Foundations of Mathematics and Precalculus – Relations and FunctionsName: Hillary Strain Date: March 19, 2014Subject: Mathematics                                                                       Grade:10Lesson Length: 1 hour

Outcome:

FP10.6

Expand and apply understanding of relations and functions including:

◦ relating data, graphs, and situations

◦ analyzing and interpreting◦ distinguishing between

relations and functions.[C, CN, R, T, V]

Indicators:

b. Explain, by providing situational and graphical examples, the relationship between the categories of “relations” .

Equipment/materials:

pencil

Advanced preparation:

Worksheet/Handout

Set: Have students complete the Construct Understanding activity on pg. 257 in the

textbook.o 1) I will write all the students’ answers in a list on the boardo 2) I will then write all the letters that are used opposite from them on the board

Development: Referring back to the activity that we did in the set we will complete the following:

o Introduce to students the following definitions Set is a collection of distinct elements (the list of words/the letters) Element is one object in the set (Star/S)

o We will then draw lines from point 1 to point 2 from the seto I will then introduce the definition of relation to them

Relation associates the elements of one set with the elements of another set (referring to the lines we drew from point 1 to point 2)

So our relation in the activity is the word chosen starts with the letter Introduce to Students that we can represent our relations in a set of ordered pairs

o {(Star,S) , (Tar,T)} Do the fruit example and have students tell me what the relation would be

o Relating the fruit to a colouro Introduce to students that they can represent the relation of our fruit using an

arrow diagram The ovals represent our set The lines represent the relation

Do the dogs example and have the students tell me what the relation would beo Relating dogs to their weighto Introduce to students that they can represent the relation of dogs from

Transferring a relation form a graph into a table The heading of each column will represent our set

Go over the 3 different ways to represent Relations with the studentsClosure:

Students will be given class time to work on their assigned homeworko Pg. 262: #3ab (ii) , 4, 5b, 7b, 9

Foundations of Mathematics and Precalculus – Properties of FunctionsName: Hillary Strain Date: March 19, 2014Subject: Mathematics                                                                       Grade:10Lesson Length: 1 hour

Outcome:

FP10.6

Expand and apply understanding of relations and functions including:

◦ relating data, graphs, and situations

◦ analyzing and interpreting◦ distinguishing between relations

and functions.[C, CN, R, T, V]

Indicators:

b. Explain, by providing situational and graphical examples, the relationship between the categories of “functions”.

Equipment/materials:

Projector

Advanced preparation:

Handout

Set: Review with the students what we did yesterday

o Different ways of representing relations (I will write these out on the board) Set of Ordered Pairs Arrow Graphs Table

o What is a set?o What is an element?

Development: Introduce to the students the following terminology, referring back to our

representations we just recalledo Domain is the set of first elements of a relation.o Range is the set of related second elements of a relation.o Function is a special type of relation where each element in the domain is

associated with exactly one element in the range. Do example #1 with the students

o Ask the students which one they believe would be a function or noto Emphasize the importance that the domain can only be associated with ONE

range.o Emphasize a relation that is NOT a function then it WON’T have a domain and

range. Do example #2 with the students

o Emphasize to look at THE FIRST ELEMENT to see if it is a function or not If it repeats=it’s not If it doesn’t repeat=it is

Explain to students what an independent variable iso If it’s represented in the table = usually the first set (which is the domain)

Explain to students what a dependent variable iso If it’s represented in the table = usually the second set (which is the range)

Do example #3 with the studentso It IS a relation because the first column represents our first set “domain” and no

numbers repeato Dependent variable = Mass of marbles.. because the mass depends on how many

marbles there is.Closure:

Students will have class time to work on the assignment They will complete it at home if they do not finish it in class

o Pg. 271&272 #4,5,8,9 and 13

5.2 Properties of Functions

Example #1: For the relations below, determine whether the relations are a function. Identify the domain and range of each function.

