Helping Children Learn Mathematics, 10th Edition
By
Reys, Lindquist, Lambdin, Smith
Chapter 1: School Mathematics in a Changing World Possible Questions for Assessment and Evaluation 1.
T or F: The text states that the prevalent view of mathematics is mainly focused on computation with numbers. Answer: F Sec Ref: What is Mathematics? Page Ref: 2
2.
All of the following are topics that should be covered in a mathematics classroom, except: A. B. C. D. E.
algebra geometry measurement geography probability
Answer: D Sec Ref: What is Mathematics? Page Ref: 2 3.
Using your own words, explain what is meant by each of the following five views of mathematics: Mathematics is a study of patterns and relationships Mathematics is a way of thinking Mathematics is an art Mathematics is a language Mathematics is a tool Answer: Answers vary. Sec Ref: What is Mathematics? Page Ref: 3
4.
Give two specific examples from the elementary school curriculum to illustrate each of the following views of mathematics: Mathematics is a study of patterns and relationships Mathematics is a way of thinking Mathematics is an art Mathematics is a language Mathematics is a tool -
Answer: Answers vary. Sec Ref: What is Mathematics? Page Ref: 3 5.
The text answers a question, “What is mathematics?” Which of the following is NOT a part of that description? A. B. C. D. E.
a language a tool a set of numbers a study of patterns and relationships a way of thinking
Answer: C Sec Ref: What is Mathematics? Page Ref: 3 6.
T or F : In terms of the time devoted to it, mathematics is second only to science. Answer: F Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 3
7.
Which of the following groups was responsible for establishing mathematics standards? A. B. C. D. E.
The National Math Council The Math Association of America The National Council of Teachers of Mathematics The North American Math Council None of the above
Answer: C Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 2-3 8.
Suppose you were asked to serve on a committee to come up with mathematics standards. Explain the three needs that you should take into consideration. Answer: Should mention and explain the needs of the subject, the needs of the child and the needs of society. Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 3
9.
T or F The three needs that should be taken into consideration when thinking about the mathematics curriculum are the needs of society, the needs of teachers, and the needs of the child. Answer: F Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 3
10.
The text states that “mathematics continues to change.” In light of that, explain how and in what ways mathematics has changed. Answer: Answer should include ideas such as shifting from social usefulness to unifying themes; new content being added at the elementary level, and other topics being introduced at earlier grade levels. Sec Ref: What Determines the Mathematics Taught? Page Ref: 3
11.
The text mentions three general factors (needs) that should be considered when thinking about the mathematics curriculum. These three include the following: A. subject, child, society B. subject, child, teacher C. parent, child, teacher D. child, society, school E. parent, teacher, society Answer: A Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 3
12.
When a teacher believes that students will need to be able to solve problems and communicate so we can advance our technological world, he or she is considering the needs of the: A. subject B. child C. government D. parent E. society Answer: E Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 4
13.
A teacher considers that she must first review whole numbers before proceeding with a lesson on fractions. Which of the following needs is she most concerned with: A. child B. society C. teacher D. subject E. international competition Answer: D Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 3
14.
T or F The Committee of Seven recommended teaching mathematics topics according to students’ mental age. Answer: T Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 4
15.
Explain what is meant by each of the following terms as they apply mathematics instruction and learning: Mental discipline – Incidental learning – Field theory – Answer: Mental discipline, working with mathematics problems strengthens the mind; incidental learning, children would learn as much mathematics as they needed and would learn it better if it was not systematically taught; field theory, emphasis on a planned program to encourage the development of insight and understanding. Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 4
16.
T or F The NCTM standards are prescriptive, meaning every state must use them. Answer: F Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 5
17.
The No Child Left Behind legislation calls for annual testing in reading and mathematics each year in grades: a. 1-3 b. 3-5 c. 3-8 d. 5-9 e. 4-9 Answer: C Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 5
18.
Explain what is meant by “high-stakes assessments”. Answer: Assessments that are used to place students, what grade to record on their report card, whether to promote students to the next grade, and so on. Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 5
19.
What is the significance of the Common Core State Standards Initiative? Answer: An attempt to have a common understanding across the United States of what students are expected to learn at each grade level. It will also bring more standardization to the mathematics education our of youth. Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6
20.
Explain what is meant by each of the following principles: Technology Curriculum Teaching Learning Assessment Equity Answer: Answers vary. Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6-7
21.
The NCTM Principles and Standards document includes some underlying principles that should guide school mathematics programs. Which of the following is NOT one of those principles? A. assessment B. learning C. teaching D. technology E. computation Answer: E Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6-7
22.
Do you agree or disagree with the following statement: Calculators should not be used because then students will not know how to compute. Explain your answer. Answer: Answers vary. Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 7
23.
One of the NCTM Principles states that we should have high expectations for all students. Which Principle is this? A. Curriculum B. Equity C. Learning D. Teaching E. Assessment Answer: B Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6
24.
What does the word equity mean? Is it synonymous with equality? Give examples to support your answer. Answer: Answers vary. Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6
25.
T or F Equity and equality are synonymous. Answer: F Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6
26.
In terms of the Curriculum Principle, explain what is meant by coherent, focused, and well articulated. Answer: Coherent, fits mathematical ideas together in a meaningful way; focused, focuses on the important mathematics topics and ideas at each grade; well articulated, builds on previous learning and grows across the grades. Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6
27.
All of the following are words associated with the Curriculum Principle, except for: A. Logical B. Coherent C. Focused D. Well articulated Answer: A Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6
28.
T or F The NCTM Learning Principle emphasizes the need for students to learn mathematics with understanding and build on prior knowledge and experiences. Answer: T Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6
29.
T or F The NCTM Learning Principle emphasizes the need for students to memorize mathematical facts. Answer: F Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 6
30.
After looking at the scores on a test, a teacher realizes that a majority of the students missed an important concept. For the next two days, the teacher reviews the concept. Which of the following Principles does this illustrate? A. learning B. teaching C. assessment D. equity E. technology Answer: C Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 7
31.
To teach mathematics effectively, what are some things that a teacher must know? Answer: Answer should include some of the following: knowledge of mathematics, knowledge of their students, and knowledge of the pedagogy. Sec Ref: Recent Influences Page Ref: 7
32.
T or F According to a meta-analysis of 54 research studies, using calculators hinder the development of mathematical skills. Answer: F Sec Ref: What Determines the Mathematics Being Taught? Page Ref: 7
33.
T or F According to a research mentioned in the text, most states have the same gradelevel expectations in the specificity and in the grade in which a skill was targeted. Answer: F Sec Ref: Where Can You Turn? Page Ref: 8
34.
Which of the following assessments is the nation’s measure of students’ achievement? A. ASCD B. NCTM C. ACT D. NAEP E. SAT
Answer: D Sec Ref: Where Can You Turn? Page Ref: 8 35.
T or F The NAEP is given to a sample of students at grades 8 and 12. Answer: F Sec Ref: Where Can You Turn? Page Ref: 8
36.
Discuss five sources of help that you can turn to when considering the mathematics program. Answer: Answers should include five of the following: state and local guidelines, research, cultural and international resources, history, textbooks, electronic materials, professional organizations, other teachers. Sec Ref: Where Can You Turn? Page Ref: 7-10
37.
Which of the following is NOT one of the resources mentioned in the text to which teachers may turn to obtain information about mathematics teaching? A. parents B. cultural and international resources C. textbooks D. professional organizations E. electronic materials Answer: A Sec Ref: Where Can You Turn? Page Ref: 7-10
38.
What do the following acronyms stand for? Explain their significance. NAEP TIMSS Answer: National Association for the Education of Young Children, American Statistical Association, National Assessment of Educational Progress, Trends in Mathematics and Science Study Sec Ref: Where Can You Turn? Page Ref: 8-9
39.
T or F According to the 2007 TIMSS, U.S. fourth graders were below the international average. Answer: F Sec Ref: Where Can You Turn? Page Ref: 9
40.
T or F According to the TIMSS study, U.S. schools are more aligned with the recommendations of the NCTM standards than Japanese schools. Answer: F Sec Ref: Where Can You Turn? Page Ref: 9
41.
A teacher belongs to NCTM and her state mathematics teacher organization. Which of the following provides reason(s) for the teacher to be part of a professional organization? A. journals B. conferences C. workshops D. sharing of ideas E. all of the above Answer: E Sec Ref: Where Can You Turn? Page Ref: 10
42.
What is your view of mathematics teaching? Support your answer with specific examples from the text. Answer: Answers vary. Sec Ref: What is Your Role Now? Page Ref: 10
43.
As you prepare to teach mathematics, what three challenges should you take into consideration? Answer: Examine one’s dispositions and beliefs towards mathematics; teaching means helping students learn, not just giving out information; realize that doing and teaching mathematics are different Sec Ref: What is Your Role Now? Page Ref: 10
Chapter 2: Helping Children Learn Mathematics with Understanding Possible Questions for Assessment and Evaluation 1.
Explain the following five strands of mathematical proficiency: Conceptual Understanding – Procedural Fluency – Strategic Competency – Adaptive Reasoning – Productive Disposition – Answer: See answer on page 14. Sec Ref: What Do We Know About Learning Mathematics? Page Ref: 13-14
2.
Which of the following is not one of the five strands of mathematical proficiency? A. Logical Reasoning B. Conceptual Understanding C. Strategic Competency D. Productive Disposition E. Procedural Fluency Answer: A Sec Ref: What Do We Know About Learning Mathematics? Page Ref: 13-14
3.
According to data from the National Center for Educational Statistics, what does a typical classroom of 25 students look like? Answer: 11 students are members of a minority group, 4 who speak another language, 10 students who qualify for free or reduced lunches, 4 students who live in poverty, and 4 students with disabilities. Sec Ref: What Do We Know About Learning Mathematics? Page Ref: 14
4.
The text mentions many different ways in which our students are different. Explain five ways in which students are different and what can you as a teacher do about it. Answer: Should mention five of the following: cognitive, physical, social, culture, family structure, background experiences, interests, motivation, learning style. Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 14
5.
T or F Equity means that all students should be treated identical. Answer: F Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 14
6.
All of the following are ways in which the text mentions that students are different, except: A. physical development B. culture C. motivation level D. learning style E. teacher Answer: E Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 14
7.
T or F Most of the students we teach will be similar in terms of their learning styles, interests, and levels of motivation. Answer: F Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 14
8.
T or F Creating a positive learning environment and establishing clear expectations are two strategies for supporting all students without having to individualize instruction. Answer: T Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 14-16
9.
Explain each of the following strategies for supporting all students without having to individualize instruction for each one: Positive learning environment Avoid negative experiences Establish clear expectations Treating all students equitably Help students retain mathematical knowledge Answer: Answers vary.
Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 14-18 10.
Which of the following are things you should do to create a positive learning environment? A. establish a safe and comfortable classroom arrangement B. establish a classroom atmosphere that is intellectually stimulating C. establish a classroom where all students receive As on tests D. all of the above E. both A and B Answer: E Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 15
11.
What are some characteristics of students who have math anxiety? Answer: Should include the following: poor performance, more than the usual number of misunderstandings, lack of confidence, take less math in second school, etc. Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 15
12.
T or F According to data from the NAEP, primary-grade children become progressively more positive as they move into middle and high school. Answer: F Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 15
13.
As a classroom teacher, discuss five strategies that you could employ to alleviate children’s math anxieties. Answer: Should include the following: emphasize mathematical meaning and understanding, model problem-solving strategies, show a positive attitude toward mathematics, provide interesting mathematical experiences, show a positive attitude toward students at all times, encourage students to tell the teacher how they feel about mathematics, and be careful not to overemphasize speed tests or drills. Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 15-16
14.
Miss Collins regularly talks with her students about math topics she finds useful and math activities she enjoys. Which of the following strategies for supporting diverse learners does this illustrate? A. Treating all students equitably. B. Establishing clear expectations. C. Avoiding negative experiences that increase anxiety. D. Creating a positive learning environment. E. None of the above Answer: C Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 16
15.
The text suggests that one strategy to help teachers establish clear expectations is to come up with a class motto. Given that, what motto would you choose for your classroom and why? Answer: Answers vary. Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 16
16.
T or F Research suggests that teachers may actually treat girls and boys differently in the mathematics classroom. Answer: T Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 17
17.
T or F Teachers tend to attribute boys’ failure to a lack of talent, whereas they tend to attribute girls’ failure to a lack of motivation. Answer: F Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 17
18.
T or F Repetition and meaningful learning are two ways in which teachers can help students retain mathematical knowledge. Answer: T Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 18
19.
Explain three ways in which teachers can help students retain their mathematical knowledge and skills. Answer: Answer should include three of the following: meaningful learning, manner in which a concept was learned, establishing connections, and repetition. Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 18
20.
Give one example to illustrate each of the following ways to help students retain mathematical knowledge and skills. Manner in which a concept was learned – Establishing connections – Reviewing key ideas – Answer: Answers vary. Sec Ref: How Can We Support the Diverse Learners in Our Classrooms? Page Ref: 18
21.
What is the difference between procedural knowledge and conceptual knowledge? Why are both necessary? Answer: Procedure - rules and steps; conceptual - understanding, meaning, connections. Children must not only learn how to do something, but also understand why. Sec Ref: Helping Children Acquire Both Procedural Knowledge and Conceptual Knowledge? Page Ref: 18
22.
T or F Conceptual knowledge is more important than procedural knowledge in regards to helping students understand mathematics. Answer: F Sec Ref: Helping Children Acquire Both Procedural Knowledge and Conceptual Knowledge? Page Ref: 18
23.
T or F A second-grade teacher spends his time teaching students the steps involved in adding two two-digit numbers. He tells a colleague that he is teaching his students conceptual knowledge. Answer: F Sec Ref: Helping Children Acquire Both Procedural Knowledge and Conceptual Knowledge? Page Ref: 18
24.
A student who has procedural knowledge without conceptual knowledge will have difficulty: A. knowing when to use it. B. remembering how to do it. C. applying it in new situations. D. judging if results are reasonable. E. all of the above. Answer: E Sec Ref: How Can We Help Children Acquire Both Procedural Knowledge and Conceptual Knowledge? Page Ref: 19
25.
Suppose you were teaching students how to add 27 + 48. What procedural and conceptual knowledge would you have to teach in order for students to understand how to add 27 + 48? Answer: Procedural - add ones first, then tens; conceptual - when you have 10 of something, you need to regroup Sec Ref: How Can We Help Children Acquire Both Procedural Knowledge and Conceptual Knowledge? Page Ref: 19
26.
Compare and contrast behaviorism and constructivism. Answer: Both have to do with our understandings of learning; behaviorism focuses on what can be observed, whereas constructivism focuses on how students think. Sec Ref: How Do Children Learn Mathematics? Page Ref: 20
27.
A teacher reasons that since multiplication can be interpreted as repeated addition, students must first be taught addition. Which of the following theories best describes her understanding of how students learn? A. Behaviorism B. Constructivism Answer: A Sec Ref: How Do Children Learn Mathematics? Page Ref: 20
28.
T or F One of the major tenets of behaviorism is the idea that behavior can be shaped through reinforcement, that is, through rewards and punishments. Answer: T Sec Ref: How Do Children Learn Mathematics? Page Ref: 20
29.
Behaviorism has some drawbacks. One of them is that it tends to not focus on learning and understanding but rules and procedures. If you were to teach from a behaviorist perspective, how would you defend your choice? Answer: Should include the following: provides instructional guidelines, allows for short-term progress, and lends itself well to the current focus on accountability. Sec Ref: How Do Children Learn Mathematics? Page Ref: 21
30.
In terms of Bruner’s theory of learning, what is meant by the following terms: enactive, iconic, symbolic? Answer: Enactive, manipulating, constructing, or arranging of real-world objects; iconic, thinking based on pictures and/or images; symbolic, manipulation of symbols. Sec Ref: How Do Children Learn Mathematics? Page Ref: 22
31.
Explain the three basic tenets of constructivism. Answer: Should include the following: knowledge is not passively received, but actively created or invented; students create (construct) new knowledge by reflecting on their physical and mental actions; learning reflects a social process in which children engage in dialogue and discussion with themselves as well as with others. Sec Ref: How Do Children Learn Mathematics? Page Ref: 22
32.
T or F Suppose you walked into Mrs. Andrews’ classroom and saw that she was showing students the rules and procedures for multiplying two fractions. After the lesson, she gives students lots of practice problems using the rules and procedures. This is an example of constructivist teaching. Answer: F Sec Ref: How Do Children Learn Mathematics? Page Ref: 22
33.
A teacher who believes it is important to identify and teach focused learning outcomes sees learning from a ________ view. A. Behaviorist B. Constructivist Answer: A Sec Ref: How Do Children Learn Mathematics? Page Ref: 22
34.
A teacher who believes that learners take new information, interpret it, and connect it to what they already know sees learning from a ________ view. A. Behaviorist B. Constructivist Answer: B Sec Ref: How Do Children Learn Mathematics? Page Ref: 22
35.
Compare Piaget’s levels of thinking with Bruner’s levels of developmental learning. Answer: Should include the idea that they both deal with how children learn and that they both go from simple thinking (abstract) to complex (abstract) thinking. Sec Ref: How Do Children Learn Mathematics? Page Ref: 22
36.
Explain what is meant by the zone of proximal development. Answer: Should include the idea that tasks should be within children’s ability to learn - if it’s too complex, they may not be able to learn it. Sec Ref: How Do Children Learn Mathematics? Page Ref: 23
37.
In your own words, explain the following four recommendations for mathematics instruction: Teach to the developmental characteristics of students Actively involve students Move learning from concrete to abstract Use communication to encourage understanding Answer: Should include the following ideas: children are at different stages of development, therefore, instruction should be adapted to meet the needs of those stages; students should be doing math and not just listening; models and manipulatives should be used, but instruction shouldn’t stop there - it should progress to abstract and symbolic ideas; students should be encouraged to talk about and discuss the mathematics they are learning. Sec Ref: How Can We Help Children Make Sense of Mathematics? Page Ref: 23-28
38.
Explain the following three types of developments: cognitive, physical and social. Answer: Cognitive, has to do with how a child reasons and thinks; physical, has to do with a child’s muscles and motors; social, has to do with how a child interacts with others. Sec Ref: How Can We Help Children Make Sense of Mathematics? Page Ref: 23
39.
Write a brief description of a primary grade learner (ages 4-7). Answer: Answers vary. Sec Ref: How Can We Help Children Make Sense of Mathematics? Page Ref: 24
40.
Write a brief description of an intermediate grade learner (ages 8 -11). Answer: Answers vary. Sec Ref: How Can We Help Children Make Sense of Mathematics? Page Ref: 24
41.
The use of perceptually different models is called: A. Multiple Perspectives B. Multiple Embodiment C. Multiple Representations D. Multiple Materials E. Multiple Perceptual Differential Models Answer: B Sec Ref: How Can We Help Children Make Sense of Mathematics? Page Ref: 27
42.
Which of the following theorist stressed the importance of communication? A. Piaget B. Bruner C. Dienes D. Vygotsky Answer: D Sec Ref: How Can We Help Children Make Sense of Mathematics? Page Ref: 28
43.
T or F Children from low SES backgrounds tend to favor problem solving in mathematics as opposed to rote memorization. Answer: F Sec Ref: Cultural Connections Page Ref: 29
44.
To help students from low SES backgrounds be successful in mathematics, Lubienski suggests the following: A. push for meaningful learning B. students should learn what was intended from the problems presented in the curriculum C. analyze achievement data to identify areas for remediation D. teachers of students from low SES backgrounds must advocate for those students E. all of the above Answer: E Sec Ref: Cultural Connections Page Ref: 29
Chapter 3: Planning and Teaching Possible Questions for Assessment and Evaluation 1.
According to the text, the instructor role involves all of the following basic functions, except: A. planning B. teaching C. disciplining D. assessing E. analyzing Answer: C Sec Ref: Introduction Page Ref: 33
2.
Discuss the following 4 roles that instructors play: planning, teaching, assessing, and analyzing. Answer: Answers vary. Sec Ref: Introduction Page Ref: 33
3.
T or F Teacher with strong content knowledge are better able to develop their students’ conceptual knowledge. Answer: T Sec Ref: Preparing to Teach: Questions to Ask Before Planning Begins Page Ref: 34
4.
Discuss three reasons why teachers with strong content knowledge are better able to develop their students’ conceptual knowledge. Answer: Should include the following things: better able to provide multiple representations of the same concept, better able to encourage student questions and comments, better able to analyze student errors and misunderstandings, and better able to connect mathematics to daily life and other content areas. Sec Ref: Preparing to Teach: Questions to Ask Before Planning Begins Page Ref: 34
5.
After correcting her students’ tests, Mrs. Wang discovers that most of her students got the problem involving fraction multiplication wrong. This is an example of which of the following functions of an instructor? A. planning B. teaching C. assessing D. analyzing E. none of the above Answer: D Sec Ref: Introduction Page Ref: 33
6.
In preparation for teaching, one of the questions that you must ask yourself is: Do I understand the mathematics I am teaching? Discuss three things that you could do to deepen your mathematical knowledge. Answer: Should include the following things: work through lesson problems and activities before teaching them to students, identify the goals of each lesson and what is important for students to learn, review the mathematics you will be teaching. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 34
7.
A teacher planning a first-grade lesson on addition realizes that his students need to manipulate concrete objects to solve addition problems and are just beginning to use symbols to represent those objects. Which of the following questions is this teacher mainly concerned with? A. Do I understand the mathematics I am teaching? B. What are the developmental characteristics of my students? C. What do my students already know? D. What kinds of tasks will I give my students? E. None of the above Answer: B Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 35
8.
T or F The research indicates that children pass through developmental stages at the same rate and therefore, all of your second-grade students should be at the same level. Answer: F Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 35
9.
During the first week of third-grade math class, Mr. Martin gives his class a pre-test. In doing this, Mr. Martin is mainly concerned with which of the following questions? A. Do I understand the mathematics I am teaching? B. What are the developmental characteristics of my students? C. What do my students already know? D. What kinds of tasks will I give my students? E. None of the above Answer: C Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 35
10.
How does knowing your students’ strengths as well as their weaknesses help you to be a better teacher? Answer: Use the strengths to compensate for the weaknesses; pair a weak student with a strong student. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 35
11.
Discuss the two things that your text suggests you could do in order to find out what your students already know about a certain math concept. Answer: Paper-and-pencil tests and by observing and talking with them. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 35
12.
