lesson 4 homework practice divide integers

Integers Worksheets

Welcome to the integers worksheets page at Math-Drills.com where you may have a negative experience, but in the world of integers, that's a good thing! This page includes Integers worksheets for comparing and ordering integers, adding, subtracting, multiplying and dividing integers and order of operations with integers.

If you've ever spent time in Canada in January, you've most likely experienced a negative integer first hand. Banks like you to keep negative balances in your accounts, so they can charge you loads of interest. Deep sea divers spend all sorts of time in negative integer territory. There are many reasons why a knowledge of integers is helpful even if you are not going to pursue an accounting or deep sea diving career. One hugely important reason is that there are many high school mathematics topics that will rely on a strong knowledge of integers and the rules associated with them.

We've included a few hundred integers worksheets on this page to help support your students in their pursuit of knowledge. You may also want to get one of those giant integer number lines to post if you are a teacher, or print off a few of our integer number lines. You can also project them on your whiteboard or make an overhead transparency. For homeschoolers or those with only one or a few students, the paper versions should do. The other thing that we highly recommend are integer chips a.k.a. two-color counters. Read more about them below.

Most Popular Integers Worksheets this Week

Integer Resources

Coordinate graph paper can be very useful when studying integers. Coordinate geometry is a practical application of integers and can give students practice with using integers while learning another related skill. Coordinate graph paper can be found on the Graph Paper page:

Coordinate Graph Paper

Integer number lines can be used for various math activities including operations with integers, counting, comparing, ordering, etc.

• Integer Number Lines Integers Number Lines from -10 to 10 Integers Number Lines from -15 to 15 Integers Number Lines from -20 to 20 Integers Number Lines from -25 to 25 OLD Integer Number Lines

Comparing and Ordering Integers

For students who are just starting with integers, it is very helpful if they can use an integer number line to compare integers and to see how the placement of integers works. They should quickly realize that negative numbers are counter-intuitive because they are probably quite used to larger absolute values meaning larger numbers. The reverse is the case, of course, with negative numbers. Students should be able to recognize easily that a positive number is always greater than a negative number and that between two negative integers, the one with the lesser absolute value is actually the greater number. Have students practice with these integers worksheets and follow up with the close proximity comparing integers worksheets.

• Comparing Integers Worksheets Comparing Positive and Negative Integers (-9 to +9) Comparing Positive and Negative Integers (-15 to +15) Comparing Positive and Negative Integers (-25 to +25) Comparing Positive and Negative Integers (-50 to +50) Comparing Positive and Negative Integers (-99 to +99) Comparing Negative Integers (-15 to -1)

By close proximity, we mean that the integers being compared differ very little in value. Depending on the range, we have allowed various differences between the two integers being compared. In the first set where the range is -9 to 9, the difference between the two numbers is always 1. With the largest range, a difference of up to 5 is allowed. These worksheets will help students further hone their ability to visualize and conceptualize the idea of negative numbers and will serve as a foundation for all the other worksheets on this page.

• Comparing Integers in Close Proximity Comparing Positive and Negative Integers (-9 to +9) in Close Proximity Comparing Positive and Negative Integers (-15 to +15) in Close Proximity Comparing Positive and Negative Integers (-25 to +25) in Close Proximity Comparing Positive and Negative Integers (-50 to +50) in Close Proximity Comparing Positive and Negative Integers (-99 to +99) in Close Proximity
• Ordering Integers Worksheets Ordering Integers on a Number Line Ordering Integers (range -9 to 9) Ordering Integers (range -20 to 20) Ordering Integers (range -50 to 50) Ordering Integers (range -99 to 99) Ordering Integers (range -999 to 999) Ordering Negative Integers (range -9 to -1) Ordering Negative Integers (range -99 to -10) Ordering Negative Integers (range -999 to -100)

Adding and Subtracting Integers

Two-color counters are fantastic manipulatives for teaching and learning about integer addition. Two-color counters are usually plastic chips that come with yellow on one side and red on the other side. They might be available in other colors, so you'll have to substitute your own colors in the following description.

