4Th grade estimating quotients: Estimating quotients | 4th grade Math Worksheet
How to Estimate Quotients | 4th Grade Math
By now, you’ve already learned about estimating sums, differences, and products.
In this lesson, we’re going to learn to estimate quotients.
A quotient is the result of division.
A dividend is the big number you divide by a divisor to get the quotient.
The steps for estimating sums, differences, products, and even quotients are mostly the same. We first use rounding.
It’s a little different for estimating quotients because the steps use compatible numbers.
What Are Compatible Numbers?
A compatible number is a number that is familiar and easy to divide, and close to the number you start with.
Let’s look at an example.
Estimate 133 ÷ 3
We can round 133 to the nearest hundred, and we get 100.
This makes the problem 100 ÷ 3, which doesn’t make things any easier.
Tip: We can quickly tell 100 is not divisible by 3, because the sum of its digits are not divisible by 3.
So instead of rounding, let’s try to find a compatible number.
First, let’s have a quick review of how to divide numbers ending in zeros.
This will come handy in a moment.
Dividing Numbers Ending In Zeros Review
When dividing numbers ending in zeros, divide the non-zero parts first and add back the zeros to the right of the quotient.
Let’s look at an example.
Divide 150 ÷ 5
First, divide the non-zero part. ✅
15 ÷ 5 = 3
Next, add back the zeros to the right of the quotient. ✅
30
So, 150 ÷ 5 = 30.
Now, you’re ready to estimate quotients.
Using Compatible Numbers To Estimate Quotients
Let’s go back to the problem.
Estimate 133 ÷ 3
The first digit of the dividend is 1.
We know 1 can’t be divided by 3.
So, we look at the first two digits instead.
They’re 1 and 3. Together they make the number 13.
But 13 is not divisible by 3.
Think of a number close to 13 that is divisible by 3.
Try to recall the multiplication facts for 3.
What number multiplied by 3 will give us a number that’s close to 13?
Let’s see:
3 × 4 = 12
3 × 5 = 15
We found two compatible numbers that are close to 13.
Which is closer?
12 is one step away from 13.
15 is 2 steps away from 13.
12 is closer.
Now, we add a zero at the end of 12. This gives us 120.
This means 120 is closer to 133 than 150.
120 is the compatible number we’re looking for.
We can now use 120 to estimate the quotient of 133 ÷ 3.
Let’s write the problem using the compatible number.
120 ÷ 3 = ?
Divide the non-zero digits first.
12 ÷ 3 = 4
Now we add back the zero that we left out.
40
Using compatible numbers, we figured out that 133 ÷ 3 is about 40. ✅
We can write this using the approximate symbol (≈):
133 ÷ 3 ≈ 40
Another Example
Estimate 694 ÷ 69.
When the divisor is a 2-digit number, we start by rounding it to the nearest 10.
The divisor in our example is 69.
We want our divisor to end in zero because that makes division much easier.
What’s 69 rounded to the nearest ten?
Yes, it’s 70.
Now, we look at our dividend.
Let’s focus on just its first two digits, 69. This makes it easier to divide.
For now, we’re just working with 69 and 7.
Let’s find compatible numbers that are close to 69, and divisible by 7.
Tip: Mastering your multiplication facts will help you find compatible numbers quickly.
So let’s think about this.
What numbers can we multiply to 7 that will give us products close to 69?
7 × 9 = 63
7 × 10 = 70
63 and 70 are close to 69.
63 is 6 steps away from 69.
70 is 1 step away from 69.
So 70 is closer.
Let’s add a zero at the end of 70 to make it a 3-digit number, like our original dividend.
This makes it 700.
This means 700 is the compatible number that’s closest to 694.
Let’s now rewrite our equation with the rounded divisor and compatible dividend.
700 ÷ 70 = ?
We can simplify the equation like this:
70 ÷ 7 = ?
Now, it’s pretty easy to get an estimate:
10
This means 694 ÷ 69 is about 10.
✅
Tip: Notice that we always write the word “about” before the estimated quotient. This tells us that the answer we got is not the exact answer, just an estimate.
We can also write our answer this way.
