Math 3230 Test 2
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? A. Distributive property of multiplication over addition B. Associative property C. Commutative property D. Identity property of multiplication
A. The distributive property is defined as a (b + c) = (a × b) + (a × c).
Language plays and important role in thinking conceptually about division. Identify the statement below that would not support students thinking about the problem 4 ÷ 583. A. Four goes into 5 how many times? B. What number times 4 will get me close to 500? C. What number times 4 will get me close to 583? D. Share 5 hundred, 8 tens, and 3 ones with 4 friends
A. The goes into language does not focus on the whole number, but on the digits and does not help students think in terms of sharing equally.
A common misconception with set models is: A. focusing on the size of the subset rather than the number of equal sets. B. exploring with a variety of models. C. determining the relative size of the numbers. D. partitioning and Iterating.
A. A common misconception with set models is to focus on the size of a subset rather than the number of equal sets in the whole. For example, if 12 counters make a whole, then a subset of 4 counters is one-third, not one-fourth; because 3 equal sets make the whole. However, the set model helps establish important connections with many real-world uses of fractions and with ratio concepts.
A fraction by itself does not describe the size of the whole. A fraction tells us only: A. the relationship between part and whole. B. the numerator is the counting number. C. the sharing of equal size groups. D. the denominator is the size of the piece being counted.
A. A fraction tells us only about the relationship between the part and the whole. Whenever two or more fractions are discussed in the same context, one cannot assume that the fractions are all parts of the same size whole. Teachers can help students understand fractional parts if they regularly ask, "What is the whole?" or "What is the unit?"
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? A. Front-end B. Compatible numbers C. Rounding D. Standard algorithm
A. 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.
Teaching fractions involves using strategies that may not have been part of a teacher's learning experience. What is a key recommendation to teachers from this chapter? A. Use multiple representations, approaches, explanations, and justifications B. Use recognized models-that is, circular and rectangular C. Use algorithms and procedures to address student misconceptions D. Use symbols early and focus on numerator and denominators definitions
A. Hopefully you have recognized that one reason fractions are not well understood is that there is a lot to know about them, from part-whole relationships to division constructs, and understanding includes representing across area, length, and set models and includes contexts that fit these models. Many of these strategies may not have been part of your own learning experience, but they must be part of your teaching experience so that your students can fully understand fractions and be successful in algebra and beyond. For students to reach high levels of understanding they need multiple representations, approaches, and opportunities to explain and justify their thinking.
What is a problem with learning only designated (standard) algorithms for fraction operations? A. Follow a procedure in a short term, but not retain B. Use it to assess whether answer makes sense C. Helps students think conceptually about the operation D. More effective and quicker to learn
A. Learning only one designated algorithm for each operation may seem quicker but is not effective. Students do not think conceptually about the operation and what they mean. When they use a procedure that they do not understand they have no means for knowing if their answer makes sense. In addition, poorly understood algorithms in short term are quickly lost and become meaningless.
Which of the following is important to do before students learn the formal algorithms? A. Address misconceptions. B. Include estimation as well as different ways in which to find the exact answer. C. Use models such as area grids and counters that illustrate the operation. D. Ensure that students have multiple experiences with various contexts.
A. Misconceptions tend to arise after the algorithm is taught and must be addressed as students make errors. The other choices should be part of instruction prior to introducing the formal algorithm.
There are multiple contexts that can guide students understanding of fractions. Which of the following would involve shading a region or a portion of a group of people? A. Part-whole B. Ratio C. Measurement D. Division
A. Part-whole can be shading a region, part of a group of people-that is, three fourths of the class went on the field trip-or part of a length. The part-whole construct is effective as a starting point for meaning of fractions.
The teachers have identified three manipulatives to use when teaching fractional concepts. Each teacher intended to select one manipulative to show each fraction model. Which teacher succeeded in selecting manipulatives for each type? A. Denise selected tangrams, color tiles, and number lines. B. Carla selected number lines, geoboards, and fraction circles. C. Bart selected fraction strips, Cuisenaire rods, and number lines. D. Angela selected 2-color counters, fraction circles, and grid paper.
