##
* *Reflection: Exit Tickets
Writing The Constant of Proportionality Equation - Section 4: Exit Ticket

After review the exit ticket, I realized I needed to do a better job making sure everyone was able to read values on a graph. Several students who seemed to be doing fine when working with a neighbor had difficulty reading the scale of the graph of JM's Mowing Rates. They do not realize that the y-axis is scaled in increments of 20. I am seeing the coordinates as (2, 99) because 99 is one line below 100, when it should be (2, 80). The other thing that I need to emphasize is that if a graph shows a proportional relationship (this one does) then each rate of y/x should evaluate to the same value. I have students correctly finding something like (10, 400) but then writing (2, 100); this is not a proportional relationship.

So in summary, I should: 1) do a better job making sure ALL students know how to interpret scales/intervals on the x- or y-axis of a graph; 2) make sure students realize that in the graph of a proportional relationship all rates of y to x should simplify to the same value.

*Improving Exit Ticket Results*

*Exit Tickets: Improving Exit Ticket Results*

# Writing The Constant of Proportionality Equation

Lesson 10 of 12

## Objective: SWBAT determine and write the constant of proportionality as an equation in the form m=y/x

## Big Idea: Students will write constant of proportionality equations from problems represented in graphs.

*40 minutes*

#### Introduction

*10 min*

I will begin with the essential question. This question will be immediately answered as we review the vocabulary for the lesson. My students should now be very familiar with unit rates but the term constant of proportionality will be brand new. When defining the term, I will provide a simple example: 15 dollars for 3 sandwiches. I'll will write it on the board as m = 15 / 3. I will ask the class what the unit rate is. They should all confidently say $5 per sandwich!

Next we will go through a model problem for the lesson. I will pick 4 ordered pairs and actually plot a point and label the ordered pairs on the graph for clarity. This will help make sure students are not confusing the x and y coordinates. In part ii, I will explicitly write the equation in the form of m equaling the y-coordinate over the x-coordinate, then I will simplify.

Part iii will be a discussion question where students must explain their thinking (**MP3**).

Part iv is meant to assess that students understand the meaning of the points on the graph. So, x = 0.5 has a y value of 2 which means half of a dollar equals 2 quarters or 0.5 / 2 = 4.

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#### Guided Problem Solving

*10 min*

The guided problem solving mirrors the example problem from the previous section. The scale of the y-axis may cause some problems for several students. Students will see labeled values of 50, 100, 150, etc but may have difficulty interpreting the unlabeled values. I will ask: how many intervals (or spaces) are there between 0 and 50? When they answer 5, I'll ask what must be the value of each interval. They should more readily say 10 now. If not I will have to show an easier example by drawing a number line by two's from 0 to 10 but only labeling the 0 and the 10.

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#### Independent Problem Solving

*15 min*

The first question of this section follows the same structure as the previous problems.

The second question put the onus of graphing on the students. I will pass out rulers here so that students can connect the points to make a neatly drawn straight line. It will be fun to explore values (2, 6) or (1, 3) as they were not actually graphed yet they have the same values as the graphed points.

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#### Exit Ticket

*5 min*

Before beginning the exit ticket, we will recall that the constant of proportionality can be written as the ratio y/x.

The first two problems are the heart of the lesson. Students should be able to successfully complete both of these.

The question in part iii, will be covered in another lesson, so this problem is not as critical for now. This question will be a bit harder than what students saw in the lesson because it is more difficult to locate the x-coordinate 1 due to the scale of the graph. Students who answer this question correctly will have a very good understanding of the constant of proportionality as it relates to graphs.

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- LESSON 1: Proportional Relationships of Whole Numbers
- LESSON 2: Proportional Relationships With Decimals
- LESSON 3: Proportional Relationships With Fractions
- LESSON 4: Finding Distances on Maps
- LESSON 5: Scaling a Recipe
- LESSON 6: Determine Equivalent Ratios - Scale Factor Between Ratios
- LESSON 7: Determine Equivalent Ratios - Scale Factor Between Terms
- LESSON 8: Determine The Graph of a Proportional Relationship
- LESSON 9: Determine Equivalent Ratios - Common Unit Rate
- LESSON 10: Writing The Constant of Proportionality Equation
- LESSON 11: Writing Equations for Proportional Relationships
- LESSON 12: The Distance Formula