# Unit 6Themes

## Equations in One Variable

### Lessons 1-5

This week your student will be learning to visualize, write, and solve equations. They did this work in previous grades with numbers. In grade 6, we often use a letter called a variable to represent a number whose value is unknown. Diagrams can help us make sense of how quantities are related. Here is an example of such a diagram:

Since 3 pieces are labeled with the same variable , we know that each of the three pieces represent the same number. Some equations that match this diagram are and

A solution to an equation is a number used in place of the variable that makes the equation true. In the previous example, the solution is 5. Think about substituting 5 for in either equation: and are both true. We can tell that, for example, 4 is not a solution, because does not equal 15.

Solving an equation is a process for finding a solution. Your student will learn that an equation like can be solved by dividing each side by 3. Notice that if you divide each side by 3, , you are left with , the solution to the equation.

Draw a diagram to represent each equation. Then, solve each equation.

Solution:

## Equal and Equivalent

### Lessons 6-11

This week your student is writing mathematical expressions, especially expressions using the distributive property.

In this diagram, we can say one side length of the large rectangle is 3 units and the other is units. So, the area of the large rectangle is . The large rectangle can be partitioned into two smaller rectangles, A and B, with no overlap. The area of A is 6 and the area of B is . So, the area of the large rectangle can also be written as . In other words, This is an example of the distributive property.

Draw and label a partitioned rectangle to show that each of these equations is always true, no matter the value of the letters.

Solution:

## Expressions with Exponents

### Lessons 12-15

This week your student will be working with exponents. When we write an expression like , we call the exponent. In this example, 7 is called the base. The exponent tells you how many factors of the base to multiply. For example, is equal to . In grade 6, students write expressions with whole-number exponents and bases that are

• whole numbers like
• fractions like
• decimals like
• variables like

Remember that a solution to an equation is a number that makes the equation true. For example, a solution to is 2, since . On the other hand, 1 is not a solution, since does not equal . Find the solution to each equation from the list provided.

List:

Solution:

1. 7, because . (Note that -7 is also a solution, but in grade 6 students aren’t expected to know about multiplying negative numbers.)
2. 3, because
3. 1, because
4. , because means
5. 0.008, because means
6. , because
7. Any number! is true no matter what number you use in place of .
8. 5, because this can be rewritten . What would we have to divide by 9 to get 27? 243, because . .

## Relationships Between Quantities

### Lessons 16-18

This week your student will study relationships between two quantities. For example, since a quarter is worth 25ȼ, we can represent the relationship between the number of quarters, , and their value in cents like this:

We can also use a table to represent the situation.

1 25
2 50
3 75

Or we can draw a graph to represent the relationship between the two quantities:

A shopper is buying granola bars. The cost of each granola bar is $0.75. 1. Write an equation that shows the cost of the granola bars, , in terms of the number of bars purchased, . 2. Create a graph representing associated values of and 3. What are the coordinates of some points on your graph? What do they represent? Solutions 1. . Every granola bar costs$0.75 and the shopper is buying of them, so the cost is .