##
* *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

**: 15 ÷ ¾**

*per hour*

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*

# Quiz + The Constant of Proportionality in Equations and Tables

Lesson 9 of 21

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

#### Quiz

*30 min*

Students enter silently. Quizzes are on their desks and they are to begin as soon as possible. They are allowed to spread out and sit at empty tables and are given the option to use cardboard dividers and noise canceling head phones. Instructions on the board notify students that they will only have 30 minutes to complete this quiz (a timer will be displayed). At the end of the quiz, all students will be asked to turn in their quizzes, and the Class Work will be distributed.

This quiz will assess student understanding of the vocabulary behind proportional relationships. Calculating unit rates will also be heavily emphasized as well as its purpose in determining the “best deal” or the “better buy”. The constant of proportionality will also be assessed as a way to calculate missing values. The last question will require students to use bar models to solve a word problem. This information will allow me to break up remediation groups into three smaller groups with whom I will need to review the answers to this quiz as well as spiral through other examples:

- Calculating Unit Rate
- Strategies using ratios and proportions to find missing values
- Students selected for this group will include those who answered questions incorrectly and did not show work using any of the strategies taught in the last week

- Reading and Creating bar models to solve problems

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#### Guided Practice

*10 min*

After having 30, all students must turn in their quizzes. Those with extended time mandates on IEPs and 504 plans will be finishing at lunch. Students must return to their original seats and class notes will be distributed. They will all be instructed to copy the 2^{nd} aim off the board.

**SWBAT identify k, the constant of proportionality given a function or table.**

Next, students are asked to read the following directions and paragraph together.

**Directions: Read and fill in the blanks wherever necessary. If you’re not sure, come see me for help**

**Function** is just a fancy word for an equation. The **function **** **describes a constant proportional relationship.** For example**, if Jainney gets $5 each week, the independent variable is __________________ and the dependent variable is _________________. I can write an equation, or a **function, **that will help me calculate the amount of money Jainney can save over any number of weeks:

After 2 – 3 minutes of doing this while I walk around to ensure students are reading through the paragraph, I will note which students have correctly identified the words to fill the blanks and ask them to share out those words so that I can copy them on the board.

I will continue by explaining that a function, or equation, can help us answer many different questions without the use of proportions or cross multiplication. It is important to note the units being compared or included in the equation. In this particular example, Jainne is saving money over a number of ** weeks**. However, some of the questions included in the class work inquire about

**. Getting students to consider and discuss the implications for substitution into our equation utilizes**

*years***MP1 , MP2**, and

**MP3**.

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#### Partnered Practice

*15 min*

Next, a timer is set and students work with partners for 10 minutes to complete the questions on the back of the class notes sheet. Those who get to the last question will be asked to come to the front to complete the question at the SMARTBoard. The work for this last question will be reviewed in the closing section of class. The first question on the back of the class notes will require a numeric expression to be written and simplified. I push ALL students to show this work by writing each of the changes in money horizontally as an expression and solve using order of operations. Answer must be given as a complete sentence:

15 + 3(5) – 40 + 10

= 15 + 15 – 40 + 10

= 30 – 40 + 10

= –10 + 10 = 0

Jainney has 0 dollars left.

In the following questions, students will use their HW answers (and I will have copies of the answers provided on the website) to further explore the location of the constant of proportionality within the equation y = kx. As I walk around the room, I can guide students by reading these questions out loud to them and showing them how to use their notes from the previous day as well as their homework to construct an equation in the form y = kx.

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#### Closing

*5 min*

In the last 5 minutes of class I hope to already have an answer for the last question worked out on the SMARTBoard by a pair of students who will discuss their solution. If this is not possible, I will have scrap pieces of paper available for students to complete the last question in the notes as an Exit Ticket. Either way, I will need to assess whether or not students are able to create an equation from a table to inform my instruction the following week.

Any other questions not completed during class will need to be completed for homework.

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Can you possibly put up the student version of Day 90 so that I can give that to my students. Thanks

| one year ago | Reply##### Similar Lessons

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###### End of Grade Review: Tables, Graphs, and Equations of Proportional Relationships

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- UNIT 1: Integers
- UNIT 2: Operations with Rational Numbers
- UNIT 3: Expressions and Equations - The Basics
- UNIT 4: Multi-step Equations, Inequalities, and Factoring
- UNIT 5: Ratios and Proportional Relationships
- UNIT 6: Percent Applications
- UNIT 7: Statistics and Probability
- UNIT 8: Test Prep
- UNIT 9: Geometry

- LESSON 1: Mastering Rational Numbers
- LESSON 2: Introducing Ratios and Rates in Word Problems
- LESSON 3: Best Buy
- LESSON 4: Ratio Tables
- LESSON 5: Quiz + Graphing Proportional Relationships
- LESSON 6: Bar Models and Ratios
- LESSON 7: Unit Rates and Ratio Tables
- LESSON 8: The Constant of Proportionality
- LESSON 9: Quiz + The Constant of Proportionality in Equations and Tables
- LESSON 10: Put it Together - Equations y = kx
- LESSON 11: Equations and Bar Models + The Constant of Proportionality
- LESSON 12: Identifying the Constant of Proportionality
- LESSON 13: Determine and Interpret Proportionality through Table and Graphs - Day 95
- LESSON 14: Quiz + Similar Figures
- LESSON 15: Scale Drawings
- LESSON 16: Map Scales
- LESSON 17: Let's Dream About a Different Place...
- LESSON 18: Practice Test
- LESSON 19: Unit 5 Test
- LESSON 20: Unit 5 Test Debrief
- LESSON 21: MAP it Out: Part 2