Sign In
| 10 Theory slides |
| 12 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.
Zain and Jordan planned an adventure across a vast desert landscape. They will cycle for three days until they reach a huge music festival! The first two days, Zain and Jordan need to cycle the same distance each day. On the third day, they will have 10 miles remaining to be cycled.
The total distance Zain and Jordan cycle during the trip is 60 miles. If they miscalculate their trip, they will miss the festival!
Equations can be named according to the minimum number of inverse operations needed to solve them.
LHS+3=RHS+3
Add terms
LHS/2=RHS/2
ca⋅b=ca⋅b
Calculate quotient
Identity Property of Multiplication
Zain and Jordan are cycling along. Already on their first day, they ran into a problem! Zain's tire got a terrible flat and they do not have a spare to replace it. They need to buy a new tire.
They head into town to buy extra tires to be better prepared.
LHS/3=RHS/3
Cross out common factors
Cancel out common factors
Calculate quotient
undoeach other. Consider the given equation. Here, the variable m is divided by 2 and then 5 is added to the result.
On the second day of this ride to the festival, Zain and Jordan begin to wonder about some of the data from their ride.
LHS⋅-45=RHS⋅-45
Commutative Property of Multiplication
ba⋅ab=1
a⋅cb=ca⋅b
Multiply
-b-a=ba
Calculate quotient
LHS−x=RHS−x
Subtract terms
LHS⋅(-1)=RHS⋅(-1)
-a(-b)=a⋅b
ca⋅b=ca⋅b
x=30
ca⋅b=ca⋅b
Multiply
Calculate quotient
Subtract term
Solve the equations by using the Properties of Equality. If necessary, give answers as decimals rounded to two decimal places.
Zain and Jordan continued cruising along their cycling trip. The beauty of the ride became even more noticeable as they could hear songbirds! Some birds sang perched atop a power line.
A few more birds flew in and joined the flock. As a result, the number of birds doubled. Then, 4 birds flew away. In the end, 8 birds were left singing on the power line.
LHS/2=RHS/2
Cross out common factors
Cancel out common factors
Calculate quotient
Zain and Jordan successfully reached the music festival! The line to enter the festival is super long. Each came up with their own way to describe the difference between the time they expected to wait in line and the time they will actually have to wait.
LHS/2=RHS/2
Cross out common factors
Cancel out common factors
Calculate quotient
An equation that models the challenge presented at the beginning of this lesson can now be written and solved. Recall that Zain and Jordan planned to cycle the same distance for the first two days of their trip, and then cycle the remaining 10 miles on the last day.
Remember, the whole trip is 60 miles long. Making these calculations will ensure they make it to the festival on time!
LHS/2=RHS/2
Cross out common factors
Simplify quotient
Calculate quotient
Translate the given sentences into equations.
We have been given a sentence and we want to translate it into an equation.
The sum of 7 and the product of 3 and s is 11.
Every equation has an equals sign and values or expressions on either side of it. Key phrases, such as is,
is equal to,
and equals
tell us about the placement of the equals sign. Let's look for such a keyword in the given sentence and replace it with the equals sign.
The sum of7 and the product of3ands is11.
⇕
The sum of7 and the product of3ands = 11.
On the left-hand side we have two keywords — sum
and product.
The word sum
tells us that addition will be used in our equation.
The sum of 7 andthe product of3ands
7 + the product of3ands
The other keyword, product,
tells us that multiplication is also used.
7 + the product of 3 and s
7 + 3 * s
On the right-hand side, we have the number 11. Putting these sides together, we have a complete equation.
7 + 3 s = 11
We want to translate the given sentence into an equation.
Twenty-one is equal to the difference of the quotient of r and three and twelve.
Like every equation, our equation will have an equals sign and values or expressions on either side of it. Key phrases, like is,
is equal to,
and equals
tell us where the equals sign should be placed.
Twenty-one is equal to the difference of the quotient ofrand three and twelve.
⇕
Twenty-one = the difference of the quotient ofrand three and twelve.
The left-hand side is the number twenty-one. On the right-hand side, we have two keywords, difference
and quotient.
The keyword difference
means that we will use subtraction in our equation.
the difference of the quotient ofr and three and twelve.
⇕
the quotient ofr and three - 12
The second keyword, quotient,
means that we will use division in our equation.
the quotient of r and three - 12 [0.5em]
r/3 - 12
Putting the two sides together, we have a complete equation.
21 = r/3 - 12
Translate the given sentences into equations.
We have been given a sentence and we want to translate it into an equation.
The difference of t and 5 is the product of 8 and t.
Every equation has an equals sign and values or expressions on either side of it. Key phrases, such as is,
is equal to,
and equals
tell us about the placement of the equals sign. Let's look for such a keyword in the given sentence and replace it with the equals sign.
The difference oftand5 is the product of8andt.
⇕
The difference oftand5 = the product of8andt.
On the left-hand side we have the keyword difference.
This means that subtraction will be used in our equation.
The difference of t and 5
t - 5
On the right-hand side, there is the keyword product.
This tells us that multiplication is used on this side of the equation.
the product of 8 and t
8 * t
Putting the two sides together, we have a complete equation.
t - 5 = 8 t
We want to translate the given sentence into an equation.
The quotient of p and 4 equals the sum of 17 and p.