694 ÷ 69 ≈ 10
Nice work. 👏
Do you remember all the steps?
1️⃣ First, round the divisor to it’s highest place value if it has two or more digits.
2️⃣ Then, find the closest compatible number to the dividend that can be divided by the divisor without remainder.
Try the practice to see if you got it.
How to Find Quotient? (Estimating Quotients Using Examples)
1. Estimate the quotient: \(463\div~5\).
Solution:
Look at the first two digits of the dividend and use basic division facts. We have to think of the numbers that are close to 463 and easily divided by 9.
Trying 450 , \(45\div~5~=~9\), So that \(450\div~5~=~90\)
Trying 500 , \(50\div~5~=~10\), So that \(500\div~5~=~100\)
Choose 450 because 463 is closer to 450.
So, \(463\div~5\) is about 90.
2. For \(5184\div~6\), find the two numbers that the quotient is in between.
Solution:
We have to think of the numbers that are close to 5184 and easily divided by 6.
Use 5100 and 5220 because they are close to 5184 and are easily divided by 6.
\(510\div~6~=~85\), so \(5100\div~6~=~850\).
\(522\div~6~=~87\), so \(5220\div~6~=~870\).
5184 is in between 5100 and 5220.
So the quotient, \(5184\div~6\) is between 850 and 870.
(Here, 5100 and 5220 are the compatible numbers which are easy to divide.)
3. A gallon contains 3,795 milliliters approximately. A gallon contains 5 times the number of milliliters as a quart. How many milliliters does a quart contain?
Solution:
We have to estimate \(3795~\text{milliliters}\div~5\) to know how many milliliters are in 1 quart.
Look at the first three digits of the dividend and use basic division facts. The first three digits of the dividend form the number 379. So we have to think of a number that is close to 379 and easily divided by 5.
Trying 3750 , \(375\div~5~=~75\), So that \(3750\div~5~=~750\)
Trying 3800 , \(380\div~5~=~76\), So that \(3800\div~5~=~760\)
Choose 3800 because 3795 is closer to 3800.
So, \(3795\div~5\) is about 760.
Therefore, there are about 760 milliliters in 1 quart.
4. For four months, a teenager works as a waiter in a restaurant and earns $3178. Every month, he earns the same amount. How much money does he make on a monthly basis?
Solution:
We have to estimate $\(3178\div~4\) to know the monthly income.
We have to think of the numbers that are close to 3178 and easily divided by 4.
Use 3080 and 3120 because they are close to 3178 and easily divided by 4.
\(308\div~4~=~77\), so \(3080\div~4~=~770\).
\(312\div~4~=~78\), so \(3120\div~4~=~780\).
3178 is in between 3080 and 3120. So the quotient, \(3178\div~4\) is between 770 and 780.
Therefore, the teenager earns between $770 to $780 per month.
5. Describe how can we come up with a better estimate for \(4250\div~5\) than the one provided.
(Round numbers 4520 to 5000. Estimating \(5000\div~5\).
\(5000\div~5~=1000\), so \(4250\div~5\) is approximately 1000).
Solution:
We have to think of the numbers that are close to 4520 and easily divided by 5.
Use 4500 and 5000 because they are close to 4520 and easily divided by 5.
\(450\div~5~=~9\), so \(4500\div~5~=~900\).
\(500\div~5~=~10\), so \(5000\div~5~=~1000\).
4520 falls between 4500 and 5000. So the quotient of \(4250\div~5\)is between 900 and 1000.
(Here, 4500 and 5000 are the compatible numbers that are easy to divide.)
Choose 4500 because 4500 is closer to 4520 as compared to 5000.
So, the quotient \(4250\div~5\) is about 900.
6. Estimate to compare: \(25\div~9 \) _ 2.
Solution: Look at the two digits of the dividend and use basic division facts.
We have to think of a number that is close to 25 and easily divided by 9.
Try 18 , \(18\div~9~=~2 \)
Try 27 , \(27\div~9~=~3 \)
Choose 27 because 27 is closer to 25.
So, \(25\div~9~=~3 \) is about 3.
