Math Exam #3
Briefly describe students' misconceptions when solving problems involving adding and subtracting fractions.
-Adding both numerators and denominators. The most common error in adding fractions is to add both the numerators and the denominators. -Failing to find common denominators. Less common, but still prevalent, is the tendency to just ignore the denominator and add the numerators. -Difficulty finding common multiples. Many students have trouble finding common denominators because they are not able to quickly come up with common multiples of the denominators. -Difficulty with mixed numbers. In a problem like 3 1/4 - 1 3/8, students subtract the smaller fraction from the larger. When given a problem like 4-7/8, students don't know what to do with the fact that one number is not a fraction. In problems like 14 1/2 - 3 1/8, students focus only on the whole-number part of the problem.
Describe the three models to represent fractions.
-Area models show the whole as the area of the defined region and the parts as equal area. The fraction is the part of the area covered as it relates to the whole unit. -Length or number lines show the whole as the unit of distance or length and the parts as equal distance/length. The fraction is the location of a point in relation to 0 and other values on the number line. -Set shows the whole as whatever value is determined as one in the set and the part as equal number of objects. The fraction is the count of objects in the subset as it relates to the defined whole.
Describe a learning progression (sequence) to develop tasks for students to learn to add and subtract fractions.
-Begin with situations using like denominators. - Continue adding and subtracting fractions with unlike denominators with tasks where only one fraction needs to be changed. -Continue with examples in which both fractions need to be changed
Mention 3 common area or region physical models to represent fractions
-Circular Fraction Pieces -Grid or Dot Paper -Rectangular Regions -Pattern Blocks -Fourths on a Geoboard -Paper Folding
Why type of tasks some researchers recommend for students to begin developing the meaning of a fraction?
-Equal sharing tasks
Mention 3 common length or measurement physical models to represent fractions.
-Fraction Strips or Cuisenaire Rods -Measurement Tools (such as a ruler) -Folded Paper Strips
Mention three research-based recommendations for teaching fractions.
-Give a greater emphasis to number sense and the meaning of fractions, rather than rote procedures for manipulating them. -Provide a variety of models and contexts to represent fractions. -Emphasize that fractions are numbers, making extensive use of number lines in representing fractions. -Spend whatever time is needed for students to understand equivalences (concretely and symbolically), including flexible naming of fractions. -Link fractions to key benchmarks and encourage estimation.
Describe a learning progression to develop tasks for students to become fluently in multiplication of fractions.
-Multiply a fraction by a whole number. You can solve by skip counting, called iterating, is the meaning behind a whole number times a fraction. -Multiply a whole number by a fraction. This type of multiplication involves partitioning (finding a part of the whole - not iterating). Stories can be paired with manipulatives to help students understand this type of fraction operation. Counters is an effective tool for finding parts of a whole. -Fractions of fractions - no subdivisions. Be careful to pick tasks where no additional partitioning is required. It is important for students to model and solve these problems in their own way, using whatever models or drawings they choose as long as they can explain their reasoning. -Subdividing the unit parts. This is the most challenging of the situations.
Mention the five constructs that a fraction symbol can represent. Provide an example of each
-Part-Whole: [Using the part-whole construct is an effective starting point for building meaning of fractions]. Part-whole can be shading a region, part of a group of people or part of a length. Ex. 3/5 of the class went on the field trip. -Measurement: [Measurement involves identifying a length and then using that length as a measurement piece to determine the length of an object]. Ex. In the fraction 5/8, you can use the unit fraction 1/8 as the selected length and then count or measure to show that it takes five of those to reach 5/8. -Division: Consider the idea of sharing $10 with 4 people. This means that each person will receive ¼ of the money, or 2 ½ dollars. -Operator: [Fractions can be used to indicate an operation]. Ex. 4/5 of 20 square feet -Ratio: [Ratios can be part-part or part-whole and can also represent the probability of an event such as ¼ can be the probability of an event is one in four]. Ex. The ratio ¾ could be the ratio of those wearing jackets (part) to those not wearing jackets (part) or it could be part-whole, those wearing jackets (part) to those in the class (whole)
To understand fractions deeply, mention the three types of activities in which students need experiences
-Partitioning -Sharing Tasks -Iterating
Describe a learning progression to develop tasks for students to become fluently in division of fractions.
-Start with a whole number divided by a whole number: 14 ÷ 5 -Continue with a fraction divided by a whole number: ½ ÷ 4 -Continue with a whole number divided by a fraction: 4 ÷ 1/3 - Continue with a fraction divided by a fraction with the same denominator: 7/8 ÷ 1/8 - Continue with fractions divided by a fraction with different denominators.
