Math Methods Test 1
Which of the following equations illustrates the associative property for addition? "2 + 5 = 7," and "7 - 5 = 2" "2 + 5 = 5 + 2" "0 + 7 = 5 + 2" "(2 + 5) + 4 = 2 + (5 + 4)"
"(2 + 5) + 4 = 2 + (5 + 4)" The associative property for addition states that when adding three or more numbers, it does not matter whether the first pair is added first or if you start with any other pair of addends. Therefore, "(2 + 5) + 4 = 2 + (5 + 4)" is the correct response.
Which of the following equations illustrates the distributive property of multiplication over addition? "2(5 + 3) = 5 + 2 × 3" "2(5 + 3) = (2 + 5) × (2 + 3)" "2(5 + 3) = 2 × 5 + 3" "2(5 + 3) = 2 × 5 + 2 × 3"
"2(5 + 3) = 2 × 5 + 2 × 3" The distributive property of multiplication over addition refers to the idea that you can split (decompose) either of the two factors in a multiplication problem into two or more parts and then multiply each of the parts by the other factor and add the results. The final product is the same as when the original factors are multiplied. Therefore, "2(5 + 3) = 2 × 5 + 2 × 3" is the correct response.
Assessing place value with the Digit Correspondence Task helps the teacher recognize the student's level of understanding. According to Ross, which of the following statements represents a full understanding of place value when using the task with 36 blocks? "36 is written without any connection to the numbers 3 and 6." "3 is matched with 3 blocks and 6 is matched with 6 blocks." "3 is identified as being in the tens position and 6 is identified as being in the ones position." "3 is correlated with 3 groups of ten blocks and 6 with 6 single blocks."
"3 is correlated with 3 groups of ten blocks and 6 with 6 single blocks." Ross's levels, in order of level of understanding, are: 1.Single numeral. The student writes 36 but views it as a single numeral. 2.Position names. The student correctly identifies the tens and ones positions but makes no connections between the individual digits and the blocks. 3.Face value. The student matches 6 blocks with the 6 and 3 blocks with the 3. 4.Transition to place value. The 6 is matched with 6 blocks and the 3 with the remaining 30 blocks, but not as 3 groups of 10. 5. Full understanding. The 3 is correlated with 3 groups of ten blocks and the 6 with 6 single blocks.
Which of the following open number sentences represents partition division? "3 × __ = 18" "3 + 6 =9" "3 × 6 = 18" "__ × 6 = 18"
"3 × __ = 18" In a partition division situation, you are trying to identify the size of the group— many times by actually sharing or dealing out a quantity into the desired number of groups. Because multiplication problems such as 3 × 6 = in the United States are interpreted as 3 groups of 6, the problem 3 × □ = 18 is the situation in which you look to identify the "size of the group"— a partition situation.
Simplify: 6×2+3÷3 5 13 10 18
13 The order of operations must be obeyed here. Remembering the saying Please Excuse My Dear Aunt Sally (PEMDAS) allows us to remember the order in which mathematical operations must be carried out, Parentheses Exponent Multiply Divide Add Subtract. Following this one will multiply 6 by 2 to obtain 12. Then one will divide 3 by 3 obtaining 1. Finally, one will add the two results together to obtain 12 + 1 = 13.
How is the number 12 represented in Base 6? 18 12 20 6
20 Base 6 has six digits, with the number six become 10. Counting in Base 6 would look like this: 0, 1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20, 21, .... Starting with 1, count up twelve units. The number in Base 6 that represents 12 is 20.
Which of the following is a composite number? 29 19 21 11
21 Composite numbers contain factors other than itself and 1. Since 21 also has factors of 3 and 7, it is composite.
What is the number 8 written in Base 3? 24 8 11 22
22 Base 3 has three digits, with the number three becoming 10. Counting in Base 3 would look like this: 0, 1, 2, 10, 11, 12, 20, 21, 22, 30, 31, .... Starting with 1, count up eight units. The number in Base 3 that represents 8 is 22.
Which of the following numbers is correctly represented by prime factorization? 30 = 2 × 3 × 5 56 = 7 × 8 24 = 2 × 2 × 3 18 = 2 × 3 + 12
30 = 2 × 3 × 5 Prime factorization is representing a number using only its prime factors.
Steve is told that milk must remain at 50 degrees F so it will not spoil and that a turkey must be cooked at 375 degrees F for 2 hours. What is the difference in temperature of the milk and the turkey (while it is cooking)? 325 335 320 315
325 In this problem, we are not concerned with negative or positive results because we are only asked about the relative difference between the two temperatures. So, the difference between 375℉ and 50℉ is 325℉.
How are 3 tens and 4 ones represented in a Base 10 system? 304 301111 34 31111
34 3 tens = 30; 4 ones =4. Put these values together to get 34.
When presenting addition problems, which of the following would you use last? 43 + 32 = 39 + 23 = 356 + 127 = 645 + 354 =
356 + 127 = First, introduce problems that do not require regrouping. Then, present problems where only one place value position requires regrouping. 356 + 127 requires regrouping in two places, making it last in the series of problems presented to students who are learning the standard algorithm.
