Saxon Investigation 1: Translating and Writing Word Problems

Saxon Investigation 1: Translating and Writing Word Problems

​In this investigation, we will study four types of word problems:

• problems about combining

• problems about separating

• problems about equal groups

• problems about comparing.

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All the problems contain three numbers.

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A problem becomes a word problem when one of its numbers is replaced with a question.

We will make three different word problems for each problem in this investigation by replacing the numbers with questions.

​Word Problems about Combining

​We combine two (or more) quantities by adding them together. We start with some and add some more. Here is a problem about combining:

a. The troop hiked 8 miles in the morning.

b. The troop hiked 7 miles in the afternoon.

c. Altogether, the troop hiked 15 miles.

​

Notice that there are three numbers. The numbers in a and b add up to the number in c. If we know any two of the numbers, we can figure out the third number. The problem is written in three sentences.

Multiple Choice

a. The troop hiked 8 miles in the morning.

b. The troop hiked 7 miles in the afternoon.

c. Altogether, the troop hiked 15 miles.

Suppose sentence b were missing from the problem. Read sentences a and c and then write a question that asks for the number in sentence b. Start with the words, “How many miles. . . .”

• How many miles did the troop hike in the afternoon?

• How many miles did the troop hike in the morning?

• Altogether, how many miles did the troop hike during the day?

• None of these above.

Multiple Choice

a. The troop hiked 8 miles in the morning.

b. The troop hiked 7 miles in the afternoon.

c. Altogether, the troop hiked 15 miles.

Suppose sentence c were missing. Read sentences a and b and then write a question that asks for the number in c. This time start the question with the words, “Altogether, how many miles. . . .”

• How many miles did the troop hike in the afternoon?

• How many miles did the troop hike in the morning?

• Altogether, how many miles did the troop hike during

  the day?

• None of these above.

Multiple Choice

a. The troop hiked 8 miles in the morning.

b. The troop hiked 7 miles in the afternoon.

c. Altogether, the troop hiked 15 miles.

Suppose sentence a were missing. Read sentences b and c and then write a question that asks for the number in sentence a. Start the question with the words, “How many miles. . . .”

• How many miles did the troop hike in the afternoon?

• How many miles did the troop hike in the morning?

• Altogether, how many miles did the troop hike during the day?

• None of these above.

​Word Problems about Separating

​We separate one quantity from a larger quantity by taking some away, or subtracting. Here is a problem about separating:

d. Jack went to the store with $28.

e. Jack spent $12 at the store.

f. Jack left the store with $16.

​

This is a problem about Jack’s money. Jack had some money; then some money “went away” at the store. There are three numbers in the problem. If one of the numbers were missing, we could figure out the missing number.

Multiple Choice

d. Jack went to the store with $28.

e. Jack spent $12 at the store.

f. Jack left the store with $16.

Suppose sentence f were missing. Read sentences d and e and then write a question that asks for the number in sentence f. Start the question with the words, “How much money did. . . .”

• How much money did Jack have when he went to the store?

• How much money did Jack spend at the store?

• How much money did Jack have when he left the store?

• None of these above.

Multiple Choice

d. Jack went to the store with $28.

e. Jack spent $12 at the store.

f. Jack left the store with $16.

Suppose sentence d were missing. Read sentences e and f and then write a question that asks for the number in sentence d. Start with the words, “How much money. . . .”

• How much money did Jack have when he went to the store?

• How much money did Jack spend at the store?

• How much money did Jack have when he left the store?

• None of these above.

Multiple Choice

d. Jack went to the store with $28.

e. Jack spent $12 at the store.

f. Jack left the store with $16.

Suppose sentence e were missing. Read sentences d

and f; and then write a question that asks for the number in sentence e.

Start with the words, “How much money. . . ."

• How much money did Jack have when he went to the store?

• How much money did Jack spend at the store?

• How much money did Jack have when he left the store?

• None of these above.

