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Elementary Education
Q:
Students should begin developing their ability to reason algebraically in elementary school.
Q:
Although algebra has few connections to real-life applications, it is highly intertwined with other areas of mathematics, and, therefore, is an important topic.
Q:
Provide a multiplication or division problem and a potential strategy that could be used to compute it. Explain why this strategy could be valuable. Describe an activity you could use to encourage the development and/or use of this method.
Q:
When estimating with rational numbers, it is best to use other rational numbers. Otherwise, the estimate is likely to be very inaccurate.
Q:
Calculators
a) Are not appropriate to use when students are learning to become more fluent estimators.
b) Don"t help students check the reasonableness of their estimates.
c) Are one of the reasons that estimation skills are so important, because students frequently make mistakes involving hitting the wrong keys.
d) Can greatly reduce student engagement.
Q:
The compatible numbers strategy
a) Is one of the least helpful strategies for estimating division problems.
b) Involves changing the numbers in the problems to make them more "friendly" to work with.
c) Is not easily applied to situations involving fraction, decimals, and rates.
d) Is not appropriate for estimating multiplication problems.
Q:
In order to be truly useful in estimation, rounding must be flexible and conceptually understood by students.
Q:
Which of the following is a true statement regarding using estimation strategies?
a) It is much more useful to focus on the estimated answer than it is to focus on the process students used to obtain the estimate.
b) The more practice students have with finding a variety of estimates for the same problem, the more confused they will become.
c) If the teacher has ELL students, he or she should ensure the students understand the context of the problem, then provide the numerical information before asking for an estimate.
d) The front-end strategy has been shown to be one of the most difficult for students to learn to use.
Q:
When developing the standard algorithm for division,
a) Teachers should avoid using a confusing algorithm based on repeated addition.
b) Teachers should have students use models after developing the written record.
c) The process of recording explicit trades can be less confusing to students than the more common bringing down.
d) The expression "goes into"is very meaningful for kids.
Q:
Because division is frequently the most onerous of all computational operations, it is better to teach students the standard algorithm rather than potentially confusing them with student-invented strategies.
Q:
Rather than using the confusing language of "carrying" the digit when multiplying, a teacher might want to encourage students to use partial products.
Q:
The _____________________________ , or connected array, can be a key visual representation of multiplication that supports students' conceptual understanding.
Q:
When using ________________________________ to compute multi-digit multiplication problems, students use facts and combinations they already know to find out more complex computation problems.
Q:
Which is an example of the compensation strategy?a) 63x 5 = 63 + 63 + 63 + 63 + 63 = 315b) 27 x4 = 20x4 + 7x 4 = 80 + 28 = 108c) 27x 4 is about 30 (27 + 3)x 4 = 120; then subtract out the extra 3 x 4, so 120 -12 = 108d) 46x 3 = 46x 2 (double) + 46 = 92 + 46 = 138
Q:
Which of the following is NOT a useful strategy for multiplying by single digits?
a) Doubling
b) Compatible numbers
c) Partitioning
d) Complete number
Q:
Although many teachers worry that multiple representations of the same problem will confuse students, research shows that, if students are shown a variety of methods from the beginning, they gain flexibility in their problem-solving strategy use and, consequently, enhanced learning.
Q:
Becoming fluent at breaking apart numbers in flexible ways is more important for multiplication than addition, and the skill is dependent on students' understanding of the __________________________ number property.
Q:
Provide an addition or subtraction problem and a potential student-invented strategy that could be used to compute it. Explain why a student-invented strategy could be valuable. Describe a method you could use to encourage the development and/or use of this method.
Q:
______________________________estimation is a strategy, most appropriate for addition or subtraction, that only takes into account the leftmost digit and pretends the remaining numbers are all zeros.
Q:
Which of the following is a true statement regarding computational estimation?
a) We can use calculators, so it's not really that necessary.
b) Its under representation in many textbooks can cause someone to incorrectly assume it's not very important.
c) Teachers should define for and accept from students only one correct estimation.
d) Examples of real-life context are not needed.
Q:
Which of the following is NOT true about strategies for developing the subtraction algorithm?a) Students should begin with models only.b) The teacher should anticipate that there will be difficulties with zeros.c) After the procedure is completely understood with models, then students should begin developing the written record.d) The general approach is completely different from that used to develop the addition algorithm.