What is another way to write 4×4×4? 12 4x3 4 to the 3rd power 3 to the 4th power
4 to the 3rd power Recalling that exponential notation is used when numbers are multiplied by themselves numerous times, we may try to simplify the expression. "3 to the 4th power" is not correct because this implies 3 is multiplied by itself 4 times. "4×3" and "12" are the same answer in different representations, but both are incorrect. Since 4 is multiplied by itself 3 times in this problem, we know it should be "4 to the 3rd power".
Which of these sets of numbers represents a true statement? 10 > 11 3 ≥ 5 5 < 7 9 ≤ 8
5 < 7 The less than symbol resembles an "alligator mouth" opening and moving away from the smaller number. "It is also helpful to notice that the smallest end in the sign will always point to the smallest number (5 < 7). "5 < 7" is the correct answer because it correctly states that: 5 is less than 7. "10 > 11" is incorrect because it incorrectly states that the number 10 is greater than 11. "9 ≤ 8" and "3 ≥ 5" are eliminated because both contain an extra line representing the concept of "equal to"; thus, the symbol in "9 ≤ 8" incorrectly states that "9 is less than or equal to 8" and "3 ≥ 5" states that "3 is greater than or equal to 5."
The benchmark numbers that are most important for very young learners are: "0 and 1." "25 and 50." "20 and 30." "5 and 10."
5 and 10. The most important benchmark numbers for young learners are 5 and 10. Because the number 10 plays such a large role in our numeration system, and because two 5s equal 10, it is very useful to develop relationships for the numbers 1 to 10 connected to the benchmarks of 5 and 10.
Which groping is in order from least to greatest? 6 to 1st power, 2 to the 3rd power, 3 to the 2nd power, 4 to the 2nd power 3 to 2nd power, 2 to the 3rd power, 6 to the 1st power, 4 to the 2nd power 6 to the 1st power, 3 to the 2nd power, 4 to the 2nd power, 2 to the 3rd power 4 to the 2nd power, 6 to the 1st power, 3 to the 2nd power, 2 to the 3rd power
6 to 1st power, 2 to the 3rd power, 3 to the 2nd power, 4 to the 2nd power 6 to the 1 power = 6, 2 to 3rd power = 8, 3 to the 2nd power = 9, 4 to the 2nd power = 16
An example of a prime number is 682 9 67 49
67 A prime number is a number whose only factors are one and itself. Even numbers greater than 2 can always be factored by 2, eliminating "682." "9" can be factored as 3 times 3 and "49" can be factored as 7 times 7. Therefore, "67" must be the right answer as it only has factors of 1 and 67.
Guadalupe Peak in Texas is 8,751 feet high. What is 8,751 rounded to the nearest hundred? 8,750 8,800 9,000 8,700
8,800 The 7 is in the hundreds place, so look at the tens place to determine rounding. Because there is a 5 in the tens place, round the 7 up to 8 and everything to the right changes to zero; the rounded number becomes 8,800.
Wright and his colleagues (2006) identified a three-level progression of children's understanding of 10. Which of the following is not one of these levels? A mastery concept of 10 An intermediate concept of 10 A facile concept of 10 An initial concept of 10
A mastery concept of 10 Wright et al.'s three-level progression includes: (1) an initial concept of 10; (2) an intermediate concept of 10; and (3) a facile concept of 10.
Which of the following is an example of a student demonstrating the skill of subitizing? A student recognizes the number 5 as an anchor number for the numerals from 3 to 7. A student recognizes the number 5 as the number 6 with 1 taken away. A student rolls a "5" on a die and is able to say it is a five without actually counting the dots. A student recognizes the number 5 as the number 4 plus 1 more.
A student rolls a "5" on a die and is able to say it is a five without actually counting the dots. Subitizing is the ability to look at an amount of objects and "see" how many there are without counting the individual items. This is a fundamental skill in the development of students' understanding of number.
Which of the following statements is false? After learning three-digit number names, students are easily able to generalize to larger numbers. Students should count and group amounts more than 1000 so that they have a feel for the size of the number. Teachers should use real-world referents to discuss larger numbers. Students struggle with three-digit numbers involving no tens (internal zero).
After learning three-digit number names, students are easily able to generalize to larger numbers. Researchers note that there are significantly more errors with four-digit number names than three-digit numbers. Students do not easily generalize to larger numbers without actually exploring additional examples and tasks.
A fourth grade teacher is working with a group of students to find a measurement in ounces that is equal to 2.5 pounds. One student answers with 25 ounces, another student answers with 40 ounces, and the third student answers with 16 ounces. What step should the teacher take next? Hand each student a calculator and ask them to do the problem again. Tell students the correct answer and have them go back to their desks. Ask each student to explain their method of solving the problem. Give student right conversation and have them work with a partner.
Ask each student to explain their method of solving the problem. Having students verbalize their methodology has many rewards: Students who are correct gain deeper understanding of the topic, students who are incorrect can talk through the problem and figure out where mistakes are made, and teachers are able to quickly assess student understanding.
Identify the mathematical properties involved in the following problem: (6 + 2) + 5 = 6 + (2 + 5). Associative property of multiplication Property of zero Associative property of addition Distributive property
Associative property of addition One of the basic properties of addition is that is can be carried out in any order. Therefore, since addition is the only operation that is performed in this exercise, one knows the correct answer is the associative property of addition.
