Deck 7: Developing Problem-Solving Strategies

Question 1

The strategy of making a table or graph is important because it:

A)makes relationships between numbers more apparent.
B)is easy for children to make these representations.
C)has connections to art.
D)has connections to spreadsheets and graphing calculators.

Answer 1

makes relationships between numbers more apparent.

Question 2

Heuristics refers to:

A)the study of calendars and time.
B)a simple form of algebra taught in elementary school.
C)a plan or general organizer for problem solving.
D)a simple form of statistics taught in elementary school.

Answer 2

a plan or general organizer for problem solving.

Question 3

In this number sequence, which are the next numbers? 
4, 8, 3, 9, 2, ____

A)1, 10, 0
B)10, 0, 1
C)10, 1, 11
D)7, 4, 11

Answer 3

10, 1, 11

Question 4

A balanced approach to mathematics means that students learn:

A)concepts and skills as they learn how to apply the skills in problems.
B)speed and accuracy in computation.
C)how to use calculators and computers at the same time they are learning math facts.
D)about geometry and number systems.

Question 5

The problem-solving strategies of acting out, drawing, and modeling are most directly related to which process standard?

A)reasoning and proof
B)connections
C)algebra
D)mathematical representation

Question 6

Building a model is a problem-solving strategy which:

A)is used by teachers to allow children to play during class.
B)allows children to use objects to represent the problem concretely.
C)keep children from feeling like failures in learning mathematics.
D)girls have more difficulty understanding spatial relationships.

Question 7

Patterning is an important strategy because:

A)children find patterns easier to work with than other strategies.
B)without patterns children cannot understand any mathematics beyond arithmetic.
C)patterns are critical for development of thinking skills.
D)learning one or two patterns is sufficient and allows time for other topics.

Question 8

Implementing a problem-solving curriculum may be difficult for all of the following reasons except:

A)parents often want their children to learn basic computational skills instead of solving problems.
B)administrators are not always accepting of the active and noisier problem-solving environment.
C)teachers have few resources for problem-solving through professional organizations.
D)preparing students for standardized tests may take away time that could be spent on solving problems.

Question 9

One value of the guess-and-check strategy is that:

A)incorrect solutions may provide clues to better ones.
B)it may be used when all the other strategies have not worked.
C)it is used by itself to solve problems.
D)it was advocated by Polya.

Question 10

The problem below is best solved using which strategy?
Pick a number, double it, add 5, double the result, subtract 6, divide by 2. If Mary's answer is 20, what was her beginning number?

A)work backwards
B)make a table or graph
C)act it out
D)solve a simpler problem

Question 11

Patterning is a thinking process in which humans find:

A)regularity and repetition in seemingly unconnected events and objects.
B)cause and effect relationships.
C)similarities among many events and objects.
D)differences between many events and objects.

Question 12

Presented with a problem such as "Which is larger, 2³⁰ or 3²⁰?" which of the following strategies would be the most effective strategy for solving the problem?

A)acting it out
B)solving a simpler problem
C)building a model
D)guess and check

Question 13

Learning about problem solving refers to learning different strategies.

Question 14

The strategy for problem-solving called Act it Out is useful because:

A)children understand the steps of a problem through play.
B)children enjoy acting out stories.
C)teachers spend more class time in problem solving.
D)it is related to the process standard of reasoning and proof.

Question 15

Problem solving in elementary school is best understood as:

A)fun activities such as puzzles and games.
B)the central focus of mathematics teaching and learning.
C)challenging problems for the mathematically talented students.
D)working word problems similar to the rote practice problems in textbooks.

Question 16

As estimation skills are developed, students need referents, or ____________, to help them make reasonable estimates.

A)hints
B)small numbers
C)heuristics
D)comparisons

Question 17

One value of drawing a picture or diagram as a problem-solving strategy is that it:

A)allows children with artistic talent to solve a problem more easily than others.
B)may obscure the problem if children get too detailed in their drawings.
C)can be difficult for girls because of their difficulties with spatial situations.
D)may develop into a solution by making the problem more apparent.

