the fractions appropriately on the number line.

Small Groups Activity

Have students create their own number lines based on BLM 4.2.



This activity reinforces for students that the intervals on a number line must be equally spaced.

Consolidate/ Debrief

Pairs Practise

Have students complete BLM 4.2.

Collect math logs. Select 5 stories to share for activation in lesson 5.

This could be projected and students could record their responses in their math journals. [note re question 1: students will need to think about thirds and then relate that to 9ths]

Home Activity or Further Classroom Consolidation

BLM 4.1: Number Cards

12 23 34 910
45 68 18 410
12 23 34 910
45 68 18 410
12 23 34 910
45 68 18 410

BLM 4.2 Using Number Lines to Answer Questions

  1. Solve this problem using one or more of the number lines below.

    Two students were throwing snowballs from their fort at a wall.

    Nick’s snowball was thrown from the fort and landed 68 of the way to the wall.

    Nanci’s snowball was thrown from the fort and landed 79 of the way to the wall.

    Which snowball landed closest to the wall?

  2. Write a problem that matches well with a number line model using the following two fractions: 56 and 910

Unit Representing Fractions: Day 5: Focusing on Area Models Junior

MO 20 min

A 20 min

C/D 20 min

60 min

Math Learning Goals

Students will:

  • connect the representation to the fraction meaning within scenarios involving fractions in order to select representations that are most appropriate
  • reason about meaning of a fraction, including as a number, as a part-whole relationship and as a quotient

Materials

Minds On...

Pairs Activity

Distribute a problem from the student generated samples for each pair of students. Have pairs solve the problem on the card.

Whole Class Discussion

Students share how they solved the problem. Have select students share their rational for selecting the strategy/model that they used.

Action!

Individual Activity

Present the following to the class:

I have a new situation for us to think about. I have a page of olden days stamps that are all joined together on a sheet. 34 of the stamps have been damaged by dampness. How many damaged stamps could there be?

Ask students to respond and draw their solutions.

Note: It is important that students understand that the stamps are attached in order to allow for interpretation to be either set(discrete) or area.

Consolidate/ Debrief

Whole Group Discussion

Have students share their responses. Probe their thinking using the following prompts:

  • How did you use fractions to solve the problem?
  • What representation did you use to help?
  • Was it a helpful representation? Why, why not?

Home Activity or Further Classroom Consolidation