Which of the following cultures is correctly paired with its mathematical contribution? A. Hindu - Pythagorean Theorem C. Mayan - Calendars B. Indian - Base 10 numbering B&C A&B
B & C Pythagoras, a Greek, developed the Pythagorean Theorem.
Which of the following should not be counted as the mathematics lesson for the day? Part-Part-Whole activities Ten Frame activities Calendar activities Hundreds Chart activities
Calendar activities The NRC Committee stated that "using the calendar does not emphasize foundational mathematics" (2009, p. 241). The committee went on to remind early childhood teachers that although the calendar may be helpful in developing a sense of time, it does not align with the need to develop mathematical relationships related to the number ten because the calendar is based on groups of seven. The NRC concluded: "Doing the calendar is not a substitute for teaching foundational mathematics" (p. 241). Ethridge and King (2005) suggested that children learn to parrot the response for the predictable questions and noted that they didn't always understand some of the concepts presented.
In teaching through problem solving, students are engaged in doing all of the following except which? Checking their answers with others or the textbook to confirm that it is correct Making connections to other concepts and examples Convincing the teacher and their peers that their solution makes sense Determining their own solution path
Checking their answers with others or the textbook to confirm that it is correct The correctness of a solution should lie in one's own justification, not in input from external sources such as the teacher, textbook, or other students. It is appropriate to have students share solutions for the purpose of critiquing each other's reasoning and making connections between strategies, but that is different from using other solutions to "check your answer."
Which of the following statements about multiplication strategies is true? Partitioning strategies rely on use of the associative property of multiplication. Cluster problems use multiplication facts and combinations that students already know in order to figure out more complex computations. Some multiplication problems can be challenging to solve with invented strategies and students should just use a calculator in those situations. Always think of complex multi-digit multiplication problems in the form of repeated addition.
Cluster problems use multiplication facts and combinations that students already know in order to figure out more complex computations. Clusters are groups of problems that are related to the target problem, but are easier to solve. They help students move toward actually solving the original problem under consideration.
Which problem structure is related to the subtraction situation "How many more?" Comparison Part-Part-Whole Take away Start unknown
Comparison The compare problem structure involves the comparison of two quantities. The third amount does not actually exist, but is the difference between the two amounts. The question is usually "How many more?" or "How many fewer?" Note that the language of "more" will often confuse students and thus will present a challenge in interpretation.
Which of the following is a good strategy for teaching computational estimation? Compatible numbers Counting on Reversibility Guess and check
Compatible numbers Nearly all computational estimations involve using easier-to-handle parts of numbers or substituting close compatible numbers for difficult-to-handle numbers so that the resulting computations can be done mentally. This is often used in division by adjusting the divisor or dividend (or both) to close numbers.
Which of the following statements is not true? Worthwhile tasks can and should be used to teach both concepts and procedures. Developing procedural knowledge requires practice and drill. Drill can provide an increased facility with a procedure that has already been learned. Practice is a meaningful engagement with mathematical ideas and should focus on helping students develop connections among mathematical ideas.
Developing procedural knowledge requires practice and drill. Developing procedural knowledge can occur without drill. Students can master their facts and learn algorithms for the procedures they need to learn through meaningful practice and solving worthwhile tasks. Drill can support learning, but caution must be taken that the drill does not result in students blindly applying rules. Teachers should also not assume that students understand because they are able to imitate a procedure they have been shown.
Marek was asked to multiply 34 × 5. He said, "30 × 5 = 150 and 4 × 5 = 20, so I can add them to get 170." Which property did Marek use to solve this multiplication problem? Identity property of multiplication Commutative property Associative property Distributive property of multiplication over addition
Distributive property of multiplication over addition The distributive property is defined as a (b + c) = (a × b) + (a × c).
A third grade teacher shows this math problem to her students: "Julianna, Margaret, Shonette, and Lisa are in a footrace. Julianna is 10 meters ahead of Margaret. Shonette is 5 meters behind Julianna, Margaret is 12 meters behind Lisa. In what order are the runners from first to last?" Which problem solving strategy is the most effective for solving this? Find a pattern Working Backward Draw a picture Try a simpler form of the problem
Draw a picture Draw a picture showing the location of each of the runners to solve the problem.
Ms. Quinones is teaching her second graders to use calculators. To successfully use calculators, however, her students must be skilled in which of the following: Estimation Finding patterns Problem solving Adding and subtracting
Estimation To successfully use calculators, students must first be able to estimate a reasonable answer. Therefore, the skill of estimation must be developed.
Delia was asked to estimate 489 + 37 + 651 + 208. She said, "400 + 600 + 200 = 1200, so it's about 1200, but I need to add about 150 more for 80 + 30 + 50 + 0. So, the sum is about 1350." Which computational estimation strategy did Delia use? Front-end Rounding Standard algorithm Compatible numbers
Front-end Delia took the numbers in the hundreds column as a front-end estimate, making 489 into 400 and so on. Then she adjusted for the numbers that were ignored by doing a front-end estimation of the numbers in the tens column to compensate.