Example #2:

Independent Variable a variable whose value is not determined by the value of another variable, and whose value determines the value of another ( the dependent) variable.Dependent Variable: a variable whose value is determined by the value of another (the independent) variable.

Example #3

Foundations of Mathematics and Precalculus – Interpreting and Sketching GraphsName: Hillary Strain Date: April 2nd, 2014Subject: Mathematics                                                                       Grade:10Lesson Length: 1 hour

Outcome:

FP10.6

Expand and apply understanding of relations and functions including:

◦ relating data, graphs, and situations◦ analyzing and interpreting◦ distinguishing between relations and

functions.[C, CN, R, T, V]

Indicators:

g. Sketch a graph to represent a situation presented orally or in writing.

Equipment/materials:

Projector Internet access

Advanced preparation:

Handout form graphing stories Handout of notes

Set: I will play videos from http://graphingstories.com

o Heighto Distance from bench

Students will fill out the handout according to the video being played (how they think the graph would be)

o Graph the story Inform students to keep an eye on the time it starts, at what height it starts at, etc

Development: Let the students know that what they were doing was interpreting the video and

sketching a graph to relate to the video Announce to the students that today they are going to be interpreting different graphs

and situations to come up with conclusions Do examples with the students Example 1

o Tell students that we are going to look at the graph and interpret the graph in order to answer the questions

o A – point out it’s the highest dot on the graph

o B – point out it’s the lowest dot on the grapho C – how do you know they’re the same? (on the same vertical line)o D – how do you know they’re the same? (on the same horizontal line)o E – D more mass and less cost

Example 2o OA – When looking at the graph – the graph goes up and to the right, so as time

increases, the distance increases - the car leaves Winnipeg and travels about 65km toward Winkler

o AB – the graph is horizontal so as the time increases the distance stays the same (not moving) - The car stops for about 15 minutes

o BC – as time increases the distance increases – The car travels about 65 km towards Winkler

o CD – the graph is horizontal so as the time increases the distance stays the same (not moving) – The car has travelled 130km (which is the distance to Winkler) so it has reached Winkler and stays there for about 2 hours

o DE – the graph goes down and to the right so as time increases the distance decreases (going home) – The car returns to Winnipeg and takes 2 hours to travel the 130km distance

Example 3o OA – since the speed increases, it’ll go up and to the righto AB – Samual cycles at 20km/h for about 30 minutes. His speed doesn’t change

so it stays horizontalo BC – samuel’s speed decreases to 5km/h so the segment goes down to the right

(still travelling, just slows down)o CD – he cycles uphill at 5km/h for 10 minutes (since speed doesn’t change for

that whole 10 minutes, line is horizontal)o DE – he slows down to 0 km/h (not moving), so his speed decreases and the

segment goes down to the righto EF – she stays stopped at 0 for 10 minutes, so the segment is horizontal.

Closure:o Students will be given class time to complete the assigned homework

o Pg. 281 & 282 #4, 5, 6, 8, 10 and 12

5.3 Interpreting and Sketching Graphs1)

2)

3)

Assignment: Pg. 4,5,6,8,10 and 12

Foundations of Mathematics and Precalculus – Graphs of Relations and FunctionsName: Hillary Strain Date: April 2nd, 2014Subject: Mathematics                                                                       Grade:10Lesson Length: 1 hour

Outcome:

FP10.6

Expand and apply understanding of relations and functions including:

◦ relating data, graphs, and situations◦ analyzing and interpreting◦ distinguishing between relations and functions.[C, CN, R, T, V]

Indicators:

Equipment/materials:

Projector Internet access

Advanced preparation:

Set: Review with the class section 5.2 (properties of functions) What is the Domain?

o Is the FIRST set of elements What is the Range?

o Is the SECOND set of elements What is a Function?

o Each element in the domain can only associate/relate with exactly one element in the range.

Show it in the set of ordered pairso {(domain,range)}o So if you’re looking at a POINT on a GRAPH the domain will be your x’s and

your range will be your y's! (TELL STUDENTS TO WRITE IT DOWN) Tell students that even if it is NOT a function, still state the domain and range.

