Reflection: Developing a Conceptual Understanding Quiz + The Constant of Proportionality in Equations and Tables - Section 2: Guided Practice

 

I’ve written reflections previously about simplifying quiz days and trying to do too much last year. This is another lesson I made sure to break apart from the quiz. This year I made many changes to my scope. I spend more than one lesson on unit rate, including its association with the constant of proportionality. I also spend longer with ratio tables, asking students to draw and complete ratio tables for each word problem they complete. It was important for me to spend more time on developing the concept and strategy used to answer these types of questions. It paid off in the end, students were making connections on their own by the end of the unit and essentially discovering the equation y = kx and graphing easily on the coordinate plane. When students have ample time to explore the concept of unit rate, they can understand the abstract and algebraic representation of a proportional relationship more easily.

I used several quotient word problems at the beginning of this unit to review and/or develop two skills: 1) to divide whole numbers and fractions/decimals, and 2) to identify the divisor and dividend in a word problem. Correctly identifying whether to multiply a ÷ b or b ÷ a is important when we develop the understanding of unit rate and the idea that two different unit rates can be calculated within one context. For example:

Sam runs 15 miles in ¾ of an hour.

Unit rate per mile:  ¾ ÷ 15
Unit rate per hour: 15 ÷ ¾

 

We also spend time using white boards to write equivalent expressions to build understanding of equivalent ratios (instead of proportions). Once we start discussing equivalent ratios I introduce ratio tables as a strategy and ask students to draw one for any ratio/rate problem they complete. As students get used to drawing ratio tables for any problem they complete, they begin to understand that relationships between the values. It also helps them identify the constant of proportionality when given a table.

Toward the end of the unit I ask them to think about an equation that will summarize all the information on the table. Why is an equation a better tool than a table to use when solving problems involving rates?

 

Differentiation: Some students struggle to find the dividend, divisor and quotient, and furthermore, struggle to understand the relationship between the three. I use highlighters so that students can identify each of these. I also have them use their highlighters and consistent colors for their notes so that they have a resource to reference. 

  Developing a Conceptual Understanding: Developing Understanding: Tables
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Quiz + The Constant of Proportionality in Equations and Tables

Unit 5: Ratios and Proportional Relationships
Lesson 9 of 21

Objective: Quiz + SWBAT identify k, the constant of proportionality given a function or table.

Big Idea: students work on a quiz and then work with partners to use equations to answer questions and create and equation by identifying the constant of proportionality in a table

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