Comparing B

# Compare familiar fraction quantities with and without benchmark referents

# Familiar Fractions and Benchmarks Comparisons

#### Description

This is a set of prompts consisting of purposefully paired fractions to elicit the use of various strategies. The prompts may be used on different occasions for either Minds-On activities or for Action tasks depending on student readiness. Repeated practice and exploration in making comparisons between fractions will deepen student understanding. These prompts are presented symbolically and without context. Encourage students to build models/representations and create contexts to support visualization of fractions, which in turn supports meaning-making.

#### Mathematics

Research shows that it is beneficial to spread fraction learning throughout the year and embed it in other strands. These prompts progress in complexity from comparisons of equivalent fractions, to examples that include same denominators, to comparisons of very close fractions with different numerators and denominators. Students are encouraged to develop a range of different strategies and to use them strategically, based on the situation.

#### Curriculum Connections

Students will:

- represent, compare and order fractional amounts using a variety of tools

#### Instructional Sequence

- Partner students and introduce the task. Post the selected prompt (select from options) on the black/whiteboard or interactive whiteboard, or distribute on a handout.
- Provide students time to complete the task. Encourage them to use graph paper, rulers, and manipulatives (concrete or virtual, such as the tools at mathies.ca).
- Have students describe their thinking. Highlight different strategies by intentionally selecting students that solved the task in different ways. Have students identify the similarities and differences between the strategies.

##### Prompt #1

Which is closer to 1 whole: ^{1}⁄_{3} or ^{2}⁄_{9}?
Describe the strategy you used to prove your thinking.

##### Prompt #2

Create a model to represent **either** ^{3}⁄_{2} and ^{2}⁄_{3}
**or** ^{5}⁄_{6} and ^{6}⁄_{5}.

Which fraction of the pair you chose is greater? How do you know?

##### Prompt #3

Order the following fractions from least to greatest:

^{2}⁄_{3}, 2 ^{5}⁄_{6},
^{11}⁄_{4}, ^{4}⁄_{8},
^{3}⁄_{5}

##### Prompt #4

Would you rather have ^{1}⁄_{4} or ^{5}⁄_{12}
of a chocolate bar? Show your thining.

#### Highlights of Student Thinking

Students may:

- construct accurate models to compare two or more fractions
- rely on the algorithm for determining equivalent fractions
- consider only the numerators or only the denominators
- use benchmarks to make estimates for comparison
- consider the size of the unit fractions (as indicated by the denominators)
- consider the proximity of the fraction to 1 by identifying the ‘missing piece’ (complement)
- be purposeful about the strategy for comparison based on the fractions given

#### Key Questions

- Did you think of contexts to help you visualize the fractions? How did this help you?
- Share how you visualized the fractions.
- How did your representation help you to compare the fractions?
- What strategy did you find most helpful? Why?
- What manipulatives could you use to help you?

#### Materials

Make tools available such as paper and markers, grid paper, paper strips for folding, and/or manipulatives such as relational rods.

## Familiar Fractions and Benchmarks Comparisons (Comp B) Teacher Notes: Anticipating Student Responses

These prompts can be used flexibly depending on student readiness, for example, as assessment for learning, activating prior knowledge, learning tasks or assessment of learning. These prompts are presented symbolically and without context in order to allow students to build models/representations and create contexts to support visualization of the meaning of the fractions. The prompts are increasingly complex and consist of purposely-paired fractions to elicit the use of various strategies.

**Prompt #1**

Which is closer to 1 whole: ^{1}⁄_{3} or ^{2}⁄_{9}?
Describe the strategy you used to prove your thinking.

__Teacher Notes:__

Students may use a variety of strategies to answer this task. One example is choosing equivalent fractions that are helpful.

For example, a student might reason that ^{1}⁄_{3} = ^{3}⁄_{9},
and that ^{2}⁄_{9} is closer to 0, therefore ^{1}⁄_{3}
is greater.

**Prompt #2**

Create a model to represent **either** ^{3}⁄_{2} and ^{2}⁄_{3}
**or** ^{5}⁄_{6} and ^{6}⁄_{5}.

Which fraction of the pair you chose is greater? How do you know?

__Teacher Notes:__

Students may construct a reasonably accurate model, such as a rectangle, to show that one fraction is more than a whole so, therefore, the greater fraction.

Here, students need to understand the proportional relationship between the numerator and denominator in each fraction (i.e., the numerator is larger than the denominator), rather than the size of the pieces (denominator), which helps them to identify the larger fraction. The students also need to understand that the size of the whole needs to be consistent when comparing fractions.

**Prompt #3**

Order the following fractions from least to greatest: ^{2}⁄_{3}, 2
^{5}⁄_{6}, ^{11}⁄_{4}, ^{4}⁄_{8}, ^{3}⁄_{5}.

__Teacher Notes:__

Students may use a number line to order fractions. Note that students can use proportional reasoning to place the fractions reasonably accurately without determining the precise location. If the number line is on a piece of paper, students can use paper folding to aid in locating the fractions.

No matter which model is selected, students need to consider the range of the set in order to recognize that they are representing values between zero and one as well as values between two and three, and will have to make a number line or other representation that accommodates these quantities. This may take some trial and error, with students realizing that they have not make their number line long enough part way through the activity (many students and even adults automatically assume that when working with fractions we are always dealing with quantities between zero and one).

**Prompt #4**

Would you rather have ^{1}⁄_{4} or ^{5}⁄_{12}
of a chocolate bar? Show your thinking.

__Teacher Notes:__

Students may use equivalence, benchmarks and/or models to show their thinking. A student may draw a rectangle and partition it in half.

They may then partition one half in half again to locate one fourth.

They may say that one fourth is the same as three twelfths. They position five twelfths between one fourth and one half, since they realize that six twelfths is one half. They may visualize one fourth partitioned into thirds (creating twelfths) to aid with the placement.

This clearly demonstrates that ^{5}⁄_{12} > ^{1}⁄_{4}.