Explain what is meant by the spiral approach to instruction. Use this approach with the concept of “angles”. Answer: The approach incorporates and builds on earlier learning to help guide the child through the increasingly intricate study of a topic. Using the approach with the “angle” concept could include having students look at different shapes and angles, finding them in the real world, then moving towards naming the different shapes and from their looking at the relationships among angles, etc. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 35
13.
T or F According to the text, in practice most teachers teach according to the spiral approach - going from simple to complex each successive year. Answer: F Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 35
14.
In Mrs. Hoffman’s second-grade class, the children work with the place values tens and ones. In third grade, they work with hundreds, tens, and ones. This is an example of: A. the spiral approach B. scaffolding C. multiple embodiment D. mathematical complexity E. none of the above Answer: A Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 35
15.
Examples of appropriate tasks include all of the following, except: A. Projects B. Problems C. Applications D. Exercises E. All of the above Answer: E Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 36
16.
What three key things must tasks accomplish? Answer: Encourage students to reason about mathematics, make connections, and solve problems. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 36
17.
T or F Memorization is an example of a lower-level task. Answer: T Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 36
18.
T or F According to research, drill and practice should follow, not precede, the development of mathematical meaning. Answer: T Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 37
19.
Suppose you are teaching a lesson on fractions, suggest three good tasks that you could have students do. Answer: Answers vary. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 37
20.
T or F According to the text, children must practice a concept before understanding it. Answer: F Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 37
21.
Summarize the four strategies mentioned in your text for creating effective classroom talk. Answer: Show students that you expect them to communicate their thinking, create a safe place for students to conjecture and to make mistakes, pose problems and assign tasks that encourage children to talk, and explicitly teach mathematics vocabulary. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 37-38
22.
Explain how teachers can benefit from listening to their students’ thinking. Answer: Listen for misconceptions and discuss/correct them. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 38
23.
During a class lecture on angles, Mr. Jones asked one of his students the following question: “When you add up the angles of a triangle, they equal what?” This is an example of: A. Low level question B. High level question Answer: A Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 38
24.
T or F Research indicates that teachers use more higher-level questions than lowerlevel ones. Answer: F Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 38
25.
Research indicates that teachers tend to ask more low level questions in class. Why might this be the case? Answer: Easiest to create during a lesson; easiest for students to answer. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 38
26.
While studying the angles in a triangle, Mrs. Janus asks what the sum of the angles is. Julie answers that it is 180 degrees. Mrs. Janus then asks her, “How could you prove that?” This type of question is a question that is primarily aimed at: A. helping students work together to make sense of mathematics B. helping students to learn to reason mathematically C. helping students learn to conjecture D. helping students connect mathematics to other areas E. none of the above Answer: B Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 38
27.
After having Alexa share her response with the class, the teacher asks the following question, “What do you think will happen to the measure of each interior angle of a polygon if we were to increase the number of angles?” This type of question is a question that is primarily aimed at: A. helping students work together to make sense of mathematics B. helping students to learn to reason mathematically C. helping students learn to conjecture D. helping students connect mathematics to other areas E. none of the above Answer: C Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 38
28.
Summarize the three basic patterns of grouping for mathematical instruction. Answer: whole class, small group, individuals working independently Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 39
29.
T or F According to the text, it is better to place low-achieving students together because then they can proceed at their own pace and help each other learn. Answer: F Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 39
30.
T or F According to the text, middle- and high-achieving students show gains when placed in homogenous groups. Answer: T Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 39
31.
Discuss four benefits of having students work in small groups. Answer: Increased communication, development of social skills, opportunities to experience success, and see more than one approach to solving a problem. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 39
32.
T or F As a beginning teacher, it is better that you develop your own original lesson plans than follow a developmentally appropriate textbook. Answer: F Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 40
33.
T or F Mr. Martinez gives his students beans to use when adding numbers together. He then claims that this is an example of a context. Answer: F Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 40
34.
T or F Students taught using standards-based texts perform better than students who are taught using conventional texts. Answer: T Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 41
35.
Your text mentions several reasons for using children’s literature in mathematics lessons. Which of the following is NOT one of those reasons? A. Provides a meaningful context for mathematics. B. Is a source of problems or a basis for generating problems. C. Provides illustrations that clearly represent mathematical concepts. D. Increases the attention to reading in the curriculum. E. None of the above. Answer: D Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 41
36.
T or F According to Bruner, learning moves from iconic to enactive to symbolic. Answer: F Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 41
37.
Which of the following is not a reason to use manipulatives? A. Meet students’ diverse learning needs B. Illustrate concepts in multiple ways C. Support problem solving D. Allow students to have fun E. None of the above Answer: D Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 42
38.
What are some commonly used materials at the elementary level? Answer: Chips or tiles, cubes, pattern blocks, tangrams, base-ten blocks, fraction models, geoboards, measuring instruments, spinners and dice, and play money. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 42
39.
Discuss three reasons why technology should be incorporated into the mathematics classroom. Answer: Should include the following: prepares students to function in our technological world, technology has the potential to increase student learning, and technology can help students learn to collaborate, to think critically, and problem solve. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 42-43
40.
T or F Research shows that preschool children do not benefit from using computers. Answer: F Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 43
41.
Explain each of the following types of computer software: drill-and-practice, tutorial, simulation, educational game, problem-solving, and tool. Answer: Drill-and-practice provides practice; tutorial provides instruction on new skills; simulation allows students to experience events and explore environments; educational game engages students in fun activities that addresses specific educational skills; problem-solving is designed to aid in the development of higher-order problem-solving strategies; tool helps with the teaching and learning process. Sec Ref: Preparing to Teach: Questions to Ask Page Ref: 43-44
42.
A fifth-grader is using a piece of computer software. The software teaches her how to add and subtract fractions. This is an example of what kind of software? A. Drill-and-practice B. Tutorial C. Educational game D. Simulation E. None of the above. Answer: B Sec Ref: Preparing to Teach: Questions to Ask Before Planning Begins Page Ref: 43
43.
Summarize the three different levels of planning. Answer: Planning for the year, planning for units, and planning for daily lessons. Sec Ref: Planning for Effective Teaching Page Ref: 45-46
44.
T or F The most familiar format for mathematics lessons is review-teach-practice. Answer: T Sec Ref: Planning for Effective Teaching Page Ref: 46
45.
T or F Japanese teachers typically follow the review-teach-practice format for mathematics lessons. Answer: F Sec Ref: Planning for Effective Teaching Page Ref: 46
46.
Discuss the differences in how U.S. teachers teach compared to Japanese teachers Answer: U.S. teachers uses the review-teach-practice approach while Japanese teachers are more interested in developing meaningful understanding of a concept. Sec Ref: Planning for Effective Teaching Page Ref: 46
47.
In Mrs. Jones’ lesson plan, she grabs the students’ attention by posing an interesting problem. This is an example of which lesson plan component? A. introduction B. launch C. instruct D. summarize E. none of the above Answer: B Sec Ref: Planning for Effective Teaching Page Ref: 47
48.
Compare and contrast the following three types of lessons: investigative, direct instruction, and exploration. Answer: They contain some of the same elements, such as an introduction, a lesson outline, and a summary. They differ in how students learn new information. Sec Ref: Planning for Effective Teaching Page Ref: 48-50
49.
Discuss the three phases of an investigative lesson. Answer: Launch, provide a motivating introduction to the lesson; investigate, students work on the problem; summarize, class discussion of findings. Sec Ref: Planning for Effective Teaching Page Ref: 48
50.
Which of the following is the correct sequence for a direct instruction lesson? A. Instruct, launch, Summarize B. Launch, Instruct, Summarize C. Summarize, Launch, Instruct D. Instruct, Summarize, Launch E. Launch, Summarize, Instruct
Answer: B Sec Ref: Planning for Effective Teaching Page Ref: 48 51.
Discuss the three tiers associated with Response to Intervention (RTI). Answer: Tier 1, instruction for all students; tier 2, supplementary instruction for some students; tier 3, individualized instruction. Sec Ref: Planning for Effective Teaching Page Ref: 53
52.
Discuss four strategies for working with English-language learners. Answer: Should include the following: give students comprehensible input, give students opportunities to increase verbal interaction during class, teach in a way that contextualizes language, use strategies and groupings that reduce the anxiety of the students, and assign activities that offer students opportunities for active involvement. Sec Ref: Meeting the Needs of All Students Page Ref: 55
53.
You want to adapt a lesson on division of whole numbers by two-digit divisors by using the level of support adaptation. Which of the following does NOT illustrate this adaptation? A. Provide base-ten blocks. B. Pair up students. C. Assign a paraprofessional to assist particular students. D. Have some students divide by one-digit divisors. E. None of the above. Answer: D Sec Ref: Meeting the Needs of All Students Page Ref: 57
54.
Explain five of the nine adaptations discussed in your text. Answer: Should include the following: size, input, participation, time, difficulty, alternate goals, level of support, output, substitute curriculum. Sec Ref: Meeting the Needs of All Students Page Ref: 57
Chapter 4: Assessment: Enhanced Learning and Teaching Possible Questions for Assessment and Evaluation 1.
Compare and contrast assessment of learning and assessment for learning. Answer: Of learning has to do with summative assessments; for learning has to do with formative assessments. Sec Ref: Introduction Page Ref: 63
2.
What are summative assessments? List some examples of summative assessments. Answer: Assessments for the purposes of public reporting and accountability. Examples include end-of-year exams and standardized tests. Sec Ref: Introduction Page Ref: 63
3.
What are formative assessments? List some examples of formative assessments. Answer: Document students’ achievement and to also help them learn more. Examples include homework, in-class assignments, and classroom tests. Sec Ref: Introduction Page Ref: 63
4.
T or F At the end of the year, Mrs. Meyers gives students a standardized math test to determine what students have learned during the school year. This is an example of assessment for learning. Answer: F Sec Ref: Introduction Page Ref: 63
5.
After a lesson on multiplication, a teacher gives out a practice sheet for students to complete at home. The practice sheet is an example of: A. a summative assessment B. a formative assessment C. none of the above D. all of the above Answer: B Sec Ref: Introduction Page Ref: 63
6.
Compare and contrast formative and summative assessments and also give one example of each. Answer: Should discuss that they are forms of assessment to gather data about student achievement; formative is for learning whereas summative is of learning. Formative - homework; summative - standardized tests. Sec Ref: Assessment for Learning: Formative Assessment Page Ref: 63
7.
T or F According to NCTM’s Assessment Standards, the four phases of the assessment process are planning, gathering, interpreting, and using. Answer: T Sec Ref: Assessment for Learning: Formative Assessment Page Ref: 63
8.
Which one of the following is not a phase associated with assessment? A. Planning B. Gathering C. Interpreting D. Using E. Grading Answer: E Sec Ref: Assessment for Learning: Formative Assessment Page Ref: 63
9.
The four phases (in sequential order) for the classroom assessment process are: A. planning, gathering, interpreting, and using B. using, interpreting, gathering, and planning C. planning, interpreting, gathering, and using D. interpreting, using, gathering, and planning E. none of the above Answer: A Sec Ref: Assessment for learning: Formative Assessment Page Ref: 63
10.
In order to assess students’ understanding of operations involving decimals, Mr. Ramirez has his students demonstrate using base-ten blocks. This is an example of which phase of the classroom assessment process? A. planning B. gathering C. interpreting D. using E. all of the above Answer: B Sec Ref: Assessment for learning: Formative Assessment Page Ref: 63
11.
Use your own example to illustrate the four phases of the assessment process. Answer: Answers vary. Sec Ref: Assessment for Learning: Formative Assessment Page Ref: 64
12.
T or F One shift in assessment practices is a move toward gathering assessment data at the end of a chapter versus collecting data continuously. Answer: F Sec Ref: Assessment for Learning: Formative Assessment Page Ref: 65
13.
T or F One shift in assessment practices is a move toward gathering assessment data continuously rather than periodically. Answer: T Sec Ref: Assessment for learning: Formative Assessment Page Ref: 65
14.
T or F One shift in assessment practices is a move toward using a single source of information as opposed to a wide variety of sources of information. Answer: F Sec Ref: Assessment for learning: Formative Assessment Page Ref: 65
15.
Summarize the three shifts in assessment practices mentioned in your text. Answer: Gathering data continuously rather than periodically, using a wide variety of sources of information as opposed to one single assessment, and keeping the needs and progress of students in mind as opposed to strictly following the curriculum. Sec Ref: Assessment for learning: Formative Assessment Page Ref: 65
16.
Discuss the three shifts associated with assessment practices. Answer: Collecting data continuously versus at the end; using a wide variety of sources of information; and keeping the needs and progress of students in mind. Sec Ref: Assessment for Learning: Formative Assessment Page Ref: 65
17.
T or F A holistic scoring rubric assigns a single score based on the overall quality of the student’s work. Answer: T Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 66
18.
T or F Analytic rubrics assign a single score based on the overall quality of the student’s work. Answer: F Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 66
19.
During a group assessment activity, Mr. Berry comes up with a rubric whereby he will assign points for each of the following categories: understanding of problem, solution to the problem, and group dynamics. This is an example of which rubric? A. Holistic B. Analytic C. None of the above Answer: Analytic Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 66
20.
T or F Mr. Smith wants to measure students’ abilities to communicate mathematically, their abilities to solve problems, and their abilities to reason and proof. Given that situation, it would be best for him to use a holistic scoring rubric. Answer: F Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 66
21.
What are some key factors associated with a successful interview? Answer: Establishing rapport with the child, accepting responses without judging, and encouraging the child to talk and explain. Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 69
22.
Mrs. Carter sat down with Amy and had her show how to solve an addition problem with base-ten blocks. When Mrs. Carter couldn’t follow Amy’s explanation, she asked Amy additional questions to be sure she understood. This is an example of which of the following types of assessments: A. interviewing B. observing C. using a rubric D. peer assessments E. none of the above Answer: A Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 69
23.
T or F Suppose you wanted to measure a student’s ability to use a ruler to find the dimensions of a box. In this case, the best assessment type would be to interview the student. Answer: F Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 70
24.
What’s a performance task? Give an example of a performance task for the concept of adding decimals. Answer: A performance task generally mirrors the real world, is open-ended and requires time for grappling with a problem. An example might be to have students add up how much it would cost to order things off of a menu. Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 70
25.
Performance tasks should contain the following attributes: (circle all that apply) A. Mirror the real world B. Open-ended C. Teacher-centered D. Collaborative group work Answer: A, B, D Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 70
26.
T or F In general, the teacher decides and chooses the artifacts that will be go into a portfolio. Answer: F Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 74
27.
Which of the following are things that a student may want to put into his portfolio? A. problem solving tasks B. writings C. reports D. none of the above E. all of the above Answer: E Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 74
28.
T or F One particular benefit of portfolios is their value as a self-assessment tool for students. Answer: T Sec Ref: Ways to Assess Students’ Abilities and Dispositions
Page Ref: 74 29.
Your text states that writing can be an invaluable tool in mathematics. Describe some ways in which you can incorporate writing into your classroom and some of its benefits. Answer: Ask students to summarize a concept, write down what they did or did not understand, how they felt about an activity, what they learned in class, etc. Writing can be used to assess children’s knowledge of and attitudes toward mathematics. Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 76
30.
T or F Criterion referenced tests are generally designed to test students on prepublished expectations. Answer: T Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 77
31.
T or F Norm referenced tests are designed to compare how a given student performed in comparison with all other students who took the same test. Answer: T Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 78
32.
On a norm-referenced test, there will always be: A. 10% of the students below average and 90% above average B. 25% of the students below average and 75% above average C. 50% of the students below average and 50% above average D. 75% of the students below average and 25% above average Answer: C Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 78
33.
T or F In a norm-referenced test, 50% of the students will be below the average and 50% will be above the average. Answer: T Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 78
34.
T or F When giving a standardized test, students should not be told the format of the test because they can then prepare for the test. Answer: F Sec Ref: Ways to Assess Students’ Abilities and Dispositions Page Ref: 78
35.
All of the following are ways to assess students’ abilities and dispositions, except: A. standardized achievement tests B. observations C. interviewing D. performance tasks E. all of the above Answer: E Sec Ref: Ways to Assess Students ‘Abilities and Dispositions Page Ref: 68-79
36.
Discuss three strategies for keeping student records. Answer: Checklists, students’ files, and class records. Sec Ref: Keeping Records and Communicating about Assessments Page Ref: 79-81
37.
Name the three groups with whom you will have to communicate assessment information. Also, for each group, include a recommendation for making the communication a positive experience. Answer: Students, parents, and administration. Sec Ref: Keeping Records and Communicating about Assessments Page Ref: 81
38.
T or F In regards to communicating with parents, a classroom teacher should only call when their child is having difficulties. Answer: F Sec Ref: Keeping Records and Communicating about Assessments Page Ref: 81
39.
T or F One way that each student’s performance on the NAEP mathematics assessment is reported is by assigning it an achievement level of below average, average, or above average. Answer: F Sec Ref: Cultural Connections Page Ref: 82
40.
T or F In general, students’ performance on the NAEP has shown improvement. Answer: T Sec Ref: Cultural Connections Page Ref: 82
41.
T or F Whites and blacks perform equally well on the NAEP. Answer: F Sec Ref: Cultural Connections Page Ref: 82
42.
What is meant by the phrase, “pedagogy of poverty”? Is this a good pedagogy? Why or why not? Answer: A pedagogy that is characterized by instruction that is very directive, controlling, and debilitating for students. Sec Ref: Cultural Connections Page Ref: 83
Chapter 5: Mathematical Processes and Practices Possible for Assessment and Evaluation 1.
T or F In many mathematics classrooms today, it is common to see the teacher teaching and students listening and bent over doing worksheets, practicing computations, rules, and formulas. Answer: F Sec Ref: Introduction Page Ref: 88
2.
Explain what is meant by each of the following five process standards: problem solving, reasoning and proof, communication, connections, and representation. Answer: Answers vary. Sec Ref: Introduction Page Ref: 88
3.
T or F The Principles and Standards for School Mathematics identifies five process standards. Answer: T Sec Ref: Introduction Page Ref: 88
4.
The CCSSM contains how many “mathematical practices”? A. 6 B. 7 C. 8 D. 9 E. 10 Answer: C Sec Ref: Introduction Page Ref: 89
5.
Compare and contrast the NCTM process standards with the CCSSM “mathematical processes”. Answer: They both deal with mathematical processes or skills. NCTM has five, whereas the CCSSM has eight. Sec Ref: Introduction Page Ref: 88-89
6.
Your text states that problem solving can be a major means of developing mathematical knowledge. Explain what is meant by this and give examples to support your answer. Answer: Answers vary. Sec Ref: Problem Solving Page Ref: 89
7.
Mrs. Matthews gives students the following question: Jerry buys 3 boxes of candy. Each box costs $2.00. What is the total cost for the 3 boxes? Is this an example of problem solving? Why or why not? Answer: No - a problem is a situation in which a person wants something and does not know immediately what to do to get it. In this case, the students would - take three times two. Sec Ref: Problem Solving Page Ref: 89
8.
T or F According to your text, understanding numbers is the foundation for all mathematical teaching. Answer: F Sec Ref: Problem Solving Page Ref: 91
9.
When students are problem solving, should calculators and other technological devices be used? Answer: At the upper elementary level, they would greatly help students to problem solve when they are working on more complex problems that don’t have “nice numbers”. Sec Ref: Problem Solving Page Ref: 91
10.
Describe the Cognitively Guided Instruction (CGI) approach to mathematics instruction. Answer: Students work individually or in small groups to solve a rich mathematical task. Afterwards, they share their approaches with each other. Sec Ref: Problem Solving Page Ref: 92
11.
T or F According to one study mentioned in your text, students who were taught using a problem-solving approach scored significantly lower when compared to students who were taught using a textbook-based approach.
Answer: F Sec Ref: Problem Solving Page Ref: 92 12.
John is given the following problem: -5 + 3 = 2. After analyzing the problem, he states that the answer is wrong because you have more negatives than positives, therefore, the answer should be -2. This is an example of which process standard? A. Problem solving B. Reasoning and proof C. Communication D. Connections E. Representations Answer: B Sec Ref: Reasoning and Proof Page Ref: 92
13.
T or F Reasoning and proof should be taught in a single unit so that students can really focus on it. Answer: F Sec Ref: Reasoning and Proof Page Ref: 94
14.
Matt comes home and tells his mom that whenever you multiply two negative numbers, you always get a positive number. This type of thinking is generally associated with: A. Problem solving B. Reasoning and proof C. Communication D. Connections E. Representations Answer: B Sec Ref: Reasoning and Proof Page Ref: 94
15.
All of the following points in regards to mathematical reasoning are true, except: A. Reasoning is about making generalizations
B. Reasoning leads to a web of generalizations C. Reasoning leads to mathematical memory built on relationships D. Learning through reasoning requires making mistakes and learning from them E. Reasoning should be reserved for students at the higher grade levels Answer: E Sec Ref: Reasoning and Proof Page Ref: 94-95 16.
T or F In order to develop students’ mathematical reasoning abilities, teachers should allow students to study flawed or incorrect reasoning. Answer: T Sec Ref: Reasoning and Proof Page Ref: 95
17.
T or F Since mathematics deals mainly with numbers, students should not be encouraged to speak and write in mathematics classrooms. Answer: F Sec Ref: Communication Page Ref: 95
18.
The following are modes of communication, except: A. charts B. speaking C. gestures D. writing E. all of the above Answer: E Sec Ref: Communication Page Ref: 96
19.
T or F Only speaking and writing are the acceptable forms of communication in a mathematics classroom. Answer: F
Sec Ref: Communication Page Ref: 96 20.
In your classroom, suggest five ways in which you could incorporate writing. Answer: Journals, open-ended writing as follow-up to a lesson, having students write their own word problems, having them describe their solution process to a problem, having them write about a process or procedure, and writing about connections between and among ideas. Sec Ref: Communication Page Ref: 97
21.
After Julie was taught fractions, decimals, and percentages, she realized that 1/4, 0.25, and 25% all represented the same quantity. Which of the following five NCTM process standards is Julie exhibiting? A. Problem solving B. Reasoning and proof C. Communication D. Connections E. Representations Answer: D Sec Ref: Connections Page Ref: 97
22.
During a lesson on fractions, Chang raises his hands and states the following: “I know how much 1/2 is because my mom uses a 1/2 cup measuring cup when she bakes cookies.” Which of the following five NCTM process standards is Chang exhibiting? A. Problem solving B. Reasoning and proof C. Communication D. Connections E. Representations Answer: D Sec Ref: Connections Page Ref: 97
23.