Q:
As with any algorithm, students must first be exposed to addition from a _____________________ approach, which must then be connected to a more abstract approach.
Q:
When students solve problems that involve ___________________________ or crossing ten, if they are using the standard algorithm, they must regroup or trade.
Q:
The take away strategy for subtraction can be easy if the subtracted number is a multiple of ten or close to one.
Q:
The concept of "think addition" to aid in subtracting large numbers works the same as when subtracting small numbers.
Q:
All of the following could be examples of student-developed strategies for obtaining the sum of two-digit numbers EXCEPTa) Adding on tens and then ones (For example, to solve 24 + 35, think 24 + 30 = 54 and 5 more makes 59.)b) Using nicer numbers to estimate (For example, to solve 24 + 47, think 24 is close to 25 and 47 is closer to 45 so 24 + 47 = 25 + 45 = 70.)c) Moving some to make 10 (For example, to solve 24 + 35, move 6 from 35 to make 24 + 6 and then add 30 to the remaining 29.)d) Adding tens and adding ones then combining (For example, to solve 24 + 35, think 20 + 30 = 50 and 4 + 5 = 9 so 50 + 9 = 59.)
Q:
When creating a classroom environment appropriate for inventing strategies
a) The teacher should immediately confirm that a student's answer is correct, in order to build his/her confidence.
b) The teacher should attempt to move unsophisticated ideas to more sophisticated thinking through coaching and questioning.
c) Student-to-student conversations should be discouraged, in order to provide students with a quiet environment to think.
d) The degree to which students feel safe to make mistakes is not an important factor.
Q:
Which of the following is a true statement about standard algorithms?
a) Students will frequently invent them on their own if they are given the time to experiment.
b) They cannot be taught in a way that would help students understand the meaning behind the steps.
c) In order to use them, students should be required to understand why they work and explain their steps.
d) There are no differences between various cultures.
Q:
Which of the following is NOT a benefit of student-developed strategies?
a) They require one specific set of steps to use them, which makes them easier to memorize.
b) They help reduce the amount of needed re-teaching.
c) Students develop stronger number sense.
d) They are frequently more efficient than standard algorithms.
Q:
Which of the following is NOT an example of a method used to compute a solution?
a) Standard algorithms
b) Student-invented strategies
c) Discourse modeling
d) Computational estimation
Q:
Modern technology has made computation easier
a) But mental computation strategies can be faster than using technology.
b) And recent studies have found that a very low percentage of adults use mental math computation in everyday life.
c) And mental computation contributes to diminished number sense.
d) So the ability to compute fluently without technology is no longer needed for most people.
Q:
Describe an activity that would help your students to better conceptualize numbers that are very large.
Describe how this activity would build conceptualization.
Q:
When helping students to conceptualize numbers with 4 or more digits, which of the following is NOT true?
a) Students should be able to generalize the idea that 10 in any one position of the number results in one single thing in the next bigger place.
b) Because these numbers are so large, teachers should just make due with the examples that are provided in textbooks.
c) Models, such unit cubes, can still be used.
d) Students should be given the opportunity to work with hands-on, real-life examples of them.
Q:
According to NCTM, it's not necessary for students to have fully developed place value understandings before giving them opportunities to solve problems with two and three-digit numbers.
Q:
Multiples of 10, 100, and sometimes 25 are called ___________________________, which work especially well with hundreds charts and number lines to help students find the distances between numbers.
Q:
A _____________________________________ is an important tool that can hang on a wall, be displayed on a smart board, or can be given to students as paper copies, which students can use to discover numerous place-value-related patterns
Q:
A student who has place value understanding at the face valuelevel, when asked to explain the digits of the number 45, would most likely
a) Be unable to identify the meaning behind the individual digits, and would see the number as one unit.
b) Be able to identify the digit in the ones place and in the tens place, but be unable to relate the meaning of the two digits to two separate amounts.
c) Match up four blocks to go with the 4 digit and five blocks to go with the 5 digit.
d) Verbalize that the 4 represents forty and the 5 represents five units.
Q:
Because students can often hide their lack of conceptual understanding, a more in-depth assessment tool, a ______________________________________, can be used to determine what they really know.
Q:
When using place-value mats, drawing ________________________________ in the ones place will make it very clear to students how many ones there are so they can avoid recounting the ones.
