Q&A Hero

Q: Which of the following is an important principle of iterating units of length? a) They must always be standard measurement units. b) There must be no overlapping or gaps between the units. c) The units can be of different lengths. d) Rulers are the best tool to measure any length.

Q: When determining the mathematics content and learning goals for a lesson, the teacher should do all of the following EXCEPT a) Consult his or her state's curriculum standards b) Ask, "What is it my students should be able to do at the end of this lesson?" c) Plan goals that will take no longer than a day for students to accomplish d) Be sure to focus on the mathematics, rather than just the activity

Q: Name two strategies or methods for helping students to develop estimation skills. Describe how these strategies/methods would contribute to conceptual understanding.

Q: Describe 4 of the steps in the process of planning for the problem-based classroom.

Q: Which does NOT describe one of the roles estimation plays in measurement? a) It can provide intrinsic motivation when students measure and see how close their previously determined estimates are. b) It helps familiarize students with standard units when they check their estimates against real measurements. c) It's really not necessary, because so many measurement tools allow precise measurements. d) There are many situations where an exact measurement is not needed and an estimate is enough.

Q: In the during phase of the lesson, teachers must do all but which of the following? a) Avoid the urge to provide too much assistance to students b) Leave students completely alone to explore their thinking c) Notice and record students' processes d) Provide any necessary extensions

Q: One method of enhancing students' familiarity with units is to compare units to common items that they can use as ________________________________.

Q: In the before phase of a lesson, the teacher has as one of her goals a) Showing students a worked example similar to what they will be doing. b) Telling them the problem solving strategies that would be best for the task. c) Answering every question the students might have about the task. d) Establishing clear expectations.

Q: Customary units of measure should always be converted to metric measures because it is so much easier to operate in metric.

Q: Describe at least 4 ways that teaching through problem solving helps ensure equity in the classroom.

Q: According to the NCTM position statement on the metric system, it is important that schools equip students to deal with diverse situations in both metric and customary systems while developing their ability to solve problems in either system.

Q: Describe at least 4 ways to use writing in the mathematics classroom.

Q: Because the United States still uses customary units, it's not necessary for U.S. students to understand the metric system.

Q: When determining how much to tell, researchers suggest that teachers tell students all of the following EXCEPT a) Certain mathematical conventions, such as the meaning behind some symbols b) Alternative strategies that don"t emerge naturally from students c) The answer to any problem that has required more than 10 minutes of students' time to solve d) Any needed clarification of students' methods

Q: A child who truly has an understanding of standard units a) Has had lots of experience using them. b) Knows that measurements always have to be calculated precisely to the nearest unit. c) Knows how to choose the unit that is appropriate for a given situation. d) Knows how units are related to one another.

Q: Which of the following is the best example of metacognition? a) Taking a timed multiplication test b) Taking two numbers in a word problem and adding them because your class worked on addition problems the previous day c) When a student looks back at problems he previously worked incorrectly to examine his mistakes d) When two students are playing a game of Integer War,during which they each flip a card and the student who calls out the higher of the two numbers first wins the round

Q: Which of the following is NOT a benefit of using nonstandard units? a) They make it easier for the student to focus directly on the attribute being measured. b) It clarifies the learning goal when the objective is about measuring an attribute and not about the unit of measurement itself. c) The fact that they are not consistent makes students realize why standardized units are so important. d) They give students practice with using simple standardized instruments.

Q: The impact of questioning on student learning is determined by the level of the question, the type of knowledge targeted, the pattern of questioning, and how you ________________________.

Q: When helping students to develop measurement concepts a) It's important to make sure they know they should always use standardized measurement units. b) You should always use precise language. c) Use as many abstract examples as possible. d) Keep in mind that simple measurement instruments are easy for kids to use and that you don"t need to devote class time to practicing using them.

Q: Provide two specific examples of research-based recommendations that contribute to effective classroom discourse and an example of what you might say or do to implement each recommendation.

Q: A line segment and double line drawing model can help students solve a variety of ________________ problems.

Q: Examples of effective use of classroom discussion include all of the following EXCEPT a) Giving students with disabilities the opportunity to leave the room for certain interventions without missing valuable new mathematics content. b) Requiring students to use mathematical vocabulary to express their ideas and take ownership of making sense of mathematics. c) Providing students with limited language skills the opportunity to articulate their thinking. d) Allowing students the opportunity to discover multiple approaches to solving problems.

Q: _________________________ are a visual representation of the connection between algebra and proportional reasoning.

