Still more to come…
Secondary Math Teacher and Intermediate Math Coach with the Greater Essex County District School Board leading a Ministry funded 1:1 iPad project called Tap Into Teen Minds. I currently teach at Tecumseh Vista Academy K-12 in the morning and focus on duties for the Middle Years Collaborative Inquiry (MYCI) Project in the afternoon.
Investigation: Going for a Jog
Video discussing the mind buster problem from our section on Direct Variation:
Identify the independent/dependent variables.
Describe the shape of the graph.
Where does it intersect the vertical axis?
Write an equation to find the distance, d, in metres, that Susan jogs in t mins.
Use the equation to determine the distance that Susan can jog in 25 mins.
Consider the distance Susan jogged in 5 minutes.
What happens to this distance when the time is doubled?
What happens to the distance when the time is tripled?
What Is Direct Variation?
A Direct Variation is a relationship between two variables in which one variable is a constant multiple of the other variable.
A video discussing the following related to Direct Variation:
From Homework – Page 242-244 #6
In the following video, Mr. Pearce and the class discuss a question from the homework. This was recorded the next day at the beginning of class with the help of the students. The question asks:
The cost of oranges varies directly with the total mass bought. 2 kg of oranges costs $4.50.
a) Describe the relationship in words.
b) Write an equation relating the cost and the mass of the oranges. What does the constant of variation represent?
c) What is the cost of 30 kg of oranges?
What Is Partial Variation?
A Partial Variation is a relationship between two variables in which the dependent variable is the sum of a constant number and a constant multiple of the independent variable.
A video discussing the following related to Partial Variation:
This video was recorded at the beginning of class. Mr. Pearce and the class take up a question from the previous day to summarize Partial Variation and discuss why the relationships involved in the question are Partial and not Direct variation. The question is:
A theatre company produced the musical Cats. The company had to pay a royalty fee of $1250 plus $325 per performance. The same theatre company also presented the musical production of Fame in the same year. For the production of Fame, they had to pay a royalty fee of $1400 plus $250 per performance.
- Write an equation that relates the total royalties and the number of performances for each musical.
- Graph the two relations on the same grid.
- When does the company pay the same royalty fee for the two productions? (Break Even Point)
- Why do you think the creators of these musicals would set royalties in the form of a partial variation instead of a direct variation?
This video was recorded during a lesson. Students were asked to work with their table groups to answer the following:
Consider the graph of line segment AB to the right.
- Is the slope positive or negative? Explain.
- Determine the rise and run by counting grid units.
- Determine the slope of the line segment AB.
This video shows students how to find the slope of a slide as well as the slope of a roof.
This video recaps the Mind Buster portion of our Sec. 4.3 (5.3) Slope (Part 2) lesson where:
A line segment has one endpoint, A(4, 5) and a slope of -2/3.
Can you find the coordinates of 2 other points on the line?
This video focuses on Slope as a Rate of Change for a linear system of equations. In this case, the question is as follows:
The distance-time graph shows two cars that are travelling at the same time.
- Which car has the greater speed, and by how much?
- What does the point of intersection of the two lines represent?
This video demonstrates How First Differences in a Table of Values Can Identify Linearity of a Relationship by comparing two relationships from a table of values. The question from the video asks:
Each table shows the speed of a skydiver before the parachute opens. Without graphing, determine whether the relation is linear or non-linear.
- Case when there is no air resistance, and
- Case when there is air resistance.
Today, students had a discussion about the four representations of a linear relationship; a description, table, graph and equation. From this, students took a table to determine the type of variation, the slope/rate of change/constant of variation, initial value and a description to match. They then created an equation in y = mx + b form to summarize their understanding.
Take the following table of values and create the following representations to match:
- A Description,
- A Graph, and
- An Equation.
Students will work in their table groups to create a table of values and graph the relationship on a grid. They will then complete the following related to the problem:
Identify the independent/dependent variables.
Describe the shape of the graph.
Where does it intersect the vertical axis?
Write an equation to find the distance, d, in metres, that Susan jogs in t mins.
Use the equation to determine the distance that Susan can jog in 25 mins.
Consider the distance Susan jogged in 5 minutes.
