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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.
Shortcuts exist to find the reciprocals of specific types of numbers such as natural numbers, integer numbers, fractions, and decimals.
Type | Reciprocal | Example |
---|---|---|
Natural number a | a1 | The reciprocal of 2 is 21. |
Integer numbers a, a=0 | a1 | The reciprocal of -6 is -61. |
Fraction ba, b=0 | ab | The reciprocal of 23 is 32. |
Decimal a | a1 | The reciprocal of 0.2 is 0.21. |
Dividing a fraction by another fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.
ba÷dc=ba⋅cd
Tearrik was gifted an heirloom by his grandfather. It is a handmade kimono.
Tearrik wants to make a box to hold this beautiful kimono. He plans to cut a piece of wood that is 5 feet long. He wants the cuts to create equal parts.
Rewrite 5 as 15
ba÷dc=ba⋅cd
Multiply fractions
Multiply
Write as a sum
Write as a sum of fractions
Calculate quotient
Rewrite 6+41 as 641
Multiply fractions
Cancel out common factors
Simplify quotient
A diagram can be used to model the division of 5 by 54. Divide each foot of the 5-foot-long wood into 5 equal pieces.
Notice that each of the smaller parts represents a 51 of a foot. Then determine how many of the 54-foot-long pieces are contained within the wood.
There are 6 of them. The length of the remaining part is a 51 of a foot. Note that the remaining part is also 41 of 54. This confirms that the result found algebraically is correct.
Tearrik is excited about making the box. A problem arises, however. He realizes a bit of paint would look cool but he does not have any in his home. Tearrik is full of energy and starts to run to the nearest paint shop.
Tearrik runs 32 of the way from the garage to the nearest paint shop.
ba÷dc=ba⋅cd
Multiply fractions
Multiply
ba÷dc=ba⋅cd
Multiply fractions
Multiply
The applet shows random divisions involving fractions. Find the corresponding quotient of the given division. Simplify the answer to its lowest terms. If the answer is a whole number, write it as a fraction with a denominator of 1.
ba÷dc=ba⋅cd
Multiply fractions
Split into factors
Cancel out common factors
Simplify quotient
Tearrik now has 6 pieces of wood. Each piece has a length of 54 feet. The total area of the pieces is 153 square feet.
ba÷dc=ba⋅cd
Multiply fractions
Split into factors
Cancel out common factors
Simplify quotient
Tearrik realizes that he cannot create a box as he imagined. He asks his mom for help. Together, they cut two of the 54-foot-long pieces of wood into squares. They manage to form a box by putting the pieces together. After that, they painted the box to match the kimono.
Think about the following question. 1121 times what number is 165?
Express the question mathematically. Can it be written as a division problem?
ba÷dc=ba⋅cd
Multiply fractions
Split into factors
Cancel out common factors
Simplify quotient
Multiply
Write as a sum
Write as a sum of fractions
Calculate quotient
Rewrite 1+139 as 1139
The applet shows a division expression that involves at least one mixed number. Find the indicated quotient. Simplify the answer. If the answer is a whole number, write it as a fraction with a denominator of 1.
fitsinto the numerator. In this example, no matter how many zeros are tried to fit in 5, the number 5 will never be reached.
Dividing by 0 then becomes impossible. |
Find the reciprocal of each number.
Recall that the reciprocal of a non-zero number is 1 divided by that number. In addition to that, their product results in the multiplicative identity of 1.
Non-zero Number | Reciprocal | Product |
---|---|---|
a | 1/a | a * 1/a=1 |
However, we want to find the reciprocal of a fraction. There is a straightforward way to find it. We switch the numerator and denominator of the fraction.
Fraction | Reciprocal |
---|---|
a/b | b/a |
We can now find the reciprocal of 2 5 using this information. We will switch its numerator and denominator. cc Fraction & Reciprocal [0.5em] 2/5 & 5/2 We can test our results by checking if the product of these numbers is 1. Let's do it!
This confirm that the reciprocal of 25 is 52.
We will find the reciprocal of the number 7. Recall that the reciprocal of a natural number is 1 divided by that number.
Natural Number | Reciprocal |
---|---|
a | 1/a |
Now that we have this information, we can write the reciprocal of 7. cc Number & Reciprocal [0.5em] 7 & 1/7 Again, we can check our result by seeing if the product of these numbers is 1.
This confirm that the reciprocal of 7 is 17.
Evaluate each expression. Simplify the answer.
Consider that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. 1/7÷9/14=1/7 * 14/9 We will now multiply the fractions. Remember that the product of two fractions equals the product of the numerators over the product of the denominators.
We can now simplify the resulting fraction. In this case, 7 is the greatest common factor of 14 and 63. Using this fact, we can rewrite the numerator and denominator of the resulting fraction.
The given division is equal to 29. 1/7÷9/14=2/9
Let's begin by rewriting the expression so that all of the numbers are fractions before we evaluate the expression. We can write any whole number as a fraction with a denominator of 1.
Next, recall that dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. 3/4÷4/1=3/4 * 1/4 We multiply the numerators with each other and the denominators with each other.
This fraction is in simplest form. The result of the division is then 316. 3/4÷4=3/16
Again, let's start by rewriting the expression so that all of the numbers are fractions.
We can now multiply the first fraction by the reciprocal of the second fraction.
The result of this division is 10. 2÷1/5=10
Find each quotient.
We will first rewrite the numbers in the expression as an improper fractions.
Next, we multiply the first fraction by the reciprocal of the second fraction to find the quotient.
Note that 12 is equal to 2 * 6. This means that we can simplify the numerator and denominator before we multiply.
The quotient is 52, which is an improper fraction. Let's write it as a mixed number.
The result of the division is then 2 12. 6 ÷ 2 25=2 12
We can find the given quotient by following the same steps we followed in the previous part. Let's first rewrite the numbers.
We can now multiply 125 by the reciprocal of 61, which is 16.
The given quotient is 25. 2 25÷ 6 =2/5
Evaluate each expression. Write the answer in simplest form.
We want to divide a mixed number by a fraction. 3 13 ÷ 5/6 We will start by rewriting the mixed number as an improper fraction.
Recall that dividing fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. Let's do it!
The result of dividing these fractions is 4.
We are given a division of a fraction into a mixed number.
14/16 ÷ 1 34
Our first step in finding this quotient is to convert the mixed number into an improper fraction. Let's do it!
The next step is to multiply the first fraction by the reciprocal of the second fraction.
The quotient is equal to 12.
In this case, we are asked to find the division of two mixed numbers.
4 39 ÷ 1 79
Let's start by converting the mixed numbers into improper fractions.
Next, we will multiply the first fraction by the reciprocal of the second fraction.
The result is an improper fraction. We will write it as a mixed number.
The diagram models the division of two numbers.
Let's only focus on the blue region. We can see that there are three units. Each of these three units is divided into three small parts.
We see that the blue region represents 2 23. We know that the diagram is used to model a division. Let's review what division means.
Division |- This operation represents the process of calculating how many times one quantity is contained within another quantity.
We see that two parts sized one and one-third fit within in a part sized two and two-thirds. We will note this on the given diagram.
Considering the definition of division and the diagram, we can say that the dividend is 2 23 and the divisor is 1 13. Then we can write the following expression. 2 23 ÷ 1 13 Note that the quotient is equal to 2 because two units of the size one and one-third can fit inside two and two-thirds.