A. Tangrams are an area model, color tiles are a set model (though they can also be used as an area model if they are used to cover a surface), and number lines are a length model.
Writing fractions in the simplest terms means to write it so: A. fraction numerator and denominator have no common whole number factors. B. fractions with larger denominators. C. fractions are reduced. D. fractions are not improper.
A. Writing a fraction in simplest terms means to write it so that numerator and denominator have no common whole-number factors. The phrase reducing fractions is not used. Because this would imply that the fraction is being made smaller, this terminology should be avoided. Fractions aresimplified, not reduced. Fraction answers can be correct even if written as an improper fraction. They do not have to be made into mixed numbers to be correct.
Which model below would not provide a clear illustration of equivalent fractions? A. Show an algorithm of multiplying the numerator and denominator by the same number B. Cut a paper strip, shade part of the strip, and ask students to use paper folding to describe what fraction of the strip is shaded C. Place a pile of 24 two-color counters with 1/4 showing red under the document camera and ask students to tell you different ways to tell what fraction is red D. Draw a rectangle on grid paper with part of it shaded and ask students to determine the fraction that is shaded while giving different possible answers
A. "Multiply the numerator and denominator by the same number" describes a procedure that results in fractions that are equivalent, but it does not illustrate the idea of equivalence. The other three answers each give one way to illustrate equivalence for each conceptual model (area, length, and set).
Using contextual problems with fraction division works in providing students with an image of what is being: A. computed. B. shared or partitioned. C. proportional to size. D. compared to unit divisors.
B. A sharing or partitioning interpretation works with whole numbers divided by whole numbers and is foundational in the progression for division involving fractions. When students can visualize sharing or dividing equal portions of a sandwich, garden plot or ribbon they begin to see what is happening as the whole or fractional parts are shared or partitioned.
The benefits of using a rectangular area to represent multiplication of fractions include all the following except which? A. They are a good connection to the standard algorithm and to applying the distributive property. B. They are easy for students to draw. C. They can illustrate fractions multiplied by whole numbers as well as multiplication involving mixed numbers. D. They readily show the concept of multiplication of a part of a part.
B. Arrays are not easy to draw because the partitioning may or may not be obvious, depending on the problem. In fact, it is important to discuss with students how they might partition an area so that it fits the multiplication problem.
Lynne used the partitioning strategy to multiply 27 x 4. Which problem below shows this strategy? A. 27 x 4, 27 + 27 + 27 + 27 = B. 27 x 4, 20 x 4 plus 7 x 4 = C. 27 x 4, 27 + 3 becomes 30 x 4 and 3 x 4 will have to be subtracted D. 27 x 4, 7 x 4 and 4 x 2 plus 2 =
B. In the second equation the factor 27 is broken into 20 and 7 and each is multiplied by 4 and then the products added together. This strategy relies on the student's knowledge of the distributive property.
One way to effectively model multiplication with large numbers is to: A. use repeated addition. B. create an area model using base-ten materials. C. use pennies to connect to money. D. use connecting cubes in groups on paper plates.
B. The area model using base-ten materials is a powerful visual that aligns well with the eventual learning of the standard algorithm.
Guiding students to develop a recording scheme for multiplication can be enhanced by the use of what tool? A. Calculator B. Base-ten materials C. Recording sheet with base-ten columns D. An open-array
C. A recording sheet with base-ten columns will support student's use of the distributive property and partial products when they are multiplying. The other materials may provide visual models, but not support the written process.
Research findings support all of the following fraction teaching ideas but one. Which of the following is the unsupported method? A. Ask students to partition rectangles, collections of counters, and paper strips. B. Ask students to connect the symbols to the visuals (showing sixths and writing 1/6) C. Give students area models that are already partitioned and ask them to record the fractional amount shaded. D. Ask students to use partitioning and describe where on a number line a particular fraction is (such as 7/8).
