Distributive Property

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Objective

SWBAT distribute positive integers to simplify expressions by working in randomly assigned pairs.

Big Idea

Students are randomly assigned to groups to distribute positive numbers

Do Now

10 minutes

Students enter silently according to the Daily Entrance Routine. They find a Do Now on their desk with 3 problems: collecting like terms, order of operations, and evaluating through substitution. I select three students with legible handwriting and ask them to do one of the problems on the board and then hurry back to their desk to copy the work. As always, I try to select students who have not yet had an opportunity to put work on the board. After students are given 5 minutes to complete the problems we review together as a class. For each problem, I run through a series of questions like these:

  • Raise your hand if you agree with this answer

This is a check for understanding. If the answer on the board is correct and many students agree, I know many students understand the concept and so I ask,

  • What do you think about the work? Evaluate it. Give your teammate some feedback about one think you like (i.e. how they showed their work step by step) or about something that is confusing. (opportunity to work on MP3 here)

If the answer on the board is incorrect and many students agree with the answer, I have just identified a common mistake and can remediate on the spot.

The first problem presents a good opportunity to discuss the use of the commutative property as well as the fact that mixed numbers have an “invisible” addition sign between them. The second problem is a good opportunity for the review of fraction operations as well as operations with positive and negative fractions. Many students got the last problem correct today and it served as a boost in their confidence and it reaffirmed the idea of practicing difficult questions until they are mastered.

Vocabulary

5 minutes

I distribute Practice Vocabulary Quizzes to students. I remind them that they will have a quiz tomorrow to review the topics we have covered this week. Students are given 4 minutes to complete the quiz and we take 4 minutes to review the answers. The top missed questions for my students were #2 – 4, and #6. Question 2 is missed because many students were under the impression that a term must have a variable. I remind them that terms are separated by addition or subtraction signs. Question 3 is missed because students often confuse the term coefficient with the term constant. I ask students to refer back to their vocab list or their cards to read the definitions for each of these words out loud. Question 4 is missed for two reasons. One is retention; many students learned this word for the first time yesterday. Most students however miss this question because they do not see a number in front of x^4 and assume that is because the coefficient is zero. I ask students what is the product of any number and 0 and they say “zero”. What would that mean about this term? (That it would be zero, which is not likely.) Question 6 is a great opportunity to review the importance of “base” and understanding that without the presence of parentheses, it is only the variable that is being raised to the 4th power. To drive this concept home, I ask students to volunteer a number for x, substitute it and evaluate the expression. This is where they can see that they don’t raise the 2 to the fourth power.

Class Notes

10 minutes

I distribute Cornell Notes. I ask students to raise their hand if they remember learning about the distributive property last year. About half of the students in each class raise their hand. Then I ask, “who remembers learning about it, but has now forgotten how to do it or when to use it?” Many students raise their hand. This tells me that students are becoming braver about being honest about their capabilities. The participation of 7th graders is notoriously low at the beginning of the year and it is a good sign that it is improving.

We begin by defining the word addend as the numbers inside the parentheses. I also point out that the numbers inside of the parentheses will not always have “plus” signs, sometimes they might see “minus” signs.

Next I evaluate the numerical expressions written along the left side-margin on their Cornell notes. First I ask students to help me evaluate using order of operations. We add the two numbers inside the parentheses (5 + 9 = 14) and then I ask, “What operation do we need to perform on 2, outside the parentheses and the 14 we just calculated?” Students correctly answer “multiplication” or I let the know this and ask them to write it into their notes. We get a result of 28. One the second example of the same numerical expression, I show students how to distribute the 2 to each of the addends inside the parentheses ( 2(5+9) = 2*5 + 2*9 = 10 + 18 = 28) and the fact that we get the same answer as the first example. Then I ask, “what if I multiplied by the first addend only, would it still be equal to 28?” This gets students thinking about one of the most common errors made when distributing, not multiplying by the first second addend. Since this is a common mistake, students are asked to always make sure they check their work when distributing by FIRST making sure they multiplied by the first addend (MP6).

We run through each of the algebraic example, showing the distribution with arrows (as shown in attached image) as well as using a visual organizer.

Lastly, students are assigned numbers 1 – 23 or 24. These numbers are entered into a random number generator to assign groups working in booths. After all booth seats have been assigned, students who are left are welcomed to work in groups in the center of the room.

Task

20 minutes

During the task, I will have answer sheets for part 1.Students will raise their hands when they are done with part 1 and receive an answer sheet and an achievement point. If students complete the second part of the task, they are to turn it in at the end of class for a possibility of earning a homework pass if all questions are answered correctly.

Question 9 is a popularly missed question in its inclusion of an exponent. Many students will square the seven rather than writing 7m^2. 

Closing

5 minutes

When there are 10 minutes of class left, students are asked to return to their seats. We take a vote of the problems they feel the most unsure about and I review the top 3 problems. Students are reminded about Monday’s lesson which showed how to write products in algebra. For example, question 9 can be worked step by step as follows: 7(m^2 +2)+7(m^2 + 2) = 7*m^2 + 7*2 + 7*m^2 + 7*2 = 7m^2 + 14 + 7m^2 + 14 = 14m^2 + 28