Unit 4: Linear Relations
4.1. Writing Equations to Describe Patterns Here is a pattern made from square tiles.
Figure 1 Figure 2 Figure 3
a) Draw the next two figures in the pattern.
b) What stays the same? c) What changes?
d) Complete the table below to show the relationship between the figure number and the number of square tiles used.
Figure Number (n) Number of Tiles 1
2 3
Investigation p. 154
A banquet hall has small square tables that seat 1 person on each side. The tables can be pushed together to form longer tables.
1 Table 2 Tables 3 Tables
The pattern continues.
a) Sketch the next 2 table arrangements in the pattern.
b) What stays the same?
c) What changes?
Note the difference between an expression and an equation:
An equation is a mathematical "sentence" that says that two things are equal. For example,
3x + 1 = 5
says that if you multiply x by 3 and add 1, you will get 5. It shows the relationship between two variables.
An expression is a mathematical "phrase" that stands for a single Number. For example,
3x + 1
is an expression whose value is three times the value of x, plus 1, whatever value the variable x might have. It contains only one variable.
An equation consists of two expressions connected by an equals sign. It can only be true or false, depending on what value x has. An expression is never true or false, but just has a numerical value.
See connect p. 155 only
See examples 1 & 2 from Practice book See examples 1 & 2 pp. 156-158
Discuss p. 158 #1 (table, picture, description, equation, graph), 3 Set Practice sheet from Practice book
4.2. Linear Relations
See skill builder and top p. 164 Investigate p. 164
A local phone company offers a cell phone plan that has a fixed cost per month and a cost related to the number of text message sent. The fixed cost is $20 and each message sent costs 10 cents.
a) Complete a table of values for this relationship.
b) Complete a graph for this relationship.
c) Write an expression for the total cost of the cell phone bill.
See Intro and examples 1 & 2 from the Homework book. See connect pp. 165-166
Notice that in some problems we form a solid straight line but in others we use a dotted line.
Note the difference between continuous and discrete data:
Discrete data only takes on particular values and no values in between. In such cases we used dotted lines. Data like the number of cars a person owns is discrete because you can either have 0 or 1 or 2 cars and so on, but you can't own 1.5 cars. Other situations include:
number of people
number of DVD’s
number of toppings
number of concert tickets
number of siblings
Continuous data can take on any value in a range. In such cases we use solid lines. Examples include:
Change in temperature over time
Change in height or weight with age
Change in distance over time
because you can be any fraction of a degree, an inch, a pound, or km. Linear Relations
We know that when the graph of a relation is a straight line, it is called a linear relation. A constant change in the independent variable means a constant
horizontal change in the graph, and a constant change in the dependent variable represents a constant vertical change. Note that we can tell from a table that we have a linear relation when there is a constant change in the independent and dependent variables. See the tables on pp. 164 & 165 for examples. Also see the tables in examples 1 & 2 from the practice book.
Set Practice sheet from Practice book See examples 1-3 pp. 167-169
Discuss p. 170 #2
4.3. Another Form of the Equation for a Linear Relation Review of Coordinate Geometry
The coordinate axes consists of two perpendicular lines:
the x-axis is the horizontal axis
the y-axis is the vertical axis.
The origin is the point where the two axes meet.
See connect p. 175 and top p. 176
Linear Relations
Oblique lines (slanted lines) are neither perpendicular nor parallel to the x or y axis. Their equations contain both and x and a y.
Equations of horizontal and vertical lines contain only one variable. As a result, x or y is always constant.
Vertical lines are perpendicular to the x-axis and have equation x = a.
Horizontal lines are perpendicular to the y-axis and have equation y = a.
See examples 1 & 2 pp. 176-177 Discuss p. 178 #2
Set pp. 176-180 #4-9, 11, 12, 14, 15cd, 17, 18
Set Mid Unit Review p. 181 or from practice book as necessary.
x y
4.4. Matching Equations and Graphs
To match graphs with equations, selected ordered pairs from graph can be tested to see if they satisfy the given equation. Select at least two points.
See connect pp. 184 & 185
Also see examples 1 & 2 pp. 186 & 187 Discuss p. 188 #2
4.5. Using Graphs to Estimate Values
We can make predictions by interpolating and extrapolating.
When we estimate values between known values we interpolate. When
graphs display discrete data, interpolation is inappropriate because there are no data values between the known data points.
When we estimate values outside of known values we extrapolate. Keep in mind that by extending the graph assumptions are made that the pattern will continue. That is not always applicable in real-life situations. In some situations there are limitations.
See connect pp. 192-193
See examples 1 & 2 from practice book & set practice sheet See examples 1-3 pp. 193-195
Discuss p. 196 #2
1. Jake is checking over his math assignment. He phones you to verify the equation for the following table of values:
x y
3 8
4 10
5 12
6 14
He thinks the equation is , since the point satisfies the equation. Is he correct? Justify your answer.
2. June stated that the equation for the graph below is , since the point satisfies the equation. Is she correct? Justify your answer.
3. Sean pays a one-time fee of $6.00 to download songs plus $0.25 for each song.
a) Write an equation to represent this situation. b) How much would it cost to download 16 songs? c) How many songs can be downloaded for $13.00?
4. Wilson is training for a 10 km race. The graph shows his times and distances at 10 minute intervals.
a) Determine how long it takes Wilson to run 3 km. b) Determine how far he can run in 45 minutes. c) Determine how fast he is running.
