Variables, Functions and Equations
An equation is a mathematical way of looking at the relationship between Thus a variable can be considered as a quantity which assumes a variety of values. Students analyze and explain precisely the process of solving an equation. Accept this, but push students to think about the factors of the numbers. Use these two statements to discuss the relationship between the terms factor and.
A quantity is, roughly speaking, something that has — or comes in — units, such as three horses or four and one-half inches or five dots.
Students think of numbers as a stand-in for quantity, an abstraction of it. They will use this concept of number as a starting place for understanding quantity, and, as they move through this unit, finally arrive at a deeper understanding of number.
Each of the activities to follow involves dots, which can be used to represent any discrete quantity. Although the students will be using numbers, surprisingly they will not be seeing the numbers as a part of arithmetic. They will just see each numeral as a symbol for representing different quantities of dots or different quantities consisting of groups of dots. In fact when most students write down the numerical symbols these are not symbols to represent mathematical concepts, but rather symbols to represent discrete things.
The students have not separated their concept of number from the concept of quantities. It is important that the students discover the link between their descriptions of objects, their own actions on objects and their concept of number.
Proportional Relationships Between Two Quantities | jogglerwiki.info
The object can be described as having two groups, where each group has three dots. However the same object can be regrouped into three groups, each having two dots. For the student to understand the process by which they can change the grouping structure and know and understand what stays the same and what changes, they need conceptual tools.
These tools allow students to construct ideas that are relevant to the problem. The ability to represent their ideas requires representational tools. These representational tools provide ways of translating their ideas into a form that can be considered and communicated. In this section the students will learn to distinguish between dots and groups of dots and groups of groups of dots.
They will learn to pay attention to the way the dots are grouped, and will learn to represent these entities using numerals the mathematical symbols for number: Conceptual Tools The basic ideas are that objects can be grouped into groups or multiple groups and that the rearrangement of the groups or the number of objects within a group does not alter the total number of objects.
The importance of dots is that every quantity can be represented by dots, a group of dots or as multiple groups of dots. The students, by rearranging the same quantity of dots in multiple ways, will become aware that the group structure does not influence the total quantity of dots.
A description of the dots requires the student to pay attention to both dots and groups of dots. There are three things that the student must be aware of when examining different groups of dots: Things are the same when they have the same number of groups and each group contains the same number of dots. Things are not the same when the total number of dots is not the same.
Things are the same but not the same when the quantity of dots is the same but the number of groups is different. Representational Tools Students will learn to represent each dot diagram, where circles indicate groups of dots, in two different ways: A number placed in front of the group symbol, e. For example, the diagram to the right can be described in the following way: This should be discouraged initially, so that the student will focus on the meaning associated with the mathematical phrase.
The students will be surprised when they eventually discover that the descriptions they have created can be interpreted in terms of multiplication! Students will be led to articulate the insight that the order followed in describing the grouping possibilities of a quantity — e. However while these both represent the same quantity of dots they are nevertheless not the same thing, this is the concept of equivalence: Thus students are asked to be creative in writing as many different ways as possible of organizing a given set of dots and representing those organizations using mathematical symbols.
It is important in the beginning to keep the different representations separated, as each representation has its own syntax and rules.
Relationships between quantities in equations and graphs (practice) | Khan Academy
Mixing the representations can cause much confusion, as it makes it extremely difficult for the student to learn the different syntaxes associated with each language. Consider the following example where there are eight dots. Eight dots, no groups Word sentence: Students should be led to articulate the insight that the order followed — e. All have exactly two factors one and itself.
Relationships Between Quantities and Reasoning with Equations and Their Graphs
Each can be represented by two rectangular arrays. All of them are prime. What is the smallest number that has both four and six as factors? This question has one right answer the least common multiple for the numbers 4 and 6 is 12 ,but students may arrive at the answer in different ways.
But because 4 and 6 both share 2 as a factor, the least common multiple is less than the product of this pair of numbers. If numbers do not have a common factor, however, then the least common multiple is their product.
To help students think about these ideas, consider presenting additional questions for them to ponder: Can you find pairs of numbers for which the least common multiple is equal to the product of the pair? Can you find pairs of numbers for which the least common multiple is less than the product of the pair?
What do you notice about the least common multiple for pairs that have common factors? What about pairs that do not have common factors? Both stations broadcast the weather at 1: When is the next time the stations will broadcast the weather at the same time?Relations and functions - Functions and their graphs - Algebra II - Khan Academy
This question provides students with a real-world context—telling time—for thinking about a situation that involves numerical reasoning. The problem also provides a problem context for thinking about multiples.
Do you think that it makes sense to split a day into twenty-four hours? Would another number have been a better choice? Why or why not? You might also ask students if they think it makes sense to have an amount other than 60 minutes in each hour, perhaps minutes, for example. What effect would such a decision have on how time is displayed on watches and clocks? How is each number below different from each of the others? Discuss the meanings of the math terms they use and the relationships among them.
This question can be asked for any set of four numbers. As an extension, ask students to choose four numbers for others to consider. Then have them list all the ways the numbers differ from one another. Use their sets of numbers for subsequent class discussions.