Adding with two-color counters is actually quite easy. You model the first number with a pile of chips flipped to the correct side and you also model the second number with a pile of chips flipped to the correct side; then you mash them all together, take out the zeros (if any) and behold, you have your answer! Need further elaboration? Read on!

The correct side means using red to model negative numbers and yellow to model positive numbers. You would model —5 with five red chips and 7 with seven yellow chips. Mashing them together should be straight forward although, you'll want to caution your students to be less exuberant than usual, so none of the chips get flipped. Taking out the zeros means removing as many pairs of yellow and red chips as you can. You can do this because —1 and 1 when added together equals zero (this is called the zero principle). If you remove the zeros, you don't affect the answer. The benefit of removing the zeros, however, is that you always end up with only one color and as a consequence, the answer to the integer question. If you have no chips left at the end, the answer is zero!

• Adding Integers Worksheets with 75 Questions Per Page (Some Parentheses) Adding Integers from -9 to 9 (75 Questions) ✎ Adding Integers from -12 to 12 (75 Questions) ✎ Adding Integers from -15 to 15 (75 Questions) ✎ Adding Integers from -20 to 20 (75 Questions) ✎ Adding Integers from -25 to 25 (75 Questions) ✎ Adding Integers from -50 to 50 (75 Questions) ✎ Adding Integers from -99 to 99 (75 Questions) ✎
• Adding Integers Worksheets with 75 Questions Per Page (All Parentheses) Adding Integers from (-9) to (+9) All Parentheses (75 Questions) ✎ Adding Integers from (-12) to (+12) All Parentheses (75 Questions) ✎ Adding Integers from (-15) to (+15) All Parentheses (75 Questions) ✎ Adding Integers from (-20) to (+20) All Parentheses (75 Questions) ✎ Adding Integers from (-25) to (+25) All Parentheses (75 Questions) ✎ Adding Integers from (-50) to (+50) All Parentheses (75 Questions) ✎ Adding Integers from (-99) to (+99) All Parentheses (75 Questions) ✎
• Adding Integers Worksheets with 75 Questions Per Page (No Parentheses) Adding Integers from -9 to 9 No Parentheses (75 Questions) ✎ Adding Integers from -12 to 12 No Parentheses (75 Questions) ✎ Adding Integers from -15 to 15 No Parentheses (75 Questions) ✎ Adding Integers from -20 to 20 No Parentheses (75 Questions) ✎ Adding Integers from -25 to 25 No Parentheses (75 Questions) ✎ Adding Integers from -50 to 50 No Parentheses (75 Questions) ✎ Adding Integers from -99 to 99 No Parentheses (75 Questions) ✎
• Adding Integers Worksheets with 50 Questions Per Page (Some Parentheses) Adding Integers from -9 to 9 (50 Questions) ✎ Adding Integers from -12 to 12 (50 Questions) ✎ Adding Integers from -15 to 15 (50 Questions) ✎ Adding Integers from -20 to 20 (50 Questions) ✎ Adding Integers from -25 to 25 (50 Questions) ✎ Adding Integers from -50 to 50 (50 Questions) ✎ Adding Integers from -99 to 99 (50 Questions) ✎
• Adding Integers Worksheets with 50 Questions Per Page (All Parentheses) Adding Integers from (-9) to (+9) All Parentheses (50 Questions) ✎ Adding Integers from (-12) to (+12) All Parentheses (50 Questions) ✎ Adding Integers from (-15) to (+15) All Parentheses (50 Questions) ✎ Adding Integers from (-20) to (+20) All Parentheses (50 Questions) ✎ Adding Integers from (-25) to (+25) All Parentheses (50 Questions) ✎ Adding Integers from (-50) to (+50) All Parentheses (50 Questions) ✎ Adding Integers from (-99) to (+99) All Parentheses (50 Questions) ✎
• Adding Integers Worksheets with 50 Questions Per Page (No Parentheses) Adding Integers from -9 to 9 No Parentheses (50 Questions) ✎ Adding Integers from -12 to 12 No Parentheses (50 Questions) ✎ Adding Integers from -15 to 15 No Parentheses (50 Questions) ✎ Adding Integers from -20 to 20 No Parentheses (50 Questions) ✎ Adding Integers from -25 to 25 No Parentheses (50 Questions) ✎ Adding Integers from -50 to 50 No Parentheses (50 Questions) ✎ Adding Integers from -99 to 99 No Parentheses (50 Questions) ✎
• Adding Integers Worksheets with 25 Large Print Questions Per Page (Some Parentheses) Adding Integers from -9 to 9 (Large Print; 25 Questions) ✎ Adding Integers from -12 to 12 (Large Print; 25 Questions) ✎ Adding Integers from -15 to 15 (Large Print; 25 Questions) ✎ Adding Integers from -20 to 20 (Large Print; 25 Questions) ✎ Adding Integers from -25 to 25 (Large Print; 25 Questions) ✎ Adding Integers from -50 to 50 (Large Print; 25 Questions) ✎ Adding Integers from -99 to 99 (Large Print; 25 Questions) ✎
• Adding Integers Worksheets with 25 Large Print Questions Per Page (All Parentheses) Adding Integers from (-9) to (+9) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-12) to (+12) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-15) to (+15) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-20) to (+20) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-25) to (+25) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-50) to (+50) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-99) to (+99) All Parentheses (Large Print; 25 Questions) ✎
• Adding Integers Worksheets with 25 Large Print Questions Per Page (No Parentheses) Adding Integers from -9 to 9 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -12 to 12 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -15 to 15 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -20 to 20 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -25 to 25 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -50 to 50 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -99 to 99 No Parentheses (Large Print; 25 Questions) ✎
• Vertically Arranged Integer Addition Worksheets 3-Digit Integer Addition (Vertically Arranged) 3-Digit Positive Plus a Negative Integer Addition (Vertically Arranged) 3-Digit Negative Plus a Positive Integer Addition (Vertically Arranged) 3-Digit Negative Plus a Negative Integer Addition (Vertically Arranged)

