In this video lesson, students consolidate their equation writing and solving skills. They solve a variety of equations with different structures. Then they match equations to situations and solve them. Students may choose any strategy to solve equations, including drawing diagrams to reason about unknown quantities, looking at the structure of the equation, or doing the same thing to each side of the equation. Students choose efficient tools and strategies for specific problems, helping them develop flexibility and fluency in writing and solving equations.
In Grade 5, students recognized volume as an attribute of solid figures. They measured volume by packing right rectangular prisms with unit cubes and found that determining volume was the same as multiplying the edge lengths of the prism (5.MD.C.3, 5.MD.C.4). Students extend this knowledge to Module 5, Topic C where they continue packing right rectangular prisms with unit cubes; however, this time the right rectangular prism has fractional lengths (6.G.A.2). In Lesson 11, students decompose a one cubic unit prism in order to conceptualize finding the volume of a right rectangular prism with fractional edge lengths using unit cubes. They connect those findings to apply the formula V = lwh and multiply fractional edge lengths (5.NF.B.4). In Lessons 12 and 13, students extend and apply the volume formula to V = The area of the base times height or simply V = bh, where b represents the area of the base. In Lesson 12, students explore the bases of right rectangular prisms and find the area of the base first, then multiply by the height. They determine that two formulas can be used to find the volume of a right rectangular prism. In Lesson 13, students apply both formulas to application problems. Topic C concludes with real-life application of the volume formula where students extend the notion that volume is additive (5.MD.C.5c) and find the volume of composite solid figures. They apply volume formulas and use their previous experience with solving equations (6.EE.B.7) to find missing volumes and missing dimensions.
In Module 4, Topic G, students are introduced to the fact that equations have a structure similar to some grammatical sentences. Some sentences are true: “George Washington was the first president of the United States.” or “2 + 3 = 5.” Some are clearly false: “Benjamin Franklin was a president of the United States.” or “7 + 3 = 5.” Sentences that are always true or always false are called closed sentences. Some sentences need additional information to determine whether they are true or false. The sentence “She is 42 years old” can be true or false depending on who “she” is. Similarly, the sentence “x + 3 = 5” can be true or false depending on the value of x. Such sentences are called open sentences. An equation with one or more variables is an open sentence. The beauty of an open sentence with one variable is that if the variable is replaced with a number, then the new sentence is no longer open: it is either clearly true or clearly false. For example, for the open sentence x + 3 = 5:
If x is replaced by 7, the new closed sentence, 7 +3 = 5 is false because 10 ≠ 5.
If x is replaced by 2, the new closed sentence, 2 + 3 = 5 is true because 5 = 5.
From here, students conclude that solving an equation is the process of determining the number(s) that, when substituted for the variable, result in a true sentence (6.EE.B.5). In the previous example, the solution for x + 3 = 5 is obviously 2. The extensive use of bar diagrams in Grades K–5 makes solving equations in Topic G a fun and exciting adventure for students. Students solve many equations twice, once with a bar diagram and once using algebra. They use identities and properties of equality that were introduced earlier in the module to solve one-step, two-step, and multistep equations. Students solve problems finding the measurements of missing angles represented by letters (4.MD.C.7) using what they learned in Grade 4 about the four operations and what they now know about equations.