A teacher is planning a lesson to introduce second graders to solving subtraction problems that require regrouping from the tens place to the ones place. Which of the math manipulatives listed below would be best for the teacher to use when presenting a lesson on subtraction with regrouping at the concrete level? Fraction circles Measuring cups Correct Base ten blocks Geoboards
Geoboards This question tests your ability to use mathematical manipulatives to develop and explore mathematical concepts with young children. Fraction circles are generally used in the elementary classroom to introduce parts of a whole or compare fractions. Measuring cups are generally used to teach how to measure capacity of liquids. Geoboards are usually used to teach how to figure the area of a shape or the characteristics of geometric shapes. Base ten blocks would be the best manipulative to use to demonstrate regrouping one ten for ten ones. Therefore, the correct answer is "base ten blocks."
The National Research Council identified all but one of the following as a foundational area in mathematics content for young children. Which area of mathematics content is not one of the NRC's foundational areas? Operations Core Number Core Relations Core Geometric Shape Core
Geometric Shape Core The three NRC foundational areas are the Number Core, the Relations Core, and the Operations Core.
Mrs. Wright asks her third graders the following question: "Does changing the order in which you subtract numbers change the answer?" Which of the following choices would allow Mrs. Wright to know that her students have employed mathematical reasoning to answer the questions? Her students could work with ordered pairs on a grid to show a pattern. Her students could employ a rubric that shows how to evaluate their own work. Her students could justify their answers by showing different subtraction problems. Her students could draw transformations showing rotations and reflections.
Her students could justify their answers by showing different subtraction problems. Justifying solution methods and results is part of mathematical reasoning. If Ms. Wright's students show new subtraction problems that justify their thinking, she will know they have employed mathematical reasoning.
Which of the following is not a common type of invented strategy for addition and subtraction situations? Shortcut strategy High-Low strategy Jump strategy Split strategy
High-Low strategy Three common types of invented strategy models include the split strategy (also thought of as decomposition), the jump strategy (similar to counting on or counting back), and the shortcut strategy (sometimes known as compensation).
Which of the following tools is useful in developing relationships of numbers to 100 and beyond? Relationship cards Dot cards Ten-frame Hundreds chart
Hundreds chart Many of the choices (ten-frames, dot cards) do not go up to 100 (unless used in multiple copies). The hundreds chart sets the pattern of the numbers to 100. Blank hundreds charts can be added so that students can fill them in, making charts that go from 200 to 300 and so on to 1000.
Which of the following is not a possible way in which to deal with a remainder in a division situation? It is made into a fraction. It forces the answer to the next whole number. It is discarded (but not left over). It is subtracted from the answer.
It is subtracted from the answer. Subtracting the remainder from the answer is not an accurate way of using the remainder to interpret the situation. All of the other choices are options of how a remainder can be accurately interpreted.
When asked to solve the division problem 143 ÷ 8, a student thinks, "What number times 8 will be close to 143 with less than 8 remaining?" Which strategy is the student using? Repeated subtraction Cluster problems Missing factor Partial products
Missing factor When students use the inverse relationship of multiplication to division, they are using the missing factor strategy.
In the problem, 5 + 6×3/2, what is the first operation which should be performed according to the order of operations? Exponent Multiply Subtract Add
Multiply Using the acronym PEMDAS as a mnemonic device to remember the order of operations, it allows us to see that the first operation required in the problems is multiplication. That is, the acronym calls for the following order: parenthesis, exponents, multiplication, division, addition, and subtraction. However, since the problem does not contain parenthesis or exponents, the first operation required in the problem is multiplication.
What are compatible pairs in addition? Numbers that easily combine to equal benchmark numbers Numbers that have the same number of digits Numbers that easily combine to equal benchmark numbers Numbers that are even Numbers that add or subtract without regrouping
Numbers that easily combine to equal benchmark numbers Compatible numbers are those that, when combined, add up to numbers like 100. For example, 25 + 75 = 100, so 25 and 75 are a compatible pair.
Which of the following assessments can be used to determine students' understanding of base-ten development? Observe students counting out a large collection of objects and see if they are grouping the objects into groups of ten. Observe students skip counting on a hundreds chart. Observe if students can immediately state the value of a quantity on a ten-frame. Observe students counting on from a number less than ten.
Observe students counting out a large collection of objects and see if they are grouping the objects into groups of ten. To demonstrate students' understanding of base ten, they should be observed counting a collection of objects. As they count, you can then see if the materials are being arranged into groups of tens. You will hear as they count if they are saying, for a collection of 32 items, "ten, twenty, thirty, thirty-one, thirty-two." Students who are beginning to integrate the accurate counting and the correct number words are already using the base-ten structure.
A pre-kindergarten teacher is sitting at a table with a small group of students. The teacher pours out a cup of blocks and asks a student to count the blocks with his fingers. The teacher is most likely assessing which skill? One-to-one correspondence Skip counting Counting on Part-whole concepts
One-to-one correspondence One-to-one correspondence is a foundational skill in pre-k and kindergarten which helps children gain meaning from numbers.
Which of the following is a common model to support invented strategies? Geoboard Open number line Sentence strip KWL chart
Open number line The open number line is a powerful tool for thinking about addition and subtraction situations. See Figures 12.14, 12.15, and 12.17 for examples.
