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Chapter 8: Developing Meanings for the Operations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
1) What statement identifies the importance of using contextual problems as a primary teaching
tool?
A) Children demonstrate procedural fluency.
B) Children identify the structure of the situations.
C) Children construct richer understanding of the operations.
D) Children connect problems to school mathematics.
2) Identify the problem that 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?
3) Which problem represents the separate, start 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 some nonfiction books and 2 fiction. She now has 8 books. How many did she
begin with?
D) Maryann had 9 library books and Jim had 4. How many more did Maryann have than Jim?
4) 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?
5) What statement below is the description of a Part-part-whole?
A) Involve comparison of two quantities.
B) Two parts that are conceptually or mentally combined into one whole.
C) The change is the amount being removed.
D) Initial action of joining start amount, a change amount and resulting amount.
6) Problems with the join and separate structures, with the start or initial amount unknown, tend
to be the hardest for students to understand and accurately solve. Identify the reason for they are
more challenging for children to use.
A) Children can model the physical action.
B) Children can act out the situation.
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C) Children cannot use counters for the initial amount.
D) Children cannot grasp a quantity represents two things at once.
7) A good lesson built on a context or related to a story would have all of the qualities listed
below EXCEPT:
A) Words, pictures and numbers are used to explain solution.
B) Students can easily find a solution with mental mathematics.
C) Students can find mistakes in other’s written solutions.
D) Designed to anticipate and develop mathematical models of the real world.
8) Identify the reason why the equal sign can confuse children.
A) Relational symbol not operations.
B) Means the answer is coming up.
C) Operations symbol not relational.
D) Means the answer is the same as.
9) The use of model-based problems that students can use counters, bar diagrams, or number
lines helps with their problem solving skills. Identify the statement below that describes what the
use of bar diagrams can demonstrate.
A) Modeling two parts in two separate piles.
B) Measuring distances from zero.
C) Generating mean-making space.
D) Breaking the shape apart to show multiple parts.
10) Identify the statement that is describes the importance for children to know the relationship
between addition and subtraction.
A) Writing symbolic equations.
B) Using the associative property.
C) Mental mathematics.
D) Using the same models or pictures.
11) Why is it significant for students to use think-addition for subtraction rather than take-away?
A) Mastering subtraction facts.
B) Mastering symbolic representations.
C) Mastering problem solving strategies.
D) Mastering model-based solutions.
12) What is the importance of students knowing 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?
13) 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
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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 one meat and one 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?
14) Which problem is an example of the comparison, product unknown (multiplication)
structure?
A) Barry’s sandwich shop offers 3 kinds of meat and 2 kinds of bread. How many different
sandwiches could he make if he uses one meat and one kind of bread for each?
B) Barry saved $12 and Jill saved $6. Barry saved how many times as much money as Jill?
C) This month, Barry saved 8 times as much as last month. Last month, he saved $3. How much
did Barry save this month?
D) Barry saved $24 total, and he saved $6 each month. For how many months had he been
saving?
15) All of the statements below are related to teaching multiplication and division EXCEPT:
A) What to do with remainders.
B) Symbolism as a way to record thinking.
C) Physical models, drawings and equations.
D) Think addition as multiplication and take away as division.
16) Remainders have an effect on all of the following EXCEPT:
A) Discarded leaving a smaller whole-number answer.
B) Rounded to the nearest whole number for an approximate answer.
C) Fractional part left over and not included in the whole-number answer.
D) Force the answer to the next highest whole number.
17) Complete this statement, “Constructing models of arrays draws attention to…”
A) Factors connection with rows and columns.
B) Factors and product.
C) Number of rows and columns.
D) Connection with measurement of area.
18) What number property is illustrated by the problem 16 × 12 = 16(10 + 2) = 160 + 32 = 192?
A) Associative.
B) Commutative.
C) Identity.
D) Distributive
19) Statements/questions from a teacher can support children’s understanding of why one can’t
divide by 0 EXCEPT:
A) “What happens when you take these 25 pennies and divide them into 0 groups?”
B) “Just memorize that you can’t divide by 0.”
C) “Can you show me how to share 8 apples between no people?”
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D) “Put 12 blocks in 0 equal groups.”
20) The key word strategy sends a wrong message about problem solving. Identify the statement
below that would be offered in support of the key word strategy.
A) Encourages children to ignore meaning and structure of problems.
B) Many problems do not have key words.
C) Encourage children to use a list of key words with corresponding operations.
D) Many problems have key words that may be misleading.
ESSAY. Write your answer in the space provided or on a separate sheet of paper.
21) Present and discuss two reasons for using contextual problems to teach addition, subtraction,
multiplication and division.
22) How do the commutative and associative properties relate to children’s understanding of the
basic facts?