C. Although this may be only a small part of teaching about partitioning, students will not learn what partitioning means if shapes have already been prepartitioned and shaded. This approach is overdone in textbooks and worksheets and teachers must plan to include the many other ideas discussed in this section.
Providing students with many contexts and visuals is essential to their building understanding of equivalence. More examples of linear situations are needed to make comparisons more visible. Which of the following would not be best to model on a number line? A. Height of plants growth B. Distance walked C. Slices of pizza eaten D. Length of hair
C. Because circles are popular fraction tools, and because sharing brings to mind food, many contexts end up being about brownies, cookies and pizza. A number line is one-dimensional; it can make comparisons more visible. The linear contexts of length of hair, growth of plants and distance walked can be shown on a number line.
Mixed numbers: A. should be added using columns, adding whole number parts and fractional parts separately (similar to place value). B. are best introduced after students understand fractions less than 1. C. can be changed into fractions or "improper" fractions and added. D. are easier to subtract than fractions less than 1.
C. Changing mixed numbers into improper fractions is one way in which to remove the need to regroup or rename the fractions, avoiding a step that leads to a lot of confusion and error with students. Mixed numbers should be mixed in with fractions less than one and with fractions written as improper fractions. When different-sized fractions are used, students can more readily generalize ways to add, rather than have a different way to add or subtract different types of numbers.
The goal is to rename a fractional amount. What is the concept that requires the use of many contexts and models? A. Multiplying by one B. Missing number equivalences C. Equivalent fractions D. Magnitude of fractions
C. Equivalence is a critical but often poorly understood concept. This is particularly true with fraction equivalence. This is the first time in students' experience that they are seeing that a fixed quantity can have more than one name (actually an infinite number of names). Area models are a good place to begin understanding equivalence.
Computational estimation refers to which of the following? A. Determining an approximate measure without making an exact measurement B. A guess of what an answer could be C. Substituting close compatible numbers for difficult-to-handle numbers so that computations can be done mentally D. Approximating the number of items in a collection
C. 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.
The way we write fractions with a top and bottom number is a convention. What method focuses on making sense of the parts rather than the symbols? A. Begin by discussing the probability of an event occurring B. Begin by talking about fractions as an operator C. Begin by using words (i.e., one-fourth) D. Begin by measuring length
C. Fraction instruction should begin with use of the words (e.g., one-fourth) rather than the symbols (e.g., ¼). This allows students to first focus on making sense of fractions as parts without the complication of also trying to make sense of the symbols.
What division approach is good for students with learning disabilities that allows them to select facts the already know? A. Explicit trade B. Cluster problems C. Repeated subtraction D. Partial quotients
C. Repeated subtraction allows students to break apart the dividend by using multiples of hundreds and tens. This is especially helpful if they have gaps in recalling some of their basic facts
Locating a fraction on a number line can be challenging but is very important. Which is a common error that students make in working with the number line? A. Parts should be the same shape and size B. Fractions are numbers C. Count the tick marks that appear without noticing any missing ones D. Visuals show all of the partitions
C. Shaughnessy (2011) found four common errors that students make in working with the number line: They use incorrect notation, change the unit (whole), count the tick marks (rather than the space between the marks), and count the tick marks that appear without noticing any missing ones. This is evidence that we must use number lines more extensively in exploring fractions.
Identify which statement below would not be considered a common or limited conception related to fractional parts? A. Knowing and being able to locate fractional parts on the number line, including using incorrect notation or incorrectly counting tick marks. B. Knowing that fractional parts must be the same size and/or that they do not have to be the same shape C. Knowing that answers can be left as fractions rather than writing them as mixed numbers D. Knowing that fractions are numbers in and of themselves (not a number over another number).
C. Students have too many experiences with pieces or parts that are already partitioned where the parts are all the same size and shape. This has resulted in all of the misconceptions here, except the correct option. It is fine for students to write a fraction greater than one either as a single fraction or a mixed number; it should not be considered wrong to leave a fraction written as an improper fraction.
