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| 13 Theory slides |
| 11 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
The value of two equivalent ratios is the same. Because of this, an equals sign can be written between two equivalent ratios to create a proportion.
In contrast, two ratios have a non-proportional relationship if they are not equivalent. Consider, for example, 32 and 23. These ratios are in their simplest form, but they are not equal. This means that they are non-proportional. This relationship is shown by writing an inequality symbol between the ratios.
Consider the given ratio. Then, analyze the values of the ratios in each answer and choose which one forms a proportion with the given ratio.
In a proportion, the product of the extremes is equal to the product of the means.
This property is also known as cross-multiplication or Means-Extremes Property of Proportion.
LHS⋅b=RHS⋅b
ca⋅b=ca⋅b
LHS⋅d=RHS⋅d
Commutative Property of Multiplication
Dylan's mom is making chocolate chip cookies. She uses a recipe which claims that for every three cups of flour, two cups of white sugar must be added. However, she wants to make a larger batch so she uses six cups of white sugar for nine cups of flour. She is unsure if she is using the correct amount.
Dylan and his brother Mark help their mother by forming a proportion with the given white sugar and flour amounts to check whether their mother has the same rate of ingredients. Although both of the guys used the Cross Products Property, their solutions were different.
Recall what the Cross Products Property states. Then identify the extremes and means in the proportion set by the boys. Are the second steps in either solution correct?
Begin by analyzing each solution separately.
Dylan began by equating the given ratios to investigate if they form a proportion. He put a question mark above the equals sign until he knows if a proportion is formed or not.
Focus on where Dylan applied the Cross Products Property.
Recall what that property states to determine whether it was correctly applied.
ba=dc ⇒ ad=bc
The property claims that the product of the extremes is equal to the product of the means. Identify the extremes and means in the proportion written by Dylan.
Next, Mark's solution will be analyzed. After writing the ratios as a likely proportion, he also chose to apply the Cross Products Property.
The family made a huge mess while baking the cookies. They need to clean the kitchen. Some instructions state that the dilution rate for cleaning concentrate is 2:9. This means 9 liters of water is used to 2 liters of cleaning concentrate.
Multiply
LHS/9=RHS/9
Rearrange equation
ba=b/3a/3
a⋅cb=ca⋅b
LHS⋅4=RHS⋅4
Cancel out common factors
Simplify quotient
Multiply
LHS/8=RHS/8
Solve the given proportion for the unknown variable x.
Direct variation is a relationship between two variables, x and y, where an increase in one variable causes the other to increase by a constant factor k. This means that if x increases, y increases, and if x decreases, y decreases. The following equation shows this relationship algebraically.
y=kx
Recall that direct variation equations can be written in the forms y=kx, k=xy, or x=ky. Identify the constant of variation k in the given direct variation equation. If the value of k is a fraction, write it in its simplest form.
A table of values can be made by substituting several random values for x and then solving the equation for y.
x | y=3x | y | (x,y) |
---|---|---|---|
-1 | y=3(-1) | -3 | (-1,-3) |
0 | y=3(0) | 0 | (0,0) |
1 | y=3(1) | 3 | (1,3) |
2 | y=3(2) | 6 | (2,6) |
3 | y=3(3) | 9 | (3,9) |
Finally, plot the ordered pairs from the table of values and connect them with a straight line.
They will use a robotic vacuum cleaner to clean the floor of not only their kitchen but their entire house!
The following graph shows the cleaned area by the vacuum cleaner in x minutes.
The graph of a proportional relationship is a line that passes through the origin. This direct variation can be represented by an equation in the form y=kx.
Notice that the given graph is a line that passes through the origin. This means that it represents a proportional relationship between the variables. With this in mind start by examining several points on the graph.
x | y=kx | y |
---|---|---|
-2 | y=0⋅(-2) | 0 |
-1 | y=0⋅(-1) | 0 |
1 | y=0⋅1 | 0 |
2 | y=0⋅2 | 0 |
Note that the number of miles traveled and the time are proportional variables. Let's start by writing a ratio between these variables. Time/Distance Traveled=12/21 Next, since the submarine continues to travel at the same rate, we can write a proportion to represent this situation. Let x be the time needed. 12/21=x/56 Now, we will solve this proportion.
The submarine needs to travel for about 1.3 hours to complete 56 miles.
Ali wants to paint the walls of his home. The graph shows the area y in square meters that can be painted using x liters of paint.
We want to find how many square meters can be painted by using 2 liters of paint. At first, we will find the unit rate, which is the point where (1,r) corresponds. Let's look at the graph to see this point!
The unit rate in this context refers to the area painted per one liter of paint. This means that Ali can paint 15 square meters with 1 liter of paint. In addition, these two variables are proportional because the line passes through the origin. We can now use this information to write a proportion to find the area y painted with 2 liters of paint. 15/1=y/2 We can use the Cross Products Property to find the value of y.
We got that Ali can paint 30 square meters of the wall using 2 liters of paint. We can graph this point on the given graph to see that this result matches the information in the graph.
This time we will find out the liters of paint needed for an area of 51 square meters. We can use the unit rate we got in the previous part. In this case, let x be the liters of paint needed for an area of 51 square meters. 15/1=51/x Once again, we will solve this proportion by applying the Cross Products Property.
Ali needs to use 3.4 liters of paint to paint 51 square meters.
Recall that the graph of a proportional relationship is a line and it passes through the origin. Also, since it represents a direct variation, its equation will be in the following form. y=kx In this form, k is the constant of variation. Let's rewrite the general form. y=kx ⇔ k=y/x Note that the ratio of y to x is equal to the constant of variation. With this in mind, we can find the constant of variation of our relation by using the coordinates of the point (45,27). k=y/x ⇒ k=27/45 Great! Next, recall that the y-coordinate of the point (1,y) represents the unit rate of a direct variation graph since its x-coordinate is 1. On the other hand, the unit rate and constant of variation are equivalent ratios. Knowing this leads us to write the following proportion. 27/45=y/1 Now, let's rewrite this proportion to find y.
We found that the unit rate is 35, which is the missing coordinate of the point (1,y).
Emily and Paulina take a cycling trip together. The following graph shows how far Emily goes in miles.
The following table shows the hours and miles related to Paulina's cycling.
Hours, x | Miles, y |
---|---|
1 | 12 |
2 | 24 |
3 | 36 |
4 | 48 |
We will examine the graph and the table one at a time to decide who is going faster. Let's start with the graph. Notice that the graph is a line that is passing through the origin. This means that it represents a direct variation. Let's find the point that x-coordinate is 1 in the graph.
Recall that the point (1,r) on a direct variation graph represents the unit rate. In other words, it shows how much Emily goes in one hour. Since the graph passes through the point (1, 10), Emily rides 10 miles in one hour. Now, we will examine the given table of values.
Hours, x | Miles, y |
---|---|
1 | 12 |
2 | 24 |
3 | 36 |
4 | 48 |
We can decide whether it represents a proportional relation between the variables x and y by checking the ratios of the numbers in each line. Let's do it!
Hours, x | Miles, y | Mile/Hour,y/x |
---|---|---|
1 | 12 | 12/1= 12 |
2 | 24 | 24/2= 12 |
3 | 36 | 36/3= 12 |
4 | 48 | 48/4= 12 |
The ratios in each row are equivalent, so the variables x and y are proportional. This means that the values in this table also represent a direct variation. With this in mind, we can say that Paulina goes 12 miles per hour by looking sy the values in the first line. c Emily & Paulina [0.5em] 10 miles/1 hour & 12 miles/1 hour We can conclude that Paulina rides faster than Emily.