Although all of these children would benefit, which of the following children would benefit the most from using a ten-frame? Pedro, who does not know that 8 is 2 away from 10 Latoya, who is having difficulty recognizing and reading the numerals 110 John, who can count 10 objects but has to recount them when asked for the total number of objects Maria, who is unable to count to 10
Pedro, who does not know that 8 is 2 away from 10 The ten-frame is simply a 2 × 5 array in which counters or dots are placed to illustrate numbers. But the ten-frame dramatically shows how much more is needed to make 10. Therefore, it is likely that Pedro will benefit most.
Pairs of students take turns throwing a die. Each time the die is thrown, each student enters that number into one of four boxes he has drawn on his paper. "After throwing the die four times, the students compare numbers. Whoever has the smallest four-digit number wins the game. This game will help students develop their understanding of:" One-to-One Correspondence Function Place Value Ordered Pairs
Place Value Students must decide which box to place each number. In order to make the smallest number possible, a large number thrown on the die should go in the ones or tens box; a small number in the thousands or hundreds box. Students are practicing place value.
For problems that involve joining (adding) or separating (subtracting) quantities, which of the following terms would not describe one of the quantities in the problem? Start Product Change Result
Product Product is a term that describes one of the quantities in a multiplicative problem, not an additive one.
An important early number concept is Part-Part-Whole. Identify the activity below that would provide children with experience in Part-Part-Whole. Create a class graph showing children's favorite ice cream flavor. Use a set of counters and cards for exploring more, less, or same. Read the book Caps for Sale and have children use connecting cubes to make all combinations of the number 6. Use dot cards and have children play War.
Read the book Caps for Sale and have children use connecting cubes to make all combinations of the number 6. Part-Part-Whole activities often focus on a single number for the entire activity. As in the Caps for Sale example, a pair of children might work on building the number 6 throughout the activity with the connecting cubes. They can either build (compose) the designated quantity in two or more parts (this is also known as a "both addends unknown" situation), or else they start with the full amount and separate it into two or more parts (decompose) to find all the options.
Mr. Gomez plans to introduce the Aztec calendar below to his students. Introducing the Aztec calendar will help his students. Understand and use standard algorithms. Recognize that different cultures have made contributions to the field of mathematics. Apply probability and statistics appropriate to the statewide curriculum. Formally assess their knowledge of the interrelationship between society and mathematics.
Recognize that different cultures have made contributions to the field of mathematics. Teaching with calendars of various cultures provides a perspective on how all cultures contribute to the math experience.
Which of the following statements is true about features of worthwhile tasks? Students build up from tasks that are considered low level to ones that are considered high level over the course of a unit on a particular topic. Relevant tasks include ones that are interesting to students and that address important mathematical ideas; they may come from literature, the media, or a textbook. Worthwhile tasks are much like story problems because they are connected to real-life contexts. Tasks that have multiple entry points mean that students have choices about which task they want to solve, and they will have different answers based on the problem they chose.
Relevant tasks include ones that are interesting to students and that address important mathematical ideas; they may come from literature, the media, or a textbook. Relevant tasks can come from any source, but they must engage the learner in studying important mathematical ideas. At all ages and for all students, high-level tasks are appropriate from day one and on every day. The concept of multiple entry points is not about choice of tasks, but about different approaches to the same task. Worthwhile tasks can be in story form, but can also be purely symbolic. Conversely, story problems may be routine and thus not engage students in mathematical inquiry.
Which of the following is not a strategy for solving contextual problems? Think about the answer before solving the problem. Rely on locating key words in the problem. Work a simpler problem. Identify the "hidden question" in a multi-step problem.
Rely on locating key words in the problem. The key-word strategy is an ineffective approach to solving contextual problems. All of the other choices are possible strategies that can be used when solving a one-step or multi-step contextual problem.
Which of the following reasons provides the best justification for why teaching through problem solving is effective for the struggling learner? Students are able to hear other solution strategies, and so have access to strategies that they did have to come up with on their own. Students do not enjoy or benefit from drill and would rather solve one challenging and interesting mathematical task. Students are able to pull from their knowledge base and use a strategy they like, which increases their chance of success and thereby motivates them to solve the problem. There are fewer exercises when each exercise is more complex; the shorter list of problems is more attractive to struggling learners.
Students are able to pull from their knowledge base and use a strategy they like, which increases their chance of success and thereby motivates them to solve the problem. Each of the other answers might have a little bit of truth to them, but there is something not true in each. The key is to have tasks that fit the learner, as this increases students' chance of success, which in turn increases their willingness to attempt the task.
Computational estimation refers to which of the following? Determining an approximate measure without making an exact measurement Approximating the number of items in a collection Substituting close compatible numbers for difficult-to-handle numbers so that computations can be done mentally A guess of what an answer could be
Substituting close compatible numbers for difficult-to-handle numbers so that computations can be done mentally Estimation can refer to three quite different ideas: measurement, quantity, and computation. Computational estimations involve using easier-to-handle parts of numbers or substituting close compatible numbers for difficult-to-handle numbers so that the resulting computations can be done mentally. Measurement estimations determine an approximate measure without making an exact measurement. Quantity estimations approximate the number of items in a collection.