Your text states that there are three types of connections that should be made. Explain these three connections and give one example for each. Answer: Ideas within mathematics are connected with one another; the
symbols and procedures of mathematics and the conceptual ideas that the symbolism represents are connected; connection between mathematics and the real world. Sec Ref: Connections Page Ref: 97
24.
Joshua reasons that 33 is referred to as “3 cubed” because it forms a cube. This demonstrates which type of connection? A. Ideas within mathematics itself B. Between symbols and procedures and the conceptual ideas that the symbolism represents C. Connections between mathematics and the real world Answer: B Sec Ref: Connections Page Ref: 97
25.
In Betsy’s science classroom, they are talking about cell reproduction. The teacher tells them that a single cell will divide into two. Those two cells will then go on to produce four cells and so on. After a few seconds, Betsy raises her hand and asks, “Is this related to exponents?”. Which of the following three types of connections is Betsy making? A. Ideas within mathematics itself B. Between symbols and procedures and the conceptual ideas that the symbolism represents C. Connections between mathematics and the real world Answer: C Sec Ref: Connections Page Ref: 98
26.
T or F The formula, A = lw, is an example of a mathematical representation. Answer: T Sec Ref: Representations Page Ref: 98
27.
Use the following five representations to illustrate the concept of 1/2: pictures, written symbols, spoken language, relevant situations, manipulatives. Answer: Answers vary. Sec Ref: Representations Page Ref: 98
28.
In her unit on fractions, Mrs. Binder orders pizza for her classroom as a culminating activity. Which of the following five modes representation is she employing? A. Picture B. Written symbol C. Spoken language D. Relevant situation E. Manipulatives Answer: D Sec Ref: Representations Page Ref: 98
29.
Students were given the following problem: “A farmer goes into his barn and counts that he has 10 animals in all. The animals consist of pigs and chickens. Furthermore, he notices that there are 32 legs total. Find the number of chickens and pigs that the farmer has.” In order to solve this problem, Mary’s group decides to draw pictures. Which of the following NCTM process standards does this best illustrate? A. Reasoning and proof B. Communication C. Connections D. Representations E. None of the above Answer: D Sec Ref: Representations Page Ref: 98
30.
T or F When students understand a concept or idea using their own representations, teachers should not show them the conventional way. Answer: F Sec Ref: Representations Page Ref: 99
31.
Your text states that “mathematical ideas can be represented in different ways.” Given that, discuss three ways to represent the number 25. Answer: Possible answers – 2 tens and 5 ones, 1 ten and 15 ones, 25 ones, one quarter. Sec Ref: Representations Page Ref: 99
32.
T or F When teaching students who are learning English, removing most of the language and context from the curriculum will help those students develop the process standards. Answer: F Sec Ref: Cultural Connections Page Ref: 101
33.
When working with students who are learning English, what are some strategies that you can employ to help them develop the mathematics process standards? Answer: Should include some of the following: use mixed-ability groups, talk through problems during whole class discussions, assign buddies, use pictures and manipulatives, speak slower or use different words, etc. Sec Ref: Cultural Connections Page Ref: 101
34.
As a beginning teacher, you want to be sure you are including the NCTM Process Standards in your teaching. Name three strategies that you will include in your instruction and tell which standards they support. Answer: Answers vary. Sec Ref: A Glance at Where We’ve Been Page Ref: 101
35.
Discuss your role in helping your students to develop their problem solving skills? Answer: Pose questions where the solution is not known and not easily identified. Sec Ref: A Glance at Where We’ve Been Page Ref: 101
36.
If you could incorporate only one of the five NCTM Process Standards, which one would you incorporate and why? Answer: Answers vary. Sec Ref: A Glance at Where We’ve Been Page Ref: 101
Chapter 6: Helping Children with Problem Solving Possible Questions for Assessment and Evaluation
1.
T or F Many of the best problems for elementary children involve new situations that they are not familiar with. Answer: F Sec Ref: Introduction Page Ref: 107
2.
What are the characteristics of a good problem solver? Answer: Plan ahead, ask themselves if what they are doing makes sense, adjust their problemsolving strategies when necessary, and reflect on the reasonableness of their solution and approach. Sec Ref: Introduction Page Ref: 107
3.
According to your text, effective problem solvers exhibit the following characteristics, except: A. Plan ahead when given the problem B. Consult the Internet for possible solutions C. Ask themselves if what they are doing makes sense D. Adjust their problem-solving strategies E. Look back afterwards to reflect on the reasonableness of their solution Answer: B Sec Ref: Introduction Page Ref: 107
4.
T or F A problem for one child may not be a problem for another child. Answer: T Sec Ref: What is a Problem and What is Problem Solving? Page Ref: 107
5.
T or F The following is an example of a problem: Three classrooms are going on a field. Each classroom has 15 students. How many students in all are going on the field trip? Answer: F Sec Ref: What is a Problem and What is Problem Solving? Page Ref: 107
6.
T or F According to your text, many traditional textbook word problems are routine problems. Answer: T Sec Ref: What is a Problem and What is Problem Solving? Page Ref: 107
7.
What is the difference between a problem and an exercise? Which is more beneficial to a child’s learning? Why? Answer: Problem – solution is not immediately obvious; exercise – solution is obvious. Problems are more beneficial because they rely on higher level thinking. Sec Ref: What is a Problem and What is Problem Solving Page Ref: 107
8.
Mr. Peters’ classroom has been learning about addition. For their homework, they get the following problem: “Jack bought 3 pairs of socks and his brother bought 2 pairs of socks. How many pairs of socks did they buy in all?” This problem is mostly an example of: A. B. C. D. E.
a routine problem a nonroutine problem a problem none of the above all of the above
Answer: A Sec Ref: What is a Problem and What is Problem Solving? Page Ref: 107 9.
Mrs. Andrew’s second-grade classroom has been learning about division. For their homework, they get the following problem: “There are 48 students who are planning on going to the park. Each bus can take 10 students. How many buses will be needed to take all 48 students to the park?” This problem is mostly an example of: A. B. C. D. E.
a routine problem a nonroutine problem a problem both B and C all of the above
Answer: D Sec Ref: What is a Problem and What is Problem Solving Page Ref: 107
10.
Whether a problem is truly a problem or merely an exercise depends on: A. The grade level of the problem B. The length of the problem C. The publisher of the problem D. The developmental level of the person trying to solve it E. All of the above Answer: D Sec Ref: What is a Problem and What is Problem Solving? Page Ref: 107
11.
T or F American students have generally been found to be successful at solving nonroutine problems. Answer: F Sec Ref: What is a Problem and What is Problem Solving Page Ref: 108
12.
What is the difference between extended-constructed-response (ECR) and short-constructedresponse (SCR) items? Answer: ECR requires students to explain their answers. Sec Ref: What is a Problem and What is Problem Solving Page Ref: 110
13.
Compare and contrast extended-constructed-response (ECR) and short-constructed-response (SCR) items. Answer: Both require students to construct their own answers; ECR requires students to also explain their answers. Sec Ref: What is a Problem and What is Problem Solving? Page Ref: 108
14.
T or F The primary goal of school mathematics instruction should be to help students master their basic facts. Answer: F Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 109
15.
T or F One of the best approaches to learning mathematics is to tell students how to do something and then give them lots of practice until they know how to do it. Answer: F Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 109
16.
Discuss the three “signposts” to guide you in teaching mathematics through problem solving. Answer: Allow mathematics to be problematic for students; focus on the methods used to solve problems; tell the right things at the right time. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 109
17.
The principal walks into Mr. Chang’s classroom and discovers that students are working together and struggling to find the solution to a problem. He later asks Mr. Chang why he didn’t just give his students the solution strategy. Which of the following “signposts” for teaching mathematics through problem solving is Mr. Chang adhering to? A. B. C. D. E.
Tell the right things at the right time Focus on the methods used to solve problems Allow mathematics to be problematic for students All of the above None of the above
Answer: C Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 109 18.
T or F According to one of the “signposts” to guide you in teaching mathematics through problem solving, mathematics should be problematic. Answer: T Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 109
19.
T or F When students are engaged in problem solving, the teacher should not show students the written symbols of mathematics and should not define technical language for them. Answer: F Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 110
20.
One of the “signposts” for teaching problem solving is to tell the right things at the right time. Given that signpost, discuss three suggestions for telling the right things at the right time. Answer: Showing students the written symbols and defining technical language; tell students about alternative strategies; and highlight the big mathematical ideas. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 110
21.
One of the “signposts” encourages students to share their approaches to solving a problem. This “signpost” is: A. B. C. D. E.
tell the right things at the right time focus on the methods used to solve problems allow mathematics to be problematic for students all of the above none of the above
Answer: B Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 110 22.
What are some of the advantages of having students share solution strategies? Answer: Realize the advantages and disadvantages of certain strategies; deepens students’ mathematical understanding by thinking about which approaches are easier and harder. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 110
23.
T or F When teaching students using a problem solving approach, it is OK to show students the written symbols of mathematics (e.g., notation for fractions, decimals, percents, etc.). Answer: T Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 110
24.
T or F When teaching students using a problem solving approach, alternative strategies should not be mentioned. Answer: F Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 110
25.
Which of the following is not something you should do when using a problem solving approach: A. B. C. D. E.
Show students the written symbols of mathematics Tell students about alternative strategies Highlight the big mathematical ideas that come up during discussion All of the above None of the above
Answer: D Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 110
26.
T or F Children should become skillful at computation before engaging in problem solving. Answer: F Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 112
27.
Explain each of the following four factors that affect students’ problem-solving skills: knowledge, beliefs and affects, control, and sociocultural. Answer: Answers vary. Sec Ref: Teaching Mathematics through Problem Solvin Page: 110-111
28.
When her students were stuck on a problem, Mrs. Daniels reminded them about a similar problem they had worked on last week. This is an example of which of the four factors that impact students’ problem-solving skills? A. knowledge B. beliefs and Affects C. control D. sociocultural E. none of the above Answer: A Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 110
29.
According to your text, students’ problem-solving abilities often correlate strongly with all of the following, except: A. students’ attitudes B. students’ self-confidence C. teacher’s belief that all students can be good problem solvers D. teacher’s belief that there is only one way to solve a problem E. students’ beliefs about themselves as problem solvers Answer: D Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 110
30.
T or F According to your text, good problem solvers jump right in and start crunching numbers so as not to waste time. Answer: F Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 111
31.
32.
Discuss some characteristics of a good classroom climate for developing students’ problem-solving skills. Answer: Should include the following: opportunities for discussion, collaboration, sharing, mutual encouragement, sufficient time for problem solving, planning aids, needed resources, technology, and good classroom management. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 111 T or F In regards to choosing appropriate problems, your text suggests that one approach is to assign a problem that involves exactly the mathematical ideas you want to emphasize before students get to those ideas in the textbook. Answer: T Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 111
33.
What is an open-ended problem and give one example of it. Answer: Problems that can have more than one correct answer - examples will vary. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 112
34.
Suggest five different sources that you could consult when searching for problems. Answer: Include five of the following: articles, books, Web sites, write problems yourself, spontaneous problems, workshops/conferences, have children write their own, use children’s literature as a context. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 114
35.
T or F Encouraging students to write their own problems is a good way to develop their problem-solving skills. Answer: T Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 115
36.
Allowing students to write their own problems leads to the following benefits: A. Students learn how problems are structured. B. Students develop critical and reasoning abilities. C. Students learn to express their ideas clearly. D. A and B only E. A, B, and C Answer: E Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 115
37.
According to your text, there are three primary benefits of having students pose their own problems. Discuss these three benefits. Answer: Students learn how problems are structured; students develop critical and reasoning abilities; and students learn to express their ideas clearly. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 115
38.
Your text suggests four principles for helping students as they learn to pose problems. Discuss those four principles. Answer: Focus students’’ attention on the various kinds of information in problems; begin with familiar topics or concepts; encourage students to use ambiguity; and teach students about the idea of domain. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 115-116
39.
Your text suggests four principles for helping students as they learn to pose problems. Discuss what is meant by the idea of domain. Answer: Numbers students are allowed to use in a particular problem. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 116
40.
Calculators are useful in problem solving when they: A. let a child solve more complex problems or problems with realistic data B. eliminate tedious computations and decrease anxiety. C. help children explore mathematical objects, concepts, and operations. D. all of the above E. only A and C Answer: D Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 117
41.
Discuss three situations where it may be beneficial to allow students to use calculators. Answer: When solving more complex problems; to eliminate tedious and time-consuming computations; and when special calculator functions can help students explore mathematical concepts and operations. Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 107
42.
T or F When problem solving, calculators should not be used so students can practice their computations. Answer: F Sec Ref: Teaching Mathematics through Problem Solving Page Ref: 117
43.
George Polya’s four-stage model of problem solving includes all of the following, except: A. understand the problem B. devise a plan for solving it C. carry out your plan D. look back to examine your solution E. turn in your solution to the teacher Answer: E Sec Ref: Strategies for Problem Solving Page Ref: 117
44.
T or F According to your text, all four stages of Polya’s problem solving model does not have to be performed every time a student engages in problem solving. Answer: F Sec Ref: Strategies for Problem Solving Page Ref: 118
45.
Write one example to illustrate each of the following problem solving strategies: act it out, make a drawing or diagram, look for a pattern, construct a table, guess and check, work backward, and solve a similar but simpler problem. Answer: Answers vary. Sec Ref: Strategies for Problem Solving Page Ref: 118-124
46.
Mr. Trim poses the following problem to his class of 10 second graders: If each of you were to shake hands with every other student, how many total handshakes would there be? Which of the following would be the best strategy to use with this problem? A. Guess and check B. Work backward C. Solve a similar problem D. Act it out E. Look for a pattern Answer: D Sec Ref: Strategies for Problem Solving Page Ref: 118
47.
T or F Guessing should not be encouraged when students are working on developing their problem-solving skills. Answer: F Sec Ref: Strategies for Problem Solving Page Ref: 121
48.
Mrs. Trim poses the following problem: If the 30 dots on a circle were connected to every other dot, how many lines would there be? After 10 minutes of working on the problem, Mrs. Trim gives her class the following clue: Think back to how we solved the problem of 10 students shaking hands with every other student? Which of the following strategies is Mrs. Trim using? A. Guess and check B. Work backward C. Solve a similar problem D. Act it out E. Look for a pattern Answer: C Sec Ref: Strategies for Problem Solving Page Ref: 122
49.
In regards to problem solving, discuss the importance of looking back. Answer: Answers vary. Sec Ref: The Importance of Looking Back Page Ref: 124
50.
T or F When developing students’ problem-solving skills, they should always work by themselves. Answer: F Sec Ref: Helping All Students with Problem Solving Page Ref: 125
51.
Discuss five specific things you could do to help meet the diverse needs of your students. Answer: Should include five of the following: assign problems that different students can approach in different ways, routinely call on individuals to rephrase problems, ask pairs or groups to share their solutions, allow students to write on the chalkboard or whiteboard, and allow students to work on problems in quiet places. Sec Ref: Helping All Students with Problem Solving Page Ref: 126
52.
Compare and contrast American and Japanese classrooms in regards to their
approaches to teaching problem solving. Answer: American classrooms have two phases and involve students doing many problems, whereas Japanese classrooms consist of four (or sometimes five) phases but typically focuses on just one (or at most two) genuine problems. Sec Ref: Cultural Connections Page Ref: 127
Chapter 7: Developing Counting and Number Sense in Early Grades Possible Questions for Assessment and Evaluation
1.
What are some of the characteristics of a child who has a good sense of numbers?
Answer: An understanding of number concepts and operation on these numbers, development of useful strategies for handling numbers and operations; facility to compute accurately and efficiently, to detect errors, and to recognize results as reasonable; ability and inclination to use this understanding in flexible ways to make mathematical judgments; and an expectation that numbers are useful and that work with numbers is meaningful and makes sense. Sec Ref: Number Sense Page Ref: 131 2.
Early number developmental skills do not include which of the following? A. Patterns B. Conservation C. Group recognition D. Comparisons E. One-to-one correspondence Answer: A Sec Ref: Number Sense Page Ref: 131
3.
Number developmental skills include all of the following, except: A. Counting forward B. Counting backward C. Skip counting D. Place Value E. Classification Answer: E Sec Ref: Number Sense Page Ref: 131
4.
T or F Number sense is a finite entity that a student either has or does not have? Answer: F Sec Ref: Number Sense Page Ref: 131
5.
Which of the following may not be an example of an observable number sense behavior? A. following rote procedures B. recognizing the reasonableness of results C. expecting numbers to make sense D. using numbers flexibly E. none of the above Answer: A Sec Ref: Number Sense Page Ref: 131
6.
Discuss the stages involved in the development of number sense. Answer: Prenumber development, early number development, and number development. Sec Ref: Number Sense Page Ref: 131
7.
Ben is working on the following problem: “Mary has 10 cookies. She gives them out equally to her five friends. How many cookies will each friend get?” After working through the problem, he comes up with an answer of 50. After thinking about his answer, he reasons that it can’t be 50 because Mary has only 10 cookies to start out with. Which of the characteristics of number sense is Ben mostly exhibiting in this case? A. an understanding of number concepts and operation on these numbers B. development of useful strategies for handling numbers and operations C. facility to compute accurately and efficiently, to detect errors, and to recognize results as reasonable D. ability and inclination to use this understanding in flexible ways to make mathematical judgments E. expectation that numbers are useful and that work with numbers is meaningful and makes sense Answer: C Sec Ref: Number Sense Page Ref: 131
8.
List five things that a child with a good sense of numbers may say about the number 7. Answer: Answers vary. Sec Ref: Number Sense Page Ref: 132
9.
Some activities to help develop prenumber concepts include the following: A. classification and patterning B. one-to-one correspondence C. comparisons D. group recognition E. establishing benchmarks of quantities, such as 5 or 10 Answer: A Sec Ref: Prenumber Concepts Page Ref: 133
10.
Suggest five classification activities that you could have your students do to help develop their prenumber concepts and skills. Answer: Answers vary. Sec Ref: Prenumber Concepts Page Ref: 133
11.
T or F A cardinal number is a number that tells how many. Answer: T Sec Ref: Prenumber Concepts Page Ref: 133
12.
A young child is placing a collection of buttons into two piles: those with two holes and those with four holes. This activity assists the child in developing which prenumber skill? A. patterns B. classification C. group recognition D. conservation E. none of the above Answer: B Sec Ref: Prenumber Concepts Page Ref: 133
13.
Michael is given a bag of M&Ms. He opens it and starts arranging the M&Ms according to color. Which of the following attributes is he using? A. Geometric B. Relational C. Physical D. Affective E. All of the above Answer: C Sec Ref: Prenumber Concepts Page Ref: 136
14.
Discuss three types of pattern activities which may be used to develop prenumber concepts and for each provide an example. Answer: Should include three of the following: copying a pattern, finding the next one, extending a pattern, and making their own patterns. Examples vary. Sec Ref: Prenumber Concepts Page Ref: 136
15.
Define conservation as it pertains to numbers and give an example to illustrate a situation where a student has NOT conserved. Answer: Conservation of number – a given number does not vary. Examples vary. Sec Ref: Early Number Development Page Ref: 137
16.
T or F By the time children are 5 or 6 years of age, most of them have learned the concept of conservation. Answer: F Sec Ref: Early Number Development Page Ref: 137
17.
A child examines two groups of five objects which are arranged in different ways. When asked which has more, the child states that there is the same number of objects in each group. Which early number concept is the child exhibiting? A. Patterns B. Conservation C. Classification D. Number sense E. None of the above
Answer: B Sec Ref: Early Number Development Page Ref: 137 18.
A child looking at the following arrangements states that row two contains more circles. Which early number concept has this particular child not yet master?
A. Patterns B. Number sense C. Conservation D. Computation E. Classification Answer: C Sec Ref: Early Number Development Page Ref: 137 19.
T or F Research shows that most children entering school can identify quantities of three or less by inspection alone without the use of counting techniques. Answer: T Sec Ref: Early Number Development Page Ref: 137
20.
Discuss four reasons why sight recognition of quantities up to five or six is important. Answer: It saves time; it is the forerunner of some powerful number ideas, it helps develop more sophisticated counting skills; and it accelerates the development of addition and subtraction. Sec Ref: Early Number Development Page Ref: 138
21.
A young child rolls a dice and without counting knows there are five dots showing. Which early number concept is the child exhibiting? A. classification B. comparison C. conservation D. group recognition E. patterns
Answer: D Sec Ref: Early Number Development Page Ref: 138 22.
From easiest to hardest, which is the correct sequence in regards to students’ abilities to recognize arrangements? A. circular, linear, rectangular, scrambled B. linear, rectangular, circular, scrambled C. rectangular, linear, scrambled, circular D. rectangular, linear, circular, scrambled E. scrambled, circular, linear, rectangular Answer: D Sec Ref: Early Number Development Page Ref: 138
23.
Explain what is meant by the word subitize. Answer: Ability to recognize how many in an arrangement of objects. Sec Ref: Early Number Development Page Ref: 138
24.
Explain the one-to-one correspondence principle and give an example to illustrate it. Answer: A concept where one object is paired with only one other object. Examples vary. Sec Ref: Early Number Development Page Ref: 138
25.
T or F Discrete objects are objects that do not lend themselves well to counting. Answer: F Sec Ref: Early Number Development Page Ref: 140
26.
T or F The amount of water in a glass is an example of a discrete object. Answer: F Sec Ref: Early Number Development Page Ref: 140
27.
Discuss the two key characteristics of a “rational counter”. Answer: Says each number name as the objects are counted and realizes that the last number named tells how many. Sec Ref: Early Number Development Page Ref: 140
28.
Summarize the four principles on which the counting process rests. Answer: Each object to be counted must be assigned one and only one number name; the number-name list must be used in a fixed order every time a group of objects is counted; the order in which the objects are counted doesn’t matter; the last number name used gives the number of objects. Sec Ref: Early Number Development Page Ref: 140-141
29.
A child counts four objects in the following manner: “one, three, four, two”. Which of the following counting principles is the child not following? A. each object to be counted must be assigned one and only one number name B. the number-name list must be used in a fixed order every time a group of objects is counted C. the order in which the objects are counted doesn’t matter D. the last number name used gives the number of objects E. none of the above Answer: B Sec Ref: Early Number Development Page Ref: 140-141
30.