What happens to this distance when the time is doubled?
What happens to the distance when the time is tripled?
The class will engage in a discussion using Apple TV as a means to quickly display different student work from across each table group.
The teacher will lead a discussion about direct variation and attempt to demystify the definition:
A Direct Variation is a relationship between two variables in which one variable is a constant multiple of the other variable.
The discussion will cover the following direct variation topics:
A video encapsulating most of the topics covered in our classroom discussion about Direct Variation.
The teacher will scaffold the students through the lesson using a gradual release of responsibility approach. The teacher will begin the first two tasks with the students and then let groups solve the problems in the way they feel most comfortable.
The direct variation task questions target the students’ ability to identify a direct variation in a word problem, from an equation, table and graph.
Tasks can be shared out via Apple TV to show multiple methods of solving each task question.
Students will answer the consolidation questions that indicate whether students can identify a direct variation in the form of an equation, table and a graph as well as whether they can determine the constant of variation when a linear equation is a direct variation.
When complete, students will submit their consolidation answers in the Google Drive Form below:
All Unit 4 Videos can be found on the Math Videos – Unit 4 Modelling With Graphs page.
Questions From Homework:
In the following video, Mr. Pearce and the class discuss a question from the homework. This was recorded the next day at the beginning of class with the help of the students. The question asked is:
The cost of oranges varies directly with the total mass bought. 2 kg of oranges costs $4.50.
a) Describe the relationship in words.
b) Write an equation relating the cost and the mass of the oranges. What does the constant of variation represent?
c) What is the cost of 30 kg of oranges?
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Learn how to solve a simple equation involving two-steps:
Solve:
4m – 6 = 12
Learn how to solve a two-step equation:
Solve:
7x – 4 = 10
Solve:
3 + 4m + 5m = 21
Solve:
4y – 13 = -6y + 7
Solve:
4(k – 3) = 2 – (2k – 6)
Solve:
(1/3)(x – 2) = 5
and
16 = [3(v + 7)]/2
Solve:
[3(z - 5)]/4 = 7
Solve:
3 = [2(n + 7)]/5
A trapezoidal backyard has an area of 100 m^2. The front and back widths are 8 m and 12 m, as shown in the diagram.
What is the length of the yard from front to back?
Rearrange the following equation for ‘h’
V = p(r^2)h
The sum of two consecutive even integers is -134. Find the numbers.
The length of Laurie’s rectangular swimming pool is triple its width. The pool covers an area of 192 m^2.
If Laurie swims across the diagonal and back, how far does she travel?
Students will complete the Problems With Homework Form to indicate any problems they had with the work from the previous day as well as to self-assess where they are in terms of their own learning in the course.
Students will have an opportunity to look at a couple questions from the previous day. Mr. Pearce will record his iPad screen and then post to YouTube after class for student reference.
Students will work in their table groups to write an equation and solve the equation that could correspond to the following problem:
The number of hours that were left in the day was one-third of the number of hours already passed.
How many hours were left in the day?
Students are instructed to answer the question in at least two ways.
Table groups will share-out via Apple TV to allow students the opportunity to view other ways to solve the problem.
Students will continue working collaboratively in their table groups to solve Unit 3 Review – Solving Equations questions.
Each group will be responsible for sharing solutions periodically throughout the duration of the class.
Students will consolidate their learning acquired during this lesson as well as throughout the unit by selecting questions from the digital textbook that target problem areas for each student.
Mr. Pearce will display possible consolidation questions over the Apple TV and he will ask students to rank the questions on a scale from 1 to 10, where 1 is easy and 10 is difficult. Students are then suggested to avoid spending too much time on questions that they find “easy” and focus on questions they find moderately to very difficult to ensure they maximize their time spent on math in preparation for the Unit Test next day.
All Unit 3 Videos can be found on the Math Videos – Unit 3 Solving Equations page.
Mr. Pearce and students during class solved #10 from the previous day’s homework:
The sum of two consecutive even integers is -134. Find the Numbers.
The length of Laurie’s rectangular swimming pool is triple its width. The pool covers an area of 192 m^2.
If Laurie swims across the diagonal and back, how far does she travel?
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