Unit 3: Unit Rates and Percentages Lesson 14: Solving Percentage Problems In …
Unit 3: Unit Rates and Percentages Lesson 14: Solving Percentage Problems
In previous lessons, students saw that a percentage is a rate per 100. They were provided with double number line diagrams to develop this understanding and to solve problems involving percentages. In this lesson, students solve similar problems but with less support. Because double number lines are not provided, students have opportunities to choose approaches that seem appropriate. Drawing a double number line is still a good strategy, but students may opt for tables or even more abbreviated reasoning methods.
Unit 3: Unit Rates and Percentages Lesson 9: Solving Rate Problems In …
Unit 3: Unit Rates and Percentages Lesson 9: Solving Rate Problems
In previous lessons, students have used tables of equivalent ratios to reason about unit rates. In this lesson, students gain fluency working with unit rates without scaffolding (MP1). They choose what unit rate they want to use to solve a problem, divide to find the desired unit rate, and multiply or divide by the unit rate to answer questions. They may choose to create diagrams to represent the situations, but the problems do not prompt students to do so. The activity about which animal ran the farthest requires students to use multiple unit rates in a sequence to be able to convert all the measurements to the same unit.
Unit 2: Introducing Ratios Lesson 13: Tables and Double Number Line Diagrams …
Unit 2: Introducing Ratios Lesson 13: Tables and Double Number Line Diagrams
In this lesson, students explicitly connect and contrast double number lines and tables. They also encounter a problem involving relatively small fractions, so the flexibility of a table makes it preferable to a double number line. Students have used tables in earlier grades to identify arithmetic patterns and record measurement equivalents. In grade 6, a new feature of working with tables is considering the relationship between values in different rows. Two features of tables make them more flexible than double number lines:
On a double number line, differences between numbers are represented by lengths on each number line. While this feature can help support reasoning about relative sizes, it can be a limitation when large or small numbers are involved, which may consequently hinder problem solving. A table removes this limitation because differences between numbers are no longer represented by the geometry of a number line. A double number line dictates the ordering of the values on the line, but in a table, pairs of values can be written in any order. 5 pounds of coffee cost $40. How much does 8.5 pounds cost?
At this point in the unit, students should have a strong sense of what it means for two ratios to be equivalent, so they can fill in a table of equivalent ratios with understanding instead of just by following a procedure. Students can also always fall back to other representations if needed.
Unit 3: Unit Rates and Percentages Lesson 10: What Are Percentages? This …
Unit 3: Unit Rates and Percentages Lesson 10: What Are Percentages?
This lesson is the first of two that introduce students to percentages as a rate per 100 (MP6) and the ways they are used to describe different types of situations.
Percentages are commonly used in two ways:
To describe a part of a whole. For example, “Jada drank 25% of the bottle of water.” In this case, the percentage expressing the amount consumed is not bigger than 100% because it refers to a part of a whole.
To describe the size of one quantity as a percentage of another quantity. For example, “Jada drank 300% as much water as Diego did.” In this case, there is no restriction on the size of the percentage, because the percentage is describing a multiplicative comparison between two quantities.
In the first usage there is a single quantity and we are describing a part of it; in the second usage we are comparing two quantities. Students may have prior exposure to percentages, but are likely to have only encountered the first usage and might not be able to make sense of percentages above 100% or those used in comparative contexts. This lesson exposes students to both applications of percentages.
Money is the main context for exploring percentages in this lesson and the warm up asks students to convert between dollars and cents providing an opportunity for the teacher to assess students’ current abilities.
For the first several lessons exploring percentages, double number lines are the primary representation presented to students. This choice is intended to strongly communicate that we are working with percent rates, and that students can and should use all of the reasoning they have developed to deal with equivalent ratios and rates when dealing with rates per 100. That said, if students prefer to reason using tables or by multiplying or dividing by unit rates, they should not be discouraged from doing so.
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