When adding fractions with like denominators it is important for students to focus which key idea? A. Connect fractions to whole numbers B. Know the meaning of numerator and denominator C. Units are the same D. Compare the two quantities
C. The key idea for addition of fractions with like denominators is that the units are the same so they can be combined. For example, this iteration allows students to think about the number of fourths and eighths.
A student says, "My answer must be wrong—my answer got bigger." Which of the following responses will best help the student understand why the answer got bigger? A. Ask the student to round the numbers to the nearest whole number and estimate. B. Tell the student that fractions work the opposite of whole numbers, so with division, the answer gets bigger. C. Ask them to explain the meaning of 8 ÷ 2, using cutting ribbon as a context, and then ask them to re-explain to you using 8 ÷ ½, still using cutting ribbon as a context. D. Show the student a picture of a rectangle partitioned into eighths; tell the student that you are dividing by eighths and they are small, like one-digit numbers, so the answer is bigger.
C. This is a common misconception that students carry from their experiences with whole numbers. You cannot tell students how to fix a misconception: They must have an opportunity to compare whole numbers to fractions and see how they are related, as in the cutting ribbon situation. This one prompt may not be enough, but it is a good start to helping the student see why division by a fraction might result in a larger number (when the divisor is less than 1).
All the following are recommendations for effective fraction computation instruction except: A. Contextual tasks B. Variety of models for each operation C. Carefully introduce procedures D. Estimation and invented methods
C. When students are only taught the procedures but do not understand them they are at risk for success with algebra and beyond. On the other hand when they are learning about fractional operations within a context they begin to build their own methods and understanding. This happens when teachers are providing area, length, and set models to strengthen the students' conceptual understanding.
Which of the following statements about multiplication strategies is true? A. Some multiplication problems can be challenging to solve with invented strategies and students should just use a calculator in those situations. B. Partitioning strategies rely on use of the associative property of multiplication. C. Always think of complex multidigit multiplication problems in the form of repeated addition. D. Cluster problems use multiplication facts and combinations that students already know in order to figure out more complex computations.
D. 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.
A critical aspect of understanding divisions of fractions is: A. the numerator is the unit. B. the partitive situations. C. inverted and multiplied. D. the divisor is the unit.
D. A critical aspect of division of fractions is understanding that the divisor is the unit and this must be understood in order to interpret the remainder. Knowing what the Unit is will help students answer what fractional part of whole they are getting in their product.
When teaching computational estimation, it is important to: A. declare that the child with the closest estimate is the winner, as a motivation tool. B. explain that there is one best way to estimate. C. point out in a class discussion the students who are the farthest "off." D. accept a range of reasonable answers.
D. 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.
All of the methods below would work to support students' knowledge about what is happening when multiplying a fraction by a whole number except: A. Use equal sets to Iterate B. Skip counting by fractional parts i.e. 1/3+1/3+1/3+1/3= C. Equal jumps of length on a ruler D. Compute with a calculator
D. Computing with a calculator would give the product, but not provide a visual or numerical model of what is happening multiplying a fraction and whole number. Skip counting and iterating is the reasoning and meaning behind a whole number times a fraction.
Which of the following best describes how to teach multiplication involving a whole number and a fraction? A. Multiplication is commutative, so these two situations should be taught together using arrays. B. Both should be taught by applying the idea that any number can be written with a one under it and then you have a fraction times a fraction—this helps students see all types of multiplication problems as the same. C. A "fraction times a whole number" and a "whole number times a fraction" are conceptually the same, so they are best taught together. D. A "fraction times a whole number" and a "whole number times a fraction" are conceptually different, so they should be taught separately.
D. Conceptually, a whole number times a fraction is more closely related to multiplying by a whole number, so it should be introduced first. In the CCSS-M, it is introduced in fourth grade; and other types of fraction multiplication is saved for fifth grade.