Q:
When students are being introduced to three-digit numbers
a) The process should be quite different from introducing students to two-digit numbers.
b) They have normally not yet mastered the two-digit number names.
c) They frequently struggle with numbers that contain no tens, like 503.
d) Their mistakes when attempting to write numeric examples should not be discussed, in order to avoid embarrassment.
Q:
Making the transition from base-ten to standard language
a) Can be made more confusing by using base-ten materials when verbalizing the number names.
b) Should not include the teacher using a mix of base-ten and standard language,
c) Should not include a discussion of the "backwards" names given to the teens, as they can be confusing.
d) Can be made less difficult by using a word wall to provide support for ELLs and students with disabilities.
Q:
When it comes to beginning grouping activities
a) Because students usually understand counting by ones, teachers should skip directly to grouping by ten.
b) Teachers should let students experiment with showing amounts in groups until they, perhaps, come to an agreement that ten is a useful-sized group to use.
c) Students should only work with very small items that can easily be bundled together.
d) Teachers should not worry about having students verbalize the amounts they are grouping.
Q:
Which of the following is NOT an example of a proportional model that can be used for place value?
a) Money
b) Beans in cups of ten and single beans
c) Base ten blocks
d) Stir straws bundled in groups of ten and with single straws
Q:
Nonproportional models should be used only after students understand that ten units makes a "ten."
Q:
Base ten blocks are the only material that should be used to model place value concepts.
Q:
In order to help students to understand the way the two digits of a number and a base ten model of it are related, models of tens should be grouped on the left, and units should be on the right, to reflect the structure of the numeric version.
Q:
Using base-ten language
a) Is demonstrated when the teacher says "We have fifty-three beans."
b) Can be helpful for students who are ELLs because many other countries routinely use base-ten language.
c) Is frequently confusing for students, and it is best avoided.
d) Looks only like this format: ____tens and ____ones.
Q:
When supporting students' ability to quickly recall basic facts, avoid
a) Only working on a few facts at once.
b) Allowing ample time before moving from reasoning to memorization.
c) Using public displays of mastery (bulletin boards, etc.).
d) Limiting the length of timed tests.
Q:
All of the following are recommendations that can support students' ability to quickly recall basic facts EXCEPT
a) Drilling for extended periods of time
b) Involving families
c) Using technology
d) Encouraging students to self-monitor
Q:
Name two key ideas or strategies that can guide a teacher's efforts to help older students who still struggle with basic facts, and describe briefly how each could be helpful.
Q:
Using games to help students practice basic facts increases student involvement, encourages student-to-student ______________, and improves communication.
Q:
Games can be appropriate to support mastery because
a) They can be less engaging than other forms of practice.
b) They frequently have aspects that allow students to self-check their answers.
c) They are frequently much more stressful than other forms of practice.
d) They rarely allow teachers methods to differentiate, based on individual student needs.
Q:
When is drill appropriate?
a) Never
b) As a practice tool, only once students are effectively using reasoning strategies that they conceptually understand
c) When every student is drilled at the same pace
d) Only when students are given for practice a set of facts that have no relationship to one another
Q:
Which of the following is a frequently used strategy to help students learn basic multiplication facts?
a) Nifty nines
b) Using unknown facts to reason and figure out the ones that are known
c) Tripling numbers and then adding one
d) Making students see there is no relationship between multiplication and division
Q:
It is of great importance that students understand the _____________________________property, because it cuts the number of multiplication/addition facts they need to memorize in half.
Q:
When using the subtraction strategy ___________________________, students use known addition facts to provide an unknown quantity or part.
Q:
Which of the following is NOT a strategy that helps students master basic facts?
a) One more than/two more than
b) Using 6 as an anchor number
c) Near doubles
d) Doubles
Q:
Story problems can provide a valuable context to help students apply strategies for computation, as long as the context is relevant to students.
Q:
Research shows that guiding students to use effective strategies can help them achieve mastery. In order to accomplish this, the teacher must be familiar with a large variety of strategies.
Q:
Despite the fact that many learners in the past learned basic facts from memorization, it is not an effective strategy for all of the following reasons EXCEPT
a) There are just too many facts to memorize, and it's an inefficient process.
b) When students are taught to simply memorize, they usually don"t go back and make sure that their work is reasonable.
c) Students don"t develop efficient reasoning strategies and use inefficient ones, such as counting by ones.
d) It greatly increases student motivation.