Subtracting with integer chips is a little different. Integer subtraction can be thought of as removing. To subtract with integer chips, begin by modeling the first number (the minuend) with integer chips. Next, remove the chips that would represent the second number from your pile and you will have your answer. Unfortunately, that isn't all there is to it. This works beautifully if you have enough of the right color chip to remove, but often times you don't. For example, 5 - (-5), would require five yellow chips to start and would also require the removal of five red chips, but there aren't any red chips! Thank goodness, we have the zero principle. Adding or subtracting zero (a red chip and a yellow chip) has no effect on the original number, so we could add as many zeros as we wanted to the pile, and the number would still be the same. All that is needed then is to add as many zeros (pairs of red and yellow chips) as needed until there are enough of the correct color chip to remove. In our example 5 - (-5), you would add 5 zeros, so that you could remove five red chips. You would then be left with 10 yellow chips (or +10) which is the answer to the question.

• Subtracting Integers Worksheets with 75 Questions Per Page (Some Parentheses) Subtracting Integers from -9 to 9 (75 Questions) ✎ Subtracting Integers from -12 to 12 (75 Questions) ✎ Subtracting Integers from -15 to 15 (75 Questions) ✎ Subtracting Integers from -20 to 20 (75 Questions) ✎ Subtracting Integers from -25 to 25 (75 Questions) ✎ Subtracting Integers from -50 to 50 (75 Questions) ✎ Subtracting Integers from -99 to 99 (75 Questions) ✎
• Subtracting Integers Worksheets with 75 Questions Per Page (All Parentheses) Subtracting Integers from (-9) to (+9) All Parentheses (75 Questions) ✎ Subtracting Integers from (-12) to (+12) All Parentheses (75 Questions) ✎ Subtracting Integers from (-15) to (+15) All Parentheses (75 Questions) ✎ Subtracting Integers from (-20) to (+20) All Parentheses (75 Questions) ✎ Subtracting Integers from (-25) to (+25) All Parentheses (75 Questions) ✎ Subtracting Integers from (-50) to (+50) All Parentheses (75 Questions) ✎ Subtracting Integers from (-99) to (+99) All Parentheses (75 Questions) ✎
• Subtracting Integers Worksheets with 75 Questions Per Page (No Parentheses) Subtracting Integers from -9 to 9 No Parentheses (75 Questions) ✎ Subtracting Integers from -12 to 12 No Parentheses (75 Questions) ✎ Subtracting Integers from -15 to 15 No Parentheses (75 Questions) ✎ Subtracting Integers from -20 to 20 No Parentheses (75 Questions) ✎ Subtracting Integers from -25 to 25 No Parentheses (75 Questions) ✎ Subtracting Integers from -50 to 50 No Parentheses (75 Questions) ✎ Subtracting Integers from -99 to 99 No Parentheses (75 Questions) ✎
• Subtracting Integers Worksheets with 50 Questions Per Page (Some Parentheses) Subtracting Integers from -9 to 9 (50 Questions) ✎ Subtracting Integers from -12 to 12 (50 Questions) ✎ Subtracting Integers from -15 to 15 (50 Questions) ✎ Subtracting Integers from -20 to 20 (50 Questions) ✎ Subtracting Integers from -25 to 25 (50 Questions) ✎ Subtracting Integers from -50 to 50 (50 Questions) ✎ Subtracting Integers from -99 to 99 (50 Questions) ✎
