How to Find the Surface Area of a Cone by Explaining How to Find the Lateral Area. Activity includes a video as well as images to use during the lesson.

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In the past, I have always engaged my students with interactive activities to help them understand how the formula for **volume of a prism** is related to the **volume of a pyramid** with the same base and height. However, until this year, I wasn’t quite sure how to explain the **surface area of a cone formula** without simply revealing it.

This year, I decided to get the students to engage in an activity to attempt finding the **formula for surface area of a cone** collaboratively and I was pleased with the result.

- Finding area of composite figures,
- finding the surface area of a cone, and
- investigating geometric relationships.

The activity portion simply involves different size pieces of paper for students to experiment making cones with. In this particular case, I created two different types of cut:

- A square piece of paper cut diagonally in half, and
- A rectangular piece of paper with a length significantly longer than the width.

Students in their table groups will use the square piece of paper to create a cone. Ensure students create the cone with paper “edge to edge” as to avoid overlapping and thus losing some of the surface area.

The teacher will then ask students to use the piece of paper to:

Find the surface area of the cone you created.

Are there any patterns or geometric relationships that you’ve found?

Some prompts for students:

- Do you believe that there is a pattern or relationship? Discuss with your elbow partner.
- What information do you feel will be important for you to find the surface area?

Often times, when using a square piece of paper, students predict that the **surface area of a cone** can be found using **Pi** times the **radius-squared** for the base plus a **quarter** of the **area of the base** since the net will appear to be a quarter of a circle. Or, algebraically:

Area of the Base + Lateral Area

= (Pi)r^2 + (Pi)(r^2)/4

However, if you give them a rectangular piece of paper, some students may realize that this will not work for every cone. This may lead to a deeper discussion about what we could do to their predicted formula to make it work for another cone. For example, if the lateral area represents 1/6 of the area of the base, then this might lead to students exploring further.

Students can then head to a Surface Area of a Cone Interactive Manipulative from Math Open Reference. The interactive widget allows users to change the height and radius of a cone to help them understand the formula.

Students may notice that when the **slant height** is equal to the **radius** of the cone, the area of the base and the lateral area are equal. The formula for surface area of a cone in this situation could be determined as follows:

SA = Area of the Base + Lateral Area

= (Pi)(r^2) + (Pi)(r^2)

= 2(Pi)(r^2)

As the **slant height** increases without changing the dimensions of the **radius**, the lateral area becomes larger than the area of the base. Rather than having a radius equal to the slant height for the lateral portion of the cone, the slant height is increased causing the cone to get taller. This changes the formula slightly:

SA = Area of the Base + Lateral Area

= (Pi)(r^2) + (Pi)(r)(s)

In this activity, we usually introduce the volume of a cone as well, since we have already demonstrated the volume relationships between prisms and pyramids. Students apply their knowledge of this volume relationship:

Volume of a Cone = Volume of a Cylinder / 3

or

Vcone = Area of the Base x Height / 3

Students can consolidate their learning by summarizing their understanding of the surface area of a cone.

Surface Area and Volume of a Cone

- MPM1D – Principles of Mathematics, Grade 9 Academic
- MFM2P – Foundations of Mathematics, Grade 10 Applied

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]]>MPM1D Unit 5 Review Videos for the unit on Analysing Linear Relations by Students of Tecumseh Vista Academy and Mr. Pearce's Mathematics Class.

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]]>Still more to come…

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]]>Math Videos - Unit 4 Modelling With Graphs in MPM1D Grade 9 Academic Math. Learn about direct variation, indirect variation and modelling with graphs.

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]]>**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**:

- How we can relate direct variation to proportional reasoning,
- what a direct variation looks like in a table and graph, and
- how to find the
**constant of variation,**.*m*

**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**:

- What a partial variation looks like in a table,
- what a partial variation looks like in a graph,
- what a partial variation looks like in an equation, y = mx + b and
- how to find the
**constant of variation,**.*m*

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 ofFamein the same year. For the production ofFame, 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
eachmusical.- 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
riseandrunby counting grid units.- Determine the
slopeof 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.

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]]>4.1 Direct Variation in MPM1D Grade 9 Academic Math. Learn how to identify linear equations that are direct variation and find the constant of variation, m.

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]]>- construct tables of values, graphs, and equations, using a variety of tools (e.g., graphing calculators, spreadsheets, graphing software, paper and pencil), to represent linear relations derived from descriptions of realistic situations;
- compare the properties of direct variation and partial variation in applications, and identify the initial value (e.g., for a relation described in words, or represented as a graph or 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:

- How we can relate direct variation to proportional reasoning,
- what a direct variation looks like in a table and graph, and
- how to find the
**constant of variation,**.*m*

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.

Videos recorded before, during or after class will be listed below:

**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?

DOWNLOAD BLANK WORKSHEET (PDF) |
DOWNLOAD WORKSHEET SOLUTIONS (PDF) |

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]]>Unit 3 Solving Equations Math Videos for MPM1D Grade 9 Math. Learn how to solve equations including multi-step, distribution, fraction coefficients.

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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?

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]]>3.6.R Solving Equations Unit 3 Review | MPM1D Grade 9 Academic Math. Learn how to solve equations including multi-step, distribution, fraction coefficients.

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]]>- simplify numerical expressions involving integers and rational numbers, with and without the use of technology;
- add and subtract polynomials with up to two variables [e.g., (2x – 5) + (3x + 1), (3x2y + 2xy2) + (4x2y – 6xy2)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
- multiply a polynomial by a monomial involving the same variable [e.g., 2x(x + 4), 2×2(3×2 – 2x + 1)], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil);
- expand and simplify polynomial expressions involving one variable [e.g., 2x(4x + 1) – 3x(x + 2)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);
- solve first-degree equations, including equations with fractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies);
- rearrange formulas involving variables in the first degree, with and without substitution (e.g., in analytic geometry, in measurement);
- solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods.

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.

Here are the two problems we solved at the beginning of class from your last day’s homework…

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?

DOWNLOAD BLANK WORKSHEET (PDF) |
DOWNLOAD WORKSHEET SOLUTIONS (PDF) |

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]]>Great YouTube spoof of the New Apple iPhone 5 which jokes about the longer, not wider, iPhone 5 screen size. Great for a laugh!

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]]>Great iPhone 5 parody…

Anyone out there have the iPhone 5 yet? Would be interested to hear your reviews. Is it worth it? Do you notice the speed of the new faster processor?

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]]>Parents and students of Tecumseh Vista Academy can watch a tutorial video demonstrating the great features of the iPad Technology Blog, Tap Into Teen Minds.

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]]>A Khan Academy video which explains why dividing by zero is left undefined by making some real world connections.

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]]>The “Dance Dare” made famous by the Ellen Show is now hitting schools across North America by storm! Check out the Assumption College Secondary version of “Dance Dare” while interviewing students about what they thought their favourite part of the year was… Here’s a shout-out to my sister, who appears at the 2:25 mark! Go […]

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]]>The “Dance Dare” made famous by the Ellen Show is now hitting schools across North America by storm!

Check out the Assumption College Secondary version of “Dance Dare” while interviewing students about what they thought their favourite part of the year was…

Here’s a shout-out to my sister, who appears at the 2:25 mark! Go Kim!

The video has even received attention from local media…

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