I will begin with the essential question: How can we use the distributive property to expand the expression? The distributive property is a 6th grade common core standard (6.EE.3 and 6.NS.4) however these focus on positive whole numbers. In 7th grade, students apply positive and negative rational numbers to the property.
It would be helpful to start with a simple problem involving constants. The area model could go along with this problem. So in the example below we might have a rectangle whose width is 5 and whose length is 8 + 2 units.
5 (8 + 2)
When we find the area we can multiply the 5 * 8 rectangle and the 5 * 2 rectangle and take the sum of the products. Students will also see this equals 5 * 10.
At our school our students spend about 90 minutes a week on a program called ST MATH. Many have already started solving distributive property problems in this program so I will bring up one of the puzzles in teacher mode from my introduction. Another similar representation can be found here. Either way, it is important to give a nice visual to help students understand how the property works.
I will then go through the examples. To tie the examples to the visual models, example 1 could be written as such: 4(2x + 7) means 4 groups of 2x + 7, so:
2x + 7
2x + 7
2x + 7
2x + 7
Which equals 8x + 28.
But we quickly want to get efficient (MP8) and just write:
4 * 2x + 4 * 7
8x + 28.
As needed I will go back to the models but they get a bit convoluted (in my opinion) when the number get large or negative.
One other note, based on my own understanding and strengths (weakness?), I generally find it easier to have differences re-written as sums before distributing. I also do this with like terms. I personally get confused when working with -7(5-4x) if I do not turn 5 - 4x to 5 + -4x. Since I am more comfortable with this method, I think I will be more effective teaching it. That being said, I need to make sure that my students can recognize all equivalent forms. So -35 + 28x is also 28x - 35, etc...
There are 7 problems here. The first 4 are similar to the first 4 of independent practice and the problems on the exit ticket. I will be very upfront and let my students know that these are the types of problems they should be able to solve in order to be successful with the lesson. The 5th and 6th problems involve a fraction and a decimal. Problem 7 involves the area of a rectangle.
I will set a short time limit for students to solving the first 4 problems. Depending how the intro goes, I may ask them to do more or less. If they are struggling, I may ask them to work on 1 at a time. Either way, each time they solve I want them to explicitly write out the products before simplifying. So for GP1, students should write 15 * 2b + 15 * 3 before simplifying to 30b + 45.
Students now work independently. Before I will be willing to answer questions, I will insist that students look at the guided problem solving section. Each problem (1-6) is very similar to the problems GP1-GP6 in guided problem solving. Of course, students may still make the common mistake of forgetting to distribute fully. For example, they may think 3(2x + 6) equals 6x + 6 when forgetting to multiply 3 * 6.
The last problem 8, is a chance for students to explain their thinking (MP3). This is an important problem because it makes sure students see all of the equivalent forms of an expression. This will be expected of them on the unit assessment and it is also important for their development as mathematicians.
The exit ticket has 4 problems. Again, these are similar to problems already solved at least twice throughout the lesson. The 4th problem brings distributive property together with combining like terms (the previous lesson).
Before we begin the exit ticket, we will briefly discuss how we used the distributive property to expand the expressions.
Students need to be able to solve at least 3 of the 4 correctly to show proficiency with the lesson.