Since \(27~> ~25 \)
Therefore, \(25\div~9> 2 \).
7. Estimate to compare: \(143\div~3 \) _ 50.
Solution:
Look at the first two digits of the dividend and use basic division facts. We have to think of a number that is close to 14 and easily divided by 3.
Trying 120 , \(12\div~3~=~4\) so \(120\div~3~=~40\)
Trying 150 , \(15\div~3~=~5\) so \(150\div~3~=~50\)
Choose 150 because 150 is closer to 143.
So, \(143\div~3\) is about 50.
Since \(143< ~50\).
Therefore, \(143\div~3~< ~50\).
Evaluation coefficients in the school electronic diary confuse parents, but give a more objective picture of knowledge
Shortly after Russian schools switched to electronic diaries, a new system for evaluating results began to be introduced in educational institutions.
Estimates for certain types of work received different “weights”, and because of this, the arithmetic mean does not coincide with the weighted average score given by the system.
Traditionally, the final grade for a quarter or trimester is given to a student based on the arithmetic mean of the grades he received during the school period, but some schools deviate from this rule and rely on another indicator – the so-called weighted average score. This is the sum of the products of the ratings and their “weight”, divided by the sum of the “weights” of these ratings. This indicator takes into account not only the assessments themselves, but also the coefficient of importance of the work for which they are delivered. For example, homework might be assigned a weight of 2, self-study 5, intermediate quiz 8, final quiz 10, and so on. Ready-made homework can be found on specialized educational projects, for example https://pomogalka.me/6-klass/russkij-yazyk/ladyzhenskaya-baranov/.
The index is set automatically, and each school determines its value independently.
There will be no freebies
Many students and their parents have already become accustomed to this and agree with this principle, but there are those who are not satisfied with it.
How is the weighted average mentioned above calculated? Each assessment of a student, which is displayed in an electronic diary, has its own coefficient depending on the complexity and importance of the work performed. The teacher himself determines the coefficient of each specific work, noting this in the electronic system.
So, for example, a grade for a control or laboratory work has a higher coefficient than for homework or an oral answer in a lesson, and, accordingly, its “weight”, and hence the impact on the final grade, will be higher. In some systems, the weight of marks is calculated on a hundred-point scale, in others – on a ten-point scale, but the principle does not change from this.
If a student received “easy” grades throughout the entire quarter, bringing essays downloaded on the Internet to lessons and doing homework using the GDZ, and wrote independent and control tests mainly in “triples”, according to the arithmetic mean, he may well count on a “four” “.
The weighted average score will not allow this – thanks to the coefficient, it will be clearly seen that the student does not know the subject at “four”, since all the main test papers are written for a lower score, and it will not be possible to correct the situation by getting good marks for simple tasks.
The system helps to solve the situation with disputable grades: according to the arithmetic mean, the question often arises of what to put the student – “four” or five”, and thanks to the weighted average score, the answer is very simple – if there are “five” for control and other complex tasks, the student deserves a higher score, and vice versa.
Konstantin Zyatkov
Konstantin Zyatkov, editor-in-chief of the weekly Nasha Versiya
School is not a job or an institution. Later, adults will have to face the fact that the fate of their studies at the institute is decided on a single exam in a semester or a year.
And in life they will not be customized at all until the deadlines come up and you do not have to answer to the customer. However, at school, students up to a certain age should be given high scores for “effort”. After all, not all students will go to the leading federal universities. For some of them, who did not happen to be gifted in all subjects, it is immediately important to maintain at least health. This whole tirade is about the fact that it remains important at school to evaluate the “effort” of children, their craving for knowledge and the desire to overcome difficult moments. The objectivity of assessing the knowledge of the subject in points is important, but not to the detriment of the life of a young person. Therefore, if the student was frightened and passed the test with a low score, it is worth evaluating his homework, and how wonderfully he stretched his hand in class. The desire to turn the school into a higher educational institution is already noticeable in the primary grades.