Q: Facilitating classroom discourse is easy for the teacher, because the students are doing all the talking.

Q: Which of the following is NOT a method that will help students develop their ability to think proportionally? a) Provide ratio and proportional tasks within many different contexts. b) Provide examples of proportional and non-proportional relationships to students and ask them to discuss the differences. c) Relate proportional reasoning to their background knowledge and experiences. d) Explain to them how important cross-multiplying is.

Q: Traditional textbooks a) Have been designed using the teaching forproblem solvingmodel. b) Should be avoided, unless they are required by your district. c) Are often not being designed around the teaching throughproblem solvingmodel. d) Rarely contain problems that can be adapted to the teaching throughproblem solvingmodel.

Q: Which of the following is an example of using the buildup strategymethod of solving proportions?a) Allison bought 3 pairs of socks for $12. To find out how much ten pairs cost, find that $12 divided by 3 is $4 a pair, and multiply $4 by 10 for a total of $40.b) A square with a length of 2 inches was enlarged by a scale factor of 4 and is now 8 inches long.c) If 5 candy bars cost $4.50, then 10 would cost $9. (Because 5 x 2 = 10, multiply $4.50 by 2.)d) If 2/3 = x/15, find the cross products, 30 = 3x, and then solve for x. x= 10.

Q: Which of the following is the weakest example of providing students with relevant contexts for problems? a) Using a piece of children's literature to introduce a situation that requires some kind of solution b) Asking students to reduce the amount of each ingredient that is needed for a recipe that is divided in half to accommodate a small family c) Asking a student to explain how he chose to divide the numerator and denominator of a fraction by 2 in order to simplify it d) Asking students to determine, given the area and population of two countries, which would be less crowded

Q: Which of the following is an example of using the unit ratiomethod of solving proportions?a) Allison bought 3 pairs of socks for $12. To find out how much 10 pairs cost, find that $12 divided by 3 is $4 a pair, and multiply $4 by 10 for a total of $40.b) A square with a length of 2 inches was enlarged by a scale factor of 4 and is now 8 inches long.c) If 5 candy bars cost $4.50, then 10 would cost $9. (Because 5 x 2 = 10, multiply $4.50 by 2)d) If 2/3 = x/15, find the cross products, 30 = 3x, and then solve for x.x= 10.

Q: Characteristics of a worthwhile problem include the level of cognitive demand, the potential of the task to have ______________________________ entry and exit points, and whether the task is relevant to students.

Q: Which of the following is NOT an example of a connection between proportional reasoning and another strand of mathematics? a) The area of a rectangle is 8 square units and the length is four units long. How long is the width? b) The negative slope of the line on the graph represents the fact that, for every 30 miles the car travels, it burns one gallon of gas. c) The triangle has been enlarged by a scale factor of 2. How wide is the new triangle if its original width is 4 inches? d) Sandy ate 1/4 of her Halloween candy and her sister ate of it. What fraction of her candy was left?

Q: Which of the following is an example of teaching throughproblem solving? a) Providing students with a list of area formulas and asking them to find the area of a given rectangle b) After students have conceptual understanding of the area of a rectangle, asking them to find the area of a triangle that was constructed by cutting a given rectangle in half and then to generalize their process to how they might find the area of any given triangle c) Teaching students the algorithm for fraction division and then asking them to find out how many servings of 1/3 pizza could be made from 3-2/3 pizzas d) Having students develop their own word problems that use a recently learned algorithm

Q: At Grocery Store A you can get 5 apples for $2. Grocery Store B has apples 3 for $1. 5 and 3 represent a ____________________ratio while $2 and $1 represent a "within" ratio.

Q: All of the following are features a problem should have EXCEPT a) Its design must take into consideration the students' current level of understanding. b) It should have an engaging context. c) It must require some kind of justification for methods and answers. d) It must be at a level that would make a student unable to solve it alone.

Q: Construct an example of when confusing additive and proportional thinking could result in an incorrect answer. Describe an incorrect process that a learner might follow. Describe a correct way to find the solution and a way you might help the learner to see his error.

Q: Which is the most accurate statement regarding posing a problem? a) Teachers should select the problems that will help make relationships between mathematical concepts explicit for students. b) Any problem used should have the potential to be solved by students using a memorized procedure. c) There should be agreement between students that there is one correct answer. d) Problems should involve words.

Q: Many learners confuse a(n) __________________________ situation, such as "Mary is 3 feet taller than Billy," for proportional situations.