A child counts five objects in the following manner: “one, two, three, four, five”. When his teacher asks him how many objects there are, he answers “three”. Which of the following counting principles is the child not following? A. each object to be counted must be assigned one and only one number name B. the number-name list must be used in a fixed order every time a group of objects is counted C. the order in which the objects are counted doesn’t matter D. the last number name used gives the number of objects E. none of the above Answer: D Sec Ref: Early Number Development Page Ref: 141
31.
Compare and contrast rote counting and rational counting. Answer: They both involve counting and using the number names; the primary distinction is that children who count rationally exhibit all four counting principles. Sec Ref: Early Number Development Page Ref: 141-142
32.
Discuss two common errors associated with rote counting. Answer: Incorrect sequence, correct correspondence and correct sequence, incorrect correspondence Sec Ref: Early Number Development Page Ref: 141
33.
T or F In order for students to count on, they need to recognize the starting number and the nested inclusion of previous numbers. Answer: T Sec Ref: Early Number Development Page Ref: 142
34.
Bailey has just finished counting 6 objects. Bailey’s teacher gives her three more objects to count. Instead of recounting all of the objects, Bailey counts, “seven, eight, nine.” Which counting strategy is Bailey using in this situation? A. counting on B. counting back C. skip counting D. none of the above E. all of the above Answer: A Sec Ref: Early Number Development Page Ref: 142
35.
T or F Counting on is preceded by counting all. Answer: T Sec Ref: Early Number Development Page Ref: 142
36.
T or F Counting back models subtraction, while counting on models addition. Answer: T Sec Ref: Early Number Development Page Ref: 143
37.
Discuss five models to help students understand counting back, including 0 and negative numbers. Answer: Some possibilities include the following: calculators, thermometers, losses and gains, pebbles in a bag, and elevators. Sec Ref: Early Number Development Page Ref: 144
38.
T or F In skip counting, students should always count by 5s and 10s. Answer: F Sec Ref: Early Number Development Page Ref: 144
39.
Matthew arranges a pile of 25 pennies into groups of five. He then proceeds to count them as 5, 10, 15, 20, 25. Which of the following counting strategies is he using? A. Counting forward B. Skip counting C. Counting on D. None of the above E. All of the above Answer: B Sec Ref: Early Number Development Page Ref: 144
40.
T or F Two early number benchmarks for students are 0 and 10. Answer: F Sec Ref: Early Number Development Page Ref: 145
41.
Discuss some connections that students should be making when learning how to count. Answer: Visual, oral, and written representations; one more and one less; and relationships between numbers. Sec Ref: Early Number Development Page Ref: 146
42.
After running a relay race, the teacher states, “Team 2 finished first.” “First” is an example of: A. a cardinal number B. an ordinal number C. a nominal number D. a negative number E. all of the above Answer: B Sec Ref: Cardinal, Ordinal, and Nominal Numbers Page Ref: 148
43.
Give an example to illustrate each of the following types of numbers: cardinal, ordinal, nominal. Answer: Answers vary. Sec Ref: Cardinal, Ordinal, and Nominal Numbers Page Ref: 148
44.
An announcer makes the following announcement: “Room 3 please go to the lunchroom.” In this case, “room 3” is an example of: A. a cardinal number B. an ordinal number C. a nominal number D. a negative number E. all of the above Answer: C Sec Ref: Cardinal, Ordinal, and Nominal Numbers Page Ref: 149
45.
T or F Young children should spend less time writing numerals, since they typically have difficulty writing numerals as well as letters. Answer: T Sec Ref: Writing Numerals Page Ref: 149
46.
What makes it difficult for young children to write numerals? Answer: Lack of development of the small muscles and the limited eye-hand coordination. Sec Ref: Writing Numerals Page Ref: 149
47.
T or F Children in different countries use different counting principles. Answer: F Sec Ref: Cultural Connections Page Ref: 151
Chapter 8: Extending Number Sense: Place Value Possible Questions for Assessment and Evaluation 1.
T or F Our number system is called the American number system. Answer: F Sec Ref: Our Numeration System Page Ref: 157
2.
In what country was our number system invented? A. United States B. India C. China D. Greece E. Italy Answer: B Sec Ref: Our Numeration System Page Ref: 157
3.
T or F According to your text, our number system was probably invented in India. Answer: T Sec Ref: Our Numeration Page Ref: 157
4.
Explain each of the following characteristics of the Hindu-Arabic numeration system: place value, base of ten, use of zero, and additive property. Answer: The position of a digit represents its value; ten is the value that determines a new collection and is represented by 10; a symbol for zero exists and allows us to represent symbolically the absence of something; numbers can be written in expanded notation and summed with respect to place value. Sec Ref: Our Numeration System Page Ref: 157
5.
All of the following are properties of the Hindu-Arabic numeration system, except: A. place value B. base of ten C. whole number property D. use of zero E. additive property
Answer: C Sec Ref: Our Numeration System Page Ref: 157 6.
Mrs. Larson tells her class that another way to write 234 is 200 + 30 + 4. Which of the following properties of the Hindu-Arabic numeration system does this represent? A. place value B. base of ten C. use of zero D. additive property E. none of the above Answer: D Sec Ref: Our Numeration System Page Ref: 157
7.
T or F In the Hindu-Arabic numeration system, five is the value that determines a new collection. Answer: F Sec Ref: Our Numeration System Page Ref: 157
8.
Contrast the Hindu-Arabic numeration system with Roman numerals. Answer: Roman numerals lack place value, have no symbol for zero, and no base. Sec Ref: Our Numeration System Page Ref: 157
9.
Explain and give examples to illustrate the two key ideas upon which the development of place value rests. Answer: Explicit grouping or trading and the position of a digit determines the number being represented. Sec Ref: Nature of Place Value Page Ref: 157
10.
T or F The Hindu-Arabic numeration system uses nine digits to represent numbers. Answer: F Sec Ref: Nature of Place Value Page Ref: 157
11.
T or F Ten tens is equivalent to 1 group of one hundred. Answer: T Sec Ref: Nature of Place Value Page Ref: 157
12.
T or F Children who can successfully count or recite numbers usually have a strong understanding of place value. Answer: F Sec Ref: Nature of Place Value Page Ref: 158
13.
Define proportional and nonproportional models and give two examples of each. Answer: In proportional models, the material for five is five times the size of the material for 1 and so on; nonproportional materials do not maintain any size relationships. Examples vary. Sec Ref: Nature of Place Value Page Ref: 158
14.
Discuss ungrouped and pregrouped materials. List two examples of each. Answer: Ungrouped, materials that children can form into groups; pregrouped, materials that are formed into groups before children use them. Examples vary. Sec Ref: Nature of Place Value Page Ref: 158
15.
T or F Base-ten blocks are examples of proportional materials. Answer: T Sec Ref: Nature of Place Value Page Ref: 158
16.
T or F Money is an example of a proportional material. Answer: F Sec Ref: Nature of Place Value Page Ref: 158
17.
Which of the following is not a nonproportional model? A. Base-ten blocks B. Abacus C. Counters D. Money E. None of the above Answer: A Sec Ref: Nature of Place Value Page Ref: 159
18.
T or F Children should be comfortable using proportional materials first before being introduced to nonproportional materials. Answer: T Sec Ref: Nature of Place Page Ref: 159
19.
In a trading game, Matt tells Jenny that he will trade his dime for her 10 pennies. This type of trading involving money is an example of: A. a proportional place value model B. a nonproportional place value model C. a semiconcrete place value model D. a symbolic place value model E. none of the above Answer: B Sec Ref: Nature of Place Page Ref: 159
20.
Which of the following common errors related to place value is Suzie making when she does the following: “eighteen, nineteen, nineteen-ten”. A. bridging the decade B. reversing digits C. writing numbers read aloud D. all of the above E. none of the above Answer: A Sec Ref: Beginning Place Value Page Ref: 160
21.
Which of the following common errors related to place value is Ian making when he says: “eight, nine, ten, tenty-one, tenty-two, tenty-three”. A. bridging the decade B. reversing digits C. writing numbers read aloud D. counting E. none of the above Answer: D Sec Ref: Beginning Place Value Page Ref: 160
22.
T or F According to your text, in developing place value and establishing number names, it is better to skip beyond the teens and start with the larger numbers. Answer: T Sec Ref: Beginning Place Value Page Ref: 160
23.
When developing children’s place value understanding, a teacher should use models in the following sequence: A. concrete, semiconcrete, symbolic B. symbolic, semiconcrete, concrete C. semiconcrete, symbolic, concrete D. concrete, symbolic, semiconcrete E. symbolic, concrete, semiconcrete Answer: A Sec Ref: Beginning Place Value Page Ref: 161
24.
123 is an example of what type of place value model? A. concrete B. semiconcrete C. symbolic D. proportional E. nonproportional Answer: C Sec Ref: Beginning Place Value Page Ref: 161
25.
T or F When developing children’s place value understanding, calculators and other technological devices should not be used. Answer: F Sec Ref: Beginning Place Value Page Ref: 164
26.
Suggest two activities that you could do on the calculator to develop children’s place value understanding. Answer: Answers vary. Sec Ref: Beginning Place Value Page Ref: 164
27.
All of the following are appropriate ways to regroup 87, except: A. 7 tens and 17 ones B. 6 tens and 17 ones C. 5 tens and 37 D. 4 tens and 47 ones E. 3 tens and 57 ones Answer: B Sec Ref: Extending Place Value Page Ref: 165
28.
T or F 132 can be regrouped as 13 tens and 2 ones. Answer: T Sec Ref: Extending Place Value Page Ref: 165
29.
Write down three equivalent representations for 132. Answer: 13 tens and 2 ones; 132 ones; 10 tens and 32 ones; one hundred and 32 ones Sec Ref: Extending Place Value Page Ref: 165
30.
Use your own example to explain the front-end approach to comparing numbers. Answer: Answers vary. Sec Ref: Extending Place Value Page Ref: 165
31.
T or F Reading and writing numbers are symbolic activities and therefore, should precede modeling and talking about numbers. Answer: F Sec Ref: Reading and Writing Numbers Page Ref: 168
32.
Suppose one of your students wrote one hundred ninety-seven as 10097. Suggest one strategy you could use to help him learn to write it correctly. Answer: Use of a place value mat. Sec Ref: Reading and Writing Numbers Page Ref: 169
33.
Give an example of a strategy or activity that you could use to help develop your students’ understanding of large numbers. Answer: Answers vary. Sec Ref: Reading and Writing Numbers Page Ref: 170
34.
Which of the following is an acceptable alternative way of writing 1,234,567? A. 1234567 B. 1-234-567 C. 1;234;567 D. 1 234 567 E. 1:234:567 Answer: D Sec Ref: Reading and Writing Numbers Page Ref: 170
35.
T or F When writing large numbers, many countries use commas to separate blocks of three digits. Answer: F Sec Ref: Reading and Writing Numbers Page Ref: 170
36.
T or F Rounding rules may vary and are not universal. Answer: T Sec Ref: Rounding Page Ref: 172
37.
According to your text, context is important when rounding. Use an example to illustrate how the context affects rounding. Answer: Answers vary. Sec Ref: Rounding Page Ref: 172
38.
Discuss two models that could be used to help students understand rounding. Answer: Base-ten blocks and roller coaster model. Sec Ref: Rounding Page Ref: 172
39.
Contrast the place value systems in the following languages: English, Spanish, and Japanese. Answer: In English and Spanish, the number names from 11 to 20 do not explicitly name the number of tens as do the naming of numbers from 20 and larger. In China and Japan, place-value patterns abound. In China and Japan, their number systems are based on ten thousand rather than one thousand. Sec Ref: Cultural Connections Page Ref: 173
40.
T or F In China and Japan, their number systems are based on ten thousand rather than one thousand. Answer: T Sec Ref: Cultural Connections Page Ref: 174
Chapter 9: Operations: Meanings and Basic Facts Possible Questions for Assessment and Evaluation 1.
What four prerequisites are important in regards to helping students understand the operations? Answer: Facility with counting, experience with a variety of concrete situations, familiarity with many problem-solving contexts, and experience using language to communicate mathematical ideas. Sec Ref: Helping Children Develop Number Sense and Computational Fluency Page Ref: 180
2.
All of the following are prerequisites for formal work on the operations, except: A. B. C. D. E.
facility with counting experience with a variety of concrete situations familiarity with many problem-solving contexts familiarity with the calculator experience using language to communicate mathematical ideas
Answer: D Sec Ref: Helping Children Develop Number Sense and Computational Fluency Page Ref: 180 3.
T or F According to research, children use counting to solve problems involving the operations long before they come to school. Answer: T Sec Ref: Helping Children Develop Number Sense and Computational Fluency Page Ref: 180
4.
You are introducing the concept of subtraction to your first graders. Choose one of the prerequisites for whole number operations mentioned in the text and describe a specific activity you would use which would provide that prerequisite for your students. Answer: Answers vary. Sec Ref: Helping Children Develop Number Sense and Computational Fluency Page Ref: 180
5.
When developing students’ understanding of the operations, why is it important for them to have experiences with a variety of concrete situations? Answer: Understanding improves if they can relate mathematical facts and symbols to an experience they can visualize; manipulative materials serve as a referent for later work with the operations as well as for constructing the basic facts; they also provide a link to connect each operation to real-world problem-solving situations; they can always use concrete materials for confirmation.
Sec Ref: Helping Children Develop Number Sense and Computational Fluency Page Ref: 180 6.
The following are reasons for using problem situations in mathematics instruction, except: A. B. C. D. E.
for developing conceptual understanding for teaching higher-level thinking and problem-solving skills for increasing students’ abilities to recall basic facts for applying a variety of mathematical ideas none of the above
Answer: C Sec Ref: Helping Children Develop Number Sense and Computational Fluency Page Ref: 181 7.
T or F Communication should be an important part of the mathematics classroom. Answer: T Sec Ref: Helping Children Develop Number Sense and Computational Fluency Page Ref: 181
8.
T or F The use of materials should follow the use of symbols so students understand what the symbols represent. Answer: F Sec Ref: Helping Children Develop Number Sense and Computational Fluency Page Ref: 181
9.
The general sequence of activities appropriate for helping children develop meaning for the four basic operations is: A. B. C. D. E.
concrete, semiconcrete, abstract abstract, semiconcrete, concrete semiconcrete, concrete, abstract abstract, concrete, semiconcrete concrete, abstract, semiconcrete
Answer: A Sec Ref: Developing Meanings for the Operations Page Ref: 181 10.
You are introducing the following problem to your students: 8 + 3 = 11. Give one specific example for each of the following types of activities: concrete, semiconcrete, abstract. Answer: Answers vary. Sec Ref: Developing Meanings for the Operations Page Ref: 181
11.
In the context of developing meanings for the operations, explain what is meant by the following terms: concrete, semiconcrete, abstract. Answer: Concrete, modeling with materials; semiconcrete, representing with pictures; abstract, representing with symbols. Sec Ref: Developing Meanings for the Operations Page Ref: 181
12.
T or F Multiplication and addition are inverse operations. Answer: F Sec Ref: Developing Meanings for the Operations Page Ref: 181
13.
T or F Multiplication and division are inverse operations. Answer: T Sec Ref: Developing Meanings for the Operations Page Ref: 181
14.
Summarize four important relationships among the four basic operations. Answer: Addition and subtraction are inverse operations, multiplication and division are inverse operations, multiplication can be viewed as repeated addition, and division can be viewed as repeated subtraction. Sec Ref: Developing Meanings for the Operations Page Ref: 182
15.
Discuss four models that could be used to help students visualize addition. Answer: Disks or counters, number line, linking cubes, and balance. Sec Ref: Developing Meanings for the Operations Page Ref: 182
16.
“Mary has a total of 10 books. Six of them are nonfiction books. How many are fiction?” This is an example of a: A. B. C. D. E.
separation problem comparison problem part-whole problem equal-groups problem combination problem
Answer: C Sec Ref: Developing Meanings for the Operations Page Ref: 182
17.
“Luke has 8 marbles. Bill has 6. How many more marbles does Luke have than Bill?” This is an example of a: A. B. C. D. E.
separation problem comparison problem part-whole problem equal-groups problem combination problem
Answer: B Sec Ref: Developing Meanings for the Operations Page Ref: 182 18.
“Nina has seven pencils. She gives three to Jerome. How many pencils does Nina have left?” This is an example of a: A. B. C. D. E.
separation problem comparison problem part-whole problem equal-groups problem combination problem
Answer: A Sec Ref: Developing Meanings for the Operations Page Ref: 182 19.
Use a balance model to illustrate the following problem: Mary has 9 books. Six of them are nonfiction and the rest are fiction. How many are fiction? Answer: Draw a balance, circle six and three on the left side of the balance beam, circle 9 on the right side of the balance beam. Sec Ref: Developing Meanings for the Operations Page Ref: 182
20.
Given 6 – ( ) = 2, what does it mean to use the strategy “think addition”? Answer: 2 plus what equals 6. Sec Ref: Developing Meanings for the Operations Page Ref: 182
21.
T or F The sequence of activities used for understanding addition and subtraction – moving from concrete, to semiconcrete, to symbolic –should NOT be followed for multiplication and division because they use other processes. Answer: F Sec Ref: Developing Meanings for the Operations Page Ref: 183
22.
The correct way to interpret 3 x 4 is: A. B. C. D. E.
3 plus 4 4 plus 3 3 groups of 4 4 groups of 3 all of the above
Answer: C Sec Ref: Developing Meanings for the Operations Page Ref: 183 23.
Jonah bought three packages of pencils. Each package contains 6 pencils. Which of the following is the correct way to represent this problem? A. 3 x 6 B. 6 x 3 C. 3 + 6 C. Both A and B D. None of the above Answer: A Sec Ref: Developing Meanings for the Operations Page Ref: 183
24.
Use the following models to illustrate 2 x 3: sets of objects, arrays, number line. Answer: Sets of objects – two groups with 3 objects in each group; arrays – a box with 2 rows and 3 columns; number line – two jumps of three each. Sec Ref: Developing Meanings for the Operations Page Ref: 191
25.
T or F Of the four types of multiplication problems, combinations and arrays are most common in elementary schools. Answer: F Sec Ref: Developing Meanings for the Operations Page Ref: 183
26.
“Gary has 3 boxes of pencils. There are 6 pencils in each box. How many pencils in all does Gary have?” This is an example of a(n): A. B. C. D. E.
equal-groups problem comparison problem combination problem area and array measurement
Answer: A Sec Ref: Developing Meanings for the Operations Page Ref: 183 27.
“Jill has 3 notebooks. Jim has twice as many. How many notebooks does Jim have?” This is an example of a(n): A. B. C. D. E.
equal-groups problem comparison problem combination problem area and array measurement
Answer: B Sec Ref: Developing Meanings for the Operations Page Ref: 183 28.
I have three pants and four shirts. How many different outfits can I make? A. B. C. D. E.
equal-groups problem comparison problem combination problem area and array measurement
Answer: C Sec Ref: Developing Meanings for the Operations Page Ref: 184 29.
What is the difference between a measurement and a partition problem? Answer: In a partition problem, you don’t know how many objects must be put in each group. Sec Ref: Developing Meanings for the Operations Page Ref: 185
30.
T or F Jill has 15 marbles. She gives five to each of her friends. How many friends will receive marbles? This is an example of a measurement problem. Answer: T Sec Ref: Developing Meanings for the Operations Page Ref: 185
31.
Why is the measurement interpretation of division also called repeated subtraction? Answer: You are taking away an equal amount each time. Sec Ref: Developing Meanings for the Operations Page Ref: 185
32.
“Jenny has 18 pieces of candy. She gives 3 pieces to each of her friends. How many friends received candies from Jenny?” This is an example of a: A. Division – measurement problem B. Division – partition problem Answer: A Sec Ref: Developing Meanings for the Operations Page Ref: 185
33.
Use 8 ÷ 2 to write two story problems to illustrate the two meanings of division. Answer: Answers vary. Sec Ref: Developing Meanings for the Operations Page Ref: 185
34.
T or F In order for students to understand the operations, it is important that they also understand the mathematical properties that pertain to teach operation. Answer: T Sec Ref: Mathematical Properties Page Ref: 185
35.
Explain the meaning of each of the following properties and give an example to illustrate its meaning: commutative, associative, distributive, and identity. Answer: Answers vary. Sec Ref: Mathematical Properties Page Ref: 185
36.
3 + 2 = 2 + 3 is an illustration of which property? A. Commutative B. Associative C. Distributive D. Identity Answer: A Sec Ref: Mathematical Properties Page Ref: 185
37.
T or F Basic addition facts involve two two-digit addends and their sum. Answer: F Sec Ref: Overview of Basic Fact Instruction Page Ref: 185
38.
T or F There are 100 basic addition facts. Answer: T Sec Ref: Overview of Basic Fact Instruction Page Ref: 185
39.
T or F There are 100 basic subtraction facts. Answer: T Sec Ref: Overview of Basic Fact Instruction Page Ref: 185
40.
T or F There are 100 basic multiplication facts. Answer: T Sec Ref: Overview of Basic Fact Instruction Page Ref: 185
41.
Several groups of number combinations make up the basic facts. Which of the following is not part of the basic facts? A. B. C. D. E.
The sums involving all one-digit addends and related subtraction differences The products involving all one-digit and related quotients 100 addition, subtraction, and multiplication problems involving one-digit numbers The addition, subtraction, multiplication, and division problems involving two-digit numbers 90 division problems involving one-digit problems
Answer: D Sec Ref: Overview of Basic Fact Instruction Page Ref: 186 42.
Explain why division only has 90 basic facts. Answer: Because division by zero is not possible. Sec Ref: Overview of Basic Fact Instruction Page Ref: 186
43.
According to your text, reasons as to why children do not master the basic facts include all of the following, except: A. B. C. D. E.
learning disability the underlying numerical understandings may not have been developed the skill of fact retrieval may not have been taught by teachers all of the above none of the above
Answer: D Sec Ref: Overview of Basic Fact Instruction
Page Ref: 187 44.
As a classroom teacher, suggest some strategies that you could use to help your students master the basic facts. Answer: Start where the children are; building understanding of the basic facts; focus on how to remember the facts. Sec Ref: Overview of Basic Fact Instruction Page Ref: 187
45.
T or F Children should develop strategies for remembering the facts before they engage in drill to develop fluency. Answer: T Sec Ref: Overview of Basic Fact Instruction Page Ref: 189
46.
T or F When counting on, students should count on from the smaller addend. Answer: F Sec Ref: Thinking Strategies for Basic Facts Page Ref: 193
47.