Third graders have to walk around and retake tests in order to get a yearly mark. This can be imagined at the institute from a person who is afraid to fly out of the university or lose his place in the hostel. But the desire to whittle young schoolchildren to the state of the highest significance and to objectify their grades, in the opinion of a layman, seems beyond the limits. Objective indicator
This principle looks quite fair, but not all parents of students are satisfied with this approach: some do not understand how the final grade is determined, some think that the transition to a new system is completely illegal .
However, this is not the case. In accordance with the 273rd Federal Law on Education, schools have the right to independently decide on what system they will use to assess students’ knowledge.
Schools that use weighted averages believe that the innovation helps to identify students’ weaknesses and show them what they need to work on.
Teachers note that children often strive to improve their quarterly marks by taking additional assignments on the subject, starting to prepare essays and memorizing material in order to answer in the lesson, but they still do not write test papers in the best way.
In addition, when evaluating the progress of students for a quarter, teachers used to rely on the results of tests. Now the calculations have simply become more convenient – now they are performed by a computer program.
Changes do not lead to a deterioration in academic performance: if a student misses a test due to illness, he will not be given a “zero” for this, which will spoil the fourth grade.
In addition, there is an opinion that such a system will help schoolchildren reduce the stress level when passing the USE: each exam task “weighs” differently, and this is not a surprise for students. The scoring system with coefficients will allow them to be better prepared for this.
Human factor
Of course, some incidents happen in practice.
So, for example, in October last year, a scandal erupted in Voronezh school No. 101: first, some grades were given for the first quarter, and then they were replaced by others, citing a changed grading system. Parents of schoolchildren were outraged that they were not warned about the innovation, and the announcement of the weighted average grading system appeared on the bulletin board a few days before the end of the term.
The director of the educational institution, in turn, said that parents were warned about the changes at the end of the last school year. According to her, she signed the corresponding order on August 31, 2019, and on September 2 she announced this on the line. The head of the school suggested that the questions arose from parents who rarely attend parent-teacher meetings. The director explained the confusion with the grades by saying that “teachers are people too”, and some of them mixed up the grades for the current and final dictations. As a result, the weighted average was calculated incorrectly, and this affected the final grade, which was subsequently corrected.
The results of the quarter are summed up in schools: parents are at a loss
December 21, 2016
19:08
Vesti-Moscow
Only a few days left before the school holidays. Schools are summing up the results of the second quarter, and many parents are at a loss. The final grades of their children do not agree with the data of electronic diaries. Some excellent students turn into good students, and poor students, on the contrary, rejoice at the results. How did it happen?
School holidays are just a few days away. Schools are summing up the results of the second quarter, and many parents, to put it mildly, are at a loss. The final grades of their children do not agree with the data of electronic diaries.
Some excellent students turn into good students, and poor students, on the contrary, rejoice at the results. How did it happen?
– In principle, I don’t understand how marks are given in the electronic diary, because there are these intermediate points.
It is difficult for Olesya Startseva to understand her daughter’s electronic diary. She does not understand why, with most of the fives in algebra, the final grade is a “solid” four.
– If the child is doing well – say, 4.60 – then the teacher will not give a 5, he will give a 4, because the child does not reach 5. That is, we pull out the losers, but the good ones? I do not understand!
But Oksana Kucherova, the mother of three schoolchildren at once, seems to have an easy task to calculate the average score.
But it’s not so simple – after all, this average score can also differ from the final grade for the quarter. Here is an example. Grades for the first trimester – seven fives and four fours.
The average score is 4.64, but the final grade is 4. Why? It turns out that the assessment of the assessment is different.
“Everyone understands that the significance of the test is still immeasurably higher than even a brilliant, but oral answer at the blackboard,” explains Elena Norenko, director of school No. 1995.
Marks for tests in the electronic diary are in bold.
– It can calculate the average score for tests – but only if you have activated this button.
How much an assessment “weighs” is up to the teachers themselves.
“Of course, first of all we are guided by the local normative act on grading. We pay attention to those works that show the child’s ability to apply the knowledge gained in practice, the extent to which he mastered the program material,” says the teacher of Russian language and literature school number 109Larisa Evstratova.
“I had such a case when I was on the verge of a three or four, and the teacher gave me a 4, because I wrote a test with a good mark,” says one of the students.