Question 18

A problem is a situation that:

A)requires an entire class period to solve.
B)teachers think the more capable students should be able to solve.
C)must have only one solution.
D)has no immediate solution or known solution strategy.

Question 19

The following are characteristics of the strategy make a table and/or a graph except :

A)learning to make a table or a graph is both a problem-solving strategy and mathematical content.
B)concepts and skills need for data collection and analysis also enable students to organize and present problem data in a table or graph.
C)recording data gives a visual display that allows students to look for patterns and relationships.
D)making a graph ensures that mathematical understanding will increase.

Question 20

The strategy break set:

A)is a strategy that should be tried when working on breaking sets of objects like base ten blocks into smaller sets of objects when solving a problem.
B)should be tried last after all other strategies have been tried to solve a problem.
C)suggests that persistence and perseverance are highly valued when solving problems.
D)suggests that having flexibility and being able to change points of view are highly regarded problem solving strategies.

Question 21

Children's literature may present situations for problem solving.

Question 22

What is the greatest strength of the strategy "solving a simpler problem"?

Question 23

Mike saw a new ebook reader advertised in the newspaper for $160. How much would he pay his credit card if he paid 1% interest each month and paid it all off after one month? If he paid half in one month and the other half the next month? If he paid it off in three month of equal installments? In four months? In twelve equal installments? Work in small groups to see if you can find a pattern in solving the problem.

Question 24

Learning for problem solving refers to developing skills to apply to real-life problems.

Question 25

Give an example of a problem that could use the "working backward" strategy, then explain why you think this is a useful strategy.

Question 26

If a store sells four flavors of ice cream and three toppings, can every child on a soccer team of 20 players have a different ice cream sundae? Using the problem-solving strategy "account for all possibilities," determine whether you have 20 combinations, more than 20, or fewer than 20. Discuss and compare how you represented your solution.

Question 27

Do you think the strategy "guess and check" or "trial and error" should be encouraged during the high school years, particularly in algebra classes? Why or why not?

Question 28

Problem solving is best taught as a way to practice computation.

Question 29

Eleven strategies are all the strategies needed for solving problems.

Question 30

Learning through problem solving means that students learn mathematics while they are solving problems.

Question 31

Learning through problem solving means that students learn the four-step Polya method and apply it in all situations.

Question 32

Compare and contrast the strategies "building a model" and "drawing a picture or diagram."

Question 33

Describe a school activity or event that a teacher might use as an opportunity for students in grades 3-6 to have an open-ended problem solving experience.

Question 34

Explain how the strategy "finding and using a pattern" progresses from kindergarten into algebra in middle school. Give an example to help you explain.

Question 35

You will throw 5 darts in a game. The bull's-eye has a value of 5. The middle circle has a value of 3, and the outer circle has a value of 1. In your game, each dart hits and sticks to the board.

(a)What is the highest score you could have in your game?
(b)The lowest score?
(c)What other scores could you have?
(d)Is there anything unique about scores for this 5-dart board?
(e)Identify your strategy or strategies and explain how you used them.
This rubric will be used to evaluate your work on this task. Credit will be awarded on a 4-point scale.

Question 36

Explain what Schroeder and Lester (1989)meant by their three approaches to problem-solving instruction (teaching about problem solving, teaching for problem solving, and teaching via problem solving).

Question 37

Learning about problem solving refers to finding real-life problems to solve.

Question 38

The "break set" or "change point of view" strategy can be difficult for some students to implement. Explain why.

Question 39

What is Polya's four-step problem-solving process?

Question 40

Explain why problem solving should be the focus of mathematics teaching and learning.

Question 41

In the Bean Bag Toss game, a player can score the points shown.

Max tossed three bean bags in the same hole and two bean bags in another hole. What is the highest score that max could receive for these five tosses? 
Show how you found your answer using pictures, words or numbers.

Question 42

Using the paper strip below, place an X in one of the squares so that if the paper strip were folded along the dashed fold line, the square with the X could cover the shaded square.

Question 43

What is the area of the shaded region of the figure below?

a. 24
b. 30
c. 36
d. 42