Comparing A

# Equivalency Comparisons #### Description

This is a set of prompts consisting of purposely 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. 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

1. 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.
2. 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).
3. Have students describe their thinking. Highlight different strategies by purposely choosing students that solved the task in different ways. Have students identify the similarities and differences between the strategies.
##### Prompt #1

Are 26 and 412 equal? Show your thinking.

##### Prompt #2

Show that 34 is the same as 1520.

##### Prompt #3

Is 26 equal to 39? Show your thinking.

#### 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

2. Share how you visualized the fractions.
4. What strategy did you find most helpful? Why?

#### Materials

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

## Equivalency Comparisons (Comp A) 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

Are 26 and 412 equal? Show your thinking.

Teacher Notes:

Using a model, students may further partition sixths into twelfths (or merge twelfths to create sixths) to generate equivalent fractions. For a number line, students may explain that two 16 hops are equivalent to four 112 hops.

##### Prompt #2

Show that 34 is the same as 1520.

Students may create representations to prove that these fractions are equal. Students who have been taught the algorithm for determining equivalent fractions by multiplying the numerator and denominator by a number may multiply both the numerator and denominator of 34 by 5 (which is equivalent to multiplying by one since they are multiplying 34 by 55).

Using a rectangular model, as in the example below, we can further partition each fourth into fifths to create twentieths. This allows students to understand how the algorithm (multiply by 55) acts on the 34. It also helps to reinforce that the two fractions, although written using different fractional units (denominators), represent the same area (or quantity) and are therefore equivalent.

##### Prompt #3

Is 26 equal to 39? Show your thinking.

Teacher Notes:

This prompt is slightly more difficult than Prompt 2, because students are not told that the fractions are equivalent. As well, there is not an obvious multiplier that will generate the second fraction.

Students may use stacked number lines to determine equivalence.