Q: One commonly used problem-solving strategy used with story problems is writing an _______________________________.

Q: According to Lamon (2006), a _________________________ thinker would understand how two amounts vary in the same way.

Q: When teaching students about problem-solving strategies, it is best not to explicitly tell students which strategy to pick and how to use it.

Q: In the scenario "Billy's dog weighs 10 pounds while Sarah's dog weighs 8 pounds", the ratio 10/8 can be interpreted in the following ways EXCEPT a) For every 5 pounds of weight Billy's dog has, Sarah's dog has 4 pounds. b) Billy's dog weighs 1-1/4 times what Sarah's dog does. c) Sarah's dog weighs 8 out of a total of 10 dog pounds. d) Billy's dog makes up 5/9 of the total dog weight.

Q: Explicitly teaching students the four-step problem solving process does not usually improve their ability to reason mathematically. It is better to let them discover the process on their own.

Q: When comparing ratios to fractions keep in mind that a) Conceptually, they are exactly the same thing. b) They have the same meaning when a ratio is of the part-to-whole type. c) They both have a fraction bar that causes students to mistakenly think they are related in some way. d) Operations can be done with fractions while they can"t be done with ratios.

Q: Which statement about the teaching throughproblem solving approach is most accurate? a) It is a less demanding process for the teacher. b) The productive struggle that it frequently provides to students leads to enhanced learning. c) Any problem-solving task will help students to develop their own strategies and learn the content. d) It is pretty much the same process as teaching forproblem solving.

Q: Part-to-_________, part-to-whole, quotients, and rates are types of ratio.

Q: Which statement about the teaching forproblem solving approach is not accurate? a) It is the method most frequently used in traditional textbooks. b) It usually involves students learning an algorithm and then being asked to apply the algorithm to a story problem. c) The approach has traditionally been a very effective way to help students gain conceptual understanding. d) It frequently results in the likelihood that a student won"t attempt a new problem without explicit instructions on how to solve it.

Q: A ______________ is a number that relates two quantities or measures within a given situation in a multiplicative relationship.

Q: Name at least two examples of a tool that could help students to do mathematics and gain relational understanding of a concept. Describe a specific example of a way each tool could help develop this understanding.

Q: Common uses for _________ percentages in real-world situations include tips, taxes, and discounts.

Q: Which of the following statements regarding teaching towards mathematical proficiency is NOT true? a) It requires students to memorize less. b) It allows students to more easily make connections to new concepts. c) It takes much less effort and time than teaching traditionally. d) It increases student enjoyment and attitudes towards mathematics.

Q: To promote conceptual understanding of percentages, teachers should do all of the following EXCEPT a) Limit percentage to those that have familiar fraction equivalents b) Require students to use models and drawings to support their solutions to problems c) Have them memorize handy percent equations such as "percent times the base equals the part" d) Provide lots of examples in contexts

Q: Which of the following is NOT a true statement regarding the five strands of mathematical proficiency? a) They are the foundation for the Standards for Mathematical Practicein the Common Core State Standards. b) Conceptual understanding is not critical to developing procedural proficiency. c) Students should be skilled at choosing, testing, and evaluating problem-solving strategies within contexts. d) Students should be skilled at reflecting, evaluating, and adapting their work, as needed.

Q: Models used to represent percentages include rational number __________, percent necklaces, and 10 10 grids.

Q: Which of the following is an example of ineffective use of manipulative materials? a) Not showing students exactly every step of how to use them b) Making a connection between the model and the mathematical concept c) Encouraging students to converse about the model without knowledge of what the mathematical goal is d) Maintaining a balance between the appropriate amounts of guidance and student exploration

Q: A percent is simply another way of expressing a fraction or decimal that is measured in ___________.

Q: _____________________________ are physical objects that students and teachers can use to illustrate and discover math concepts. Sometimes they are created specifically for mathematical purposes (Unifix cubes), and sometimes they are originally created for other purposes (buttons).

Q: Contrary to those of whole numbers, the algorithms for decimal multiplication and division have no connection.

Q: In contrast to a student who has understanding closer to the instrumental end of the continuum, students with understanding closer to the ______________________ end are more likely to be able to provide their own real-life examples of a concept.

Q: Which is true about the algorithm for decimal multiplication? a) Students can discover the method by being given a series of multiplication problems with factors that have the same digits, but decimals in different places. b) It is too complicated for students to discover on their own, so they should just explicitly be shown how to do it. c) They should be shown how to estimate after they are shown the algorithm. d) The repeated addition strategy that works for whole number multiplication is not applicable.