Describe four misconceptions that students may have about fraction concepts. Describe how to help student to overcome the misconceptions
-Students think that the numerator and denominator are separate values and have difficulty seeing them as a single value. To help, find fraction values on a number line and avoid saying "three out of four" or "three over four". Instead say "three fourths" -Students do not understand that 2/3 means two equal-sized parts. To help, ask students to create their own representations of fractions across various manipulatives and on paper. -Students think that a fraction such as 1/5 is smaller than a fraction such as 1/10 because 5 is less than 10. Conversely, students may be told the reverse-the bigger the denominator, the smaller the fraction. This may cause students to overgeneralize that 1/5 is more than 7/10. To help, use many visuals and contexts that show parts of the whole. -Students mistakenly use the operation "rules" for whole numbers to compute with fractions. For example, ½ + ½ = 2/4. To help, use many visuals and contexts that emphasize estimation and focus on whether answers are reasonable or not.
Mention two reasons why fractions are concepts are difficult to understand.
-There are many meanings of fractions -Fractions are written in a unique way -Students overgeneralize their whole-number knowledge
Find four ways to explain that 4/6= 2/3 Describe which of the two show more evidence of conceptual understanding.
-They are the same because you can simplify 4/6 and get 2/3 **-If you have a set of 6 items and you take 4 of them, that would be 4/6. But you can make the 6 into 3 groups, and the 4 would be 2 groups out of the 3 groups. That means it's 2/3 -If you start with 2/3, you can multiply the top and bottom numbers by 2 and that will give you 4/6, so they are equal. **-If you had a square cut into 3 parts and you shaded 2, that would be 2/3 shaded. If you cut all 3 of these parts in half, that would be 4 parts shaded and 6 parts in all. That's 4/6, and it would be the same amount.
List the four errors that Shaughnessey found students make in working with the number line as a model for fractions.
-They use incorrect notation -Change the unit -Count the tick marks rather than the space between the marks -Count the tick marks that appear without noticing any missing ones
Briefly explain the three main misconceptions/difficulties related to multiplication of fractions.
-Treating the denominator the same as in addition/subtraction problems. In adding, the process is counting parts of a whole, so those parts must be the same size. In multiplication, you are actually finding a part of a part, so the part may change size. -Inability to estimate approximate size of the answer. Some students have been told that multiplication makes bigger so they have difficulty deciding whether their answer make sense. Estimation, contexts, and visuals are needed to better understand fraction multiplication. -Matching multiplication situations with multiplication (and not division). Multiplication and division are closely related, and our language is sometimes not as precise as it needs to be. In the question, "What is 1/3 of $24?" students may correctly decides to divide by 3 or multiply by 1/3 but they may incorrectly divide by 1/3, confusing the idea that they are finding a fraction of the whole.
List the four instructional strategies that some researchers (e.g., Siegler et al., 2010) recommend to effective fraction computation instruction.
-Use contextual tasks -Explore each operation with a variety of models -Let estimation and invented methods play a big role in the development of strategies -Address common misconceptions regarding computational procedures
List the three most important benchmarks for estimating fractions.
0, 1/2, and 1
What does it mean that the "fraction size is relative"?
A fraction by itself does not describe the size of the whole or the size of the parts. A fraction only tells about the relationship between the part and the whole.
Use appropriate fraction language to describe the meaning of a fractional part (e.g., 2/3)
A fractional part is the part that results when the whole or unit has been partitioned into equal-sized portions or fair shares.
Provide the definition of measurement
A measurement is a number that indicates a comparison between the attribute of the object (or situation, or event) being measured and the same attribute of a given unit of measure.
Are middle school students generally better at computing with fractions or estimating?
At computing. [Nearly two-thirds of middle school students could find the exact answer to a problem, but only one-fourth could correctly estimate. Notice that computing an answer requires finding the common denominator, but to estimate requires no computation whatsoever.]
What type of manipulative is the most effective for representing fractions greater than 1?
Connecting cubes
What is the main difference between modeling story problems involving addition and subtractions of fractions with Cuisenaire rods and drawing circles?
Cuisenaire rods are linear models. The first decision that must be made is what strip to use as the whole. That decision is not required with a circular model, where the whole is already established as the circle. The whole must be the same for both fractions.
How do the textbook's authors determine equivalent fractions by multiplying by 1?
Equivalence is based on the multiplicative identity (any number multiplied by 1 remains unchanged). Any fraction of the form n/n can be used as the identity element. Therefore, 3/4 = 3/4 x 1 = 3/4 x 2/2 = 6/8.
