Math 391 - Test 3

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.

List the three most important benchmarks for estimating fractions.

0, 1/2, and 1

Mention the 5 constructs that a fraction symbol can represent. Provide an example of each.

1. Part-Whole 2. Measurement 3. Division 4. Operator 5. Ratio

Find 4 ways to explain that 4/6 = 2/3. Describe which of the two show more evidence of conceptual understanding.

1. They are the same because you can simplify 4/6 and get 2/3. 2. 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. 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. 4. 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. The 2nd and 4th responses are more conceptual.

List the 4 errors that Shaughnessy found students make in working with the number line as a model for fractions.

1. They use incorrect notation 2. Change the unit 3. Count the tick marks rather than the space between the marks 4. Count the tick marks that appear without noticing any missing ones

Describe the three models to represent fractions.

Area models Length or number lines Set

Are middle school students generally better at computing with fractions or estimating?

At computing.

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.

Estimation of fraction computation.

Benchmarks. Relative size of unit fractions.

List the three main steps involved in the measurement process.

Decide on the attribute to be measured. Select a unit that has that attribute. Compare the units with the attribute of the object being measured. The number of units required to match the attribute of the object is the measure.

Mention 3 common length or measurement physical models to represent fractions.

Fraction Strips or Cuisenaire Rods Measurement Tools (such as ruler) Folded Paper Strips

Mention 7 tips for teaching estimation.

Help students learn strategies by having them first try a specific approach. Discuss how different students made their estimates. Accept a range of estimates. Do not promote a winning estimate. Encourage students to give a range of estimates that they believe includes the actual measure. Make measurement estimation an ongoing activity. Be precise with your language.

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).

List the 3 experiences in the sequence to support children's development of the measurement process.

Make comparisons Use physical models of measuring units Use measuring instruments

To understand fractions deeply, mention the three types of activities in which students need experiences.

Partitioning Sharing Tasks Iterating

What is perhaps the most familiar and common real context for adding or subtracting fractions?

Ruler

Briefly describe three of the characteristics of the van Hiele Levels.

Sequential: To move on a particular level beyond level 0, students need to move through previous levels in order. Developmental: Developing the characteristics of reasoning of a particular level is a gradual process over a period of time based on experience. Age independent: Developing the characteristics of reasoning of a particular level does not depend completely on age. Experience dependent: Advancement through the levels requires geometric experience.

Why should instruction on addition and subtraction initially focus on solving contextual tasks using area and linear models?

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 describe the four content goals of geometry education.

Shapes and properties: A study of the properties of shapes in two and three dimensions, as well as the relationships build on properties. Transformation Location 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.

Describe a learning progression to develop tasks for students to become fluent in division of fractions.

Start with whole number divided by whole number. Continue with a fraction divided by a whole number. Continue with a whole number divided by a fraction. Continue with a fraction divided by a fraction with the same denominator. Continue with fractions divided by a fraction with different denominators.

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.

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).

What does the textbook recommend for partitioning activities involving area models?

The fractional parts must be the same size, but not necessarily the same shape The number of equal-sized parts that can be partitioned within the unit determines the fractional amount Use different area models.

What does the textbook recommend 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 but any common denominator will work. 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.

Mention two reasons why fractions are concepts that are difficult to understand.

There are many meanings of fractions. Fractions are written in a unique way. Students overgeneralize their whole-number knowledge.

Briefly explain the four misconceptions related to division of fractions.

Thinking the answer should be smaller. Connecting the illustration with the answer. Knowing what the unit is. Writing remainders.

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 needed to fill or match the attribute being measured.

Problems involving the partitive (fair sharing) interpretation of division.

Whole number divided by whole number. Fractions divided by whole number. Whole numbers divided by fractions

Problems involving the measurement interpretation of division.

Whole numbers divided by fractions. Fractions divided by fractions.

Mention 3 advantages of using the area model for multiplying fractions.

works for problems in which partitioning a length can be challenging. provides a powerful visual good model for connecting to the standard algorithm for multiplying fractions.

Briefly describe the three strategies that can be used to teach estimation in measuring activities.

Develop benchmarks or referents. Use "chunking" or subdivisions. Iterate units.

Provide the definition of measurement.

A measurement is a number that indicates a comparison between the attribute of the object being measured and the same attribute of a given unit of measurement.

Briefly describe students' misconceptions when solving problems involving adding and subtracting fractions.

Adding both numerators and denominators. Failing to find common denominators. Difficulty finding common multiples. Difficulty with mixed numbers.

Describe two common misconceptions that students may have about area.

Confusing linear and square units Difficulty conceptualizing the meaning of height and base

What type of manipulative is the most effective for representing fractions greater than 1?