• Subtracting Integers Worksheets with 50 Questions Per Page (All Parentheses) Subtracting Integers from (-9) to (+9) All Parentheses (50 Questions) ✎ Subtracting Integers from (-12) to (+12) All Parentheses (50 Questions) ✎ Subtracting Integers from (-15) to (+15) All Parentheses (50 Questions) ✎ Subtracting Integers from (-20) to (+20) All Parentheses (50 Questions) ✎ Subtracting Integers from (-25) to (+25) All Parentheses (50 Questions) ✎ Subtracting Integers from (-50) to (+50) All Parentheses (50 Questions) ✎ Subtracting Integers from (-99) to (+99) All Parentheses (50 Questions) ✎
• Subtracting Integers Worksheets with 50 Questions Per Page (No Parentheses) Subtracting Integers from -9 to 9 No Parentheses (50 Questions) ✎ Subtracting Integers from -12 to 12 No Parentheses (50 Questions) ✎ Subtracting Integers from -15 to 15 No Parentheses (50 Questions) ✎ Subtracting Integers from -20 to 20 No Parentheses (50 Questions) ✎ Subtracting Integers from -25 to 25 No Parentheses (50 Questions) ✎ Subtracting Integers from -50 to 50 No Parentheses (50 Questions) ✎ Subtracting Integers from -99 to 99 No Parentheses (50 Questions) ✎
• Subtracting Integers Worksheets with 25 Large Print Questions Per Page (Some Parentheses) Subtracting Integers from -9 to 9 (Large Print; 25 Questions) ✎ Subtracting Integers from -12 to 12 (Large Print; 25 Questions) ✎ Subtracting Integers from -15 to 15 (Large Print; 25 Questions) ✎ Subtracting Integers from -20 to 20 (Large Print; 25 Questions) ✎ Subtracting Integers from -25 to 25 (Large Print; 25 Questions) ✎ Subtracting Integers from -50 to 50 (Large Print; 25 Questions) ✎ Subtracting Integers from -99 to 99 (Large Print; 25 Questions) ✎
• Subtracting Integers Worksheets with 25 Large Print Questions Per Page (All Parentheses) Subtracting Integers from (-9) to (+9) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-12) to (+12) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-15) to (+15) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-20) to (+20) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-25) to (+25) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-50) to (+50) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-99) to (+99) All Parentheses (Large Print; 25 Questions) ✎
• Subtracting Integers Worksheets with 25 Large Print Questions Per Page (No Parentheses) Subtracting Integers from (-9) to 9 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-12) to 12 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-15) to 15 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-20) to 20 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-25) to 25 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-50) to 50 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-99) to 99 No Parentheses (Large Print; 25 Questions) ✎
• Vertically Arranged Integer Subtraction Worksheets 3-Digit Integer Subtraction (Vertically Arranged) 3-Digit Positive Minus a Positive Integer Subtraction (Vertically Arranged) 3-Digit Positive Minus a Negative Integer Subtraction (Vertically Arranged) 3-Digit Negative Minus a Positive Integer Subtraction (Vertically Arranged) 3-Digit Negative Minus a Negative Integer Subtraction (Vertically Arranged)

