# Text Reflections

### Chapters 14, 15, 16, & 17

## Chapter 14 Algebraic Thinking: Generalizations, Patterns, and Functions

**Five Forms of Algebraic Reasoning**

*Generalizations from arithmetic and from all of mathematics*

-Algebra is a useful tool for generalizing arithmetic and representing patterns and regularities in our world.

*Meaningful use of symbolism*

-Symbolism, especially involving equality and variables, must be well understood conceptually for students to be successful in mathematics, particularly algebra.

*Study of structure in the number system*

-Methods we use to compute and the structures in our number system can and should be generalized. For example, the generalization that a+b=b+a tells us that 83+27=27+83 without computing the sums on each side of the equal sign.

*Study of patterns and functions*

-Patterns, both repeating and growing, can be recognized, extended, and generalized.

-Functions in k-8 mathematics describe in concrete ways the notion that for every input, there is an output.

*Process of mathematical modeling*

-Understanding of functions is strengthened when they are explored across representations, as each representation provides a different view of the same relationship.

## Chapter 15 Developing Fraction Concepts

**Fraction Constructs**

*Part-Whole*- Part-whole goes beyond shading a region. For example, it could be part of a group of people (3/5 of the class went on the field trip), or it could be part of a length ( we walked 31/2 miles).

*Measuremen*t- Involves identifying a length and then using that length as a measurement piece to determine the length of an object. For example, in te 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*- Should be connected to fractions; understand the idea of sharing $10 with 4 people.

*Operator*- Fractions that are used to indicate an operation, 4/5 of 20 sqaure feet, or 2/3 of an audience.

*Ratio*- can be part-part or part-whole. They are context fractions that are used. For example, 1/4 can mean probability of an event 1 in 4.

**Partitioning**- when you section a shape into equal-sized pieces and it is a major part of developing fraction concepts with young students. Students can partition not only regions or shapes, but lengths and sets too. For example, students can partition sets of objects such as coins, counters, or baseball cards.

**Iteration**- refers to the counting or repetition of a piece. It is an important part of being able to understand and use fractions. Students should be able to understand the concept that 3/4 can be thought of as a count of three parts called fourths.

## Chapter 16 Developing Strategies for Fraction Computation

-It is important to give students ample opportunity to develop fraction number sense prior to and during instruction about common denominators and other procedures for computation.

-Use contextual tasks

-Explore each operation with a variety of models

-Let estimation and informal methods play a big role in the development of strategies.

Address common misconceptions regarding computational procedures.

-For addition and subtraction, the numerator tells the number of parts and the denominator the unit. The parts are added or subtracted.

-Repeated addition and area models support development of concepts and algorithms for multiplication of fractions.

-Partition and measurement models lead to two different thought processes for division of fractions.

-Students need to know how to multiply and divide fractions as well as making estimations.

A Problem-Based Number-Sense ApproachA Problem-Based Number-Sense Approach

-Use contextual tasks

-Explore each operation with a variety of models

-Let estimation and informal methods play a big role in the development of strategies.

Address common misconceptions regarding computational procedures.

-For addition and subtraction, the numerator tells the number of parts and the denominator the unit. The parts are added or subtracted.

-Repeated addition and area models support development of concepts and algorithms for multiplication of fractions.

-Partition and measurement models lead to two different thought processes for division of fractions.

-Students need to know how to multiply and divide fractions as well as making estimations.

## Chapter 17 Developing Concepts of Decimals and Percents

-The decimal point is a convention that has been developed to indicate the units position.

-Decimals are simply another way to write fractions.

-Every decimal idea can be related to an existing fraction idea.

-Percents are simply hundreths and as such are a third way of writing both fractions and decimals.

-Multiplication and division of two numbers will produce the same digits, regardless of the positions of the decimal point.

-Students should become adept at estimating decimal computations well before they learn to compute with pencil and paper. For many decimal computations, rough estimates can be made by rounding the numbers to whole numbers or simple base-ten fractions.-

-Connection Between Fractions and Decimals

-Use familiar fraction concepts and models to explore rational numbers that are easily represented by decimals: tenths, hundredths, thousandths

-Base 10 system can be extended to include numbers less than 1 as well as large numbers

-Use models to make meaningful transitions between fractions and decimals

-Limit the percents to familiar fractions (halve, thirds, fourths, fifths) or easy percents (1/10, 1/100) and use numbers compatible with those fractions.

-Do not suggest any rules or procedures for different types of problems.

-Do use terms part, whole and percent (or fraction). Percent and fraction are interchangeable. Require students to use models or drawings to explain their solutions.

-Decimals are simply another way to write fractions.

-Every decimal idea can be related to an existing fraction idea.

-Percents are simply hundreths and as such are a third way of writing both fractions and decimals.

-Multiplication and division of two numbers will produce the same digits, regardless of the positions of the decimal point.

-Students should become adept at estimating decimal computations well before they learn to compute with pencil and paper. For many decimal computations, rough estimates can be made by rounding the numbers to whole numbers or simple base-ten fractions.-

-Connection Between Fractions and Decimals

-Use familiar fraction concepts and models to explore rational numbers that are easily represented by decimals: tenths, hundredths, thousandths

-Base 10 system can be extended to include numbers less than 1 as well as large numbers

-Use models to make meaningful transitions between fractions and decimals

**Teaching Percents**-Limit the percents to familiar fractions (halve, thirds, fourths, fifths) or easy percents (1/10, 1/100) and use numbers compatible with those fractions.

-Do not suggest any rules or procedures for different types of problems.

-Do use terms part, whole and percent (or fraction). Percent and fraction are interchangeable. Require students to use models or drawings to explain their solutions.