​Word Problems about Equal Groups

​Some problems are about items that are clustered in groups of equal size. These problems might describe the number of groups, the number in each group, and/or the total number in all groups. By multiplying the number in each group by the number of groups, we can find the total in all groups. Here is an example of an “equal groups” problem:

​

At Lincoln School there are the same number of students in each fifth grade class.

g. At Lincoln School there are 4 classes of fifth grade students.

h. There are 30 students in each fifth grade class.

i. Altogether, there are 120 fifth grade students at Lincoln School.

​

Again we see three numbers in the problem. If we know two of the numbers, we can figure out the third number.

Multiple Choice

At Lincoln School there are the same number of students in each fifth grade class.

g. At Lincoln School there are 4 classes of fifth grade students.

h. There are 30 students in each fifth grade class.

i. Altogether, there are 120 fifth grade students at Lincoln School.

Suppose sentence g were missing. Read sentences h and i and then write a question that asks for the number in sentence g. Start with the words, “How many classes. . . .”

• How many classes of fifth grade students are there at Lincoln School?

• How many students are in each fifth grade class?

• Altogether, how many fifth grade students are at Lincoln School?

• None of these above.

Multiple Choice

At Lincoln School there are the same number of students in each fifth grade class.

g. At Lincoln School there are 4 classes of fifth grade students.

h. There are 30 students in each fifth grade class.

i. Altogether, there are 120 fifth grade students at Lincoln School.

Suppose sentence h were missing. Read sentences g and i and then write a question that asks for the number in sentence h. Start with the words, “How many students . . .”

• How many classes of fifth grade students are there at Lincoln School?

• How many students are in each fifth grade class?

• Altogether, how many fifth grade students are at Lincoln School?

• None of these above.

Multiple Choice

At Lincoln School there are the same number of students in each fifth grade class.

g. At Lincoln School there are 4 classes of fifth grade students.

h. There are 30 students in each fifth grade class.

i. Altogether, there are 120 fifth grade students at Lincoln School.

Suppose sentence i were missing. Read sentences g and h and then write a question that asks for the number in sentence i. Start with the words, “Altogether, how many . . .”

• How many classes of fifth grade students are there at Lincoln School?

• How many students are in each fifth grade class?

• Altogether, how many fifth grade students are at Lincoln School?

• None of these above.

​Word Problems about Comparing

​One way to compare two numbers is to find how much larger or how much smaller one number is than the other. By subtracting the smaller number from the larger number, we find the difference of the numbers.

​

Consider this problem about comparing:

j. Abe is 5 years old.

k. Gabe is 11 years old.

l. Gabe is 6 years older than Abe.

​

A comparison may be stated two ways. For example, sentence l could

have been written, “Abe is 6 years younger than Gabe.”

Multiple Choice

Consider this problem about comparing:

j. Abe is 5 years old.

k. Gabe is 11 years old.

l. Gabe is 6 years older than Abe.

Suppose sentence j were missing. Read sentences k and l and then write a

question that asks for the number in sentence j.

• How old is Abe?

• How old is Gabe?

• Abe is how much younger than Gabe?

• Gabe is how much older than Abe?

Multiple Choice

Consider this problem about comparing:

j. Abe is 5 years old.

k. Gabe is 11 years old.

l. Gabe is 6 years older than Abe.

Suppose sentence k were missing. Read sentences j and l and then write a question that asks for the number in sentence k.

• How old is Abe?

• How old is Gabe?

• Abe is how much younger than Gabe?

• None of these above.

Multiple Select

Consider this problem about comparing:

j. Abe is 5 years old.

k. Gabe is 11 years old.

l. Gabe is 6 years older than Abe.

Suppose sentence l were missing. Read sentences j and k and then write a question that asks for the number in sentence l. You should be able to phrase the question two different ways.

• How old is Abe?

• How old is Gabe?

• Abe is how much younger than Gabe?

• Gabe is how much older than Abe?