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| 12 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.
Diego is going through some old stuff in storage. He discovers an entire rack of vinyls! There are 200 in total and they belong to his dad. Diego asks his father how long this collection took to gather.
Multiplication and division are inverse operations. They can be used to solve equations by the following properties of equality.
Given an equation, multiplying each side of the equation by the same number yields an equivalent equation. Let a, b, and c be real numbers.
If a = b, then a * c = b * c.
The Multiplication Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider the following example. x÷4&=2 x÷4 * 4&=2 * 4 x&=8
Here, by multiplying both sides of the equation by 4, the variable x was isolated and the solution of the equation was found.Dividing each side of an equation by the same nonzero number yields an equivalent equation. Let a, b, and c be real numbers.
If a = b and c≠ 0, then a ÷ c = b ÷ c.
The Division Property of Equality is an axiom, so it does not need a proof to be accepted as true. This property is one of the Properties of Equality that can be used when solving equations. 5x&=10 5x ÷ 5&=10 ÷ 5 x&=2
As can be observed, by dividing both sides of the equation by 5, the variable x was isolated and the solution of the equation was found..LHS /5.=.RHS /5.
Cross out common factors
Simplify quotient
Calculate quotient
undoeach other. Consider the given equation.
LHS * 5=RHS* 5
a/5* 5 = a
Multiply
The Multiplication Property of Equality can be used instead of the Division Property of Equality anytime when solving an equation with a coefficient that is a rational number. Consider an equation in the form abx = c. a/bx = c The coefficient ab is a fraction. For that reason, the equation can be solved by multiplying both sides by the reciprocal of ab. As an example, consider the following equation. 3/5 x = 6 The coefficient next to the variable is 35. Interchange the numerator and denominator to find its reciprocal. The reciprocal of 35 is 53, so the equation can be solved by multiplying both sides by 53.
LHS * 5/3=RHS* 5/3
Commutative Property of Multiplication
a/b* b/a=1
a * 1=a
a*b/c= a* b/c
Multiply
Calculate quotient
Diego finds himself wondering about the speed at which a record rotates. His father tells Diego that the record rotates 190 times while playing a song on the album. Diego later reads online that a record rotates 59 of a full rotation every second it plays.
Identify the coefficient of the variable. Multiply both sides of the equation by the reciprocal of the coefficient.
LHS * 9/5=RHS* 9/5
Commutative Property of Multiplication
a/b* b/a=1
a * 1=a
a*b/c= a* b/c
Multiply
Calculate quotient
Solve the equations using the Multiplication Property of Equality or the Division Property of Equality. If necessary, round answers to two decimal places.
Many real-life situations can be algebraically modeled by equations. These equations can involve a variable that represents an unknown quantity. Consider modeling the following situation.
Diego categorizes his father's record collection by genre. He finds out that there are 8 different genres, each with the same number of records. The collection contains 200 records. How many records are there in each genre? |
.LHS /8.=.RHS /8.
Cross out common factors
Simplify quotient
Calculate quotient
Diego's father's old turntable is broken. Diego is so eager to listen to the records that he decides to make some money to buy the replacement parts. The parts that Diego needs to buy costs $96. Diego's neighbor offers him $8 per walk to walk her dog.
8 x = 96 This equation models the given situation.
.LHS /8.=.RHS /8.
Cross out common factors
Simplify quotient
Calculate quotient
x = 12 ⇓ Diego needs to walk the dog12times.
The dog Diego walked loved him so much — just look at the two of them!
His neighbor, the dog owner, was so impressed that affter 12 walks, she decided to pay Diego handsomely. Diego received $10 for each walk.
t/12 = 10 Note that different equations, equivalent to this one, can also be used to model this situation. An example of such an equation is t = 12 * 10.
LHS * 12=RHS* 12
a/12* 12 = a
Multiply
The challenge presented at the beginning of the lesson can be solved by writing an equation that models the situation and then solving the equation. It stated that Diego's father collected 200 vinyl records and bought 10 records every month.
Number of Months: m Diego's father bought 10 records every month. Then, the number of records Diego's father collected in m months is 10 m. There are 200 records in the collection, so 10 m must be equal to 200. 10 m = 200 Diego figured out the equation!
.LHS /10.=.RHS /10.
a/10* 10 = a
Calculate quotient
.LHS /1.6.=.RHS /1.6.
a/1.6* 1.6 = a
Calculate quotient
Zain loves running in the park.
This week, Zain went running five times. In total, they ran 15 miles.
We know that Zain went running 5 times this week. In total, they ran 15 miles. We want to write an equation for the number of miles they ran each time. They went running 5times this week. In total, they ran 15miles. In this case, the unknown quantity is the number of miles Zain ran each time. We can represent this unknown quantity with a variable. Let's use m as the variable. Miles Ran Each Time: m We know that Zain went running 5 times this week. Then, the total number of miles they ran this week is equal to the product of the number m of miles they ran each time and 5. Product of 5 and m is equal to 15. ⇓ 5 * m = 15 The equation 5m=15 can be used to find the number of miles Zain ran each time.
Next, we need to solve the equation from Part A.
5m = 15
When solving equations, we can use inverse operations and Properties of Equality to undo
the operations applied to the variable. In this case, the variable m is multiplied by 5.
5 m=15
We use the inverse operation of multiplication to undo this operation. That would be division. The Division Property of Equality lets us divide both sides of the equation by 5. Let's do it!
The solution to our equation is m = 3.
Finally, we want to determine how many miles Zain ran every time they went running this week. We know from Part A that the number of miles they ran is represented by the equation 5m = 15. 5m = 15 Here, m represents the number of miles Zain ran every time. We know from Part B that the solution to our equation is 3. Therefore, Zain ran 3 miles every time they went running this week.
Ali went on a long ski trip.
The skiing trip lasted five days. Ali skied eight miles each day. Answer the following questions to find out how many miles Ali skied in total during the trip.
We know that Ali went on a 5-day skiing trip. Every day, he skied 8 miles. We want to write an equation for the number of miles he skied in total. During a 5-day skiing trip,Ali skied 8miles daily. In our case, the unknown quantity is the total number of miles Ali skied. Let's use d as the variable. Miles Skied in Total: d We know that Ali's trip lasted 5 days. The quotient of the total distance he skied, d, and the number of days the trip lasted is equal to the distance he skied every day, which is 8 miles. The quotient of d and 5 is equal to 8. ⇓ d/5 = 8 The equation d5=8 can be used to find the number of miles Ali skied on his trip.
Next, we need to solve the equation from Part A.
d/5 = 8
When solving equations, we can use inverse operations and Properties of Equality to undo
the operations applied to the variable. In this case, the variable d is divided by 5.
d/5 = 8
We use the inverse operation of division to undo this operation. That would be multiplication. The Multiplication Property of Equality lets us divide both sides of the equation by 5. Let's use this property and then simplify.
The solution to our equation is d = 40.
Finally, we want to determine how many miles Ali skied on his trip. We know from Part A that the number of miles he skied is represented by the equation d5 = 8. d/5 = 8 Here, d represents the total number of miles Ali skied. We know from Part B that the solution to our equation is 40. Therefore, Ali skied 40 miles on his trip.