The worksheets in this section include addition and subtraction on the same page. Students will have to pay close attention to the signs and apply their knowledge of integer addition and subtraction to each question. The use of counters or number lines could be helpful to some students.

• Adding and Subtracting Integers Worksheets with 75 Questions Per Page (Some Parentheses) Adding & Subtracting Integers from -9 to 9 (75 Questions) ✎ Adding & Subtracting Integers from -10 to 10 (75 Questions) ✎ Adding & Subtracting Integers from -12 to 12 (75 Questions) ✎ Adding & Subtracting Integers from -15 to 15 (75 Questions) ✎ Adding & Subtracting Integers from -20 to 20 (75 Questions) ✎ Adding & Subtracting Integers from -25 to 25 (75 Questions) ✎ Adding & Subtracting Integers from -50 to 50 (75 Questions) ✎ Adding & Subtracting Integers from -99 to 99 (75 Questions) ✎
• Adding and Subtracting Integers Worksheets with 75 Questions Per Page (All Parentheses) Adding & Subtracting Integers from (-5) to (+5) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-9) to (+9) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-12) to (+12) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-15) to (+15) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-20) to (+20) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-25) to (+25) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-50) to (+50) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-99) to (+99) All Parentheses (75 Questions) ✎
• Adding and Subtracting Integers Worksheets with 75 Questions Per Page (No Parentheses) Adding & Subtracting Integers from -9 to 9 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -12 to 12 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -15 to 15 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -20 to 20 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -25 to 25 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -50 to 50 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -99 to 99 No Parentheses (75 Questions) ✎
• Adding and Subtracting Integers Worksheets with 50 Questions Per Page (Some Parentheses) Adding & Subtracting Integers from -9 to 9 (50 Questions) ✎ Adding & Subtracting Integers from -12 to 12 (50 Questions) ✎ Adding & Subtracting Integers from -15 to 15 (50 Questions) ✎ Adding & Subtracting Integers from -20 to 20 (50 Questions) ✎ Adding & Subtracting Integers from -25 to 25 (50 Questions) ✎ Adding & Subtracting Integers from -50 to 50 (50 Questions) ✎ Adding & Subtracting Integers from -99 to 99 (50 Questions) ✎
• Adding and Subtracting Integers Worksheets with 50 Questions Per Page (All Parentheses) Adding & Subtracting Integers from (-9) to (+9) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-12) to (+12) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-15) to (+15) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-20) to (+20) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-25) to (+25) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-50) to (+50) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-99) to (+99) All Parentheses (50 Questions) ✎
• Adding and Subtracting Integers Worksheets with 50 Questions Per Page (No Parentheses) Adding & Subtracting Integers from -9 to 9 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -12 to 12 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -15 to 15 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -20 to 20 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -25 to 25 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -50 to 50 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -99 to 99 No Parentheses (50 Questions) ✎
• Adding and Subtracting Integers Worksheets with 25 Large Print Questions Per Page (Some Parentheses) Adding & Subtracting Integers from -9 to 9 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -12 to 12 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -15 to 15 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -20 to 20 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -25 to 25 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -50 to 50 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -99 to 99 (Large Print; 25 Questions) ✎
• Adding and Subtracting Integers Worksheets with 25 Large Print Questions Per Page (All Parentheses) Adding & Subtracting Integers from (-9) to (+9) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-12) to (+12) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-15) to (+15) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-20) to (+20) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-25) to (+25) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-50) to (+50) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-99) to (+99) All Parentheses (Large Print; 25 Questions) ✎
• Adding and Subtracting Integers Worksheets with 25 Large Print Questions Per Page (No Parentheses) Adding & Subtracting Integers from (-9) to 9 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-12) to 12 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-15) to 15 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-20) to 20 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-25) to 25 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-50) to 50 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-99) to 99 No Parentheses (Large Print; 25 Questions) ✎

