Standard #: MA.5.AR.1.2

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.5.FR.2.1
• MA.5.FR.2.2
• MA.5.M.1.1
• MA.5.GR.2.1
• MA.5.DP.1.1

 

Terms from the K-12 Glossary

• NA

 

Vertical Alignment

Previous Benchmarks

• MA.4.AR.1.2
• MA.4.AR.1.3

 

Next Benchmarks

• MA.6.NSO.2.3

 

Purpose and Instructional Strategies

The purpose of this benchmark is to continue the work from grade 4 where students began solving real-world with fractions, and prepares them for grade 6 (MA.6.NSO.2.3) where they will solve real-world fraction problems using all four operations with fractions (MTR.7.1). 

• Students need to develop an understanding that when adding or subtracting fractions, the fractions must refer to the same whole.
• During instruction, teachers should provide opportunities for students to practice solving problems using models or drawings to add, subtract or multiply with fractions. Begin with students modeling with whole numbers, have them explain how they used the model or drawing to arrive at the solution, then scaffold using the same methodology using fraction models. 
• Models to consider when solving fraction problems should include, but are not limited to, area models (rectangles), linear models (fraction strips/bars and number lines) and set models (counters) (MTR.2.1). 
• Please note that it is not expected for students to always find least common multiples or make fractions greater than 1 into mixed numbers, but it is expected that students know and understand equivalent fractions, including naming fractions greater than 1 as mixed numbers to add, subtract or multiply. 
• It is important that teachers have students rename the fractions with a common denominator when solving addition and subtraction fraction problems in lieu of the “butterfly” method (or other shortcut/mnemonic) to ensure students build a complete conceptual understanding of what makes solving addition and subtraction of fractions problems true.

 

Common Misconceptions or Errors

• When solving real-world problems, students can often confuse contexts that require subtraction and multiplication of fractions. 
  • For example, “Mark has $\frac{\text{3}}{\text{4}}$ yards of rope and he gives half of the rope to a friend. How much rope does Mark have left?” expects students to find $\frac{\text{1}}{\text{2}}$ of $\frac{\text{3}}{\text{4}}$, or multiply $\frac{\text{1}}{\text{2}}$ × $\frac{\text{3}}{\text{4}}$ to find the product that represents how much is given to the friend. On the other hand, “Mark has $\frac{\text{3}}{\text{4}}$ yards of rope and gives $\frac{\text{1}}{\text{2}}$ yard of rope to a friend. How much rope does Mark have left?” expects students to take $\frac{\text{1}}{\text{2}}$ yard from $\frac{\text{3}}{\text{4}}$ yard, or subtract $\frac{\text{3}}{\text{4}}$ − $\frac{\text{1}}{\text{2}}$ to find the difference. Encourage students to look for the units in the problem (e.g., $\frac{\text{1}}{\text{2}}$ yard versus $\frac{\text{1}}{\text{2}}$of the whole rope) to determine the appropriate operation. 
• Students may believe that multiplication always results in a larger number. Using models when multiplying with fractions will enable students to generalize about multiplication algorithms that are based on conceptual understanding (MTR.5.1). 
• Students can have difficulty with word problems when determining which operation to use, and the stress of working with fractions makes this happen more often. 
  • For example, “Mark has $\frac{\text{3}}{\text{4}}$ yards of rope and he gives a third of the rope to a friend. How much rope does Mark have left?” expects students to first find $\frac{\text{1}}{\text{3}}$ of $\frac{\text{3}}{\text{4}}$, or multiply 1/3 × $\frac{\text{3}}{\text{4}}$, and then to find the difference to find how much Mark has left. On the other hand, "Mark has $\frac{\text{3}}{\text{4}}$ yards of rope and gives $\frac{\text{1}}{\text{3}}$ yard of rope to a friend. How much rope does Mark have left?” only requires finding the difference $\frac{\text{3}}{\text{4}}$ − $\frac{\text{1}}{\text{3}}$.

