Standard #: MA.3.FR.1.2

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.3.FR.1.1 
• MA.3.FR.1.3 
• MA.3.FR.2.1 
• MA.3.FR.2.2

Terms from the K-12 Glossary

• Number Line

Vertical Alignment

Previous Benchmarks

• MA.2.FR.1.1 
• MA.2.FR.1.2

Next Benchmarks

• MA.4.FR.2.1 
• MA.4.FR.2.2

Purpose and Instructional Strategies

• During instruction, teachers should have students practice representing fractions using manipulatives (e.g., fraction strips, circles, relationship rods), visual area models (e.g., partitioned shapes) and on a number line. Manipulatives, visual models and number lines must extend beyond 1 so that students can represent fractions greater than one (MTR.2.1, MTR.5.1). 
• In instruction of MA.3.FR.1.1, students learn that unit fractions are the foundation for all fractions. MA.3.FR.1.2 builds understanding that all fractions, including fractions equal to and greater than one, decompose as the sum of unit fractions. 
• In understanding fractions are numbers, students make connections about whole number operations that will allow them to perform operations with fractions in later grades. For example, understanding fractions as numbers allows students to reason that $\frac{\text{2}}{\text{3}}$ + $\frac{\text{2}}{\text{3}}$ = $\frac{\text{4}}{\text{3}}$ in Grade 4 because we are adding together a total of 4 parts that are each one-third in size (MTR.5.1).

Common Misconceptions or Errors

• Students can misconceive that fractions equal to and greater than 1 can also be represented as the sum of unit fractions (e.g.,$\frac{\text{5}}{\text{2}}$ = $\frac{\text{1}}{\text{2}}$ + $\frac{\text{1}}{\text{2}}$+ $\frac{\text{1}}{\text{2}}$ + $\frac{\text{1}}{\text{2}}$ + $\frac{\text{1}}{\text{2}}$ ). Flexible representations of models (e.g., rectangular area models that align with number lines) help students connect their understanding of fractions and how they are decomposed into unit fractions.

Strategies to Support Tiered Instruction

• Instruction includes modeling how fractions are decomposed. Using fraction circles, students build $\frac{\text{4}}{\text{4}}$  and then see that there are 4 pieces that make up the whole circle. 
  • Example: 

• Instruction includes more than one model so that students can experience and connect fractions in multiple ways. Flexible representations of models (e.g., rectangular area models that align with number lines) help students connect their understanding of fractions and how they are decomposed into unit fractions. 
  • Example: 

• Students then apply this understanding to fractions greater than one. Using fraction circles, students build $\frac{\text{8}}{\text{4}}$  and then see that there are 8 pieces that make up two whole circles. 
  • Example: 

• Instruction includes folding and/or cutting pre-made shapes into halves. Students physically bend the paper into halves and then label the pieces. Instruction includes relating the pieces back to the numerator and denominator and then connecting it to the equation. Using multiple shapes with the same denominators will solidify basic fraction understanding. Instruction should progress with other denominators. 
  • Example:

Instructional Tasks

Instructional Task 1 

• Part A. How many one-fifth sized parts are added together to equal 1 whole? Prove your thinking with a visual model or number line. 
• Part B. How many one-fifth sized parts are added together to equal 2 wholes? Prove your thinking with a visual model or number line.

Instructional Items

Instructional Item 1 

• Represent the fraction  8/3 as the sum of unit fractions. 

Instructional Item 2 

• Which of the following expressions models  7/4 ?
  • a. $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ 

• b. $\frac{\text{1}}{\text{7}}$ + $\frac{\text{1}}{\text{7}}$ + $\frac{\text{1}}{\text{7}}$ + $\frac{\text{1}}{\text{7}}$ 

• c. $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$+$\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ + $\frac{\text{1}}{\text{4}}$ 

• d. $\frac{\text{1}}{\text{7}}$ + $\frac{\text{1}}{\text{7}}$ + $\frac{\text{1}}{\text{7}}$ + $\frac{\text{1}}{\text{7}}$+$\frac{\text{1}}{\text{7}}$ + $\frac{\text{1}}{\text{7}}$ + $\frac{\text{1}}{\text{7}}$