These worksheets include groups of questions that all result in positive or negative sums or differences. They can be used to help students see more clearly how certain integer questions end up with positive and negative results. In the case of addition of negative and positive integers, some people suggest looking for the "heavier" value to determine whether the sum will be positive of negative. More technically, it would be the integer with the greater absolute value. For example, in the question (−2) + 5, the absolute value of the positive integer is greater, so the sum will be positive.

In subtraction questions, the focus is on the subtrahend (the value being subtracted). In positive minus positive questions, if the subtrahend is greater than the minuend, the answer will be negative. In negative minus negative questions, if the subtrahend has a greater absolute value, the answer will be positive. Vice-versa for both situations. Alternatively, students can always convert subtraction questions to addition questions by changing the signs (e.g. (−5) − (−7) is the same as (−5) + 7; 3 − 5 is the same as 3 + (−5)).

• Scaffolded Integer Addition and Subtraction Positive Plus Negative Integer Addition (Scaffolded) ✎ Negative Plus Positive Integer Addition (Scaffolded) ✎ Mixed Integer Addition (Scaffolded) ✎ Positive Minus Positive Integer Subtraction (Scaffolded) ✎ Negative Minus Negative Integer Subtraction (Scaffolded) ✎

Multiplying and Dividing Integers

Multiplying integers is very similar to multiplication facts except students need to learn the rules for the negative and positive signs. In short, they are:

In words, multiplying two positives or two negatives together results in a positive product, and multiplying a negative and a positive in either order results in a negative product. So, -8 × 8, 8 × (-8), -8 × (-8) and 8 × 8 all result in an absolute value of 64, but in two cases, the answer is positive (64) and in two cases the answer is negative (-64).

Should you wish to develop some "real-world" examples of integer multiplication, it might be a stretch due to the abstract nature of negative numbers. Sure, you could come up with some scenario about owing a debt and removing the debt in previous months, but this may only result in confusion. For now students can learn the rules of multiplying integers and worry about the analogies later!

• Multiplying Integers with 100 Questions Per Page Multiplying Mixed Integers from -9 to 9 (100 Questions) ✎ Multiplying Positive by Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying Negative by Positive Integers from -9 to 9 (100 Questions) ✎ Multiplying Negative by Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying Mixed Integers from -12 to 12 (100 Questions) ✎ Multiplying Positive by Negative Integers from -12 to 12 (100 Questions) ✎ Multiplying Negative by Positive Integers from -12 to 12 (100 Questions) ✎ Multiplying Negative by Negative Integers from -12 to 12 (100 Questions) ✎ Multiplying Mixed Integers from -20 to 20 (100 Questions) ✎ Multiplying Mixed Integers from -50 to 50 (100 Questions) ✎
• Multiplying Integers with 50 Questions Per Page Multiplying Mixed Integers from -9 to 9 (50 Questions) ✎ Multiplying Positive by Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying Negative by Positive Integers from -9 to 9 (50 Questions) ✎ Multiplying Negative by Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying Mixed Integers from -12 to 12 (50 Questions) ✎ Multiplying Positive by Negative Integers from -12 to 12 (50 Questions) ✎ Multiplying Negative by Positive Integers from -12 to 12 (50 Questions) ✎ Multiplying Negative by Negative Integers from -12 to 12 (50 Questions) ✎
• Multiplying Integers with 25 Large Print Questions Per Page Multiplying Mixed Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Positive by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Negative by Positive Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Negative by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Mixed Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying Positive by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying Negative by Positive Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying Negative by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎

Luckily (for your students), the rules of dividing integers are the same as the rules for multiplying:

Dividing a positive by a positive integer or a negative by a negative integer will result in a positive integer. Dividing a negative by a positive integer or a positive by a negative integer will result in a negative integer. A good grasp of division facts and a knowledge of the rules for multiplying and dividing integers will go a long way in helping your students master integer division. Use the worksheets in this section to guide students along.

