Chapter 9: Developing Meanings for the Operations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
1) Which problem represents the join, result unknown structure?
a) Maryann had 3 library books before she checked out 2 more. How many did she have all together?
b) Maryann had 5 library books before she returned 2 of them. How many does she have now?
c) Maryann had 4 nonfiction books and 2 fiction. How many more nonfiction books does she have?
d) Maryann had 9 library books and Jim had 4. How many more did Maryann have than Jim?
2) Which problem represents the separate, result unknown structure?
a) Maryann had 3 library books before she checked out 2 more. How many did she have all together?
b) Maryann had 5 library books before she returned 2 of them. How many does she have now?
c) Maryann had 4 nonfiction books and 2 fiction. How many books does she have?
d) Maryann had 9 library books and Jim had 4. How many more did Maryann have than Jim?
3) Which problem represents the compare, difference unknown structure?
a) Maryann had 3 library books before she checked out 2 more. How many did she have all together?
b) Maryann had 5 library books before she returned 2 of them. How many does she have now?
c) Maryann had 4 nonfiction books and 2 fiction. How many books does she have?
d) Maryann had 9 library books and Jim had 4. How many more did Maryann have than Jim?
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
4) Most curricula place a heavy emphasis on join and separate structures, with the result unknown.
5) Problems with the join and separate structures, with the start or initial amount unknown, tend to be the
easiest for students to understand and accurately solve.
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
6) The ____________________ equation for a problem lists the numbers in the order of the meaning in the
problem.
7) An equation that isolates the unknown is the _______________________form.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
8) Which of the following is a true statement about contextual problems?
a) They are connected more to what is referred to as “school mathematics” than they are to children’s
lives.
b) Their origin could result from recent experiences in the classroom.
c) Their language orientation can make them inappropriate for ELL students.
d) Enhancing them with visual aids or students acting them out would be inappropriate.
9) The relationship between addition and subtraction
a) Can confuse children and should not be addressed in class.
b) Should be developed for children by providing them with examples of the same number sets
appearing in different contexts.
c) Is not an example of an inverse relationship.
d) Can prevent students from developing flexibility in the methods they use to solve problems.
10) Which of the following is NOT a true statement regarding mathematical symbols?
a) It is important that very young children understand the symbols +, –, and = to learn about addition
and subtraction concepts.
b) Great care should be taken to assure students understand = means “is equal to” rather than “the
answer is.”
c) By first grade, symbolic conventions are important for children to know.
d) Frequently, the meaning of the equal sign confuses students.
SHORT ANSWER. Write the word or phrase that best completes each statement or
answers the question.
11) Bar diagrams, counters, and number lines are examples of ___________ that can be used to help solve
story problems.
12) When thinking about subtraction as __________________________ rather than “take away,” students ask
themselves “What goes with the part I see to make the whole I need?”
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
13) The commutative property
a) Applies to addition and subtraction.
b) Helps students master basic facts because, if they really understand it, it reduces the number of
individual facts they have to memorize.
c) Should be demonstrated with problems that have the same sums but different addends.
d) Is a term that even very young students should memorize.
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
14) Adding and subtracting the number 0 can be confusing for students because they assume the value of a
number must change when adding or subtracting another number (even 0) to/from it.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
15) Which problem is an example of the equal groups, number of groups unknown structure?
a) This month, Barry saved 8 times as much as last month. Last month, he saved $3. How much did
Barry save this month?
b) Barry’s sandwich shop offers 3 kinds of meat and 2 kinds of bread. How many different
sandwiches could he make if he uses 1 meat and 1 kind of bread for each?
c) Barry saved $12 and Jill saved $6. Barry saved how many times as much money as Jill?
d) Barry saved $24 total, and he saved $6 each month. For how many months had he been saving?
16) Which problem is an example of the comparison, product unknown (multiplication) structure?
a) This month, Barry saved 8 times as much as last month. Last month, he saved $3. How much did
Barry save this month?
b) Barry’s sandwich shop offers 3 kinds of meat and 2 kinds of bread. How many different
sandwiches could he make if he uses 1 meat and 1 kind of bread for each?
c) Barry saved $12 and Jill saved $6. Barry saved how many times as much money as Jill?
d) Barry saved $24 total, and he saved $6 each month. For how many months had he been saving?
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
17) The usual mathematical convention says that 4 × 8 represents “four sets of eight.”
18) Because real-life division usually results in a simple whole number, it is not important for students to
understand remainders as anything more than the amount “left over.”
ESSAY. Write your answer in the space provided or on a separate sheet of paper.
19) Describe at least four kinds of models one could use to model a multiplication or division situation.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
20) What number property is illustrated by the problem 16 × 12 = 16(10 + 2) = 160 + 32 = 192?
a) Associative
b) Commutative
c) Distributive
d) Identity
21) Which would NOT be a good statement/question from a teacher to help a student realize why he or she
can’t divide by 0?
a) “What happens when you take these 25 pennies and divide them into 0 groups?”
b) “Can you show me how to share 8 apples between no people?”
c) “What can you multiply by 0 to get 7?”
d) “Just memorize that you can’t divide by 0.”
TRUE/FALSE. Write “T” if the statement is true and “F” if the statement is false.
22) The key word problem solving strategy is effective, because it makes the process of choosing an operation
less complicated for students.
