Math Methods Test 1

An important early number concept is Part-Part-Whole. Identify the activity below that would provide children with experience in Part-Part-Whole.

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. )

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 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. (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.)

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:

Reciter. A reciter is a child who verbally counts using number words, but not always in the correct order.

not a strategy for solving contextual problems

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.)

provides the best justification for why teaching through problem solving is effective for the struggling learner?

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.

Computational estimation

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. )

statements about standard algorithms

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.)

not a common type of invented strategy for addition and subtraction situations

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). )

correct way to say 32 using base-ten language

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.

instructional activities would be an important component of a lesson on addition with regrouping

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. )

statement about multiplication strategy

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- (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.)

One way to effectively model multiplication with large numbers is to:

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. )

When adding 10 on a hundreds chart, the most efficient strategy that demonstrates place value understanding is to:

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 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. )

Proficiency with division requires understanding:

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. )

equation illustrates the distributive property of multiplication over addition

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.)

Which of the following open number sentences represents partition division?

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.)

The benchmark numbers that are most important for very young learners are:

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.

Wright and his colleagues (2006) identified a three-level progression of children's understanding 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.)

A false statement?

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.

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?

Geometric Shape Core The three NRC foundational areas are the Number Core, the Relations Core, and the Operations Core.

following tools is useful in developing relationships of numbers to 100 and beyond?

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.

a good strategy for teaching computational estimation

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.

What are compatible pairs in addition?

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.)

a common model to support invented strategies?

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 (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. )

following statements about names for numbers is true

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).

When teaching computational estimation, it is important to

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. )

Teachers and students should orally refer to the manipulatives for ones, tens, and hundreds as:

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:

oral names for numbers, written names for numbers, and base-ten concepts. ( 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.)

Invented strategies are:

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. (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. )

Young children tend to have more difficulty learning the relationship of:

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. )

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?

"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.

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. 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

a highly successful strategy for students solving subtraction situations?

Think Addition (Think Addition is a powerful strategy that works well for all students and in particular for students with disabilities. )

A child with number sense is best defined as having:

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. )

One-to-one correspondence allows young children to easily:

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.)

equations illustrates the associative property for addition

(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.)

When presenting addition problems, which of the following would you use last?

645 + 354 = square " ("356 + 127 = square ") "39 + 23 = square " "43 + 32 = square " 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.

an example of a student demonstrating the skill of subitizing?

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)

a FALSE statement?

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.

Multiples of 10, 100, 1000, and occasionally other numbers, such as multiples of 25, are referred to as ____________ numbers.

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.

should not be counted as the mathematics lesson for the day?

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 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."

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?

Distributive property of multiplication over addition The distributive property is defined as a (b + c) = (a × b) + (a × c).

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 (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. )

not a possible way in which to deal with a remainder in a division situation ( You can do these but not the one on the back 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. (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

Missing factor (When students use the inverse relationship of multiplication to division, they are using the missing factor strategy. )

Which 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. (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.)

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?

Product (Product is a term that describes one of the quantities in a multiplicative problem, not an additive one.)

A statement that is true about features of worthwhile tasks

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.

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. (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.)

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. (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?

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.)

A pre-place value understanding of number relies on children:

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.

Graphing activities are particularly valuable because they give children opportunities to:

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 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). 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).

Computational estimation is best described as:

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.)