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

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 *not* a solution, because

**Solving** an equation is a process for finding a solution. Your student will learn that an equation like

Here is a task to try with your student:

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

Solution:

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

This is an example of the distributive property.

Here is a task to try with your student:

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

5x+2x=(5+2)x 3(a+b)=3a+3b

Solution:

Answers vary. Sample responses:

This week your student will be working with **exponents**. When we write an expression like **base**. The exponent tells you how many factors of the base to multiply. For example,

- whole numbers like
74 - fractions like
(17)4 - decimals like
7.74 - variables like
x4

Here is a task to try with your student:

Remember that a solution to an equation is a number that makes the equation true. For example, a solution to

n2=49 4n=64 4n=4 (34)2=n 0.23=n n4=116 1n=1 3n÷32=33

List:

Solution:

- 7, because
72=49 . (Note that -7 is also a solution, but in grade 6 students aren’t expected to know about multiplying negative numbers.) - 3, because
43=64 - 1, because
41=4 916 , because(34)2 means(34)⋅(34) - 0.008, because
0.23 means(0.2)⋅(0.2)⋅(0.2) 12 , because(12)4=116 - Any number!
1n=1 is true no matter what number you use in place ofn . - 5, because this can be rewritten
3n÷9=27 . What would we have to divide by 9 to get 27? 243, because27⋅9=243 .35=243 .

This week your student will study relationships between two quantities. For example, since a quarter is worth 25?, we can represent the relationship between the number of quarters,

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:

Here is a task to try with your student:

A shopper is buying granola bars. The cost of each granola bar is $0.75.

- Write an equation that shows the cost of the granola bars,
c , in terms of the number of bars purchased,n . - Create a graph representing associated values of
c andn . - What are the coordinates of some points on your graph? What do they represent?

Solutions

c=0.75n . Every granola bar costs $0.75 and the shopper is buyingn of them, so the cost is0.75n .- Answers vary. One way to create a graph is to label the horizontal axis with "number of bars" with intervals, 0, 1, 2, 3, etc, and label the vertical axis with "total cost in dollars" with intervals 0, 0.25, 0.50, 0.75, etc.
- If the graph is created as described in this solution, the first coordinate is the number of granola bars and the second is the cost in dollars for that number of granola bars. Some points on such a graph are
(2,1.50) and(10,7.50)

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.