Q:
Which of the following is NOT one of the phases in the process of learning basic facts?
a) Drawing strategies
b) Counting strategies
c) Mastery
d) Reasoning strategies
Q:
What statement most accurately describes a person who has mastered basic facts?
a) He or she can use a model or a picture to figure out the answer to a basic fact problem.
b) He or she has a conceptual understanding of the basic facts and can access her background knowledge to, with some time, figure out the answer to a basic fact problem.
c) He or she has obtained a set of essential understandings and progressed through a set of stages so that she just "knows" the answers to certain basic fact problems.
d) He or she can use a set of manipulatives and reasoning to figure out the answer to a basic fact problem.
Q:
The key word problem solving strategy is effective, because it makes the process of choosing an operation less complicated for students.
Q:
Which would NOT be a good statement/question from a teacher to help a student realize why he or she can"t divide by 0?
a) "What happens when you take these 25 pennies and divide them into 0 groups?"
b) "Can you show me how to share 8 apples between no people?"
c) "What can you multiply by 0 to get 7?"
d) "Just memorize that you can"t divide by 0."
Q:
What number property is illustrated by the problem 16 12 = 16(10 + 2) = 160 + 32 = 192?
a) Associative
b) Commutative
c) Distributive
d) Identity
Q:
Describe at least four kinds of models one could use to model a multiplication or division situation.
Q:
Because real-life division usually results in a simple whole number, it is not important for students to understand remainders as anything more than the amount "left over."
Q:
The usual mathematical convention says that 4 8 represents "four sets of eight."
Q:
Which problem is an example of the comparison, product unknown (multiplication) structure?
a) This month, Barry saved 8 times as much as last month. Last month, he saved $3. How much did Barry save this month?
b) Barry's sandwich shop offers 3 kinds of meat and 2 kinds of bread. How many different sandwiches could he make if he uses 1 meat and 1 kind of bread for each?
c) Barry saved $12 and Jill saved $6. Barry saved how many times as much money as Jill?
d) Barry saved $24 total, and he saved $6 each month. For how many months had he been saving?
Q:
Which problem is an example of the equal groups, number of groups unknown structure?
a) This month, Barry saved 8 times as much as last month. Last month, he saved $3. How much did Barry save this month?
b) Barry's sandwich shop offers 3 kinds of meat and 2 kinds of bread. How many different sandwiches could he make if he uses 1 meat and 1 kind of bread for each?
c) Barry saved $12 and Jill saved $6. Barry saved how many times as much money as Jill?
d) Barry saved $24 total, and he saved $6 each month. For how many months had he been saving?
Q:
Adding and subtracting the number 0 can be confusing for students because they assume the value of a number must change when adding or subtracting another number (even 0) to/from it.
Q:
The commutative property
a) Applies to addition and subtraction.
b) Helps students master basic facts because, if they really understand it, it reduces the number of individual facts they have to memorize.
c) Should be demonstrated with problems that have the same sums but different addends.
d) Is a term that even very young students should memorize.
Q:
When thinking about subtraction as __________________________ rather than "take away," students ask themselves "What goes with the part I see to make the whole I need?"
Q:
Bar diagrams, counters, and number lines are examples of ___________ that can be used to help solve story problems.
Q:
Which of the following is NOT a true statement regarding mathematical symbols?a) It is important that very young children understand the symbols +, -, and = to learn about addition and subtraction concepts.b) Great care should be taken to assure students understand = means "is equal to" rather than "the answer is."c) By first grade, symbolic conventions are important for children to know.d) Frequently, the meaning of the equal sign confuses students.
Q:
The relationship between addition and subtraction
a) Can confuse children and should not be addressed in class.
b) Should be developed for children by providing them with examples of the same number sets appearing in different contexts.
c) Is not an example of an inverse relationship.
d) Can prevent students from developing flexibility in the methods they use to solve problems.
Q:
Which of the following is a true statement about contextual problems?
a) They are connected more to what is referred to as "school mathematics" than they are to children's lives.
b) Their origin could result from recent experiences in the classroom.
c) Their language orientation can make them inappropriate for ELL students.
d) Enhancing them with visual aids or students acting them out would be inappropriate.
Q:
An equation that isolates the unknown is the _______________________form.
Q:
The ____________________ equation for a problem lists the numbers in the order of the meaning in the problem.
Q:
Problems with the join and separate structures, with the start or initial amount unknown, tend to be the easiest for students to understand and accurately solve.