Discuss and give an example to illustrate each of the following thinking strategies for addition facts: commutativity, adding doubles and near doubles, counting on. Answer: Answers vary. Sec Ref: Thinking Strategies for Basic Facts Page Ref: 191-195
48.
Facts for the “1-2-3 family” include all of the following, except: A. B. C. D. E.
1+ 2 = 3 2 +1 = 3 2–1=1 3–1=2 3–2=1
Answer: C Sec Ref: Thinking Strategies for Basic Facts Page Ref: 195 49.
Your text states that the primary thinking strategy for division is “think multiplication”. Use that strategy on the following problem: 18 ÷ 6 = ?. Answer: Ask yourself: what number times 6 equals 18. Sec Ref: Thinking Strategies for Basic Facts Page Ref: 199
50.
T or F The idea of adding zero is one of the hardest strategies for children to learn. Answer: T Sec Ref: Thinking Strategies for Basic Facts Page Ref: 193
51.
The following is an example of which thinking strategy? 7 + 6 = ? “I don’t know what 7 + 6 is so I’ll take 3 from the six and add it to the 7 to make that 10. Then I’ll add 10 to the 3 that is left of the six.” A. B. C. D. E.
commutativity counting on doubles adding to 10 and beyond adding near doubles
Answer: D Sec Ref: Thinking Strategies for Basic Facts Page Ref: 194 52.
The following is an example of which thinking strategy? 13 – 6 = ? “I don’t know what 13 minus 6 is, but I know what must be added to 6 will equal 13. Six plus seven gives me 13, therefore, the answer is 7.” A. B. C. D. E.
commutativity subtracting one and zero doubles counting back think addition
Answer: E Sec Ref: Thinking Strategies for Basic Facts Page Ref: 195 53.
Give an example to illustrate each of the following thinking strategies for subtraction: think addition, doubles, counting back, counting on. Answer: Answers vary. Sec Ref: Thinking Strategies for Basic Facts Page Ref: 195-196
54.
The following is an example of which thinking strategy? 4 x 5 = ? “Four times five means that I have four sets with five objects in each set. Therefore, instead of multiplying, I’ll add five plus five plus five plus five.” A. B. C. D. E.
skip counting repeated addition splitting the product commutativity multiplying by one and zero
Answer: B Sec Ref: Thinking Strategies for Basic Facts Page Ref: 197 55.
Give an example to illustrate each of the following thinking strategies for multiplication: skip counting, repeated addition, splitting the product, commutativity. Answer: Answers vary. Sec Ref: Thinking Strategies for Basic Facts Page Ref: 196
56.
T or F Traditionally, teaching division has taken a large portion of time in elementary schools. Answer: T Sec Ref: Thinking Strategies for Basic Facts Page Ref: 199
57.
Give an example to illustrate the concept of “think multiplication”? Answer: Answers vary. Sec Ref: Thinking Strategies for Basic Facts Page Ref: 199
58.
T or F Dividing zero by some number and dividing some number by 0 results in the same answer. Answer: F Sec Ref: Thinking Strategies for Basic Facts Page Ref: 201
59.
T or F The facts considered “basic” in some countries are not the same as those we call “basic facts” in the United States. Answer: T Sec Ref: Cultural Connections Page Ref: 202
60.
T or F Basic addition facts in Korea focus on combinations that total 15 or less. Answer: F Sec Ref: Cultural Connections Page Ref: 202
Chapter 10: Computation Methods: Calculators, Mental Computation, and Estimation Possible Questions for Assessment and Evaluation 1.
T or F Historically, elementary school mathematics has emphasized mental computation far more than other methods. Answer: F Sec Ref: Introduction Page Ref: 207
2.
T or F Children tend to use written computation even when they could use a more efficient method. Answer: T Sec Ref: Introduction Page Ref: 207
3.
According to your text, which of the following should always be used to check the reasonableness of the result? A. written computations B. calculators C. teachers D. estimations E. classmates Answer: D Sec Ref: Introduction Page Ref: 207
5.
Discuss the two essential decisions involved in every computation. Answer: Deciding on the type of result needed and deciding on the best method for getting that result. The first decision involves the question of whether an estimate is appropriate or an exact answer is needed and the second involves selecting the best method to get the result. Sec Ref: Introduction Page Ref: 207
6.
Instructional goals for teaching computation include teaching children to: A. master one method for solving multiple problems B. develop competence with each of the computational methods C. determine the reasonableness of a result D. all of the above E. only B and C Answer: E Sec Ref: Introduction Page Ref: 207
7.
As a mathematics teacher, what are some instructional goals that you should have when teaching computation? Answer: Help students to develop competence with each of the computational methods; help students to choose a method that is appropriate for the computation at hand; help students to apply the chosen method correctly; and help students to use estimation to determine the reasonableness of the result. Sec Ref: Introduction Page Ref: 207
8.
T or F More than 80% of all mathematical computations in daily life involve written computation and estimation, rather than mental computation. Answer: F Sec Ref: Balancing Your Instruction Page Ref: 208
9.
According to research, what percentage of instructional time is devoted to written computation? A. 10 – 30% B. 30 – 50% C. 50 – 70% D. 70 – 90% E. 90 – 100% Answer: D Sec Ref: Balancing Your Instruction Page Ref: 208
10.
T or F According to research, 70-90% of the instructional time is devoted towards written computation. To balance this out, the CCSSM recommends more time devoted to mental computation. Answer: F Sec Ref: Balancing Your Instruction Page Ref: 208
11.
In terms of teaching mental computation, written computation, and estimation, how much time would you devote to each and why? Answer: Answers vary. Sec Ref: Balancing Your Instruction Page Ref: 208
12.
T or F In general, the amount of time spent on estimation has increased over time. Answer: T Sec Ref: Balancing Your Instruction Page Ref: 208
13.
Upon learning that the children in your classroom will be using calculators, a parent expresses concern. Summarize three points you might make to defend the use of calculators in your classroom. Answer: Students still have to think when using the calculator; calculators can raise students’ achievement; calculators are useful as instructional tools. Sec Ref: Calculators Page Ref: 209
14.
T or F Using calculators always makes computations go faster. Answer: F Sec Ref: Calculators Page Ref: 209
15.
Give one example to illustrate the idea that using calculators requires thinking. Answer: Answers vary. Sec Ref: Calculators Page Ref: 209
16.
Which of the following statements is not a fact regarding the use of calculators? A. When using calculators, students must still do the thinking. B. Calculators can raise students’ achievement. C. Using calculators always makes computations faster. D. It is sometimes faster to compute mentally. E. Calculators can be useful as instructional tools. Answer: C Sec Ref: Calculators Page Ref: 209
17.
Give one example to illustrate a situation where calculators are not always the fastest way of doing computations. Answer: Answers vary. Sec Ref: Calculators Page Ref: 209
18.
What does it mean to use the calculator as an instructional tool, as compared to a computational tool? Answer: Computation is mainly to get the result; instructional is to help students understand the concepts, processes, and patterns associated with computation. Sec Ref: Calculators Page Ref: 210
19.
When would it be appropriate to consider using a calculator? Answer: When computational skills are not the main focus of instruction. Sec Ref: Calculators Page Ref: 210
20.
Which of the following is an appropriate way to use the calculator as an instructional tool? A. to facilitate problem solving B. to facilitate a search for patterns C. to ease the burden of doing tedious computations D. to remove anxiety about doing computations incorrectly E. to provide motivation and confidence Answer: B Sec Ref: Calculators
Page Ref: 210 21.
All of the following are appropriate ways to use the calculator as a computational tool, except: A. to facilitate problem solving B. to facilitate a search for patterns C. to ease the burden of doing tedious computations D. to remove anxiety about doing computations incorrectly E. to provide motivation and confidence Answer: B Sec Ref: Calculators Page Ref: 210
22.
Compare and contrast mental computation and written computation. Answer: Mental is done “all in the head”. Sec Ref: Mental Computation Page Ref: 211
23.
T or F When we talk about mental computation, we are talking about computation that is done mostly in the head – with the use of a calculator or paper and pencil in some instances. Answer: F Sec Ref: Mental Computation Page Ref: 211
24.
Barry explained how he mentally solved the problem, 24 + 23. He said, “I added 20 plus 20 which makes 40 , added 4 plus 3 which makes 7, and then added 40 plus 7 to get 47.” Which mental computation strategy is Barry using? A. adding from the left B. counting on C. making tens D. doubling E. bridging Answer: A Sec Ref: Mental Computation Page Ref: 212
25.
Katrina explained how she mentally solved the problem, 37 + 44. She said, “I added 3 plus 37 which makes 40, and then added 40 to 40 which makes 80 and then added 1 more to get 81.” Which mental computation strategy is Katrina using? A. adding from the left B. counting on C. making tens D. doubling E. bridging Answer: C Sec Ref: Mental Computation Page Ref: 212
26.
Jimmy explained how he mentally solved the problem, 11 + 12 + 19 + 18. He said, “I added 11 and 19 to get 30 and added 12 and 18 to get 30. Finally, I added 30 and 30 together to get 60.” Which mental computation strategy is Jimmy using? A. Counting on B. Adding from the left C. Making compatibles D. Doubling E. Bridging Answer: C Sec Ref: Mental Computation Page Ref: 212
27.
Melissa explained how she mentally solved the problem, 36 + 23. She said, “I counted by tens, 36…46…56 and then counted by ones 57…58…59.” Which mental computation strategy is Melissa using? A. Counting on B. Adding from the left C. Making compatibles D. Doubling E. Bridging Answer: A Sec Ref: Mental Computation Page Ref: 212
28.
Give an example to illustrate each of the following strategies for whole number addition: adding from the left, counting on, making tens, doubling, making compatibles, and bridging. Answer: Answers vary. Sec Ref: Mental Computation Page Ref: 212
29.
What are some of the benefits of using mental computation? Answer: Encourages flexible thinking, promotes number sense, and encourages creative and efficient work with numbers. Sec Ref: Mental Computation Page Ref: 213
30.
T or F When teaching mental computation, students should all be encouraged to use the same strategy to solve problems. Answer: F Sec Ref: Mental Computation Page Ref: 213
31.
What are some guidelines that you should keep in mind when developing students’ mental computation skills? Answer: Encourage students to do computations mentally, learn which computations students prefer to do mentally, find out if students are applying written algorithms mentally, plan to include mental computation systematically and regularly as an integral part of instruction, keep practice sessions short, develop children’s confidence, and encourage inventiveness. Sec Ref: Mental Computation Page Ref: 214-215
32.
Which of the following statements about the teaching of mental computation is not a recommendation? A. encourage students to do computations mentally B. learn which computations students prefer to do mentally C. find out if students are applying written algorithms mentally D. encourage all students to use the same strategy for solving a problem E. encourage inventiveness Answer: D Sec Ref: Mental Computation
Page Ref: 214-215 33.
What is the difference between mental computation and estimation? Answer: Computation is getting an exact answer. Sec Ref: Mental Computation Page Ref: 216
34.
Use an example to discuss the three points at which estimation can be used to monitor computations. Answer: Examples vary. Three points are before starting the computation, while doing the computation, and after completing the computation. Sec Ref: Estimation Page Ref: 216
35.
T or F According to your text, students who are proficient at written computations are also usually good estimators. Answer: F Sec Ref: Estimation Page Ref: 216
36.
Some words used in connection with estimation include all of the following, except: A. about B. approximate C. exact D. close enough E. almost Answer: C Sec Ref: Estimation Page Ref: 217
37.
Give an example to illustrate each of the following types of estimation strategies: frontend, front-end with adjustment, compatible numbers, and clustering. Answer: Answers vary. Sec Ref: Estimation Page Ref: 217-220
38.
A student solved the problem, 132 + 222 + 348 + 435, in the following manner: 100 + 200 + 300 + 400. Which estimation strategy is he using? A. front-end B. compatible numbers C. clustering D. front-end with adjustment E. none of the above Answer: A Sec Ref: Estimation Page Ref: 217
39.
T or F The front-end estimation strategy will produce an answer that is always less than the actual. Answer: T Sec Ref: Estimation Page Ref: 217
40.
A student solved the problem, 24 x 9, in the following manner: 24 is close to 25 and 9 is close to 10, therefore, 25 x 10 is 250. Which estimation strategy is she using? A. front-end B. compatible numbers C. clustering D. front-end with adjustment E. none of the above Answer: B Sec Ref: Estimation Page Ref: 218
41.
A student solved the problem, 213 ÷ 7, in the following manner: round 213 to 210 and take 210 ÷ 7, which comes out to be 30. Which estimation strategy is he using? A. front-end B. compatible numbers C. clustering D. front-end with adjustment E. none of the above Answer: B Sec Ref: Estimation
Page Ref: 218 42.
T or F The clustering estimation strategy is used to estimate a difference. Answer: F Sec Ref: Estimation Page Ref: 220
43.
A student solved the problem, 18 + 22 + 21 + 19 + 20, in the following manner: all the numbers are close to 20, therefore, 5 x 20 = 100. Which estimation strategy is he using? A. front-end B. compatible numbers C. clustering D. front-end with adjustment E. none of the above Answer: C Sec Ref: Estimation Page Ref: 220
44.
T or F Different estimation strategies can work for the same problem. Answer: T Sec Ref: Estimation Page Ref: 230
45.
Your text states that “different estimation strategies can work for the same problem.” Use an example to illustrate the truthfulness of that statement. Answer: Answers vary. Sec Ref: Estimation Page Ref: 221
46.
What are five guidelines that should be followed when teaching estimation? Answer: Give students problems that encourage and reward estimation; make sure students are not computing exact answers and then rounding to produce estimates; ask students to tell how they made their estimates; fight the one-right-answer syndrome; and encourage students to think of real-world situations that involve making estimates. Sec Ref: Estimation Page Ref: 221-222
47.
T or F Estimation is often held in low esteem by Japanese and Taiwanese students.
Answer: T Sec Ref: Cultural Connections Page Ref: 222
Chapter 11: Standard and Alternative Computational Algorithms Possible Questions for Assessment and Evaluation 1.
Discuss some of the changes in regards to the teaching and learning of computation. Answer: Instead of teaching procedures, it’s become more of a problem-solving process; also, instead of focusing entirely on accuracy and efficiency, instruction should aim towards understanding of computation. Sec Ref: Introduction Page Ref: 228
2.
T or F According to research, there is much consensus among the states as to when children should master the four basic operations. Answer: F Sec Ref: Introduction Page Ref: 228
3.
T or F With the advent of calculators and various other technologies, the ability to use paper-and-pencil algorithms is considered not essential. Answer: F Sec Ref: Introduction Page Ref: 230
4.
In order to develop computational fluency, all of the following recommendations should be followed, except: A. fostering a solid understanding of and proficiency with simple calculations B. abandoning the teaching of tedious calculations C. developing the skills necessary to use appropriate technology D. eliminating the teaching of paper-and-pencil algorithms E. fostering the use of a wide variety of computation and estimation techniques Answer: D Sec Ref: Introduction Page Ref: 230
5.
Suppose that you will begin a unit on developing computational fluency, what are some things you should keep in mind? Answer: Foster a solid understanding of and proficiency with simple calculations; abandon the teaching of tedious calculations using paper-and-pencil algorithms in favor of exploring more mathematics; foster the use of a wide variety of computation and estimation techniques; develop the skills necessary to use appropriate technology; and provide students with ways to check the reasonableness of computations. Sec Ref: Introduction Page Ref: 230
6.
When teaching algorithms, which of the following two things must be taken into consideration? (circle two) A. materials B. textbooks C. place values D. assessments E. technologies Answer: A and C Sec Ref: Teaching Algorithms with Understanding Page Ref: 231
7.
Use an example to discuss the difference between conceptual and procedural understanding. Answer: Answers vary. Sec Ref: Teaching Algorithms with Understanding Page Ref: 231
8.
List five materials that are commonly used to help students with computation. Answer: Unifix cubes, base-ten blocks, rods, popsicle sticks, bean sticks, and buttons. Sec Ref: Teaching Algorithms with Understanding Page Ref: 231
9.
T or F Place value understanding is important for each of the algorithms for wholenumber computation. Answer: T Sec Ref: Teaching Algorithms with Understanding Page Ref: 231
10.
Give an example to illustrate what it means to regroup. Answer: Answers vary. Sec Ref: Teaching Algorithms with Understanding Page Ref: 232
11.
All of the following are appropriate ways to rename 143, except: A. 13 tens and 13 ones B. 14 tens and 3 ones C. 15 tens and 3 ones D. 12 tens and 23 ones E. 1 hundred and 43 ones Answer: C Sec Ref: Teaching Algorithms with Understanding Page Ref: 232
12.
List five other ways to rename 132. Answer: Answers vary. Sec Ref: Teaching Algorithms with Understanding Page Ref: 232
13.
T or F Alternative algorithms should never be end goals, rather they should be transitional steps on the way to learning the standard algorithms. Answer: F Sec Ref: Addition Page Ref: 235
14.
T or F Children should be encouraged to use the standard algorithms only, as these are widely used by adults. Answer: F Sec Ref: Addition Page Ref: 235
15.
Give an example to demonstrate the partial-sum addition algorithm and the higherdecade addition strategy. Answer: Answers vary. Sec Ref: Addition Page Ref: 235-236
16.
Jimmy solved 43 + 38 in the following manner: 40 plus 30 equals 70, 3 plus 8 equals 11, therefore, the answer is 70 plus 11 equals 81. In solving the problem, which strategy did Jimmy use? A. higher-decade B. partial-sum C. Partial-difference D. partial-products E. all of the above Answer: B Sec Ref: Addition Page Ref: 235
17.
Model the following problem using base-ten blocks: 26 + 17. Answer: Add 6 ones to 7 ones to get 13 ones. Trade 10 ones for one ten. Add this one ten to the 2 tens from 26 and the 1 ten from 17 to get 4 a total of 4 tens. Final answer is 4 tens and 3 or 43. Sec Ref: Addition Page Ref: 235
18.
T or F 8 + 3 is an example of a higher-decade combination. Answer: F Sec Ref: Addition Page Ref: 236
19.
T or F Decomposition is another name for the standard subtraction algorithm. Answer: T Sec Ref: Subtraction Page Ref: 237
20.
Explain why decomposition is another name for the standard subtraction algorithm. Answer: It involves decomposing or renaming the sum (the number you are subtracting from). Sec Ref: Subtraction Page Ref: 237
21.
Model the following problem using base-ten blocks: 33 – 15. Answer: Trade 1 ten from 33 for 10 ones, leaving you with 2 tens; you now have 13 ones; take 5 ones away from 13 and you have 8 ones left; subtract 1 ten from the 2 tens and you have 1 ten left; the answer is 1 ten and 8 ones or 18. Sec Ref: Subtraction Page Ref: 237
22.
Regrouping is an important concept that students must understand if they are to understand the subtraction algorithm. To that extent, 483 can be regrouped as all of the following, except: A. 48 tens and 3 ones B. 4 hundreds, 7 tens, and 13 ones C. 48 hundreds, 3 ones D. 47 tens and 13 ones E. none of the above Answer: C Sec Ref: Teaching Algorithms with Understanding Page Ref: 239
23.
Suppose your students did the following: 246 – 184 = 142. What mistake did they make and how can you help them to understand the mistake? Answer: Subtracted the smaller number from the larger in all the columns; go back and do some more modeling using manipulatives. Sec Ref: Teaching Algorithms with Understanding Page Ref: 239
24.
Give an example to illustrate the partial-difference subtraction algorithm. Answer: Answers vary. Sec Ref: Teaching Algorithms with Understanding Page Ref: 239
25.
Which of the following is not a prerequisite for understanding multiplication algorithms? A. expanded notation B. place value understanding C. distributive property D. basic facts of multiplication E. all of the above
Answer: E Sec Ref: Multiplication Page Ref: 240 26.
To solve, 13 x 19, Jill thought the following: “9 times 3 is 27, 9 times 10 is 90, 10 times 3 is 30, and 10 times 10 is 100; to get the answer, I add 27 plus 90 plus 30 plus 100.” Which of the following strategies is Jill using? A. partial-sum B. partial-difference C. partial-products D. lattice E. higher-decade Answer: C Sec Ref: Multiplication Page Ref: 240
27.
T or F Lattice multiplication is an algorithm that was invented in the late 1990s. Answer: F Sec Ref: Multiplication Page Ref: 241
28.
Diagram how you would solve 234 x 12 using the lattice algorithm. Answer: Set up a lattice with 3 columns and 2 rows. Multiply each row with each column. Add up the diagonals. The answer is 2808. Sec Ref: Multiplication Page Ref: 241
29.
T or F According to your text, multiplying by 10 comes easily to most children. Answer: T Sec Ref: Multiplication Page Ref: 242
30.
Discuss three strategies you can use to help students who make multiplication mistakes involving zeros? Answer: Estimation to get a ballpark answer, using partial products, using place-value charts, and using expanded notation. Sec Ref: Multiplication Page Ref: 242
31.
Illustrate how you would show 13 x 12 on a grid. Answer: 13 rows and 12 columns; the partial sums should be 10 x 10 (100), 3 x 10 (30), 10 x 2 (20), and 3 x 2 (6). Sec Ref: Multiplication Page Ref: 242
32.
T or F Of all the traditional algorithms, multiplication is the most difficult for children to master. Answer: F Sec Ref: Division Page Ref: 243
33.
The most difficult computational algorithm for children is: A. addition B. subtraction C. multiplication D. division E. none of the above Answer: D Sec Ref: Division Page Ref: 243
34.
Your text states that division is difficult for which of the following reasons? A. computation begins at the left B. the computation involves subtraction C. the computation involves multiplication D. all of the above E. only A and B Answer: D Sec Ref: Division Page Ref: 243
35.
T or F When children are first introduced to division, they should not encounter problems with remainders. Answer: F Sec Ref: Division Page Ref: 244
36.
Give an example to illustrate the subtractive algorithm for division. Answer: Answers vary. Sec Ref: Division Page Ref: 245
37.
Use the subtractive algorithm for division with the following problem: 345 ÷ 8. Answer: Solution methodology varies; answer is 43r1. Sec Ref: Division Page Ref: 245
38.
T or F According to your text, when introducing students to division, they should only work with problems that come out even. Answer: T Sec Ref: Division Page Ref: 248
39.
T or F The computational algorithms typically taught in the United States are universal. Answer: F Sec Ref: Cultural Connections Page Ref: 249
40.