 

Strategies to Support Tiered Instruction

• Instruction includes opportunities to identify the appropriate operation to use in a real- world problem that requires addition, subtraction, or multiplication of fractions. The teacher guides students to identify the units in the problem for clarification on which operation is appropriate. 
  • For example, the teacher displays and reads the following two problems: 
    • “Ganie has $\frac{\text{7}}{\text{8}}$ of a bar of chocolate left and gives half of what she has to her friend Sarah. How much of a whole chocolate bar does she have left?” 
    • “Ganie has $\frac{\text{7}}{\text{8}}$ of a bar of chocolate left and she gives $\frac{\text{1}}{\text{2}}$ of the original bar of chocolate to her friend Sarah. How much of her chocolate bar does she have left?” (See illustration below) 
  • The teacher uses questioning and prompting to have students identify what operations must be used to solve each problem. The teacher asks students to share what they notice about each problem (e.g., the similarities and the differences), placing emphasis on the units (e.g., “half of the amount of chocolate that Janie has in the first problem vs. $\frac{\text{1}}{\text{2}}$ of the whole chocolate bar” in the second problem). The teacher guides students to identify that in the first problem, they will need to multiply $\frac{\text{7}}{\text{8}}$ × $\frac{\text{1}}{\text{2}}$ and in the second problem, they will need to subtract $\frac{\text{7}}{\text{8}}$ − $\frac{\text{1}}{\text{2}}$ to solve. Students solve using models. 

$\frac{\text{7}}{\text{8}}$×$\frac{\text{1}}{\text{2}}$=$\frac{\text{7}}{\text{16}}$

• For example, the teacher displays and reads the following problem: "Tia has $\frac{\text{3}}{\text{8}}$ yards of ribbon and she gives half of the ribbon to a friend. How much ribbon does Tia have left?” The teacher uses questioning and prompting to have students identify what operation must be used to solve the problem. The teacher asks students, “Did Tia give half of the ribbon or half a yard of ribbon to her  friend?” Emphasis is placed on the units (e.g., half of the whole ribbon vs. $\frac{\text{1}}{\text{2}}$ yard of ribbon) while guiding students to identify that they will need to multiply $\frac{\text{3}}{\text{8}}$ × $\frac{\text{1}}{\text{2}}$ to solve. Students solve using the area model and counters. The cells with both color counters indicate the numerator in the solution. This is repeated with similar word problems, using frequent guiding questions to support student understanding.  

• Instruction includes opportunities to use models when solving problems that involve multiplication of fractions to increase understanding that multiplication does not always result in a larger number. The use of models when multiplying with fractions will enable students to generalize about multiplication algorithms that are based on conceptual understanding. 
  • For example, the teacher displays and reads aloud the following problem: “Rosalind spent $\frac{\text{2}}{\text{3}}$ of an hour helping in the garden. Her sister spent $\frac{\text{1}}{\text{2}}$ the amount of time as Rosalind did helping in the garden. How much time did Rosalind’s sister spend helping in the garden?” Students solve the problem using an area model. The teacher uses questioning to help students draw a model to represent the problem. This is repeated with similar word problems involving multiplication of fractions. 

• For example, the teacher displays and reads aloud the following problem: "Astrid spent $\frac{\text{7}}{\text{8}}$ of an hour reading her book. Elliot spent $\frac{\text{1}}{\text{3}}$ the amount of time as Astrid did reading. How much time did Elliot spend reading?” Students solve using the area model and counters. The cells with both color counters indicate the numerator in the solution. The teacher uses questioning to help students draw a model to represent the problem. This is repeated with similar word problems involving multiplication of fractions, using frequent guiding questions to support student understanding.

 

Instructional Tasks

Instructional Task 1 (MTR.7.1) 

Rachel wants to bake her two favorite brownie recipes. One recipe needs 1 $\frac{\text{1}}{\text{2}}$ cups of flour and the other recipe needs $\frac{\text{3}}{\text{4}}$ cups of flour. How much flour does Rachel need to bake her two favorite brownie recipes? 

 

Instructional Task 2 (MTR.7.1) 

Shawn finished a 100 meter race in $\frac{\text{3}}{\text{8}}$ of one minute. The winner of the race finished in $\frac{\text{1}}{\text{3}}$ of Shawn’s time. How long did it take for the winner of the race to finish? 

 

Instructional Items

Instructional Item 1 

Monica has 2 $\frac{\text{3}}{\text{4}}$ cups of berries. She uses $\frac{\text{5}}{\text{8}}$ cups of berries to make a smoothie. She then uses $\frac{\text{1}}{\text{2}}$ cup for a fruit salad. After she makes her smoothie and fruit salad, how much of the berries will Monica have left? 

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.