Connecting cubes

Mention 6 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

Provide an example of each of the three activities to support children's development of the measurement of area.

Comparison activities: help students distinguish between size (or area) and shape, length, and other dimensions Using physical models of area units: activities involving "covering" the surface of two-dimensional shapes Developing formulas for area: develop area formulas for some shapes such as a rectangle, parallelogram, etc.

Provide an example of each of the three activities to support children's development of the measurement of length.

Comparison activities: shorter/longer Using physical models of lengths units: asking how long an object is Making and using rulers: students making their own rules and then comparing the results with standard rulers

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.

What type of tasks do some researchers recommend for students to begin developing the meaning of a fraction?

Equal sharing tasks

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.

Mention and briefly discuss the three instructional goals for teaching standard units of measure.

Familiarity with the unit. Ability to select an appropriate unit. Knowledge of relationship between units.

Describe two benefits of the use of nonstandard units for beginning measurement activities.

Focus directly on the attribute being measured. Provide a good rationale for using standard units.

Briefly describe 4 reasons for engaging students in tasks involving estimation in measurement activities.

Focuses on the attribute being measured and the measuring process. Provides an intrinsic motivation for measurement activities. Estimation helps to develop familiarity with the unit. The use of a benchmark to make an estimate promotes multiplicative reasoning.

Mention 5 research based recommendations for teaching fractions.

Give a greater emphasis to number sense and the meaning of fractions Provide a variety of models and contexts to represent fractions. Emphasize that fractions are numbers Spend whatever time is needed for students to understand equivalence Link fractions to key benchmarks and encourage estimation.

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.

How should students solve story problems when learning addition and subtraction with fractions?

Incorporate different addition situations Use a mix of area and linear contexts Use a mix of whole numbers, mixed numbers, and fractions Addition and subtraction situtations Sometimes involve more than two addends Ask students to select a picture or tool to illustrate it and write the symbols that accurately model the situation.

Why should instruction in addition and subtraction of fractions 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.

Level 0: Visualization. The objects of thought are at level 0 are (individual) shapes and what they look like. 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. 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. The products of thought at level 2 are relationships between properties of geometric figures. Level 3: Deduction. Level 4: Rigor

Mention two types of activities that can help children develop familiarity with standard units.

Making comparisons that focus on a single unit Activities that develop personal referents or benchmarks for single units.

Mention three common misconceptions and difficulties students might have about measuring length.

Measuring from the wrong end of the ruler or beginning at 1 instead of 0. Counting the hash marks rather than the spaces (units) Not aligning two objects when comparing them.

Describe a learning progression to develop tasks for students to become fluent in multiplication of fractions.

Multiply a fraction by a whole number. Multiply a whole number by a fraction. Fractions of fractions - no subdivisions. Subdividing the unit parts.

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. Estimation can strengthen understanding of fractions.

Explain the meaning of partitioning. Provide an example.

Partitioning is sectioning a shape into equal-sized parts.

Describe the meaning of spatial sense.

Spatial sense is an intuition about shapes and the relationship between them. It includes the ability to mentally visualize objects and spatial relationships.

Mention 4 benefits of guiding students to develop formulas for measurement, particularly area.

Students gain conceptual understanding of the ideas and relationships involved Students engage in doing mathematics Students who understand where formulas come tend to remember them or are able to derive them Students develop beliefs that mathematics makes sense, see mathematics as an integrated whole, etc.

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.

Describe four misconceptions that students may have about fraction concepts. Describe how to help students 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 fractions 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 manipulative 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 denominators, 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, 1/2 + 1/2 = 2/4. To help, use many visuals and contexts that emphasize estimation and focus on whether answers are reasonable or not.

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 tell 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.

Mention two common errors made by students solving problems involving perimeter and area.

They confuse perimeter and area. They may not understand the "relationship" between area and perimeter.

What does the book recommend about teaching and learning addition and subtraction 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.

Briefly explain the three main misconceptions/ difficulties related to multiplication of fractions.

Treating the denominator the same as in addition/ subtraction problems. Inability to estimate approximate size of the answer. Matching multiplication situations with multiplication (and not division).

Why is it important to use estimation when teaching and learning division of fractions?

Understanding division can be greatly supported by using estimation.

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.

Describe the meaning of unitizing. Provide an example.

Unitizing is the assignment of a unit to an arbitrary quantity or set of objects that may or may not be represented physically.

List the 4 instructional strategies that some researchers 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

What does the textbook recommend 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.

Describe briefly the meaning of measurement estimation.

the process of using mental and visual information to measure or make comparisons without using measuring instruments.