Give an example to illustrate the equal-additions algorithm and also explain why it works. Answer: Examples vary. It works because you are adding the same amount to both numbers. Sec Ref: Cultural Connections Page Ref: 250
Chapter 12: Fractions and Decimals: Concepts and Operations Possible Questions for Assessment and Evaluation 1.
T or F The CCSSM places more emphasis on fractions at Grades 1-3, but less in the higher grades. Answer: F Sec Ref: Introduction Page Ref: Page 255
2.
T or F Students should be introduced to decimals only after they have mastered fractions. Answer: F Sec Ref: Introduction Page Ref: Page 255
3.
In order for students to understand fractions, they must understand the following two concepts: A. equivalence and the word name B. partitioning and equivalence C. models and partitioning D. models and the word name E. none of the above Answer: B Sec Ref: Introduction Page Ref: Page 255
4.
Discuss the following two concepts associated with fractions: partitioning and equivalence. Answer: Partitioning, sharing equally; equivalence, two fractions representing the same amount. Sec Ref: Conceptual Development of Fractions Page Ref: Page 255
5.
Explain and give an example to illustrate the three meanings of fractions. Answer: Part-whole – a whole has been partitioned into equal parts; quotient- involves the idea of dividing equally; ratio – comparing one set of something to another set. Examples vary. Sec Ref: Conceptual Development of Fractions
Page Ref: Page 3 6.
Jenny has a circle that’s been divided into six equal parts. She then shades in five of those parts. Which meaning of fractions does this best represent? A. part-whole B. quotient C. ratio D. all of the above E. none of the above Answer: A Sec Ref: Conceptual Development of Fractions Page Ref: Page 256
7.
T or F The fraction 2/3 can be restated as two one-thirds. Answer: T Sec Ref: Conceptual Development of Fractions Page Ref: Page 256
8.
What is a unit fraction? Give an example of one. Answer: A fraction with a numerator of 1. Examples vary. Sec Ref: Conceptual Development of Fractions Page Ref: Page 256
9.
Billy has four cookies and he wants to share them equally among his five friends. Which meaning of fractions does this best represent? A. part-whole B. quotient C. ratio D. all of the above E. none of the above Answer: B Sec Ref: Conceptual Development of Fractions Page Ref: Page 256
10.
T or F All three meanings of fractions involve the idea of partitioning. Answer: F Sec Ref: Conceptual Development of Fractions
Page Ref: Page 256 11.
T or F A circle is the easiest region model to partition. Answer: F Sec Ref: Conceptual Development of Fractions Page Ref: Page 257
12.
According to your text, which of the following region models is easiest for children to partition? A. Circle B. Rectangle C. Triangle Answer: B Sec Ref: Conceptual Development of Fractions Page Ref: Page 257
13.
Mrs. Lye took out a circle and divided it equally (same size and shape) into five sections. Which part-whole model best illustrates this situation? A. region B. length C. set D. area E. none of the above Answer: A Sec Ref: Conceptual Development of Fractions Page Ref: Page 257
14.
Jill has 2 feet of string. She wants to cut it into fourths. Which part-whole model will best illustrate this situation? A. region B. length C. set D. area E. none of the above Answer: B Sec Ref: Conceptual Development of Fractions Page Ref: Page 257
15.
Mario has 10 marbles that he would like to share equally with his 5 friends. Which partwhole model will best illustrate this situation? A. region B. length C. set D. area E. none of the above Answer: C Sec Ref: Conceptual Development of Fractions Page Ref: Page 258
16.
Use the set model to represent three-fifths of 15 marbles. Answer: The 15 marbles should be partitioned into 5 equal parts (3 marbles per part). Circle or shade in 3 of these 5 equal parts. Sec Ref: Conceptual Development of Fractions Page Ref: Page 5
17.
What is the difference between the region model and the area model? Answer: In the area model, the parts must be equal in area but not necessarily congruent. Sec Ref: Conceptual Development of Fractions Page Ref: Page 258
18.
T or F The simplest meaning and the simplest model for fractions is the part-whole meaning and the region model. Answer: T Sec Ref: Conceptual Development of Fractions Page Ref: Page 258
19.
Suggest a strategy for helping your students make sense of fractions. Answer: Allow students to partition models, introduce fraction words, count the fractional parts, and show them the fraction symbols. Sec Ref: Conceptual Development of Fractions Page Ref: Page 258-260
20.
When helping to develop students’ understanding of fractions, which is the most appropriate developmental sequence? A. partitioning, counting, words, symbols B. symbols, counting, words, partitioning C. symbols, partitioning, counting, words D. partitioning, words, counting, symbols E. partitioning, symbols, counting, words Answer: D Sec Ref: Conceptual Development of Fractions Page Ref: Page 260
21.
In regards to fraction symbols, which connection do most textbooks concentrate on? A. model and words B. model and symbol C. words and symbol D. words and drawing E. drawings and model Answer: B Sec Ref: Conceptual Development of Fractions Page Ref: Page 261
22.
T or F In regards to fraction symbols, most textbooks usually concentrate on going from a model to a symbol. Answer: T Sec Ref: Conceptual Development of Fractions Page Ref: Page 261
23.
Explain why one-half and two-fourths are equivalent fractions. Answer: They both represent the same amount. Sec Ref: Conceptual Development of Fractions Page Ref: Page 261
24.
Explain why your authors make the following statement: “If we could make one change in curricula and assessment, it would be to throw out all the early references to out of.” Answer: Encourages the thinking of fractions as two whole numbers. Sec Ref: Conceptual Development of Fractions Page Ref: Page 262
25.
T or F Children tend to think of fractions as two whole numbers, the numerator and the denominator. Answer: T Sec Ref: Conceptual Development of Fractions Page Ref: Page 262
26.
Which of the following fractions is not considered a friendly fraction? A. 1/2 B. 1/3 C. 3/4 D. 2/4 E. 3/7 Answer: E Sec Ref: Conceptual Development of Fractions Page Ref: Page 262
27.
Three key benchmarks include the following: (circle three) A. 0 B. 1/4 C. 1/2 D. 3/4 E. 1 Answer: A, C, E Sec Ref: Conceptual Development of Fractions Page Ref: Page 263
28.
If you have 2/7 of a piece of chocolate, how many more pieces will be needed to make a whole? A. 3 B. 4 C. 5 D. 6 E. 7 Answer: C Sec Ref: Conceptual Development of Fractions Page Ref: Page 263
29.
What models could you use to help students determine which is bigger, 2/3 or 3/4? Answer: Concrete models, pictorial models, and symbolic representation. Sec Ref: Conceptual Development of Fractions Page Ref: Page 264
30.
Using a region model, illustrate how 2/3 and 6/9 are equivalent. Answer: Both of them can be represented with a rectangle containing nine equal parts. For 2/3, 6 of those nine parts would be shaded in – just like 6/9. Sec Ref: Conceptual Development of Fractions Page Ref: Page 265
31.
T or F To find equivalent fractions, you add the same number to both the numerator and denominator. Answer: F Sec Ref: Conceptual Development of Fractions Page Ref: Page 265
32.
T or F It is easier to compare two fractions that are represented by the same partitioning than two fractions that are represented by different partitioning. Answer: T Sec Ref: Conceptual Development of Fractions Page Ref: Page 265
33.
T or F Between any two fractions, there are an infinite number of fractions. Answer: T Sec Ref: Conceptual Development of Fractions Page Ref: Page 266
34.
Which of the following models would best illustrate the density property of fractions? A. Region B. Area C. Set D. Measurement Answer: D Sec Ref: Conceptual Development of Fractions Page Ref: Page 267
35.
The following is a pictorial model for an improper fraction. Which improper fraction matches the model?
A. 1/2 B. 2 1/2 C. 3/4 D. 5/2 E. none of the above Answer: D Sec Ref: Conceptual Development of Fractions Page Ref: Page 267 36.
The following is a pictorial model for a mixed number. Which mixed number matches the model?
A. 1/2 B. 2 1/2 C. 3/4 D. 5/2 E. none of the above Answer: B Sec Ref: Conceptual Development of Fractions Page Ref: Page 267 37.
What is the relationship between improper fractions and mixed numbers? Answer: They are equivalent – different ways of representing the same amount. Sec Ref: Conceptual Development of Fractions Page Ref: Page 267
38.
Model 7/4 pictorially. Answer: Draw two circles (or rectangles, etc.). Partition both of them into 4 equal parts. Shade all 4 parts for the first circle, shade in 3 parts for the second circle. Sec Ref: Conceptual Development of Fractions Page Ref: Page 267
39.
T or F One of the keys to helping children understand operations with fractions is to help them understand the concept of equivalence. Answer: T Sec Ref: Operations with Fractions Page Ref: Page 267
40.
T or F Multiplication of fractions can be thought of as repeated addition – just as in multiplying whole numbers. Answer: F Sec Ref: Operations with Fractions Page Ref: Page 267
41.
What are some differences between operations involving whole numbers and fractions? Answer: Multiplication of fractions cannot be thought of as repeated addition and when you multiply fractions, the product is always less than either fraction; when you divide a whole number by a fraction, the quotient will be larger than the whole number. Sec Ref: Operations with Fractions Page Ref: Page 268
42.
What general strategy should be used when teaching students how to add and subtract fractions? Answer: Begin with easy problems that involve joining and separating; also, encourage students to use models and diagrams. Sec Ref: Operations with Fractions Page Ref: Page 268
43.
Draw a model/diagram to illustrate how you would solve 2/3 + 1/4. Answer: Answers vary. Sec Ref: Operations with Fractions Page Ref: Page 269
44.
You have 3 jars, each with 2/3 of water. How much water do you have in all? Draw a picture/model to illustrate the solution to this problem. Answer: Draw three jars or boxes; each should be 2/3 full. The answer is 2 or 6/3. Sec Ref: Operations with Fractions Page Ref: Page 269
45.
When you add and subtract fractions, why must the denominators be the same? Answer: So that the parts that you are adding or subtracting are the same. Sec Ref: Operations with Fractions Page Ref: 269
46.
The shaded area in the following model illustrates what multiplication problem?
A. 3 x 1/4 B. 3/4 x 3 C. 3/4 x 3/4 D. 1/4 x 1/4 E. 3 x 3 Answer: C Sec Ref: Operations with Fractions Page Ref: 271
47.
The shaded area in the following model illustrates what multiplication problem?
A. 3 x 4 B. 3/5 x 4/5 C. 5/3 x 5/4 D. 3 x 4/5 E. 4 x 3/5 Answer: B Sec Ref: Operations with Fractions Page Ref: Page 271 48.
Why does the product get smaller when you take a fraction times a fraction? Answer: Because you are taking a part of a part. Sec Ref: Operations with Fractions Page Ref: Page 271
49.
The following is a pictorial model for a division problem. Which problem represents the model?
A. 1 ÷ 1/2 B. 2 ÷ 1/2 C. 3 ÷ 1/2 D. 4 ÷ 1/2 E. 5 ÷ 1/2 Answer: D Sec Ref: Operations with Fractions Page Ref: Page 272
50.
T or F To develop decimal concepts, you should relate decimals to fractions and place value. Answer: T Sec Ref: Conceptual Development of Decimals Page Ref: Page 272
51.
T or F Decimals are just another notation for fractions. Answer: T Sec Ref: Conceptual Development of Decimals Page Ref: Page 273
52.
The mathematically appropriate way to say 1.27 is: A. one hundred twenty-seven B. one point two seven C. one and twenty-seven hundred D. one point twenty-seven hundredths E. one and twenty-seven hundredths Answer: E Sec Ref: Conceptual Development of Decimals Page Ref: Page 273
53.
The place-value grid below contains a decimal. The decimal should be identified as:
A. B. C. D. E.
Tens
Ones
Tenths
Hundredths
4
3
0
7
43 and 70 hundredths 43 point zero 7 43 and 7 hundredths 43 and 7 hundred none of the above
Answer: C Sec Ref: Conceptual Development of Decimals Page Ref: Page 275
54.
Why is computation involving decimals easier than computation involving fractions? Answer: It is easier for decimals because the computation follows the same rules as for whole numbers. Sec Ref: Operations with Decimals Page Ref: Page 276
55.
The following is a pictorial model for a decimal problem. Which problem represents the model?
A. 23 x 23 B. 2.3 x 2.3 C. 1.3 x 1.3 D. 2.3 x 1.3 E. .23 x .23 Answer: B Sec Ref: Operations with Decimals Page Ref: Page 278 56.
Draw a pictorial model for the following decimal problem, 1.3 x 1.2. Answer:
Sec Ref: Operations with Decimals Page Ref: 278
57.
T or F The Chinese were one of the first cultures to make use of fractions. Answer: F Sec Ref: Cultural Connections Page Ref: Page 278
58.
T or F Some countries use the comma for the decimal point. Answer: T Sec Ref: Cultural Connections Page Ref: Page 279
59.
T or F According to your text, some countries say the denominator before they say the numerator. Answer: T Sec Ref: Cultural Connections Page Ref: Page 279
Chapter 13: Ratio, Proportion, and Percent: Meanings and Applications Possible Questions for Assessment and Evaluation 1.
What is a ratio? Give an example to support your answer. Answer: A ratio is a comparison between two or more things. Examples vary. Sec Ref: Ratios Page Ref: 284
2.
Jeremy’s teacher gave him four stickers for one good behavior note. Which of the following is not an appropriate way to express this situation? A. “four to one” B. 4 to 1 C. 4;1 D. 4:1 E. 4/1 Answer: C Sec Ref: Ratios Page Ref: 284
3.
A ratio is a ________ between two or more numbers in a given order. A. divisional relationship B. subtractive relationship C. additive relationship D. multiplicative relationship E. all of the above Answer: D Sec Ref: Ratios Page Ref: 285
4.
T or F Suppose you had 3 red marbles and 4 blue marbles. The following ratios are both acceptable ways of representing this relationship: 3:4 and 4:3. Answer: T Sec Ref: Ratios Page Ref: 285
5.
T or F A ratio is a multiplicative relationship between two or more numbers. Answer: T Sec Ref: Ratios Page Ref: 285
6.
Suppose you had a bracelet with 4 red beads followed by 7 green beads. What would be the part-to-part and part-to-whole ratios for this situation? Answer: Part-to-part: 4:7 or 7:4; part-to-whole: 4:11 or 7:11. Sec Ref: Ratios Page Ref: 286
7.
The following question was on an exam: Write the ratio for a classroom that has 7 boys and 4 girls. Jim wrote the following ratio: 7:11. However, his teacher said the correct answer is 7:4. Who’s correct in this situation – Jim or his teacher? Answer: Both are correct; Jim wrote the part-to-whole ratio while his teacher wrote the part-to-part ratio. Sec Ref: Ratios Page Ref: 286
8.
Suppose you had 6 boys and 7 girls in your class. Write the four possible ratios for this situation. Answer: 6:7, 7:6, 6:13, 7:13 Sec Ref: Ratios Page Ref: 286
9.
T or F When we combine ratios, it is the same process as adding fractions. Answer: F Sec Ref: Ratios Page Ref: 286
10.
Give an example to illustrate why combining ratios is not the same as adding fractions. Answer: Answers vary. Sec Ref: Ratios Page Ref: 287
11.
Mary has 3 white stickers and 4 red stickers. Jimmy has 4 white stickers and 6 red stickers. If they were to combine their stickers together, what would the ratio be? Answer: 7 white stickers to 10 red stickers Sec Ref: Ratios Page Ref: 287
12.
What is a proportion? Give an example to support your answer. Answer: A statement that two or more ratios are equal. Examples vary. Sec Ref: Proportions Page Ref: 287
13.
Which of the following ratios is not proportional to 3:4? A. 4:3 B. 6:8 C. 12:16 D. 15:20 E. 18:24 Answer: A Sec Ref: Proportions Page Ref: 296
14.
Find five ratios that are proportional to 2:3. Answer: Answers vary. Sec Ref: Proportions Page Ref: 287
15.
T or F According to your text, learning to add and subtract has been called the “capstone” of elementary school mathematics. Answer: F Sec Ref: Proportions Page Ref: 287
16.
T or F Proportional reasoning requires making comparisons using relative and not absolute terms. Answer: T Sec Ref: Proportions
Page Ref: 287 17.
Give an example to illustrate the idea of proportional reasoning. Answer: Examples vary. Sec Ref: Proportions Page Ref: 287
18.
Find the measure of the missing side. 4 2
X
6
Answer: X = 12 Sec Ref: Proportions Page Ref: 287 19.
T or F Ratios always involve like quantities – such as 3 beads to 4 beads. Answer: F Sec Ref: Proportions Page Ref: 288
20.
Mary went to one store and noticed the following price sticker: 5 candies for 10 cents. She then went to another store and noticed the following price sticker for the same candies: 21 candies for 63 cents. This is an example of which kind of proportional problem? A. numerical-comparison B. missing-number C. scaling Answer: A Sec Ref: Proportions Page Ref: 289
21.
Suppose 3 people need 500 square feet to live comfortably. How many square feet would 7 people need? This is an example of which kind of proportional problem? A. numerical-comparison B. missing-number C. scaling Answer: B Sec Ref: Proportions Page Ref: 289
22.
Give an example to illustrate each of the following types of proportional problems: numerical-comparison, missing-number, and scaling. Answer: Examples vary. Sec Ref: Proportions Page Ref: 289
23.
Which of the following are examples of real-life situations that could be used to help students understand proportions. A. recipes B. maps C. scale drawings D. prices E. all of the above Answer: E Sec Ref: Proportions Page Ref: 290
24.
T or F The understanding of percent requires no new skills or concepts beyond those used in mastering fractions, decimals, ratios, and proportions. Answer: T Sec Ref: Percents Page Ref: 292
25.
T or F The topic of percent is not really a mathematical topic, but rather the application of a particular type of notational system. Answer: T Sec Ref: Percents Page Ref: 292
26.
If percent is not really a mathematical topic, then why study it?
27.
Answer: Because of its social utility. Page Ref: 292 Percents are special ratios based on: A. 1 B. 25 C. 50 D. 75 E. 100 Answer: E Sec Ref: Percents Page Ref: 292
28.
The ratio equivalent of 33% is: A. .33 B. 33:100 C. 100:33 D. 33 E. None of the above Answer: B Sec Ref: Percents Page Ref: 292
29.
T or F According to your text, percent should only be introduced after students thoroughly understand fractions and decimals. Answer: T Sec Ref: Percents Page Ref: 292
30.
All of the following are common percent benchmarks, except for: A. 10% B. 15% C. 25% D. 33% E. 50% Answer: B Sec Ref: Percents
Page Ref: 292 31.
The shaded area in the following grid shows what percent?
A. 10% B. 30% C. 50% D. 70% E. 90% Answer: D Sec Ref: Percents Page Ref: 293 32.
Discuss five models that you can use with your students to develop their understanding of percent. Answer: Should include the following: money, rectangular grids, percent bars, meter sticks, fraction circles. Sec Ref: Percents Page Ref: 293
33.
Which of the following is not a correct way to express “78 percent”? A. 78% B. 78/100 C. .78 D. 0.78 E. 7.8 Answer: E Sec Ref: Percents Page Ref: 293
34.
Write the percent, fraction, and decimal notations for the following diagram:
Answer: 53%, 53/100, .53 Sec Ref: Percents Page Ref: 293 35.
T or F One particular troublesome aspect of percents involves small percents between 0% and 1%. Answer: T Sec Ref: Percents Page Ref: 294
36.
Use a grid to represent 223%. Answer:
37.
Sec Ref: Percents Page Ref: 294 According to your text, there are only how many basic types of percent problems? A. 1
B. 2 C. 3 D. 4 E. 5 Answer: C Sec Ref: Percents Page Ref: 294 38.
Give an example to illustrate the equation and ratio methods for solving percent problems. Answer: Examples vary. Sec Ref: Percents Page Ref: 295
39.
Melissa’s team won 12 of their 17 games. This is an example of which type of percent problem? A. finding the percent of a given number B. finding what percent one number is of another number C. find the total when only a percent is known Answer: B Sec Ref: Percents Page Ref: 295
40.
Mark’s team won 25% of their 20 baseball games. How many games did his team win? This is an example of which type of percent problem? A. finding the percent of a given number B. finding what percent one number is of another number C. find the total when only a percent is known Answer: A Sec Ref: Percents Page Ref: 295
41.
The price on a pair of pants was $25 and it was marked down 15%. What was its original price? This is an example of which type of percent problem? A. finding the percent of a given number
B. finding what percent one number is of another number C. find the total when only a percent is known Answer: C Sec Ref: Percents Page Ref: 296 42.
Use both the equation and ratio methods, along with an informal method to solve the following problem: Mr. Matthews has 20 pieces of candy. He wants to give 25% of the candies to his first hour class. How many candies does this represent? Answer: Equation - .25 x 20 = 5; ratio – 25%/100% = x/20; informal – 25% is 1/4, therefore, 1/4 of 20 is 5. Sec Ref: Percents Page Ref: 296
Chapter 14: Algebraic Thinking Possible Questions for Assessment and Evaluation 1.
T or F Research has shown that children should not be introduced to algebra until they are in high school. Answer: F Sec Ref: Introduction Page Ref: 300
2.
What is algebra? Answer: Should include the following ideas: a study of patterns and relationships, a way of thinking, an art, characterized by order and internal consistency, a language that uses carefully defined terms and symbols, a tool. Sec Ref: Introduction Page Ref: 301
3.
What three topics are important building blocks for developing algebraic understanding? Answer: Problems, patterns, and relations. Sec Ref: Problems, Patterns, and Relations Page Ref: 302
4
What is the difference between a repeating pattern and a growing pattern? Answer: A repeating pattern has a core element that is repeated over and over; a growing pattern changes by the same amount from the preceding term. Sec Ref: Problems, Patterns and Relations Page Ref: 303
5.
What is the core repeating element in the following pattern?
A. B. C. D. E.
circle, rectangle circle, rectangle, circle circle, rectangle, circle, rectangle circle, rectangle, circle, rectangle, rectangle none of the above
Answer: D Sec Ref: Problems, Patterns, and Relations Page Ref: 303
6.
What shape should be inserted in the space in the following pattern?
Answer: Sec Ref: Problems, Patterns, and Relations Page Ref: 303 7.
T or F According to your text, children should begin recognizing and thinking about patterns using numbers. Answer: F Sec Ref: Problems, Patterns, and Relations Page Ref: 303
8.
The core in the following pattern contains how many elements?