Research recommends that teachers use one of the following to support students' 39; understanding that fractions are numbers and they expand the number system beyond whole numbers. A. Cuisenaire rods B. Circular number pieces C. Color counters D. Number lines
D. Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades on (Siegler et al., 2010, p. 1).
Which of the following best describes the relationship between iterating and partitioning? A. Iterating and partitioning are inverses of each other. B. Partitioning is the denominator (the size of the parts) and iterating is the numerator (how many parts). C. Iterating is counting by unit fractions and partitioning is grouping unit fractions together. D. Partitioning is finding the parts of a whole, whereas iterating is counting the fractional parts.
D. Partitioning is the physical activity of finding parts of a whole (or several wholes). Iterating is the act of counting, and with fractions that means counting fractional parts.
Proficiency with division requires understanding: A. a mnemonic such as Dead Monkeys Smell Bad (divide, multiply, subtract, and bring down). B. fraction sense. C. properties of two-dimensional shapes. D. place value, multiplication, and the properties of the operations.
D. 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.
All of the following statements are research-based recommendations for teaching and learning about fractions except one. Which one? A. Emphasize the meanings of fractions, rather than rote procedures B. Provide a variety of models and context to represent fractions C. Invest time for students to understand equivalence concretely and symbolically D. Give greater emphasis to specific algorithms for finding common denominators
D. Specific algorithms for comparing two fractions, such as finding a common denominator or using cross-multiplication are procedures and used with no thought about the size of the fractions, and often they are not the most efficient strategy. If students are taught these rules before they have had the opportunity to think about the relative sizes of various fractions, they are less likely to develop number sense (conceptual knowledge) or fluency (procedural knowledge) to effectively compare fractions.
When students use the break apart of decomposition strategy with division, what must they remember? A. Remember that you can decompose the dividend and the divisor B. Remember that you may still have remainders C. Remember that you must record each calculation D. Remember that you cannot break apart the divisor
D. Students must be reminded that the divisor must remain stable. You can only break apart the dividend.
Which of the following strategies would you like students to use when determining which of these fractions is greater 7/8 or 5/6? A. Compare to benchmark of ½ B. Find cross products C. Find a common denominator D. Compare how far from 1
D. The best choice is a reasoning strategy: 7/8 is only 1/8 away from 1 and 5/6 is 1/6 away from one. Therefore, 7/8 is closer. Using a common denominator strategy is not as efficient, so it is not the strategy you would want students to select.
Which of the following is a good explanation for how to add fractions? A. Find the common denominator because one only adds the numerators. B. Join parts and find a common denominator in order to combine correctly. C. Put together through combining the numerators. D. Add equal-sized parts—finding a common denominator can help to solve the problem.
D. The key to addition (and subtraction) is the notion of equal-sized parts. Common denominators are a strategy, but are not required to solve a problem. Also, you cannot combine numerators unless the denominators are the same.
Which of the following options would be misleading for student understanding of fractions? A. Emphasize conceptual understanding by connecting to visuals. B. Use examples that are not just part-whole, but are also measurement and operator situations. C. Design situations to address student misconceptions and help them make distinctions between whole numbers and fractions. D. Tell students that fractions are different from whole numbers, so the procedures are also different.
D. The key word here is "Tell." You cannot tell students to understand something. The "design situations to address student misconceptions and help them make distinctions between whole numbers and fractions" option addresses the issue of whole numbers and fractions and is an effective way to help students understand fractions.
What is the primary reason to not focus on specific algorithms for comparing two fractions? A. Cross-multiplication is too easy B. Not the most efficient strategies C. Common denominators are too hard D. Developing number sense about relative size of fractions is less likely
D. Too often these procedures are done with no thought about the size of the fractions, and often they are not the most efficient strategy. If students are taught these rules before they have had the opportunity to think about the relative sizes of various fractions, they are less likely to develop number sense (conceptual knowledge) or fluency (procedural knowledge) to effectively compare fractions.
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? A. Cluster problems B. Repeated subtraction C. Partial products D. Missing factor
D. When students use the inverse relationship of multiplication to division, they are using the missing factor strategy.
