Lesson 13

Integer Quotients

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SWBAT use multiplication and division fact families to determine the signs of integer quotients

Big Idea

Students use their previous knowledge of signed products and fact families to discover the sign of quotients.


10 minutes

In the introduction I will emphasize the essential question.  In short, we will use what we already know about multiplication and multiplication of signed values to determine the signs for integer quotients.


The problems on this page should be a review for the majority of students.  The problems require students to make multiplication and division fact families.  I want to make sure students can make fact families with familiar values so that they can be successful in the problem solving section of the lesson.  It may not be necessary to complete every problem on the page.  I may start by asking students to complete problem one in 15-20 seconds.  If this is achieved with ease, I may ask them to try problem 4 which includes a decimal.  The same goes for problems 5-8.  I may only need to ask students to do problem number 5 and see how successful they are.  If the majority have an easy time with the fact family I will ask them to do problem 8.  

Problems 9-12 provide more practice with fact families.  If time is short, I may not ask students to solve these problems at all.

At the end of the introduction we review the signs of products.   This review will serve as a reminder of the conclusion from the previous day's lesson.  It will also help lead into the next section of this lesson.

Problem Solving

15 minutes

Students work with partners to write multiplication and division fact families for three problems given one of the multiplication facts.  These problems are designed so that students see every combination of positive and negative in a division problem.  Students then answer questions about their findings.  While students are working, it will be very important for me to walk around to make sure that they are creating correct fact families.  Students often forget or neglect to include the negative sign for negative values.  Otherwise, I have purposely chosen simple single digit factors to help make it easier to get to the essential question.  

At the start of this section, I will set a timer for about 5-6 minutes to create a sense of urgency.  Also, I know that I can extend this part of the lesson for 2-3 more minutes.

When nearly everyone has completed the page we will discuss the answers.  I will cold call students based on observations that I have made from their work.  I will initially call on either an incorrect answer or an answer that is somewhat vague.  Taking question for example, a student may answer "They are the same sign". I would then ask either that student or another student to refine the answer by asking "what do you mean by they".  This is to lead students to have an answer with more detail like "The division problems with positive products have a dividend and a divisor that are either both positive or both negative."  (MP3, MP6)


At the end of this part of the lesson, we summarize a general rule for dividing integers.  We will see that it is the same as the rule for multiplication.  (MP7, MP8).

Independent Practice

20 minutes

The first 4 problems of independent practice are problems that should be evaluated only for the sign of the quotient but no calculations are required.  Students put to practice the "discovered" rules of integer quotients.


Problems 5-12 do require evaluation.  Problems 11 and 12 include the division of 3 values.  I expect some students may forget that division should be evaluated from left to right. 


Problems 13-16 required students to evaluate algebraic expressions.

Problem 17 is to get students to see that the planes change in altitude and can be seen as -1200 feet over an average of 4 minutes, resulting in an average of -300 feet per minute.


Problems 18-20 have students apply the mean to integers.  This provides a nice review of integer addition while having students make a real-world use of integer division.

The extension has 8 one-step equations to be solved.  Students are asked to write the equation using fact families and then solve using one of the facts.  These instructions are to help students who reach the extension but are not confident solvers of one-step equations - especially now that negative values are included.  That being said, I will accept any method that a student uses - mental math, inverse operations, guess and check - as long as the student can explain their process.

Exit Ticket

5 minutes

The exit ticket has two open ended short questions that allow students to make up their own problem that result in a positive quotient and a negative quotient.  The third question gives a multiplication fact of positive values and then asks students to determine values that make quotients.  One thing I like about question 3 is that the two blanks for the quotient of -x can be solved in two different ways:  -z / y  or z / -y