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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 |
Determine whether the given scenario represents a proportional or a non-proportional relation.
Recall that two variables, x and y, are proportional if their values increase or decrease at a constant rate. This means that the ratio yx equals a constant k. y/x=k ⇔ y=kx With this in mind, let's consider the ratio of Heichi's age to his brother's. 8/15 We can now check whether this ratio stays the same when boys are growing. Randomly calculate the ratio after 3 years. 8+ 3/15+ 3=11/18 Note that the first ratio, 815, and the resulting ratio after 3 years, 1118, are in their simplest form. These ratios are not equivalent. 8/15 ≠ 11/18 This means that the scenario related to the boys' ages is a non-proportional situation.
We are told that a car travels 90 kilometers in one hour. This means that the car will travel the same kilometer in every hour. Let's visualize this situation.
Since every one unit increase in time results in the same amount of distance traveled, there is a constant rate of change between the time and distance traveled. 90 km/1 h=180 km/2 h=270 km/3 h=360 km/4 h This means that the given variables, time and distance traveled, are proportional.
One way to know if two rates are equivalent is to write them as fractions. We can check if the relationship between the two quantities is the same when we compare these fractions in their simplest form. Let's see if the given rates are equivalent! c|c First Package & Second Package [0.8em] 6pairs of socks/$15 & 4pairs of socks/$8 We need to simplify the above expressions. Let's start by simplifying the rate for the first package. We will only look at the numbers to make the math a little bit simpler.
Let's now simplify the rate for the second package.
Finally, we can compare the simplified rates. 2/3 ≠ 1/2 Since the fractions are not equivalent, the rates are not equivalent.
Another way to know if two rates are equivalent is to check if we can divide or multiply the numerator and denominator of one rate by the same number to obtain the other rate.
The numerator and the denominator are not divided by the same number, so the fractions are not equivalent.
The table shows the number of white and black drops used while mixing them to get a specific tone of dark gray.
Drops of White, x | Drops of Black, y |
---|---|
1 | 3 |
3 | 9 |
5 | 15 |
10 | 30 |
We will check whether there exists a constant ratio between the variables x and y to determine if they are proportional. Let's start with the first and second rows of the table.
Drops of White, x | Drops of Black, y |
---|---|
1 | 3 |
3 | 9 |
5 | 15 |
10 | 30 |
The ratio of the number of white drops to the number of black drops is 13 in the first row. In the second row, the ratio is 39. Are these ratios the equivalent? 1/3 and 3/9 We can check this by writing them in their simplest forms. Note that 13 is already in its form but 39 can be reduced by 3.
We ended with the same ratio. Next, we will examine the ratios for the numbers in the third and fourth rows by proceeding in the same way.
Drops of White, x | Drops of Black, y | Ratio of White Drops to Black Drops |
---|---|---|
1 | 3 | 1/3 |
3 | 9 | 3/9= 1/3 |
5 | 15 | 5/15= 1/3 |
10 | 30 | 10/30= 1/3 |
Since the numbers in each line can be written as the same ratio, the number of white drops and black drops are proportional.
Identify the constant of variations of the given equations.
Consider the general form of a direct variation equation. y= kx In this form, k is the constant of variation or proportionality between the variables x and y. We can rewrite the given equation in this form to find out its constant of variation. y=2x/5 ⇔ y= 2/5 * x We find that k= 25, which is the constant of variation.
Once again, we will apply the same process like we did in Part A. Let's rewrite the given equation in the form y=kx to identify the value of k.
The constant of variation k equals 37.
Heichi examines the price of apple juice packs in three different shopping centers. He makes a table to compare the prices.
Shop | Cans in a Pack | Packs | Price |
---|---|---|---|
Shopping Center I | 9 | 4 | $10 |
Shopping Center II | 12 | 3 | $8 |
Shopping Center III | 6 | 4 | $9 |
We are given the prices of apple juice packs from three different stores. We will find the unit prices for the cans of apple juice at each store to find the cheapest offer. Let's start by examining the given data in the table.
Shop | Cans in a Pack | Packs | Price |
---|---|---|---|
Shopping Center I | 9 | 4 | $10 |
Shopping Center II | 12 | 3 | $8 |
Shopping Center III | 6 | 4 | $9 |
First, we will find the total number of cans that each store offers at the given price. Let's multiply the number of cans in a pack by the number of packs. Shopping Center I:& 9 * 4 = 36 Shopping Center II:& 12 * 3 = 36 Shopping Center III:& 6 * 4 = 24 Next, we will write the rate of the price of each option to the number of cans each gives. This leads us to calculate the unit rate by dividing its numerator by its corresponding denominator. In this context, the unit rate refers to the price of one can. Let's do it!
Shop | Rate | Unit Price |
---|---|---|
Shopping Center I | $10/36 cans | $10÷ 36/36cans÷ 36≈$0.28/1can |
Shopping Center II | $8/36 cans | $8÷ 36/36cans÷ 36≈$0.22/1can |
Shopping Center III | $9/24 cans | $9÷ 24/24cans÷ 24=$0.38/1can |
Now that we know the unit prices, we can determine from which store Heichi should purchase the cans of apple juice. Consider the unit prices obtained in the table. Shopping Center I:& $0.28/1can [1.5ex] Shopping Center II:& $0.22/1can [1.5ex] Shopping Center III:& $0.38/1can The unit prices inform us that for 1 can of apple juice, we need to pay $0.28 at the first shopping center, $0.22 at the second one, and $0.38 at the third one. This means that Shopping Center II has the lowest price for a can of apple juice, so Heichi should buy apple juice there.
Jordan, Emily, and Paulina use the Cross Product Property to check if the given ratios form a proportion.
We can see that each guy began by equating the ratios but put a question mark in the equals sign because they wanted to find out if the ratios formed a proportion. 3/9? =12/40 Then, they used different methods in the second step. We will examine each work one at a time. Let's examine how Jordan applied the Cross Products Property in the second step.
Notice that she multiplied the numerator and denominator of each fraction. However, the Cross Products Property states that in any proportion, the product of the extremes is equal to the product of the means. a/b=c/d ⇒ ad= bc This means that Jordan didn't apply the property in a correct way. Let's see how Emily worked.
On the other hand, Emily multiplied the numerators with each other and the denominators with each other, which is not the same as multiplying the extremes and means. This means that Jordan's work is also not correct. Lastly, let's have a look at how Paulina did this multiplication.
Notice that Paulina multiplied the extremes and means. Then, she ended with different numbers, which means the ratios do not represent a proportion. Since the factors are chosen correctly, we can conclude that Paulina applied the Cross Products Property correctly.