## Task Teacher Guide

Be sure to read the teacher guide prior to running the task. When you’re ready to run the task, use the tabs at the top of the page to navigate through the lesson.

### In This Task…

Students will continue exploring representing, comparing and ordering fractions through a curious task and a consolidation prompt that will act as purposeful practice.

### Intentionality…

The purpose of the Day 2 activities is to reinforce key concepts from Day 1. Students will engage in a string of related problems through a math talk and will have an opportunity to complete independent purposeful practice.

The math talk and purposeful practice serve to develop a deeper understanding of the following * big ideas*.

**Number:**

- Fractions can be represented in a variety of ways;
- How you partition the whole determines the fractional unit (i.e.: partitioning a whole into 6 parts will result in 6 sixth parts);
- As you partition a whole into more parts, the smaller the size of each part;
- As you partition a whole into more parts, the larger the unit fraction (called the denominator in later grades);
- In order to compare two or more fractional quantities, the whole must be the same;
- The count (called the numerator in later grades) indicates the number of parts relative to the number of parts in the whole indicated by the unit fraction (the denominator); and,
- Fractional amounts exist between whole numbers.

## Math Talk

### Related String of Problems

Ask students to cut a piece of paper into strips of equal length.

One-by-one, we’ll lead students through a string of related problems that will give them an opportunity to build their fluency and flexibility with partitioning a whole into fractional parts.

Partition each bar model into...

halvesthirdsfourthsfifths

Note that this activity may take longer than you anticipate, especially as students continue to grapple with folding paper into thirds and fifths. This is an important skill for students to build their spatial reasoning and their conceptual understanding related to partitioning - a key concept for fractions.

You might encourage students to have more strips ready and to abandon strips that have been folded incorrectly to avoid any unnecessary confusion or misconceptions.

Note that the fraction strips created through paper folding in this math talk will be useful for the Sense Making portion of the lesson.

Next, prompt students to:

Create a “truth” or a “lie” about two fractions and then ask a friend whether it is a truth or a lie.

For example:

“3 fourths is greater than 4 fifths.”

Is this a truth or is it a lie?

Use your fraction strips to help you convince your friend.

You can decide how many “truths” and “lies” you’d like students to create about fractions.

## Spark

### Setting The Context

Show students * this video*:

While you’re welcome to do a notice and wonder routine with your students, you might also simply build the storyline of the Wooly Worm Race to another category involving hares.

### Estimation: Prompt

After building the storyline about how four hares were competing in the * longest jump* competition, you can now share our prompt to get students estimating:

How far do you think the winning hare jumped?

We can now ask students to make an estimate (not a guess) as we want them to be as strategic as they can possibly be. This will force them to use spatial reasoning alone to try and come up with an initial estimate and to share it with their neighbours by trying to articulate why they believe their prediction is reasonable.

As we did in Day 1, we have intentionally left out the length of the jumping area in order to encourage students to use fractional thinking rather than measuring in standard units. It is clear that the winning hare jumped less than the total distance of one whole jumping area.

Consider asking students to think about a distance that would be "too low" and a distance that would be "too high" before asking for their best estimate in order to help them come up with a more reasonable estimate.

Let them chat with their neighbours and challenge them to an estimation duel or a math fight.

### While Students Are Estimating...

**Monitor** student thinking by circulating around the room and listening to the mathematical discourse.

Encourage students to use precise mathematical language and positional language (in front, behind, on top…) to articulate their defense.

If students’ estimates are unreasonable, encourage them to select a manipulative to represent the length of the jumping pit (perhaps linking cubes, a paper strip, etc.).

While we hope to engage in learning around fractions including partitioning a whole to represent fractional amounts for the purpose of comparing, we will * welcome *students who share their estimates in decimal form and

*impose the use of fractions on them. Because the calculator is introduced to students so early in their mathematical journey, many students default to expressing fractional amounts as decimals without necessarily having a deep conceptual understanding of how they work. Later in this lesson and in this unit, we hope to continue building the conceptual underpinnings necessary for becoming flexible with fractions and we encourage you to attempt making connections to decimals and percentages if students bring them up when estimating or sharing their thinking. If students*

**will not***estimate using decimals, be sure to help students by using appropriate language utilizing place value and the fractional language associated.*

**do**Encourage students to share.

After students have had an opportunity to share their best guess, tell them that you will now be sharing the distances jumped by the top 4 finishers from the championship competition.

## Sense Making

### Crafting A Productive Struggle: Prompt

Share the distances jumped by the top 4 finishers from the championship competition:

Notice that the unit fraction is * expressed in words* rather than standard-notation. We will work towards standard-notation throughout this unit, but for the time being, we want to emphasize the fractional unit. You will also notice that we are only working with a limited number of denominators, and they have been strategically selected.