It is a challenge for students to understand that larger units will produce a smaller measure and vice versa. Design a task to confront students with this idea
Have students measure a length with a specific unit. Then have them predict the measure of the same length with a unit that is, for example, half or twice as long as the original unit. Discussion of their predictions should follow
Mention three advantages of using the area model for multiplying fractions.
It works for problems in which partitioning a length can be challenging -It provides a powerful visual to show that a result can be quite a bit smaller than either of the fractions used or that if the fractions are both close to 1, then the result is also close to 1. -It is a good model for connecting to the standard algorithm for multiplying fractions.
Explain the meaning of iterating. Provide an example.
Iterating is counting fractional parts, which helps students understand the relationship between the parts (numerator) and the whole (denominator). For example, students need to understand that ¾ can be thought of as a count of three parts called fourths
Why instruction in addition and subtraction of fractions should start with invented strategies?
Just like with whole numbers, invented strategies are important for students because they build on student understanding of fractions and fraction equivalence, and they can eventually be connected to the standard algorithm in such a way that the standard algorithm makes sense.
Succinctly describe the first four van Hiele levels of geometric thought (levels 0-3)
Level 0: Visualization. The objects of thought are at level 0 are (individual) shapes and what they look like. At this level, students only recognize and name figures based on the global characteristics of the figure rather than by their parts or properties. Appearance is dominant at this level, so students will sort or classify shapes based on their physical appearance. The products of thought are classes or grouping of shapes that seem to be "alike." Level 1: Analysis: The objects of thought at level 1 are classes of shapes rather than individual shapes. Students reasoning at this level begin to discern properties of figures. The figures are recognized as having parts and are recognized by their parts. The irrelevant features (such as size and orientation) fade into the background). [Students operating at level 1 may be able to list all the properties of squares, rectangles, and parallelograms, but may not see that these are subclasses of one another - that all squares are rectangles and all rectangles are parallelograms]. The products of thought at level 1 are properties of figures. Level 2: Informal deduction: The objects of thought at level 2 are the properties of figures. At his level students are able to develop relationships among properties of figures. They can engage in if-then reasoning and thus classify figures using only a minimum set of defining characteristics [Definitions are meaningful, class inclusion is understood, and informal arguments are developed and followed. Formal proofs can be followed but students can't construct arguments based on different premises]. The products of thought at level 2 are relationships between properties of geometric figures.
Describe briefly the meaning of measurement estimation
Measuring estimation is the process of using mental and visual information to measure or make comparisons without using measuring instruments.
Explain why it is important to engage students in tasks involving estimating with fractions.
Number sense with fractions means that students have some intuitive feel about the relative size of fractions (knowing "about" how big a particular fraction is). Estimation can strengthen understanding of fractions.
Explain the meaning of partitioning. Provide an example.
Partitioning is sectioning a shape into equal-sized parts. For example, when a brownie is divided into 4 equal shares, the parts are called fourths.
What does the textbook recommends for partitioning activities involving area models?
Provide opportunities for students to conceptualize that a) the fractional parts must be the same size, but not necessarily the same shape b) the number of equal-sized parts that can be partitioned within the unitdetermines the fractional amount. Use different area models. [Area should be the first types of models to use in teaching fractional parts. Young students tend to focus on shape, when the focus should be on equal-sized parts.]
What is perhaps the most familiar and common real context for adding or subtracting fractions?
Ruler
How should students solve story problems when learning addition and subtraction with fractions?
Solve problem that incorporate different addition situations (join, compare, etc.); (2) use a mix of area and linear contexts; (3) use a mix of whole numbers, mixed numbers, and fractions; (4) include both addition and subtraction situations; and (5) sometimes involve more than two addends. With each story you use, it is very important to ask students to select a picture or tool to illustrate is and write the symbols that accurately model the situation.
Describe the meaning of spatial sense
Spatial sense is an intuition about shapes and the relationships between them. It includes the ability to mentally visualize objects and spatial relationships.
Explain why addition and subtraction with fractions should begin with situations using like denominators.
Students should focus on the key idea that the units are the same so they can be combined. Iteration connects fraction operations to whole-number operations and explains why the denominator stays the same.
Explain the common-denominator approach to division of fractions.
The common-denominator algorithm relies on the measurement or repeated subtraction concept of division. Once each number is expressed in terms of the same fractional part, the answer is exactly the same as the whole-number problem. The name of the fractional part (the denominator) is no longer important and the problem is one of dividing the numerators.
