How should students show their work for math word problems?
Someone recently asked me about showing work in math word problems and I thought others might enjoy hearing about this topic also.
Personally, in the lower grades, I'd ask the child to EXPLAIN their thought processes orally, and then gradually teach them to write something on paper. The main thing students in grades 1-3 need to write is the actual calculations they did, not only the final answer.
For example, if they added 23 and 87 to get the answer, they should write 23 + 87 = 110 and include the units of whatever it was, such as $23 + $87 = $110 or 23 cm + 87 cm = 110 cm.
In the upper elementary grades (4-6) I'd like to see students write sentences and/or words in addition to the calculations so that another person can follow their solution.
I'll give some examples.
(from 4th grade)
Mr. Jefferson travels from Paducah to Lexington and back, three times a month. What is his total mileage? (A map shows that the distance in question is 255 miles.)
An example solution showing the work:
One round trip is 255 + 255 = 510 miles.
Three round trips are 510 + 510 + 510 = 1,530 miles.
(from grade 4)
Mick earned $345 from strawberry picking, and Jeanine earned three times as much. How much did they earn in total?
A concise solution showing the work:
Jeanine: 3 × $345 = $1,035
Total: $345 + $1,035 = $1,380
A bit more wordy solution showing the work:
Jeanine earned 3 × $345 = $1,035.
In total, Mick and Jeanine earned $345 + $1,035 = $1,380
Both should allow another person to follow the reasoning.
(from grade 5)
A T-shirt cost $10.50, but now it is discounted by 2/5 of its price.
Annie buys ten shirts with the discounted price. What is her total bill?
An example concise solution showing the work:
Cost of one shirt:
$10.50 ÷ 5 = $2.10
3 × $2.10 = $6.30
Total bill:
10 × $6.30 = $63.00
A bit more wordy solution:
One shirt costs 3/5 of $10.50, which is $10.50 / 5 × 3 = $6.30.
Then ten shirts cost 10 × $6.30 = $63.00.
David pays a 20% income tax on his $2,100 salary.
a. How many dollars is the tax?
b. How much money does he have left after paying the tax?
These are very simple questions so this time writing the calculations is enough.
a. $2,100 / 10 = $210
2 × $210 = $420
or
$2,100 / 5 = $420.
b. Again, all that's necessary to show the work is to write a subtraction.
$2,100 − $420 = $1680
(from 6th grade)
14. A herd of 40 horses had some bay, some chestnut, and some white
horses. Thirty percent of them are bay, and 45% are chestnut.
How many horses are white?
An example solution showing the work:
100% − 30% − 45% = 25%.
So 25% of the horses are white.
25% is 1/4.
And 1/4 of 40 horses is 10 horses.
OR
Percentage of white horses:
100% − 30% − 45% = 25%
25% of 40 horses is 10 horses.
So 10 horses are white.
In a nutshell:
The purpose of writing down the work allows someone else to follow the person's thought processes. This is of course important for students to learn no matter what their future occupation: they need to be able to explain to others how they solve a problem, whether a math problem or a problem in some other field of life!
One more tip: You could ask a fellow student or sibling to read the student's work to check if it can be followed and understood!
Personally, in the lower grades, I'd ask the child to EXPLAIN their thought processes orally, and then gradually teach them to write something on paper. The main thing students in grades 1-3 need to write is the actual calculations they did, not only the final answer.
For example, if they added 23 and 87 to get the answer, they should write 23 + 87 = 110 and include the units of whatever it was, such as $23 + $87 = $110 or 23 cm + 87 cm = 110 cm.
In the upper elementary grades (4-6) I'd like to see students write sentences and/or words in addition to the calculations so that another person can follow their solution.
I'll give some examples.
(from 4th grade)
Mr. Jefferson travels from Paducah to Lexington and back, three times a month. What is his total mileage? (A map shows that the distance in question is 255 miles.)
An example solution showing the work:
One round trip is 255 + 255 = 510 miles.
Three round trips are 510 + 510 + 510 = 1,530 miles.
(from grade 4)
Mick earned $345 from strawberry picking, and Jeanine earned three times as much. How much did they earn in total?
A concise solution showing the work:
Jeanine: 3 × $345 = $1,035
Total: $345 + $1,035 = $1,380
A bit more wordy solution showing the work:
Jeanine earned 3 × $345 = $1,035.
In total, Mick and Jeanine earned $345 + $1,035 = $1,380
Both should allow another person to follow the reasoning.
(from grade 5)
A T-shirt cost $10.50, but now it is discounted by 2/5 of its price.
Annie buys ten shirts with the discounted price. What is her total bill?
An example concise solution showing the work:
Cost of one shirt:
$10.50 ÷ 5 = $2.10
3 × $2.10 = $6.30
Total bill:
10 × $6.30 = $63.00
A bit more wordy solution:
One shirt costs 3/5 of $10.50, which is $10.50 / 5 × 3 = $6.30.
Then ten shirts cost 10 × $6.30 = $63.00.
David pays a 20% income tax on his $2,100 salary.
a. How many dollars is the tax?
b. How much money does he have left after paying the tax?
These are very simple questions so this time writing the calculations is enough.
a. $2,100 / 10 = $210
2 × $210 = $420
or
$2,100 / 5 = $420.
b. Again, all that's necessary to show the work is to write a subtraction.
$2,100 − $420 = $1680
(from 6th grade)
14. A herd of 40 horses had some bay, some chestnut, and some white
horses. Thirty percent of them are bay, and 45% are chestnut.
How many horses are white?
An example solution showing the work:
100% − 30% − 45% = 25%.
So 25% of the horses are white.
25% is 1/4.
And 1/4 of 40 horses is 10 horses.
OR
Percentage of white horses:
100% − 30% − 45% = 25%
25% of 40 horses is 10 horses.
So 10 horses are white.
In a nutshell:
The purpose of writing down the work allows someone else to follow the person's thought processes. This is of course important for students to learn no matter what their future occupation: they need to be able to explain to others how they solve a problem, whether a math problem or a problem in some other field of life!
One more tip: You could ask a fellow student or sibling to read the student's work to check if it can be followed and understood!
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