Equations and Bar Models + The Constant of Proportionality

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Objective

SWBAT apply functions and bar models to solve word problems.

Big Idea

students play a game of showdown and discuss solutions that involve bar models and equations in the form y = kx to solve ratio problems

Do Now

10 minutes

Students enter silently according to the Daily Entrance Routine. The  Do Now includes three questions around the concept of proportionality.

The first question will require students to use a proportion or equivalent ratios to find the amount of pool water being leaked per hour. Since the information given is in the form of fractions, I expect some students to struggle putting together the initial ratio of 1/3 gallon : 1/6 hour. For these students I will be prepared with the following feedback:

  • Write a ratio that includes the information given. First draw a fraction bar. Label the top gallons and the bottom hours. Now fill in the fractions given in the problem next to each unit.

Next, I can help students by asking them to set up a proportion. If they do not know how to do this, I will ask them to refer back to their notes to figure this out along with the following guiding questions:

  • What is a proportion made out of? What does it include? (two ratios with an equal sign in between)
  • You are given the rate of water leaving for a sixth of an hour… if you want to know how much is leaving in one hour, ask yourself, what do you multiply by to get from 1/6 to 1 whole hour?

The second question asks students to identify the constant of proportionality from a table. This is a great opportunity to shout kids out if they are referring back to their notes from the previous two lessons. The notes from 3 lessons back detail the way to calculate k, the constant of proportionality. I try to wait until a student identifies these notes on their own, shout them out for doing so independently, and ask them to share with the class the date at the top right hand corner of these notes. By doing this I am placing importance on the independent action of referring back to notes, a skill that I try to have students use to get ready for high school.

The last problem asks students to identify the graph which shows a proportional relationship. The answer is letter D, “a straight line that goes through the origin”. When reviewing this answer, I have all students write this phrase next to this problem. We reviewed this concept through homework and classwork in the past three days. Writing and re-writing this phrase will help students remember and study this concept. It is equally important to push students to explain why the graph is a straight line and why it must begin at the origin. One question I use to push this discussion is:

A proportional relationship can be described using the equation y = kx, where k is the constant of proportionality. Explain how this equation justifies the reason why the graph must be a straight line that goes through the origin.

Answer: In the equation y = kx, no matter what k, the constant of proportionality is, if x = 0, y will also equal 0. Which means the graph of the line should always go through the origin, (0,0).

Guided Practice

10 minutes

I distribute the guided practice paper to students and begin by explaining that there are two strategies we can use when being asked to solve word problems involving ratios:

  • We can use bar models (which we have reviewed before in class but are reviewing once again due to the low performance on last week’s quiz for the question asking students to use bar models)
  • Or we can write an equation that will help us determine future values

 

I model the use of both of these methods with one problem. The tables completed with red fonts included in the worksheet are meant to be completed and drawn by students with assistance from me and any student identified from the results to the Quiz as available helpers. These students will be asked to walk around the room to help others set up their bar models.

In the second example students we will be filling in values into the table together, and then I will ask students to work on the graph, first by identifying the proper scale to be used as well as the proper locations for the number of cats vs. dogs. This will require students recall the x-axis as the horizontal axis and y-axis as the vertical axis. Once a student or group of students have correctly identified these axes, pairs will be given 3 – 4 minutes to graph the values we determined together in the table and draw a line through the points.

We will come back together as a larger group to recall from our notes (link) the process for writing an equation in the form y = kx.

Showdown

20 minutes

After reviewing writing an equation in the form y = kx and answering any student questions, all students will be randomly placed in groups of 4 using a random group generator on the SMARTBoard. Each group will receive white boards, dry erase markers, and erasers for a game of Showdown. Here are the rules:

  • Each person must have a dry erase marker and board. The eraser should be placed in the center of the table.
  • Group members must sit around a table so that they face each other.
  • Each person will also receive a copy of the task which includes four problems on the front and 2 on the back of the sheet.
  • We will play the game with the first 4 problem included on the front of the sheet.
  • At the beginning of each round students will have 3 minutes to independently and silently work out the solutions to each problem on the whiteboard. A timer will be displayed.
  • At the end of these 3 minutes, I will shout “Showdown!” and students must stop writing and show each of their group members their answers on the white board. Markers must be capped and students will have2 minutes to discuss their answers and determine the right answer. They will have 1 minute to write their solution on their paper and place it flat on their table.
  • In each subsequent round, I will be walking around to check the work and answers to the previous problem, awarding achievement points to individuals or groups who have the correct work. This is a great opportunity to reward groups with extra points if they are doing a good job in teamwork. Often t this age, drama can take over and impact the way students work together. I have found that positive reinforcement is best when attempting to encourage cooperation within groups, especially when they are formed randomly.

 

As students work through the showdown game they are using MP3 to discuss and critique each others’ arguments about the solutions to each problem. Fifteen minutes ought to be given to these 4 problems and the game while the remaining 5 minutes can go to answering the 2 multiple choice problems on the back of the Class Work sheet. 

Closing

10 minutes

As we close out the Task section of class I inform students that they must complete all class work for homework which will be turned in for a grade the following day. After making this announcement and giving students the opportunity to write it down into their agendas, I will distribute exit tickets as a final check for understanding. The results to this Exit Ticket will inform my instruction moving forward on students’ abilities to identify proportional relationships in a table, calculate the constant of proportionality given x and y, and writing an equation to describe a proportional relationship.