• Dividing Integers with 100 Questions Per Page Dividing Mixed Integers from -9 to 9 (100 Questions) ✎ Dividing Positive by Negative Integers from -9 to 9 (100 Questions) ✎ Dividing Negative by Positive Integers from -9 to 9 (100 Questions) ✎ Dividing Negative by Negative Integers from -9 to 9 (100 Questions) ✎ Dividing Mixed Integers from -12 to 12 (100 Questions) ✎ Dividing Positive by Negative Integers from -12 to 12 (100 Questions) ✎ Dividing Negative by Positive Integers from -12 to 12 (100 Questions) ✎ Dividing Negative by Negative Integers from -12 to 12 (100 Questions) ✎
• Dividing Integers with 50 Questions Per Page Dividing Mixed Integers from -9 to 9 (50 Questions) ✎ Dividing Positive by Negative Integers from -9 to 9 (50 Questions) ✎ Dividing Negative by Positive Integers from -9 to 9 (50 Questions) ✎ Dividing Negative by Negative Integers from -9 to 9 (50 Questions) ✎ Dividing Mixed Integers from -12 to 12 (50 Questions) ✎ Dividing Positive by Negative Integers from -12 to 12 (50 Questions) ✎ Dividing Negative by Positive Integers from -12 to 12 (50 Questions) ✎ Dividing Negative by Negative Integers from -12 to 12 (50 Questions) ✎
• Dividing Integers with 25 Large Print Questions Per Page Dividing Mixed Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Positive by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Negative by Positive Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Negative by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Mixed Integers from -12 to 12 (25 Questions; Large Print) ✎ Dividing Positive by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎ Dividing Negative by Positive Integers from -12 to 12 (25 Questions; Large Print) ✎ Dividing Negative by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎

This section includes worksheets with both multiplying and dividing integers on the same page. As long as students know their facts and the integer rules for multiplying and dividing, their sole worry will be to pay attention to the operation signs.

• Multiplying and Dividing Integers with 100 Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (100 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (100 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (100 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (100 Questions) ✎
• Multiplying and Dividing Integers with 75 Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (75 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (75 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (75 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (75 Questions) ✎
• Multiplying and Dividing Integers with 50 Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (50 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (50 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (50 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (50 Questions) ✎
• Multiplying and Dividing Integers with 25 Large Print Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (25 Questions; Large Print) ✎

All Operations with Integers

In this section, the integers math worksheets include all of the operations. Students will need to pay attention to the operations and the signs and use mental math or another strategy to arrive at the correct answers. It should go without saying that students need to know their basic addition, subtraction, multiplication and division facts and rules regarding operations with integers before they should complete any of these worksheets independently. Of course, the worksheets can be used as a source of questions for lessons, tests or other learning activities.

• All Operations with Integers with 50 Questions Per Page (Some Parentheses) All operations with integers from -9 to 9 (50 Questions) ✎ All operations with integers from -12 to 12 (50 Questions) ✎ All operations with integers from -15 to 15 (50 Questions) ✎ All operations with integers from -20 to 20 (50 Questions) ✎ All operations with integers from -25 to 25 (50 Questions) ✎ All operations with integers from -50 to 50 (50 Questions) ✎ All operations with integers from -99 to 99 (50 Questions) ✎
• All Operations with Integers with 50 Questions Per Page (All Parentheses) All operations with integers from (-9) to (+9) All Parentheses (50 Questions) ✎ All operations with integers from (-12) to (+12) All Parentheses (50 Questions) ✎ All operations with integers from (-15) to (+15) All Parentheses (50 Questions) ✎ All operations with integers from (-20) to (+20) All Parentheses (50 Questions) ✎ All operations with integers from (-25) to (+25) All Parentheses (50 Questions) ✎ All operations with integers from (-50) to (+50) All Parentheses (50 Questions) ✎ All operations with integers from (-99) to (+99) All Parentheses (50 Questions) ✎
• All Operations with Integers with 50 Questions Per Page (No Parentheses) All operations with integers from -9 to 9 No Parentheses (50 Questions) ✎ All operations with integers from -12 to 12 No Parentheses (50 Questions) ✎ All operations with integers from -15 to 15 No Parentheses (50 Questions) ✎ All operations with integers from -20 to 20 No Parentheses (50 Questions) ✎ All operations with integers from -25 to 25 No Parentheses (50 Questions) ✎ All operations with integers from -50 to 50 No Parentheses (50 Questions) ✎ All operations with integers from -99 to 99 No Parentheses (50 Questions) ✎