Ms. Dale asked her students to solve the following problem using a 4-step problem solving process: Helen is helping her mother make aprons as Christmas presents for her six aunts. Helen is in charge of buying the buttons for the aprons at the craft store. Each apron needs eight buttons, and her mother wants an extra four buttons. How many total buttons should Helen buy?" As she was grading the students' work, Ms. Dale noticed Justin made the following mistake: How does this example demonstrate the importance of instituting a standard problem-solving process in a math classrooms? Teachers can spot areas of student confusion and plan appropriate reteaching strategies. Students get tired of answering questions and start making mistakes. Students are forced to spend too much time solving a problem. Because students work longer on fewer problems, less practice time for basic skills is available.
Teachers can spot areas of student confusion and plan appropriate reteaching strategies. Because students must show their work and prove their answers in a four-step problem-solving process, teachers are better able to see where they make mistakes and which skills need reteaching.
Which of the following statements about standard algorithms is true? Most countries use the same standard algorithms in mathematics. Teachers should spend a significant amount of time with invented strategies before introducing a standard algorithm. Standard algorithms should be taught without the use of models (such as completely on a symbolic level). Standard algorithms are the only method for adding and subtracting multi-digit numbers.
Teachers should spend a significant amount of time with invented strategies before introducing a standard algorithm. It is important for students to have many experiences with the development and discussion of invented strategies. In every case, the State Standards present the expectation for knowledge of the standard algorithm long after the topic has been introduced through the use of materials and a variety of strategies.
Why is teaching students about the structure of word problems important? The structures help students focus on sense making and the development of the meaning of the operations. The structures help students develop a key-word strategy. The structures will be on the end-of-year test. The structures help students memorize their basic facts.
The structures help students focus on sense making and the development of the meaning of the operations. These categories help students develop a schema to separate important information and to help them make sense of the problems. In particular, students should be explicitly taught these underlying structures so that they can identify important characteristics of the situations and determine when to add or subtract. Students' thinking can be supported by identifying whether a problem fits a "join" or "separate" classification, which helps them understand the meaning of the operations—in this case addition and subtraction. Then, when students are exposed to new problems, the familiar characteristics will assist them in generalizing from similar problems on which they have practiced.
Based on your interpretation of implementing classroom discussions, which of the following teacher actions do you think best supports student learning? The teacher hears an incorrect solution and asks students what they think about the idea. The teacher hears a correct and novel solution and compliments the student for thinking in a different way. The teacher sees an efficient and clever solution during work time and begins the discussion by having that student share her ideas. The teacher hears an incorrect solution and asks the student to re-explain his or her idea.
The teacher hears an incorrect solution and asks students what they think about the idea. Errors and misconceptions, as well as correct solutions, are part of what is discussed and negotiated in the classroom. When a teacher confirms that a solution is correct or implies that it is incorrect, students learn to focus on the teacher for confirmation that a strategy works, rather than relying on their own mathematical reasoning.
Which of the following is a highly successful strategy for students solving subtraction situations? Think Addition Borrowing Partial Sums Take Away
Think Addition Think Addition is a powerful strategy that works well for all students and in particular for students with disabilities.
What is the correct way to say 32 using base-ten language? Thirty-two ones Three and two Three tens and some more Three tens and two ones
Three tens and two ones Base-ten language includes the place value of each digit in the reading of the number. So 32 would be three tens and two ones.
A first-grade teacher wants to use manipulatives to demonstrate basic addition and how numbers are constructed. Which manipulative would be best suited for this lesson? Unifix cubes Five color spinners Pattern blocks Interlocking fraction bars
Unifix cubes Unifix cubes are a standard size, shape and weight so they can be used to demonstrate adding one by one or can interlock to combine numbers in different groupings. Although "pattern blocks" and "interlocking fraction bars" are plausible, as any object can be used to show basic numbers and addition, the teacher risks confusing the student by using pattern blocks or fraction bars of different sizes to represent standard values. "Five color spinners" is incorrect because students can be confused by the use of a spinner divided but not separated into portions to represent separate values.
An example of an extension of students' knowledge of basic facts and place value to solving two-digit addition problems is the: Up Over Ten strategy. Large Addends strategy. Functions strategy. Difference strategy.
Up Over Ten strategy. Students continue to use the Up Over Ten strategy as they make connections between 8 + 4 and 80 + 40, or 8 + 4 and 28 + 34.
What is the best way to help students see the equal sign as a relational symbol? Say it is like a calculator-you see it and it and it gives you the answer. Call it "the answer is" symbol. Tell students it is just like an addition or subtraction symbol. Use the language "is the same as" when you read an equal sign.
Use the language "is the same as" when you read an equal sign. The equal sign is a relational symbol indicating that one side of an equation is equal to the other side of the equation. By using the language "is the same as" or "equals" when you read the symbol, you are reinforcing this definition and this relationship.
Which of the following instructional activities would be an important component of a lesson on addition with regrouping? Reviewing the concept of greater than and less than Using base-ten materials to model the problem Adding basic facts with sums to ten Demonstrating the commutative property of addition
Using base-ten materials to model the problem When teaching regrouping, important components include linking the problem to concrete materials, such as base-ten materials (including real-world contexts), and reinforcing place value with those materials.
