• MA.5.AR.1 Algebraic Reasoning

Solve problems involving the four operations with whole numbers and fractions.

MA.5.AR.1.2 Solve real-world problems involving the addition, subtraction or multiplication of fractions, including mixed numbers and fractions greater than 1. Example: Shanice had a sleepover, and her mom is making French toast in the morning. If her mom had 2 1/4 loaves of bread and used 1 1/2 loaves for the French toast, how much bread does she have left?

Benchmark Clarifications:
Clarification 1: Instruction includes the use of visual models and equations to represent the problem.

Purpose and Instructional Strategies
The purpose of this benchmark is to continue the work from Grade 4 (MA.4.AR.1.2/1.3) 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 ¾ yards of rope and he gives half of the rope to a friend. How much rope does Mark have left?” expects students to find ½ of ¾, or multiply ½ x ¾ to find the product that represents how much is given to the friend. On the other hand, “Mark has ¾ yards of rope and gives ½ yard of rope to a friend. How much rope does Mark have left?” expects students to take ½ yard from ¾ yard, or subtract ¾ - ½ to find the difference. Encourage students to look for the units in the problem (e.g., ½ yard versus ½ 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.
o For example, “Mark has 3/4 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 1/3 of 3/4, or multiply 1/3 x 3/4 , and then to find the difference to find how much Mark has left. On the other hand, “Mark has 3/4 yards of rope and gives 1/3 yard of rope to a friend. How much rope does Mark have left?” only requires finding the difference 3/4 - 1/3 .

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