Development: Give students handout

o Referring to our first graph, can it be a function? Transfer the graph into a arrow diagram

First set will be out x-values Second set will be our y-values Students will see that the one x value has 3 associated y values NOT A FUNCTION

o Referring to our second graph, can it be a function? Transfer into an arrow diagram It is a function

Introduce to the students the vertical line test Tell students that we can use this technique to see if a graph is a function or not!

o If our vertical line touches only ONE point or ONE spot on a line then it will be a function!

o Go back to the little activity we did before and use the vertical line test to show how it works.

Complete examples with the students Example 1

o Remind students to use the vertical line test! Example 2

o Announce to students that the Domain and Range written out for graphs is different

o Show students the dot on the end of the graphs. The dot indicates that the graph stops at that point So in our first example, our graph ends at the x value of 3

o The domain is the set of X values of the function SO the domain in this case will be the set of all real numbers that are less

than or equal to 3. Because it stops at 3, so it can still be three that’s why it’s “equal to” Since there is no dot on the other side that means it will go on forever so

all the numbers less that 3 D={x≤3 }

o The range is the set of Y values of the function SO the range in this case will be the set of all real numers greater than or

equal to -1 Because when you look at the y-axis, the y-values stop at -1 (graph

doesn’t go any furture than that Since there is no dot on the other side of the parabola it will go on

forever R={y ≥−1 }

Example 3o A - Emphasize that the time won’t change because of the number of boats there

are. So the boats are the dependent variable. Because the number of boats present will vary depending on the time of day it is.

o B – You only connect the points on a graph if a value is continuous. Ex: if we know something is ALWAYS going to be going up by 1 then

we can connect the dots. IN THIS CASE: Since we won’t ever really know how many boats will

be at anchored at a time (it can vary) we DO NOT connect themo C

Domain is our x-values Range is our y-values

Example 4o Using our graph, since we know our x value we will work our way up the graph

to see where our y value is ato Using our graph, since we know our y value we will work out way down to see

where our x-value is at

Closure: Students will be given class time to work on the assignment

o Assignment: Pg. 294-296 #4, 6, 7, 8, 14, 16, 17

5.5 Graphs of Relations and Functions

1)

2)

3)

4)

Assignment: Pg. 294-296 #4, 6, 7, 8, 14, 16, 17

Friday the students had the class to complete this hand in assignment.

Name: April 4th, 2014 5.1-5.3 & 5.5 Assignment

/251) Represent this relation as a set of ordered pairs and as an arrow diagram. (5.1)

Animal Class

ant Insecta

eagle Aves

snake Reptilia

turtle Reptilia

whale Mammalia

/2

2) The table shows the costs of student bus tickets, C dollars, for different numbers of tickets, n. (5.2)

a) Why is this relation also a function?b) Identify the independent and dependent variable. Explain your answer.c) Write the domain and range.

Number of Tickets, n Cost, C ($)

1 1.75

2 3.50

3 5.25

4 7.00

5 8.75

/4

3) Each point on this graph represents a person. Explain your answer to each question below. (5.3)

a) Which person is the oldest? What is his/her age?b) Which person is the youngest? What is his/her age?c) Which two people have the same height? What is this height?d) Which two people have the same age? What is this age?e) Which of person B or C is taller for her or his age?

/5

4) This graph represents a day trip from Athabasca to Kikino in Alberta, a distance of approximately 140 km. Describe the journey for each segment of the graph. (5.3)

/5

5) Which of these graphs represents a function? Justify your answer. (5.5)

/2

6) This graph shows the approximate height of the tide, h meters, as a function of time, t, at Port Clements, Haida Gwaii on June 17, 2009. (5.5)

a) Identify the dependent variable and the independent variable. b) Determine the domain and range of the graph.

/3

7) Here is a graph of the function g(x) = 4x-3 (5.5)a) Determine the range value when the domain value is 3.b) Determine the domain value when the range value is -7.

/4