A. 1 B. 2 C. 3 D. 4 E. 5 Answer: D Sec Ref: Problems, Patterns, and Relations Page Ref: 304 9.
Is the following pattern an example of a repeating pattern or a growing pattern? 1, 1, 2, 3, 5, 8, 13, 21… A. repeating pattern B. growing pattern Answer: Growing pattern. Sec Ref: Problems, Patterns, and Relations Page Ref: 304
10.
T or F The equal sign can be interpreted as “same as”. Answer: T Sec Ref: Problems, Patterns, and Relations Page Ref: 305
11.
What is a function and give an example to illustrate it. Answer: A function is a way of expressing the relationship between two things. Examples vary. Sec Ref: Problems, Patterns, and Relations Page Ref: 306
12.
Complete the following function table.
Bicycles Number of Wheels
1 2
5
3
6 12
2
4
Answer: 5=10, 3=6, 2=4, 4=8 Sec Ref: Problems, Patterns, and Relations Page Ref: 306 13.
Determine the rule for the following function table. Input Out
3 7
7 19
12 34
14 40
16 46
20 58
Answer: Multiply the input by 3 and subtract 2. Sec Ref: Problems, Patterns, and Relations Page Ref: 306 14.
T or F The equal sign means “get an answer”. Answer: F Sec Ref: Problems, Patterns, and Relations Page Ref: 308
15.
On a math test, Joan did the following: 3 + 6 = 9 + 2. Why might she have written a 9 instead of 7? Answer: She probably does not understand the equal sign as meaning “same as”; instead, she interpreted the equal sign to mean calculate the answer. Sec Ref: Language and Symbols of Algebra Page Ref: 309
16.
Summarize the three uses of variables. Give an example to illustrate each use. Answer: The three uses of variables are as a placeholder, for generalization, and used in formulas and functions. Examples vary. Sec Ref: Language and Symbols of Algebra Page Ref: 10
17.
The following equation represents which use of variables? 13 – b = 8 A. placeholder B. generalization C. formulas and functions Answer: A Sec Ref: Language and Symbols of Algebra Page Ref: 309
18.
The following equation represents which use of variables? A = πr2
A. placeholder B. generalization C. formulas and functions Answer: C Sec Ref: Language and Symbols of Algebra Page Ref: 309 19.
The following equation represents which use of variables? a + b = b + a A. placeholder B. generalization C. formulas and functions Answer: B Sec Ref: Language and Symbols of Algebra Page Ref: 309
20.
What is the difference between an expression and an equation? Answer: Equations have the equal sign; expressions don’t. Sec Ref: Language and Symbols of Algebra Page Ref: 309
21.
T or F When you have an open sentence, there is only one number that will satisfy the sentence. Answer: F Sec Ref: Language and Symbols of Algebra Page Ref: 310
22.
Which of the following is the appropriate sequence to model algebraic problems? A. B. C. D. E.
concrete, abstract, pictorial abstract, pictorial, concrete concrete, pictorial, abstract abstract, concrete, pictorial pictorial, concrete, abstract
Answer: C Sec Ref: Representing, Generalizing, and Justifying Page Ref: 310 23.
For the following problem, describe/illustrate what you would do to help students model it concretely, pictorially, and abstractly. “Jenny had 12 books. She gave some to her friend. Now she has 5 books. How many books did she give to her friend?” Answer: Concretely – start out with 12 actual books and take away until you have 5 left, you gave away seven so the answer would be seven; pictorially – use pictures of books or boxes representing books and go through the process mentioned previously; abstractly – have students learn to write 12 = 5. Sec Ref: Representing, Generalizing, and Justifying Page Ref: 311
24.
Write a story problem for the following open sentence: 22 -
= 13
Answer: Answers vary. Sec Ref: Representing, Generalizing, and Justifying Page Ref: 311 25.
T or F Even when students have given you the correct answer, you should ask them to justify their answer. Answer: T Sec Ref: Representing, Generalizing, and Justifying Page Ref: 312
26.
What is the difference between a recursive expression and an explicit equation? Answer: A recursive expression allows you to find the value of a term given the value of the previous term; an explicit equation lets you calculate the value of one term given the number of the term. Sec Ref: Representing, Generalizing, and Justifying Page Ref: 312
27.
T or F A recursive expression is more powerful than an explicit equation. Answer: F Sec Ref: Representing, Generalizing, and Justifying
Page Ref: 312 28.
Is the following statement an example of a recursive expression or an explicit equation? “If n is the value of one term, then n + 3 is the value of the next term.” A. recursive B. explicit Answer: A Sec Ref: Representing, Generalizing, and Justifying Page Ref: 312
29.
Is the following an example of a recursive expression or an explicit equation? N=2n + 1 A. recursive B. explicit Answer: B Sec Ref: Representing, Generalizing, and Justifying Page Ref: 312-313
30.
T or F In order to prove that a conjecture is not true for all numbers, you only need one example. Answer: T Sec Ref: Representing, Generalizing, and Justifying Page Ref: 317
31.
Use models to illustrate the commutative and distributive properties. Answer: Commutative – use an array (such as 3 x 4), to show the property, just flip the array on its side and you should have 4 x 3; distributive- use an array (such as 3 x 7), to show the property, break up the 7 into 3 and 4 (or any combination leading to 7). Sec Ref: Representing, Generalizing, and Justifying Page Ref: 317-318
32.
Use a grid or an array to represent the following equation: 3 x 5 = (3 x 2) + (3 x 3) Answer:
=
+
Sec Ref: Representing, Generalizing, and Justifying Page Ref: 318
33.
In order to help students make generalizations, the following instructional sequence should be followed: A. B. C. D. E.
closed sentence, open sentence, generalizations open sentence, closed sentence, generalizations generalizations, closed sentence, open sentence generalizations, open sentence, closed sentence all of the above
Answer: A Sec Ref: Representing, Generalizing, and Justifying Page Ref: 318 34.
Elizabeth justified her generalization that whenever you add 0 to a number, you always end up with that same number by stating, “it works because my teacher told me so.” This is an example of which kind of justification? A. appeal to authority B. justification by example C. deductive arguments Answer: A Sec Ref: Representing, Generalizing, and Justifying Page Ref: 318
35.
Martinez justified his generalization that whenever you subtract zero from any number, you always end up with that number by stating, “if I subtract 0 from 20, I end up with 20.” This is an example of which kind of justification? A. appeal to authority B. justification by example C. deductive arguments Answer: B Sec Ref: Representing, Generalizing, and Justifying Page Ref: 318
36.
Summarize the three types of justifications and give examples to illustrate each type. Answer: Appeal to authority, justification by example, and deductive arguments. Examples vary. Sec Ref: Representing, Generalizing, and Justifying Page Ref: 318
37.
T or F The word algebra comes from the Latin variant of an Arab word meaning mathematics. Answer: F Sec Ref: Cultural Connections Page Ref: 319
Chapter 15: Geometry Possible Questions for Assessment and Evaluation 1.
Discuss three reasons why it’s important to teach geometry. Answer: Should include three of the following: it’s a different topic that engages children differently both in performance and persistence, it is a natural topic for including other skills, children can make and verify conjectures about geometric figures, and it’s a topic that will help a teacher teach many other mathematical topics. Sec Ref: Introduction Page Ref: 2
2.
T or F According to the NCTM, geometry should not be introduced to students until they are in middle school. Answer: F Sec Ref: Introduction Page Ref: 323
3.
T or F Geometry is seen as the second most important area in pre-K-2 and in grades 3-5. Answer: T Sec Ref: Introduction Page Ref: 323-324
4.
Geometry is seen as the second most important area in which grade span? A. 3-5 B. 6-8 C. 9-11 D. 12 E. all of the above Answer: A Sec Ref: Introduction Page Ref: 323-324
5.
Explain the van Hiele levels of geometric thought. Answer: Five levels (0-4); level 0 – children focus on some visual cues; level 1 – children view a geometric shape as a whole; level 2 – children focus on the relationships between parts of a shape and defining attributes; level 3 – children establish relationships among properties and among figures; level 4 – children use deduction to prove statements.
Sec Ref: Shapes Page Ref: 325 6.
A student looks at a triangle and says: “It looks like my roof.” The statement, “it looks like my roof” indicates that this student is at what van Hiele level of geometric thought? A. 0 B. 1 C. 2 D. 3 E. 4 Answer: B Sec Ref: Shapes Page Ref: 325
7.
If a student was at level 2 of the van Hiele levels of geometric thought, what would she say about a triangle? Answer: Possible answer – it has three sides and three angles. Sec Ref: Shapes Page Ref: 325
8.
T or F Even though a ball may be hollow, it is still referred to as a solid. Answer: T Sec Ref: Shapes Page Ref: 325
9.
Suggest two activities that could be done to develop students’ abilities to describe and sort. Answer: Answers vary. Sec Ref: Shapes Page Ref: 326-327
10.
T or F The vertex is a point at which three or more edges come together. Answer: T Sec Ref: Shapes Page Ref: 326
11.
Determine the number of faces, edges, and vertices in the following shape.
Answer: Faces – 6; edges – 12; vertices 8. Sec Ref: Shapes Page Ref: 326 12.
Determine the number of faces, edges, and vertices in the following shape.
Answer: Faces – 5; edges – 9; vertices – 6 Sec Ref: Shapes Page Ref: 326 13.
What are face and edge models? What do they emphasize? Answer: Face models are construction paper tubes and polyhedral shape and they emphasize the faces while edge models are models using sticks (i.e. toothpicks) that emphasize the edges. Sec Ref: Shapes Page Ref: 328
14.
T or F Any solid whose faces are all polygons is a polyhedron. Answer: T Sec Ref: Shapes Page Ref: 328
15.
T or F A sphere is an example of a polyhedron. Answer: F Sec Ref: Shapes Page Ref: 328
16.
Euler’s formula is a relationship between which of the following three attributes? (Circle three) A. Area B. Edge C. Vertex D. Perimeter E. Face Answer: B, C, E Sec Ref: Shapes Page Ref: 328
17.
T or F Children should begin to recognize types of shapes through formal definitions. Answer: F Sec Ref: Shapes Page Ref: 329
18.
One of the first properties that children focus on is: A. number of sides B. symmetry C. sizes of angles D. parallel and perpendicular lines E. altitude Answer: A Sec Ref: Shapes Page Ref: 331
19.
Students are making heart-shaped cards for Valentine’s Day. One student notices that if she were to fold the heart in half, the two sides match. Which of the following properties would this be a teachable moment for? A. number of sides and corners B. symmetry C. sizes of angles D. parallel and perpendicular lines E. altitude Answer: B Sec Ref: Shapes Page Ref: 332
20.
The butterfly below has which kind of symmetry?
A. Line B. Rotational Answer: A Sec Ref: Shapes Page Ref: 332 21.
How many lines of symmetry does a circle have? Answer: An infinite number Sec Ref: Shapes Page Ref: 333
22.
Students are examining features of objects in the room, such as the opposite sides of a book, the top and bottom of a chalkboard, and the edge of the wall that touches the edge of the floor. Which property of shape is the focus? A. number of sides and corners B. symmetry C. sizes of angles D. parallel and perpendicular lines E. altitude Answer: D Sec Ref: Shapes Page Ref: 334
23.
T or F A convex shape is a polygon with all angles less than 180 degrees. Answer: T Sec Ref: Shapes Page Ref: 334-335
24.
T or F The altitude (or height) of a geometric shape depends on what is specified as the base. Answer: T Sec Ref: Shapes Page Ref: 335
25.
Give an example to illustrate why the altitude of a geometric shape depends on what is specified as the base. Answer: Answers vary. Sec Ref: Shapes Page Ref: 335
26.
Common names for quadrilaterals include all of the following, except: A. squares B. rectangles C. triangles D. trapezoids E. parallelograms Answer: C Sec Ref: Shapes Page Ref: 336
27.
T or F Quadrilaterals may fit into several categories. Answer: T Sec Ref: Shapes Page Ref: 336
28.
T or F Pentagons are named according to the number of sides. Answer: T Sec Ref: Shapes Page Ref: 337
29.
T or F The polygon below is a hexagon.
Answer: T Sec Ref: Shapes Page Ref: 337 30.
Describe an activity that could be done to develop children’s abilities to find locations and places. Answer: Ways to do this include using maps, coordinate grids, and number lines. Sec Ref: Space Page Ref: 338-339
31.
Describe the three types of transformations. Answer: Translations (slides), rotations (turns), and reflections (flips). Sec Ref: Transformations Page Ref: 340
32.
What transformation has occurred in the following diagram?
A. translation B. rotation C. reflection Answer: C Sec Ref: Transformations Page Ref: 340
33.
What transformation has occurred in the following diagram?
A. translation B. rotation C. reflection Answer: B Sec Ref: Transformations Page Ref: 340 34.
What transformation has occurred in the following diagram?
A. translation B. rotation C. reflection Answer: A Sec Ref: Transformations Page Ref: 340 35.
T or F Students develop a deeper understanding of transformations if they are asked to make conjectures and justify their thinking. Answer: T Sec Ref: Transformations Page Ref: 341
36.
T or F If two shapes have the same area, then they are also congruent. Answer: F Sec Ref: Transformations Page Ref: 341
37.
What is the difference between congruence and similarity? Answer: Congruence – shapes are the same; similarity – shapes are similar. Sec Ref: Transformations Page Ref: 341
38.
T or F The following two shapes are similar. 3
4
5
4
Answer: F Sec Ref: Transformations Page Ref: 341 39.
Describe an activity that could be done to develop students’ abilities to visualize and reason geometrically. Answer: Answers vary. Sec Ref: Visualization and Spatial Reasoning Page Ref: 342
40.
T or F Research shows that the use of physical materials has not been effective in developing geometric representations. Answer: F Sec Ref: Visualization and Spatial Reasoning Page Ref: 342
Chapter 16: Measurement Possible Questions for Assessment and Evaluation 1.
Your text gives several reasons for teaching measurement. Which of the following is not one of those reasons? A. application to students’ daily lives B. helps students to compute answers quickly and efficiently C. helps students learn about other topics in mathematics D. measurement is an effective way to engage many types of students E. measurement is useful in other areas of the curriculum Answer: B Sec Ref: Introduction Page Ref: 348
2.
Your text gives four reasons for teaching measurement. Discuss three of those four reasons. Answer: Should include three of the following: application to students’ daily lives, measurement uses many other topics in mathematics, measurement engages many types of students, useful in other areas of the curriculum. Sec Ref: Introduction Page Ref: 348
3.
T or F Research shows that U.S. children perform better in measurement than their counterparts in other countries. Answer: F Sec Ref: Introduction Page Ref: 348
4.
Wilson and Osborne recommend four things that should be done to help develop students’ measurement skills. What are those recommendations? Answer: Children must measure frequently and often; children must develop estimation skills with measurement; children should encounter activity-oriented measurement situations by doing and experimenting rather than by passively observing; instructional planning should emphasize the important ideas of measurement that transfer or work across measurement systems. Sec Ref: The Measurement Process Page Ref: 349
5.
Which of the following is not a recommendation of Wilson and Osborne for the teaching of measurement? A. Children must master computation before they can measure B. Children must measure frequently and often C. Children must develop estimation skills with measurement D. Children should encounter activity-oriented measurement situations E. Children must learn the important ideas that transfer across measurement systems Answer: A Sec Ref: The Measurement Process Page Ref: 349
6.
What does it mean to measure? Answer: The process by which a number is assigned to an attribute of an object or event. Sec Ref: The Measurement Process Page Ref: 349
7.
Which of the following are common attributes that are measured in most elementary mathematics programs? A. length B. weight/mass C. temperature D. all of the above E. A and B only Answer: D Sec Ref: The Measurement Process Page Ref: 349
8.
Discuss the five steps involved in the measuring process. Answer: Identify the attribute by comparing objects; choose a unit; compare the object to the unit by iterating the unit; find the number of units; report the number of units. Sec Ref: The Measurement Process Page Ref: 349
9.
T or F Nonstandard units should only be introduced after students have experience working with standard units. Answer: F
Sec Ref: The Measurement Process Page Ref: 350 10.
In order to understand measurable attributes, your text suggests making three types of comparisons. Discuss these three types and give examples of each. Answer: Comparing two objects perceptually (they look the same or they look different); comparing two objects directly (they are placed next to each other); comparing two objects indirectly (a third object is used to compare objects). Examples vary. Sec Ref: The Measurement Process Page Ref: 350
11.
Of the following attributes, which one is most easily perceived? A. length B. capacity C. weight D. area E. time Answer: A Sec Ref: The Measurement Process Page Ref: 351
12.
Mary and Jim are trying to compare the widths of a window and door. In order to do so, they use strings to measure the widths. They then compare the string lengths to see which one is wider. This is an example of which type of comparison? A. comparing two objects perceptually B. comparing two objects directly C. comparing two objects indirectly Answer: C Sec Ref: The Measurement Process Page Ref: 351
13.
Julie has two crayons. She puts them next to each other and proclaims, “the red one is longer than the blue one”. This is an example of which type of comparison? A. comparing two objects perceptually B. comparing two objects directly C. comparing two objects indirectly Answer: B Sec Ref: The Measurement Process Page Ref: 351
14.
Mark looks out the classroom window and notices two cars. After a few seconds, he makes the following statement, “the two cars are about the same height.” This is an example of which type of comparison? A. comparing two objects perceptually B. comparing two objects directly C. comparing two objects indirectly Answer: A Sec Ref: The Measurement Process Page Ref: 531
15.
Why is it harder for children to understand distance than the length of a long, thin object? Answer: Because you have to imagine the straight path between the two endpoints. Sec Ref: The Measurement Process Page Ref: 532
16.
T or F The perimeter of an object is a special type of length. Answer: T Sec Ref: The Measurement Process Page Ref: 532
17.
Compare and contrast perimeter and circumference. Answer: They both measure the distance around an object. Circumference is a special word for the distance around a circle. Sec Ref: The Measurement Process Page Ref: 532
18.
T or F Capacity is the attribute that tells how much a two-dimensional container can hold. Answer: F Sec Ref: The Measurement Process Page Ref: 533
19.
Two elementary students are trying to determine which of two jars hold more sand. In order to determine which holds more, they poor the sand from one container to the other. This is an example of which type of comparison? A. comparing two objects perceptually B. comparing two objects directly C. comparing two objects indirectly Answer: B Sec Ref: The Measurement Process Page Ref: 533
20.
Jack, a Kindergartener, looks at the following two containers and states that container A has more liquid. Explain why Jack would think container A has more liquid.
A
B
Answer: Jack is focused on the height. Sec Ref: The Measurement Process Page Ref: 533 21.
T or F Weight and mass mean the same thing and can be used interchangeably. Answer: F Sec Ref: The Measurement Process Page Ref: 533
22.
Dinah puts an orange in her right hand and an apple in her left. She then lifts them both up and proclaims that the orange is heavier than the apple. This is an example of which type of comparison? A. comparing two objects perceptually B. comparing two objects directly C. comparing two objects indirectly Answer: A Sec Ref: The Measurement Process Page Ref: 533
23.
When comparing the weight of two objects, young children often think the larger object weighs more. One way to help them understand that this is not always the case, you should have students: A. compare small and heavy with large and light objects B. weigh the objects in a pan balance scale C. look carefully at the two objects D. all of the above E. both A and B Answer: E Sec Ref: The Measurement Process Page Ref: 534
24.
What is meant by the concept of conservation of area? Answer: A region can be cut and rearranged without changing the area. Sec Ref: The Measurement Process Page Ref: 534
25.
T or F A plane region’s area remains constant when cut or rearranged. Answer: T Sec Ref: The Measurement Process Page Ref: 534
26.
Students in Mrs. Jones class are trying to determine how much paper they will need to cover a bulletin board. Which of the following attributes will they need to know to determine how much paper is needed? A. capacity B. weight C. volume D. mass E. area Answer: E Sec Ref: The Measurement Process Page Ref: 354
27.
T or F According to your text, volume should receive little formal attention until fifth grade. Answer: T Sec Ref: The Measurement Process Page Ref: 355
28.
Michael has two objects in his hands. After a few seconds, he proclaims that the object in his right hand is warmer than the object in his left hand. This is an example of which type of comparison? A. comparing two objects perceptually B. comparing two objects directly C. comparing two objects indirectly Answer: A Sec Ref: The Measurement Process Page Ref: 355
29.
Name the two attributes of events that can measured. Answer: Time of occurrence and length of duration. Sec Ref: The Measurement Process Page Ref: 355
30.
Which of the following is an example of a nonstandard unit of measure? A. inch B. acre C. book edge D. centimeter E. gallon Answer: C Sec Ref: The Measurement Process Page Ref: 356
31.
Explain the nine measurement concepts. Answer: A unit must remain constant; a measurement must include both a number and the unit; two measurements may be easily compared if the same unit is used; one unit may be more appropriate than another to measure an object; there is an inverse relationship between the number of units and the size of the unit; standard units are needed to communicate effectively; a smaller unit gives a more exact measurement; units may be combined or subdivided to make other units; units must match the attribute that is being measured. Sec Ref: The Measurement Process Page Ref: 356
32.
After measuring the length of his book, Jim says: “It is 6.” Which measurement concept is Jim violating? A. a unit must remain constant B. a measurement must include both a number and the unit C. two measurements may be easily compared if the same unit is used D. one unit may be more appropriate than another to measure an object E. standard units are needed to communicate effectively Answer: B Sec Ref: The Measurement Process Page Ref: 356
33.
In order to develop students’ understanding of the measurement concepts, Mr. Chang asks the following question before asking his students to measure the distance between their classroom and the gym: “Should we use a paper clip or a yard stick to measure the distance between our classroom and the gym?” This illustrates which of the following measurement concepts? A. a unit must remain constant B. a measurement must include both a number and the unit C. two measurements may be easily compared if the same unit is used D. one unit may be more appropriate than another to measure an object E. standard units are needed to communicate effectively Answer: D Sec Ref: The Measurement Process Page Ref: 356
34.
Jenny measures the length of her book by using a paper clip, three buttons, and five stickers. She then reports that her book is 9. Which measurement concept is Jenny not understanding? A. a unit must remain constant B. a measurement must include both a number and the unit C. two measurements may be easily compared if the same unit is used D. one unit may be more appropriate than another to measure an object E. a smaller unit gives a more exact measurement Answer: A Sec Ref: The Measurement Process Page Ref: 356
35.