What does the denominator of a fraction tell us?
The denominator tells what is being counted [indicates the number of equal-sized parts in which the whole has been divided]
Describe the first step (comparison) in the sequence of the three experiences to support children's development of the measurement process.
The goal of the first step is to understand the attribute to be measured by making comparisons based on the attribute (e.g., longer/shorter, heavier/lighter, etc).
Describe the second step (using physical models of measuring units (including estimation) in the sequence of the three experiences to support children's development of the measurement process.
The goal of the second step is for students to understand how filling, covering, or matching of an attribute with physical models of measuring units produces a number called a measure.
What does the numerator of a fraction tells us?
The numerator counts [the number of fractional parts]
Explain why the term reducing fractions is not appropriate.
The phrase reducing fractions implies that the fraction is being made smaller. Fractions are simplified, not reduced.
What does the textbook recommends regarding the use of least common multiple for addition and subtraction computation?
The skill of finding the least common multiple requires having a good command of multiplication facts. Least common denominators are preferred because the computation is more manageable with smaller numbers but any common denominator will work, whether it is the smallest or not. As students' skills improve, finding the smallest multiple is more efficient.
Describe the third step (using measurement instruments) in the sequence of the three experiences to support children's development of the measurement process.
The third step involves using measuring instruments to compare them with physical models to compare how the measuring tool performs the same function.
Why instruction on addition and subtraction should initially focus on solving contextual tasks using area and linear models?
There are area, length, and set models for illustrating fractions. Set models can be confusing in adding fractions, as they can reinforce the adding of the denominator. Therefore, instruction should initially focus on area and linear models.
Briefly explain the four misconceptions related to division of fractions.
Thinking the answer should be smaller. Based on their experiences with whole-number division, students think that when dividing by a fraction, the answer should be smaller. -Connecting the illustration with the answer. Students may understand that 1 1/2 ÷ 1/4 means "How many fourths are in 1 ½?" so they may set out to count how many fourths and get 6. But in recording their answer, they can confuse the fact that they were using fractions and instead record 6/4. -Knowing what the unit is. Students might get an answer such as 3/8, but when you say "3/8 of what?" they don't know. To make sense of division, students must know what the unit is. -Writing remainders. Knowing what the unit is (the divisor) is critical and must be understood in giving the remainder. In a problem like 3 3/8 ÷ 1/4, students are likely to count 4 fourths for each whole (12 fourths) and on more for 2/8 but then not know what to do with the extra eighth. I is important to be sure they understand the measurement concept of division.
Explain the difference between the measurement processes of tiling and iteration
Tiling involves using as many copies of the unit as are needed to fill or match the attribute measured. Iteration, on the other hand, involves using a single copy of a unit as many times as need to fill or match the attribute being measured.
What does the book recommends about teaching and learning addition and subtractions of fractions involving fractions greater than one?
To include mixed numbers in all of your stories and examples and encourage students to solve them in ways that make sense to them.
Why is it important to use estimation when teaching and learning division of fractions?
Understanding division can be greatly supported by using estimation. Will the answer be greater than 1? Will the answer be less 1? The answer should be obvious to someone who understand the meaning of the this operation. Consider what 12 divided by ¼ means. You are actually answering the questions, "How many fourths in 12?" (there are 48 fourths in 12 wholes). Ask students to estimate by asking questions such "About how many halves in 4 1/3?"Reinforce answers of 8 or 9.
What are the two types of prior knowledge that students need to develop the algorithm for adding and subtracting fractions?
Understanding the meaning of fraction symbols and having a strong conceptual foundation of equivalence is critical to operations of fractions.
What does the textbook recommends for partitioning activities with length models?
Use paper strips and number lines for students to partition length models. Provide examples where shaded sections are in different positions and where partitioning isn't already shown to strength students' understanding of equal parts.
List the three experiences in the sequence to support children's development of the measurement process.
a. Make comparisons b. Use physical models of measuring units c. Use measuring instruments
Mention two benefits of guiding students to develop formulas for measurement, particularly area
a) Students gain conceptual understanding of the ideas and relationships involved; b) Students engage in "doing" mathematics; c) Students who understand where formulas come tend to remember them or are able to derive them. d) Students develop beliefs that mathematics makes sense, see mathematics as an integrated whole, etc.
Mention two types of activities that can help children develop familiarity with standard units.
a. Making comparisons that focus on a single unit (variety of ways familiar items can be measured) b. Activities that develop personal referents or benchmarks for single units.