Like every equation, our equation will have an equals sign and values or expressions on either side of it. Key phrases, like is,
is equal to,
and equals
tell us where the equals sign should be placed.
The quotient ofpand4 equals the sum of17 andp.
⇕
The quotient ofpand4 = the sum of17 andp.
On the left-hand side there is the keyword quotient.
This means that division will be used in this part of the equation.
the quotient of p and 4 [0.5em]
p/4
On the right-hand side, we have the keywords sum.
This tells us that we will use addition in our equation.
the sum of 17 and p
17 + p
Putting the two sides together, we have a complete equation.
p/4 = 17 + p
Solve the given equations. Check your answers.
We have to isolate the variable x on one side to solve the given equation. 3x + 6 = 24 In this equation, x is multiplied by 3 and then 6 is added to the result. To solve this equation, we use the inverse operations of the operations applied to the variable in reverse order. This means first subtracting 6 from both sides of the equation using the Subtraction Property of Equality.
Next, we can undo the multiplication. We divide both sides of the equation by 3. This does not change the solution by the Division Property of Equality.
Our calculations show that the solution to the given equation is x=6. To check that this solution is correct, let's substitute x=6 into the original equation and simplify.
The left-hand side and right-hand side are equal, so x=6 is the correct solution.
For this equation, we need to isolate t on one side. t/5 - 2 = 6 Here, the variable t is divided by 5 and then 2 is subtracted from the quotient. We can solve the equation by undoing the operations applied to the variable in reverse order. First, we use the Addition Property of Equality to add 2 to both sides of the equation.
Next, the division can be undone by multiplying both sides of the equation by 5. The Multiplication Property of Equality ensures that the solution remains the same.
The solution to the given equation is t=40. Finally, we substitute t=40 into the original equation and check if our answer is correct.
Substituting t=40 resulted in a true statement. We can conclude that t=40 is the correct solution.
Solve the given equations. Check the answers.
We have to isolate the variable s on one side to solve the given equation. 2s - 2 = 8 In this equation, s is multiplied by 2 and then 2 is subtracted from the result. To solve this equation, we use the inverse operations of the operations applied to the variable in reverse order. This means first adding 2 to both sides of the equation using the Addition Property of Equality.
Next, we can undo the multiplication. We divide both sides of the equation by 2. This does not change the solution by the Division Property of Equality.
Our calculations show that the solution to the given equation is s=5 . To check that this solution is correct, let's substitute s=5 into the original equation and simplify.
The left-hand side and right-hand side are equal, so s=5 is the correct solution.
For this equation, we need to isolate p on one side. p/3 + 2 = 23 Here, the variable p is divided by 3 and then 2 is added to the quotient. We can solve the equation by undoing the operations applied to the variable in reverse order. First, we use the Subtraction Property of Equality to subtract 2 from both sides of the equation.
Next, the division can be undone by multiplying both sides of the equation by 3. The Multiplication Property of Equality ensures that the solution remains the same.
The solution to the given equation is p=63. Finally, we substitute p=63 into the original equation and check if our answer is correct.
Substituting p=63 resulted in a true statement. We can conclude that p=63 is the correct solution.
Solve the given equations. Check your answers.
In this equation, there are variable terms on both sides. To solve the equation, we first transfer all the variable terms to one side using inverse operations and the Properties of Equality. Here, we will use the Subtraction Property of Equality to subtract x from both sides. Then, we combine like terms.
This results in a regular equation. We will solve it by using the Division Property of Equality to divide both sides by 3.
Our calculations show that the solution to the given equation is x=5. To check that this solution is correct, let's substitute x=5 into the original equation and simplify.
The left-hand side and right-hand side are equal, so x=5 is the correct solution.
For this equation, we need to move all the variable terms to one side. Let's use the Subtraction Property of Equality to subtract v from both sides. Then we will combine like terms.
This turns the given equation into a regular equation with one variable term. We can solve it by using the Division Property of Equality to divide both sides by 2.
The solution to the given equation is v=- 2. Finally, we substitute v=- 2 into the original equation and check if our answer is correct.
Substituting v=- 2 resulted in a true statement. We can conclude that v=- 2 is the correct solution.
Solve the given equations. Check your answers.
In this equation, there are variable terms on both sides. To solve the equation, we first transfer all the variable terms to one side using inverse operations and the Properties of Equality. Here, we will use the Subtraction Property of Equality to subtract z from both sides. Then, we combine like terms.
This results in a regular equation with one variable. The coefficient next to the variable is a fraction. We can solve the equation by using the Multiplication Property of Equality to multiply both sides by the reciprocal of the coefficient, or 6- 5.
Our calculations show that the solution to the given equation is z=- 6. To check that this solution is correct, let's substitute z=- 6 into the original equation and simplify.
The left-hand side and right-hand side are equal, so z=- 6 is the correct solution.
For this equation, we need to move all the variable terms to one side. Let's use the Subtraction Property of Equality to subtract o from both sides. Then we will combine like terms.
This turns the given equation into a regular equation with one variable. Here, the coefficient next to the variable is a fraction. We can solve the equation by using the Multiplication Property of Equality to multiply both sides by the reciprocal of the coefficient, or 3- 2.
The solution to the given equation is o = 9. Finally, we substitute o = 9 into the original equation and check if our answer is correct.
Substituting o = 9 resulted in a true statement. We can conclude that o = 9 is the correct solution.