What are standard units and why are they important? Answer: Standard units are either customary (e.g., inch, pint, pound) or metric (e.g., meter, liter, gram). They are important for communication purposes. Sec Ref: The Measurement Process Page Ref: 357
36.
T or F According to recent national assessments, children have a better understanding of metric units. Answer: F Sec Ref: The Measurement Process Page Ref: 357
37.
A child measures the top of her desk and the class book case with paper clips and with pencils. In each case, she notices that it takes more paper clips than pencils to measure. Which of the measurement concepts is she demonstrating? A. a measurement must include both a number and the unit B. two measurements may be easily compared if the same unit is used C. there is an inverse relationship between the number of units and the size of the unit D. standard units are needed to communicate effectively E. units may be combined or subdivided to make other units Answer: C Sec Ref: The Measurement Process Page Ref: 357
38.
Discuss and give examples of the three techniques that are used to find the number of units. Answer: Counting units, using an instrument, formulas. Examples vary. Sec Ref: The Measurement Process Page Ref: 357
39.
T or F Beginning activities for counting units should be with multiple copies of a nonstandard unit that are put end to end long enough to represent the object students are measuring. Answer: T Sec Ref: The Measurement Process Page Ref: 357
40.
List six common measurement instruments used in elementary schools. Answer: Rulers, scales, graduated containers, thermometers, protractors, and clocks. Sec Ref: The Measurement Process Page Ref: 358
41.
What are some difficulties that students have with using rulers? Answer: Not knowing what unit they are using; lining up the ruler properly; measuring something that is longer than the ruler; difficulty in measuring to the nearest fourth, eight, or sixteenth. Sec Ref: The Measurement Process Page Ref: 358-359
42.
When teaching time with ordinary dial clocks, what are some difficulties that you as the classroom teacher should keep in mind? Answer: Two or more ways to read the scale and the hands move in a circular, not linear, fashion. Sec Ref: The Measurement Process Page Ref: 360
43.
T or F It is easier for children to use a digital clock than a dial clock to solve problems involving elapsed time. Answer: F Sec Ref: The Measurement Process Page Ref: 360
44.
Which area formula is often the first formula children encounter? A. triangle B. rectangle C. parallelogram D. pentagon E. hexagon Answer: B Sec Ref: The Measurement Process Page Ref: 361
45.
Why is the area formula for a parallelogram similar to the area formula for a rectangle? Answer: If you were to cut a triangle out from one of the sides of a parallelogram and attach it to the other side, you would form a rectangle. Sec Ref: The Measurement Process Page Ref: 362
46.
Why is the area formula for a triangle half the area formula for a rectangle? Answer: If you put two triangles together, you form a rectangle. Sec Ref: The Measurement Process Page Ref: 362
47.
1
Explain why the formula for a trapezoid is, 𝐴 = 2 [𝑎 𝑥 (𝑏 + 𝐵)]. Answer: Two congruent trapezoids can be put together to make a parallelogram.
Sec Ref: The Measurement Process Page Ref: 362 48.
T or F If students can measure a certain line segment, then they will also be able to draw that line segment as well. Answer: F Sec Ref: The Measurement Process Page Ref: 363
49.
Explain why conversion in the metric system is easier than in the customary system. Answer: The metric system uses prefixes and is based on 10. Sec Ref: Comparing Measurements Page Ref: 363
50.
Discuss and give examples to illustrate the two types of measurement estimations. Answer: First – the attribute and object are named and the measurement is unknown; two – the measurement is known and the object is to be chosen. Examples vary. Sec Ref: Estimating Measurements Page Ref: 365
51.
Mark measures himself and finds out that he is 3 feet 6 inches tall. Using this knowledge, he thinks that the table in the classroom must be about 4 feet since it is a little longer than he is tall. Which of the following estimation strategies is Mark using? A. compare to a referent B. chunking C. unitizing D. compensation E. clustering Answer: A Sec Ref: Estimating Measurements Page Ref: 365
52.
Yesterday Maria went from her house to her friend’s house and the park and then came back home. She estimates that it must have been about six miles total since it’s about a mile from her house to her friend’s house and then about two miles from there to the park. Which of the following estimation strategies is Maria using? A. compare to a referent B. chunking C. unitizing D. compensation E. clustering Answer: B Sec Ref: Estimating Measurements Page Ref: 365
53.
Suggest two activities that could be done to connect the attributes. Answer: Answers vary. Sec Ref: Connecting Attributes Page Ref: 367
Chapter 17: Data Analysis, Statistics, and Probability Possible Questions for Assessment and Evaluation 1.
Data analysis, statistics, and probability is an important topic because: A. it provides a meaningful context for promoting problem solving B. it enhances communication C. it develops number sense D. it Involves computations E. all of the above Answer: E Sec Ref: Introduction Page Ref: 373
2.
Discuss four reasons why data analysis, statistics, and probability should be studied. Answer: Children encounter ideas of statistics and probability outside of school every day; data analysis, statistics, and probability provide connections to other mathematics topics or school subjects; data analysis, statistics, and probability provide opportunities for computational activity in a meaningful context; data analysis, statistics, and probability provide opportunities for developing critical thinking skills. Sec Ref: Introduction Page Ref: 373-374
3.
What steps are involved in a statistical investigation? Answer: Formulate a question, collect data, analyze data, and interpret data. Sec Ref: Introduction Page Ref: 375
4.
Which of the following has the steps involved in a statistical investigation in the correct order? A. Formulate question, analyze data, collect data, interpret results B. Interpret results, formulate question, collect data, analyze data C. Formulate question, collect data, analyze data, interpret results D. Collect data, analyze data, interpret data, formula question E. None of the above Answer: C Sec Ref: Introduction Page Ref: 375
5.
If you were teaching at the elementary level, suggest two possible questions that would lead to a good statistical investigation. Answer: Answers vary. Sec Ref: Formulating Questions Page Ref: 375
6.
T or F When posing questions, the answer should immediately be obvious. Answer: F Sec Ref: Formulating Questions Page Ref: 375
7.
When posing questions, which of the following things should be taken into consideration? A. answer is not immediately obvious B. students identify their own questions C. questions should be relevant D. all of the above E. only B and C Answer: D Sec Ref: Formulating Questions Page Ref: 375
8.
Data may be collected from surveys, experiments, and simulations. Write a question that would be suitable to each type. Answer: Answers vary. Sec Ref: Collecting Data Page Ref: 376
9.
Critique the following survey: Mr. Thompson’s students are collecting data on students’ soda preferences. In order to collect the data, they go and talk to students who are on the football team. Answer: The data may be biased because they are only talking to athletes. Sec Ref: Collecting Data Page Ref: 376
10.
Students in Mrs. Abrams class wants to collect data to answer the following question, “What color do students like the most?” In order to collect the appropriate data, which of the following techniques should be used? A. Survey B. Experiment C. Simulation Answer: A Sec Ref: Collecting Data Page Ref: 376
11.
Students in Mrs. Abrams class wants to collect data to answer the following question, “What brand of light bulb lasts the longest?” In order to collect the appropriate data, which of the following techniques should be used? A. Survey B. Experiment C. Simulation Answer: B Sec Ref: Collecting Data Page Ref: 376
12.
Students in Mrs. Abrams class wants to collect data to answer the following question, “If we were to flip a coin 50 times, how many heads and tails will we get?” In order to collect the appropriate data, which of the following techniques should be used? A. Survey B. Experiment C. Simulation Answer: C Sec Ref: Collecting Data Page Ref: 376
13.
T or F In statistics, a sample is a subset of a population. Answer: T Sec Ref: Collecting Data Page Ref: 376
14.
Discuss the relationship between the population and a sample. Give an example to support your answer.
Answer: Population is the whole group you are studying; sample is a subset of the population. Examples vary. Sec Ref: Collecting Data Page Ref: 376 15.
T or F Knowledge in graph construction should not begin until students are in middle school. Answer: F Sec Ref: Analyzing Data: Graphical Organization Page Ref: 377
16.
Discuss how you would help students move from concrete to symbolic graphs. Answer: Start out with concrete objects, then use pictures, then use symbols. Sec Ref: Analyzing Data: Graphical Organization Page Ref: 377
17.
What are “real graphs”? Give an example of one. Answer: Graphs in which real, concrete objects are used. Examples vary. Sec Ref: Analyzing Data: Graphical Organization Page Ref: 377
18.
What are “sketch graphs” and give an example of one. Answer: Sketch graphs don’t require labels or titles and don’t require time-consuming attention to construction. They may be made with concrete materials or with paper and pencil. Examples vary. Sec Ref: Analyzing Data: Graphical Organization Page Ref: 378
19.
Use the following data to make a stem-and-leaf plot: 13, 15, 14, 12, 16, 18, 17, 13, 21, 22, 24, 23. Answer: 2 1
124 35426873
Sec Ref: Analyzing Data: Graphical Organization Page Ref: 379
20.
All of the following are components of a box plot, except: A. minimum B. maximum C. mode D. lower quartile E. whiskers Answer: C Sec Ref: Analyzing Data: Graphical Organization Page Ref: 380
21.
What kind of graph is used mostly for discrete, or separate and distinct, data? A. picture graph B. line graph C. pie graph D. bar graph E. all of the above Answer: D Sec Ref: Analyzing Data: Graphical Organization Page Ref: 380
22.
Students in Mr. Lee’s first-grade class have collected data regarding students’ favorite colors. Which of the following types of graphs would be most appropriate to represent the data? A. line graph B. bar graph C. stem-and-leaf plot D. box-and-whisker plot E. histogram Answer: B Sec Ref: Analyzing Data: Graphical Organization Page Ref: 380
23.
Mr. Lee wants to know how his students did on their last math exam. Specifically, he wants to know the following things: minimum and maximum scores, the variance in the scores, and a comparison of how the boys did compared to the girls. Which of the following types of graphs would provide Mr. Lee with this information? A. line graph B. bar graph C. stem-and-leaf plot D. box-and-whisker plot E. histogram Answer: D Sec Ref: Analyzing Data: Graphical Organization
24.
Compare and contrast bar graphs and histograms. Answer: They both use bars to indicate the quantity of something. Difference is that a histogram is used with continuous data, not discrete data. Histograms are more appropriate for middle school students. Sec Ref: Analyzing Data: Graphical Organization Page Ref: 381
25.
Which type of graph looks like a bar graph but is used with continuous data rather than discrete data? A. line graph B. histogram C. picture graph D. pie graph E. none of the above Answer: B Sec Ref: Analyzing Data: Graphical Organization Page Ref: 381
26.
What kind of graph is easy to interpret but is limited by the fact that it represents only a fixed moment in time, and it cannot exceed 100 percent? A. picture graph B. line graph C. bar graph D. pie graph E. histogram
Answer: D Sec Ref: Analyzing Data: Graphical Organization Page Ref: 381 27.
T or F Pie graph construction is more appropriate for middle school students. Answer: T Sec Ref: Analyzing Data: Graphical Organization Page Ref: 381
28.
Jeremy wants to know how much time he devotes to certain activities in a 24-hour time period. Which of the following types of graphs would be most appropriate for this situation? A. line graph B. picture C. stem-and-leaf plot D. real graph E. pie graph Answer: E Sec Ref: Analyzing Data: Graphical Organization Page Ref: 381
29.
What kind of graph is particularly useful for showing trends over time? A. picture graph B. line graph C. bar graph D. pie graph E. histogram Answer: B Sec Ref: Analyzing Data: Graphical Organization Page Ref: 382
30.
Students in Mrs. Anderson’s class collected weather data for 6 hours. What kind of graph would be most appropriate for this type of data? A. picture graph B. line graph C. bar graph D. pie graph E. histogram
Answer: B Sec Ref: Analyzing Data: Graphical Organization Page Ref: 382 31.
T or F Two of the most common types of descriptive statistics include measures of variation and measures of central tendency. Answer: T Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 384
32.
Which of the following is not an average? A. mean B. median C. mode D. range E. none of the above Answer: D Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 385
33.
T or F The concept of average should be taught meaningfully with concrete materials first before proceeding to algorithms. Answer: T Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 385
34.
T or F The mode is affected very little by extreme scores. Answer: T Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 385
35.
T or F In a given data, there can only be one mode. Answer: F Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 385
36.
Discuss a situation where you can have more than one mode.
Answer: Answers vary. Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 385 37.
Describe an activity that you could do with your students to illustrate the concept of mode. Answer: Answers vary. Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 385
38.
Describe an activity that you could do with your students to illustrate the concept of median. Answer: Answers vary. Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 385
39.
A student builds unifix cube towers to represent the ages of the five children in her family. She places them in order form smallest to largest. She removes the shortest and tallest towers; then she again removes the next shortest and tallest towers. The tower in the middle is remaining. This student has just found which average? A. mean B. median C. mode D. range E. none of the above Answer: B Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 385
40.
T or F In order to determine the median of a set of scores, the scores must be ordered. Answer: T Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 385
41.
Why is the mean called the arithmetic average? Answer: Because it is determined by adding all the values involved and dividing by the number of addends. Sec Ref: Analyzing Data: Descriptive Statistics
Page Ref: 386 42.
T or F The mean can only be used with numerical data. Answer: T Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 386
43.
Describe an activity that you could do with your students to illustrate the concept of mean. Answer: Answers vary. Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 386
44.
Which average is most affected by extreme scores? A. mean B. median C. mode D. range E. all of the above Answer: A Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 387
45.
Which of the following three types of averages would be most representative for the following scores: 25, 36, 93, 35, 27, 30, 28. A. mean B. median C. mode Answer: B Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 388
46.
T or F The range measures the distance that a data point is away from the mean.
47.
Answer: F Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 389 What does it mean for a student to have data sense?
Answer: Students who have data sense are able to determine how data should be interpreted; they are able to read and evaluate statistical information being presented; they are also able to use statistical language when reasoning about data. Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 389 48.
Explain procedural and conceptual knowledge. Answer: Procedural knowledge – how to do something; conceptual knowledge – understanding something. Sec Ref: Interpreting Results Page Ref: 389
49.
Explain the three levels of graph comprehension. Answer: Reading the data – the student is able to answer specific questions for which the answer is prominently displayed; reading between the data – the student is able to find relationships in the data; reading beyond the data – the student is able to predict or make inferences. Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 390
50.
A child asks, “Which ice cream flavor got the most votes?”. This is an example of which of the following type of graph comprehension? A. reading the data B. reading between the data C. reading beyond the data D. all of the above E. none of the above Answer: A Sec Ref: Analyzing Data: Descriptive Statistics Page Ref: 390
51.
A child says, “Based on this graph, I predict that the temperature will be colder in the evening”. This is an example of which of the following type of graph comprehension?
A. reading the data B. reading between the data C. reading beyond the data D. all of the above E. none of the above Answer: C Sec Ref: Interpreting Results Page Ref: 390 52.
Critique the following graph: Amount of Homework
4
Hrs. 2
2010
2011 Years
Answer: The bar for 2011 is distorted. Sec Ref: Interpreting Results Page Ref: 20 53.
T or F Probability assigns a number (from -1 to 1) to an event. Answer: F Sec Ref: Probability Page Ref: 391
54.
T or F The probability of an event occurring can also be designated with percent. Answer: T Sec Ref: Probability Page Ref: 392
55.
What are some key words used in association with probability? Answer: Certain, uncertain, impossible, likely, unlikely. Sec Ref: Probability Page Ref: 393
56.
T or F In order to calculate the probability of an event, the sample space must be known. Answer: T Sec Ref: Probability Page Ref: 394
57.
Name two things that you could use with students to help illustrate the concept of sample space. Answer: Dice and spinners. Sec Ref: Probability Page Ref: 394
58.
In the context of probability, what is meant by “fair”. Answer: No inherent biases exist that would affect the outcome of an event. Sec Ref: Probability Page Ref: 396
59.
Give an example of a situation where the outcome wouldn’t be fair? Answer: Answers vary. Sec Ref: Probability Page Ref: 396
60.
A teacher has students draw and replace colored cubes from a bag. The class has a discussion that since they shook the bag after replacement, the cube that is drawn first has no impact on the cube that is drawn second. The teacher is reinforcing the concept of: A. unfairness B. outcomes C. sample space D. independence of events E. all of the above Answer: D Sec Ref: Probability
Page Ref: 396 61.
What are some common misconceptions that students have about probability? Answer: They make predictions based on preferences; they hold biases against certain numbers; they may expect all outcomes in an experiment to be equally likely. Sec Ref: Probability Page Ref: 397
Chapter 18: Number Theory Possible Questions for Assessment and Evaluation 1.
T or F Number theory is a branch of mathematics that deals primarily with the integers. Answer: T Sec Ref: Introduction Page Ref: 402
2.
Discuss the four reasons why number theory should be taught in elementary schools. Answer: Number theory is a prime source to show that numbers can be fascinating; number theory opens the doors to many mathematical conjectures; number theory provides an avenue to extend and practice mathematical skills; number theory offers a source of recreation. Sec Ref: Introduction Page Ref: 402-404
3.
The number 24 is what type of number? A. abundant B. deficient C. perfect Answer: A Sec Ref: Introduction Page Ref: 403
4.
T or F The number 21 is an abundant number. Answer: F Sec Ref: Introduction Page Ref: 403
5.
The number 28 is what type of number? A. abundant B. deficient C. perfect Answer: C Sec Ref: Introduction Page Ref: 403
6.
Define the following types of numbers and give examples of each: abundant, deficient, perfect.
Answer: Abundant, sum of the factors of a number is more than the number; deficient, sum of the factors of a number is less than the number; perfect, sum of the factors of a number is equal to the number. Examples vary. Sec Ref: Introduction Page Ref: 403 7.
T or F One of the first number theory topics children encounter is prime and composite numbers. Answer: F Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 404
8.
T or F 0 is an even number. Answer: T Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 404
9.
What makes a number even or odd? Answer: Even – divisible by 2 (no remainder when divided by 2); odd- has a remainder of 1. Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 405
10.
Define the term, factor. Answer: A factor is a number that divides another number with no remainder. Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 405
11.
Which of the following is not a factor of 12? A. B. C. D. E.
1 2 3 4 5
Answer: E Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 405
12.
List all of the factors of 48.
Answer: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 405 13.
If students were working with tiles or graph papers to find the factors of 16, how many different rectangle representations would there be? Draw the rectangle representations. Answer: Five (1 x 16, 2 x 8, 4 x 4, 8 x 2, 16 x 1) Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 405
14.
T or F A multiple of a number is the sum of that number and any other whole number. Answer: F Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 407
15.
Explain the relationship between multiples and factors. Answer: If a number is a factor of a multiple, you can take that factor times a whole number to get that multiple. Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 407
16.
Jimmy is working on finding the factors to 12 and 18. After he’s found all the factors, he notices that they both have a 6. Which of the following number theory topics does this best illustrate? A. B. C. D. E.
odds and evens factors and multiples primes and composites prime factorization divisibility
Answer: B Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 407 17.
T or F A prime number is a number with more than two factors. Answer: F Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 407
18.
T or F A composite number is a number with more than two factors.
Answer: T Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 407 19.
The number 21 is which kind of number? A. prime B. composite Answer: B Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 407
20.
T or F Prime numbers are always odd numbers. Answer: F Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 407
21.
How can the use of rectangle representations be used to help students understand prime numbers? Answer: If only two rectangle representations can be made, then a number is prime. For example, for 11, you have 1 x 11 and 11 x 1. Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 408
22.
T or F Every composite number can be expressed as a product of primes. Answer: T Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 408
23.
Which of the following is the correct prime factorization for 160? A. B. C. D. E.
1 x 160 4x8x5 2 x 8 x 10 2x2x2x2x2x5 all of the above
Answer: D Sec Ref: Number Theory in Elementary School Mathematics Page Ref: 408
24.
Should students be required to memorize the divisibility rules?
Answer: With the advent of calculators and technology, it’s not necessary that they memorize the rules. However, students should understand why the rules work. Sec Ref: Divisibility Page Ref: 410 25.
Using base-ten blocks, how would you help students to understand why 42 would be divisible by 3? Answer: Since you have 4 rods, they can be shared with 3 students, leaving 1 rod. The one rod and 2 units gives you 12 units. The 12 units can then be divided equally among 3 students. Sec Ref: Divisibility Page Ref: 411
26.
Discuss the term relatively prime and give an example to support your answer. Answer: Two numbers are relatively prime if they have no common factors other than 1. Examples vary. Sec Ref: Other Number Theory Topics Page Ref: 412
27.
Give an example to illustrate two numbers that are relatively prime. What makes them relatively prime? Answer: Two numbers are relatively prime if they have no common factors other than 1. Examples vary. Sec Ref: Other Number Theory Topics Page Ref: 412
28.
Which of the following sets of numbers is relatively prime? A. 3 and 8 B. 6 and 21 C. 4 and 12 D. 9 and 12 E. none of the above Answer: A Sec Ref: Other Number Theory Topics Page Ref: 412
29.
Which of the following is not an example of a square number?
A. B. C. D. E.
4 9 13 16 25
Answer: C Sec Ref: Other Number Theory Topics Page Ref: 412 30.
Why are 4, 9, 16, 25, etc. considered square numbers? Answer: You can represent them with a square figure. Sec Ref: Other Number Theory Topics Page Ref: 413
31.
Using clock arithmetic, what would the answer be for (8 + 4) mod 10? A. B. C. D. E.
1 2 3 4 5
Answer: B Sec Ref: Other Number Theory Topics Page Ref: 413 32.
T or F When using mod 5, we use the numbers 1, 2, 3, 4, 5. Answer: F Sec Ref: Other Number Theory Topics Page Ref: 413
33.
T or F Pascal’s Triangle is most closely associated with probability. Answer: T Sec Ref: Other Number Theory Topics Page Ref: 414
34.
Explain what is meant by a Pythagorean triple. Answer: A triple of numbers (a, b, c) such that a2 + b2 = c2. Sec Ref: Other Number Theory Topics Page Ref: 414
35.
How could you visually show that 32 + 42 = 52?
Answer: Use squares. For 3 square, it would be a square 3 x 3 or 9 square units. For 4 square, it would be a square 4 x 4 or 16 square units. 9 square units plus 16 square units is equal to a square that is 5 x 5, or 25 square units. Sec Ref: Other Number Theory Topics Page Ref: 415 36.
What is significant about the Fibonacci sequence? Answer: It is a sequence that is rich – with examples of the pattern in nature, art, and even our bodies. Sec Ref: Other Number Theory Topics Page Ref: 416
37.
T or F The study of number theory is a 20th century phenomenon. Answer: F Sec Ref: Cultural Connections Page Ref: 416