Which of the following statements about names for numbers is true? "106" should be read as "one hundred and six." There are many more errors saying the names of three-digit numbers than four-digit numbers. Whenever you refer to a number in the tens, hundreds, or thousands (or beyond), make sure you just say "six," rather than referring to it with its place-value location, such as 6 tens (or 60). When a student writes "three hundred fifty-eight" as "300508," the student may be at an early stage in moving accurately between oral three-digit numbers and written three-digit numbers.
When a student writes "three hundred fifty-eight" as "300508," the student may be at an early stage in moving accurately between oral three-digit numbers and written three-digit numbers. In kindergarten and first grade, students need to connect the base-ten concepts with the oral number names they have repeatedly used. They know the words but have not thought of them in terms of tens and ones. In fact, early on they may want to write twenty-one as 201. The connections between oral and written numbers is not straightforward; some researchers suggest that these early expanded number writing attempts are an early milestone on the route to full understanding (Byrge, Smith, & Mix, 2013).
Use this venn diagram to answer the question that follows. a c b d
a Factors of 28 include both 2 and 4, but not 3, so it would be placed in the space where only 2 and 4 overlap.
A child with number sense is best defined as having: a flexibility with thinking about numbers and their relationships. math as his or her favorite subject. the ability to link math and reading. the ability to write all numbers to 100.
a flexibility with thinking about numbers and their relationships. Children who have number sense have a good intuition about numbers and their relationships. Number sense develops gradually as a result of exploring numbers, flexibly visualizing them in a variety of contexts, and relating them in multiple ways. Number sense develops as students understand the size of numbers, develop multiple ways of thinking about and representing numbers, use numbers as referents, and develop accurate perceptions about the effects of operations on numbers.
When teaching computational estimation, it is important to: declare that the child with the closest estimate is the winner, as a motivation tool. accept a range of reasonable answers. explain that there is one best way to estimate. point out in a class discussion the students who are the farthest "off."
accept a range of reasonable answers. Always accept a range of answers, as that is what making an estimate is about. Students should not guess an answer; rather, they should use strategies and reasoning to come up with an approximate answer.
Multiples of 10, 100, 1000, and occasionally other numbers, such as multiples of 25, are referred to as ____________ numbers. benchmark counting base-ten grouping
benchmark Benchmark numbers are special numbers that focus on ten-structured thinking, that is, flexibility in using the structure of tens in our number system. They are multiples of 10, 100, and 1000 (place value positions) and occasionally other special numbers, such as multiples of 25.
One-to-one correspondence allows young children to easily: recognize two-dimensional shapes. divide numbers. compare quantities. find patterned sets.
compare quantities.. In one-to-one correspondence, the child can match one object to one other object or counting word. So, for each object they count, they match it with a counting word in order.
A pre-place value understanding of number relies on children: counting by tens and ones. counting using teddy bear counters. counting by ones. counting up to 100 with accuracy.
counting by ones. A pre-place value understanding of number is based on a count-by-ones approach to quantity, which means, for example, that the number 18 to them means 18 ones. They are not able to separate the quantity into place-value groups: after counting 18 teddy bears, a young child might tell you that the 1 stands for 1 teddy bear and the 8 stands for 8 teddy bears. Such students have not had enough experiences to realize that we are always grouping by tens. Recall Wright and his colleagues' three levels of understanding: (1) children understand ten as ten ones; (2) children see ten as a unit; and (3) children easily work with units of ten.
One way to effectively model multiplication with large numbers is to: create an area model using base-ten materials. use pennies to connect to money. use repeated addition. use connecting cubes in groups on paper plates.
create an area model using base-ten materials. The area model using base-ten materials is a powerful visual that aligns well with the eventual learning of the standard algorithm.
As reported by the National Mathematics Advisory Panel, what a 5- or 6-year-old child knows about mathematics not only predicts the child's future math achievement, but also forecasts: future social studies achievement. future reading achievement. future ability to write well. future science achievement.
future reading achievement. Research shared by the National Mathematics Advisory Panel has revealed that what a 5- or 6-year-old knows about mathematics also predicts that child's reading achievement.
When asking children to make estimates, it is often helpful to: give three possible ranges of estimates and ask them to pick the one that is reasonable. give a prize to the child who is closest. remind them of the importance of precision. suggest that they should guess any number.
give three possible ranges of estimates and ask them to pick the one that is reasonable. Producing an estimate is a difficult task for young children. They do not easily grasp the concept of "estimate" or "about." To support them, provide a range of options for them to select from. This will help them see that the term "about" can embrace a group of numbers and not a single focused answer. The more diverse the three choices of possible ranges, the easier it will be for students to make a decision on an initial estimation.
Young children tend to have more difficulty learning the relationship of: less than. equal to. more than. greater than.
less than. Though the concept of less is logically related to the concept of more (selecting the set with more is the same as not selecting the set with less), the concept of less proves to be more difficult for children than more. A possible explanation is that young children have many opportunities to use the word more but may have limited exposure to the word less.
Graphing activities are particularly valuable because they give children opportunities to: use different colors to show different choices. pick a favorite. make comparisons of numbers that have meaning to them. vote on important issues.
make comparisons of numbers that have meaning to them. When students create graphs of data, the comparisons made in the responses (How many more students like pizza than quesadillas?) provide opportunities for them to think about the size of quantities that have meaning to them.
