Distributing Constants over Linear Expressions
In this activity students will use the integers one through nine along with linear expressions x, x+1, x+2, …, x+9 to figure out how to distribute those integers over the expression. It will satisfy the following Common Core Standard:
CCSS.Math.Content.6.EE.A.3
Apply the properties of operations to generate equivalent expressions.
For each group you will need to print one copy of the numbers one through nine and up to nine copies of the linear expressions. It was fairly simple to make these manipulatives. In TinkerCad I used a square for the numbers and a rectangle for the expressions. I copy and pasted to get ten copies of the square (for the numbers 1-9 and an ‘x’) and nine copies of the rectangle. I placed the numbers and the x on the squares and used the text generator for the expressions.
Start by working through one example with students. Pick a number and an expression and place them together as shown here:
Write your new expression here: 3(x+5)
Now take three copies of the (x+5) as shown here:
How many x letters do you see? Count them to get 3 x letters.
How many constants do you count? 5+5+5 = 15
What is the solution? 3x + 15
Then have students make up three of their own examples to distribute the constant over the expression. To make sure they understand the concept of distributing, ask students to describe how they can re-write 4(x+2) directly without any intermediate steps and how they can re-write any number times (x+ any number) directly without any intermediate steps. A student worksheet and solutions can be found here.
Dividing Fractions
This activity is for sixth grade when students are learning how to divide fractions. It will satisfy the following standard:
CCSS.Math.Content.6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
Before getting to dividing a fraction by another fraction, we will start with dividing a fraction by a whole number. To create a model I used TinkerCad to get rectangles of different sizes. The original “whole” square is 50 mm by 50 mm. Then I made a half, a fourth, and eighth, and a sixteenth of that whole. If you would like to use examples with fractions other than these you can do similar steps with different size fractions.
Depending on what examples students are figuring out will determine how many of each shape you will need to give them. For my examples I need 4 halves, 6 fourths, and 4 eighths. The first example we’ll look at is ½ divided by 4. You can talk students through this first example and then give them additional problems to try in groups. Start with the ½ rectangle and think of dividing it into 4 equal parts. We want to know how big one of those parts is. You can figure out that four of the ⅛ths will fit on the ½. This means ½ divided by 4 = ⅛.
Once students are comfortable using a model to div