Q: If a student makes the statement "I can simplify 6/8 to 3/4,"the teacher can conclude the student probably conceptually understands the idea of equivalent fractions.

Q: After students have become fluent at solving story problems with decimals, it's important to see if they can reason without a context.

Q: Because student learning should build upon previous knowledge and life experiences, teachers should become familiar with and respect the various forms of diversity in their classrooms.

Q: Decimal estimation a) Should be something that students can do well before they begin to compute decimals with pencil and paper. b) Is not very useful in everyday life. c) Does not help students determine whether or not their computation results are reasonable. d) Is not a domain in the Common Core State Standards.

Q: The process of scaffolding involves providing students with a learning task that requires them to seek assistance outside of the classroom.

Q: Which of the following is a misconception some students have about comparing decimals? a) The shorter the decimal number is, the bigger the amount it represents. b) .003 and .03 are different amounts. c) .4 is bigger than .6 because 1/4 is bigger than 1/6. d) 0 is smaller than .36 because it had a digit in the ones place, while .36 does not.

Q: Which of the following is a true statement regarding the practice of exposing students to multiple approaches to solving problems? a) The most useful aspect of the strategy is that students should see a variety of inferior methods so that they can better appreciate the one best method to solve a problem. b) Class discussions are not a valuable way for students to investigate alternative problem-solving strategies and make connections between mathematical ideas. c) The strategy is not very useful for very simple mathematical ideas, such as basic computation facts. d) Exposure to multiple approaches and the subsequent connections that develop can help students to recall the steps to complete mathematical processes.

Q: The main purpose of activities that require comparing and ordering fractions and decimals is to create a better understanding of place value and numeration concepts.

Q: One of the educational implications of the learning theory discussed in Chapter 2of the text is a) Students' mistakes should be minimized in order to build their confidence and enjoyment in the mathematics classroom. b) Most students learn best in a quiet class that consists primarily of direct-instruction, so that they can focus and won"t be distracted by others. c) Class activities and lessons should be designed with students' prior experiences in mind. d) New concepts should only be presented to students through teacher-centered presentations, in order to help them build the necessary background knowledge.

Q: Which of the following statements regarding students' use of calculators to find decimal equivalents from fractions is NOT true? a) Students can look for patterns from which fractions result in repeating versus terminating decimals. b) They always become too dependent on the technology. c) This should only be done after students have a conceptual understanding of the connection between the two formats. d) They can generalize a pattern about fractions that have only nine digits in their denominators.

Q: The _________________________________ refers to a range of knowledge that may be out of reach for a learner to learn on her own but that can be accessible with support.

Q: 0, 1/2, and 1 are very good _____________________________ to which students can compare decimal numbers, in order to better approximate their size.

Q: One of the big ideas behind ___________________________ is that learning occurs through interactions that are influenced by classroom culture.

Q: The use of ___________________fractions, those to which students have had lots of exposure, can help them see the connection between non-base-ten fractions and decimals.

Q: The big idea behind ___________________________ is that learners are creators of their own learning.

Q: Results of NAEP exams reveal that students perform quite well when asked to convert between fractions and decimals.

Q: Which of the following represents an example of an effective way a teacher may help facilitate students' construction of mathematical relationships? a) Asking students, "How is today's topic related to the fraction multiplication we investigatedlast week?" b) Prompting a student with a hint as soon as he pauses while offering an explanation of a process c) Conveying the message to families and students that, as long as the concepts are taught correctly, math is usually easy d) Never intervening in students' construction of meaning. The process is much more effective if they are just left alone

Q: A good early method to help students see the connection between fractions and decimals is to a) Show them how to use a calculator to divide the fraction numerator by the denominator to find the decimal. b) Be sure to use precise language when speaking about decimals, such as "point seven two." c) Show them how to round decimal numbers to the closest whole number. d) Have them use base-ten models to build models of base-ten fractions.

Q: In order to avoid distractions, teachers should minimize talking between students during math class.

Q: Name two methods that could help students develop the connection between fractions and decimals. Then describe how these methods develop conceptual understanding.

Q: To set up an environment for "doing" mathematics, teachers need to a) Develop and demonstrate rules. b) Efficiently manage time and materials. c) Quickly provide corrected answers, so students are not embarrassed by mistakes. d) Allow students to make engage in "productive struggle."