When adding 10 on a hundreds chart, the most efficient strategy that demonstrates place value understanding is to: move to the right 10 spaces. move up one row directly above the number. move down one row directly below the number. move to the left 10 spaces.
move down one row directly below the number. Moving down one row directly below the number shows that the student can add 10 to the number without having to use the inefficient count-by-ones strategy.
An effective way in which to support young children's learning of numbers between 10 and 20 and to begin the development of place value is to have the children think of the teen numbers as: numbers that are doubles of other known numbers less than 10. numbers that are less than 100. numbers that are words that are difficult to remember. numbers that are 10 and some more.
numbers that are 10 and some more. A set of 10 should figure prominently in the discussion of the teen numbers. Initially, children do not see a numeric pattern in the numbers between 10 and 20 (especially with the confusion from the names of the teen numbers). Each number between 10 and 20 should be thought about as 10 and some more. This will establish a continuing pattern so that numbers between 20 and 30, for example, are two 10s and some more, and so on.
Teachers and students should orally refer to the manipulatives for ones, tens, and hundreds as: singles, rods, and rafts. ones, tens, and hundreds. little blocks, sticks, and big squares. cubes, longs, and flats.
ones, tens, and hundreds. We may use the terms cube, long, and flat to describe the shape of base-ten materials, because students notice the shape pattern made as each gets 10 times larger. In fact, it is still critical to call these representations "ones, tens, and hundreds," particularly for students with disabilities. We need to consistently name them by the number they represent rather than their shape. This reinforces conceptual understanding and is less confusing for students who may struggle with these concepts.
The three components of relational understanding of place value integrate: standard names for numbers, base-ten names for numbers, and base-ten concepts. counting by ones, counting by tens, and counting by hundreds. unitary, base ten, and counting. oral names for numbers, written names for numbers, and base-ten concepts.
oral names for numbers, written names for numbers, and base-ten concepts. As is shown in Figure 11.3, the three components of a relational understanding of place value are base-ten concepts, oral names for numbers, and written names for numbers. This knowledge must be integrated for students to demonstrate a full understanding of place value.
The following bar diagram is an example of: area model. cluster problems. partial quotients missing factor.
partial quotients The bar diagram represents the recording of a set of repeated subtractions from the dividend through a partial quotients model.
Proficiency with division requires understanding: a mnemonic such as Dead Monkeys Smell Bad (divide, multiply, subtract, and bring down). place value, multiplication, and the properties of the operations. fraction sense. properties of two-dimensional shapes.
place value, multiplication, and the properties of the operations. Place value, multiplication, and the properties of the operations of multiplication and division are the foundational skills of finding whole-number quotients, as also indicated in the Common Core State Standards.
When introducing place value concepts, it is most important that base-ten models for ones, tens, and hundreds be: proportional (model for a ten is 10 times larger than the model for a 1). used in a pocket chart. virtual models (such as computer representations of base-ten blocks). proportional (model for a ten is 10 times larger than the model for a 1). pregrouped (models cannot be taken apart or put together).
proportional (model for a ten is 10 times larger than the model for a 1). An effective base-ten model for ones, tens, and hundreds is one that is proportional. That is, a model for ten is physically 10 times larger than the model for a one, and a hundred model is 10 times larger than the ten model. Proportional materials allow students to check that ten of any given piece is equivalent to one piece in the column to the left (10 tens equals 1 hundred, and so on).
According to the learning trajectory for counting by Clements and Sarama (2009), a child who can count verbally in an accurate order, but not consistently, is called a: counter. producer. reciter. precounter.
reciter A reciter is a child who verbally counts using number words, but not always in the correct order.
Invented strategies are: "right-handed" rather than "left-handed" (students start on the right). generally slower than standard algorithms. digit-oriented rather than number-oriented. the basis for mental computation and estimation.
the basis for mental computation and estimation. Invented strategies build on number sense, which enhances the ability to use computational estimation and do mental math. They are number oriented, left handed, and generally faster than the standard algorithm.
Children who know that the last count word indicates the amount of the set understand: the concept of the cardinality principle. the concept of numeral recognition. the skill of subitizing. the skill of counting on and counting back.
the concept of the cardinality principle When children understand that the last count word they say (when matching counting words with objects) indicates the amount of the set, they know the set's cardinality. That is, they have grasped the cardinality principle.
The following model (grid paper) can be used to represent: the product of two 2-digit factors as the sum of partial products. the open-array model. lattice multiplication. combinations problems.
the product of two 2-digit factors as the sum of partial products. Grid paper is another way to represent the area model, as it shows the partial products in a multi-digit multiplication problem.
Computational estimation is best described as: determining an approximate measure without making an exact measurement. using easy-to-handle parts of numbers or substituting close compatible numbers for difficult-to-handle numbers so that computations can be done mentally or answers can be assessed for reasonableness. a way to figure how many people are in a stadium for the game. guessing the number of items in a collection.
using easy-to-handle parts of numbers or substituting close compatible numbers for difficult-to-handle numbers so that computations can be done mentally or answers can be assessed for reasonableness. Computational estimation is about estimating multiplications and divisions using numbers that can be computed mentally. Computational estimation is used to evaluate answers and carry out mental mathematics in real-world situations.