Provide an example of each of the three activities to support children's development of the measurement of area
a. Comparison activities (Answers will vary but the focus should be on comparison activities to help students distinguish between size (or area) and shape, length, and other dimensions. b. Using physical models of area units (Answers will vary but the focus should be on activities involving "covering" the surface of two-dimensional shapes. c. Developing formulas for area (Answers will vary but the focus on activities to develop area formulas for some shapes such a rectangle, parallelogram, rectangle, etc.)
Provide an example of each of the three activities to support children's development of the measurement of length
a. Comparison activities: (Answers vary, but the focus should be on shorter/longer) b. Using physical models of lengths units: (Answers vary but the focus should be on asking how long an object is) c. Making and using rulers (Answers vary but the focus should be on students making their own rules and then comparing the results with standard rulers)
Describe two common misconceptions that students may have about area
a. Confusing linear and square units: One of the major issues with area measurement is thinking about area as the length of two segments rather than the measure of a surface. b. Difficulty conceptualizing the meaning of height and base because the heights of two-dimensional figures are not always measured along an edge.
List the three main steps involved in the measurement process
a. Decide on the attribute to be measured. b. Select a unit that has that attribute. c. Compare the units - by filling, covering, matching, or using some other method- with the attribute of the object being measured. The number of units required to match the attribute of the object is the measure.
Briefly describe the three strategies than can be used to teach estimation in measuring activities.
a. Develop benchmarks or referents. Referents should be things that are easily envisioned by the student. b. Use "chunking" or subdivisions. It is often easier to estimate a chunk than the attribute of the whole object. c. Iterate units. It consists of marking off single units physically or mentally.
Briefly describe two reasons for engaging students in tasks involving estimation in measurement activities.
a. Estimation focus on the attribute being measured and the measuring process. b. Estimation provides an intrinsic motivation for measurement activities. It is interesting to see how close we can come in our estimates. c. When standard units are used, estimation helps to develop familiarity with the unit. d. The use of a benchmark to make an estimate promotes multiplicative reasoning.
Mention and briefly discuss the three instructional goals for teaching standard units of measure
a. Familiarity with the unit. Students should have a basic idea of the size of commonly used units and what they measure. b. Ability to select an appropriate unit. Students should know both what is a reasonable measurement unit in a given situation and the required level of precision. c. Knowledge of relationships between units. Students should know the relationship between commonly used units.
Mention three tips for teaching estimation
a. Help students learn strategies by having them first try a specific approach. b. Discuss how different students made their estimates. c. Accept a range of estimates. d. Do not promote a winning estimate. e. Encourage students to give a range of estimates that they believe includes the actual measure. f. Make measurement estimation an ongoing activity. g. Be precise with your language.
Mention three common misconceptions and difficulties students might have about measuring length.
a. Measuring from the wrong end of the ruler or beginning at 1 instead of 0. b. Counting the hash marks rather than the spaces (units) c. Not aligning two objects when comparing them.
Describe two benefits of the use of nonstandard units for beginning measurement activities.
a. Nonstandard units focus directly on the attribute being measured. b. The use of nonstandard units avoids conflicting objectives in introductory lessons (is the objective about the meaning of the measuring process or to understand meters?) c. Nonstandard units provide a good rationale for using standard units.
Briefly describe three of the characteristics of the van Hiele Levels
a. Sequential: To move to a particular level beyond level 0, students need to move through previous levels in order. The products of thought of one level become the objects of thought of the next level. b. Developmental: Developing the characteristics of reasoning of a particular level is a gradual process over a period of time based on experience. c. Age independent: Developing the characteristics of reasoning of a particular level does not depend completely on age. Thus, a first grader and a high school student could be at level 0. d. Experience dependent: Advancement through the levels requires geometric experience. Student should explore, talk about, and interact with content at the next level while increasing experiences at their current level.
Briefly describe the four content goals of geometry education
a. Shapes and properties: A study or the properties of shapes in two and three dimensions, as well as the relationships built on properties. b. Transformation: A study of translations, reflections, rotations, dilations, the study of symmetry, and the concept of similarity. (You are not responsible for the description) c. Location: A study of coordinate geometry or other ways of specifying how objects are located in plane or space. (You are not responsible for the description) d. Visualization: Recognizing of shapes in the environment, developing relationships between two- and three dimensional objects, and the ability to draw and recognize objects from different viewpoints.
Mention two common errors made by students solving problems involving perimeter and area
a. They confuse perimeter and area. b. They may not understand the "relationship" between area and perimeter. (e.g., they may think that if the perimeter increases then the area increases, which is false)