**Prompt students to:**

Represent the distance each hare jumped to help you compare and order from longest jump to shortest jump.

## During Moves

### While Students Are Productively Struggling....

**Monitor** student thinking by circulating around the room and listening to the mathematical discourse. **Select** and **sequence** some of the student solution strategies and ask a student from the selected groups to share with the class from:

- most accessible to least accessible solution strategies and representations;
- most common/frequent to least common/frequent strategies and representations; or,
- choose another approach to selecting and sequencing student work.

The tools and representations you might see students using to convince their peers and/or the teacher include:

- Paper folding
- Concrete manipulatives such as fraction strips or relational rods
- A pictorial representation such as a bar model
- A number line (possible, but very unlikely if in a primary classroom)

Have students share their strategies and reasoning for how to represent the distance each hare jumped and how they know the ranking order of all four contestants. Ask them to convince you and their peers that their answer is correct by sharing mathematical models.

Discuss their strategies and elicit student thinking during your consolidation to build off of their current prior knowledge and understanding rather than “fixing” or “funnelling” student thinking to a strategy and/or model that does not connect to their strategy and/or approach.

### Student Approach #1: Paper Folding

I used the fraction strips that we created during the math talk to help me quickly determine how far each hare jumped and then to rank them.

It was easy to just shade in how many equal parts of the whole were shown in the table.

### Student Approach #2: Drawing a Bar Model

I drew 4 of the same length rectangles to represent the distance of the jumping area for the 4 hares.

I used my markers to help me figure out where to draw the lines to make equal parts for each of the jumpers.

After colouring in the number of parts for each jumper, I noticed that the list in the table was already in order from 1st to 4th!

## Next Moves

### Consolidation

During today’s consolidation, the goal is to revisit the power of the concrete and pictorial linear model as a tool to represent, order, and compare fractions. Rather than leading students towards the idea of finding a common denominator (which could be beyond their level of conceptual understanding at this point), these models are an effective tool to reason through this scenario and demonstrate the skill of comparing and ordering fractions with unlike denominators.

The consolidation is your opportunity to once again address any misconceptions that may exist such as students who may have struggled to equi-partition the whole, or to be mindful of the consistently congruent distance of the whole jumping area. Students may need more time to develop this spatial awareness in order to use the pictorial models effectively.

Consider viewing this silent solution animation video to help you prepare for consolidation:

Here’s the final frame of the animation:

### Reveal

After consolidating learning by making connections using student generated solutions, you can share the following reveal video:

A still frame of all four jumpers is as follows:

Go back to student estimates from the **Spark** portion of the lesson to determine who had the closest estimate. Be sure to celebrate the student or students who were closest so that students realize that there was value in sharing with the group.

Did students use decimal notation and/or decimal language and/or percentages to share estimates at the beginning of the lesson?

Consider extending from here to give students an opportunity to explore and make connections with comparing and converting between benchmark fractions, decimals and percentages.

### Reflect

Provide students an opportunity to reflect on their learning by offering this consolidation prompt to be completed independently.

**Consolidation Prompt:**

In the grasshopper category of the longest jump competition, there were 3 contestants. The length of their jumps are recorded in the table below.

Represent the fraction of the whole jumping area each grasshopper jumped to help you compare and order from longest jump to shortest jump.

We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

## Resources and Downloads

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## PowerPoint Slides

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## Explore The Entire Unit of Study

This Make Math Moments Task was designed to spark curiosity for a multi-day unit of study with built in purposeful practice, and extensions to elicit and emerge mathematical models and strategies.

Click the links at the top of this task to head to the other related lessons created for this unit of study.

### Cut a piece...

*Cut a piece of paper into strips.*

*6 to 10 strips would be helpful.*

### Fold...

*Fold one of the strips into halves.*

### Fold...

*Fold another strip into thirds.*

### Fold...

*Fold another strip into fourths.*

### Fold...

*Fold another strip into fifths.*

### Truths and lies...

*Create a “truth” or a “lie” about two fractions and then ask a friend whether it is a truth or a lie.*

*For example:*

*“3 fourths is greater than 4 fifths.”*

*Is this a truth or is it a lie?*

*Use your fraction strips to help you convince your friend.*

### Consolidation Prompt

*In the grasshopper category of the longest jump competition, there were 3 contestants. The length of their jumps are recorded in the table below.*

*Represent the fraction of the whole jumping area each grasshopper jumped to help you compare and order from longest jump to shortest jump.*