Order of operations with integers can be found on the Order of Operations page:

Order of Operations with Integers

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Division of integers

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Negative numbers & division

Division of or by negative numbers can be conceptually difficult; these integer worksheets provide extra practice in both normal and long division form .

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Long division:

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Division of Integers Exercises

Dividing integers practice problems with answers.

There are ten (10) practice questions below about dividing integers . Keep practicing until you master the skill! Good luck!

For your convenience, I included a summary on how to divide integers. The main idea is that if you divide two integers with the same sign , the answer is positive . However, if the signs of the two integers are different , the answer is negative .

Problem 1: Divide the integers: [latex]21 \div 7[/latex]

[latex]3[/latex]

Problem 2: Divide the integers: [latex]\left( { – 42} \right) \div 6[/latex]

[latex]-7[/latex]

Problem 3: Divide the integers: [latex]36 \div \left( { – 4} \right)[/latex]

[latex]-9[/latex]

Problem 4: Divide the integers: [latex]\left( { – 54} \right) \div \left( { – 9} \right)[/latex]

[latex]6[/latex]

Problem 5: Divide the integers: [latex]\left( { – 144} \right) \div 6[/latex]

[latex]-24[/latex]

Problem 6: Divide the integers: [latex]\left( { – 19} \right) \div \left( { – 19} \right)[/latex]

[latex]1[/latex]

Problem 7: Divide the integers: [latex]132 \div 12[/latex]

[latex]11[/latex]

Problem 8: Divide the integers: [latex]189 \div \left( { – 9} \right) \div \left( { – 7} \right)[/latex]

Problem 9: Divide the integers: [latex]120 \div \left( { – 2} \right) \div 5[/latex]

[latex]-12[/latex]

Problem 10: Divide the integers: [latex]\left( { – 96} \right) \div \left( { – 4} \right) \div \left( { – 6} \right)[/latex]

[latex]-4[/latex]

You might also like these tutorials:

• Dividing Integers

Division of Integers

Related Topics: Lesson Plans and Worksheets for Grade 7 Lesson Plans and Worksheets for all Grades More Lessons for Grade 7 Common Core For Grade 7

Examples, videos, and solutions to help Grade 7 students recognize that division is the reverse process of multiplication, and that integers can be divided provided the divisor is not zero.

New York State Common Core Math Grade 7, Module 2, Lesson 12

Worksheets for Grade 7

Lesson 12 Student Outcomes

• Students recognize that division is the reverse process of multiplication, and that integers can be divided provided the divisor is not zero.
• Students understand that every quotient of integers (with a non-zero divisor) is a rational number and divide signed numbers by dividing their absolute values to get the absolute value of the quotient. The quotient is positive if the divisor and dividend have the same signs and negative if they have opposite signs.

Lesson 12 Summary

• The rules for dividing integers are similar to the rules for multiplying integers (when the divisor is not zero). The quotient is positive if the divisor and dividend have the same signs, and negative if they have opposite signs. The quotient of any 2 integers (with a non-zero divisor) will be a rational number.

NYS Math Module 2 Grade 7 Lesson 12 Classwork

Exercise 1: Recalling the Relationship Between Multiplication and Division a. List examples of division problems that produced a quotient that is a negative number. b. If the quotient is a negative number, what must be true about the signs of the dividend and divisor? c. List your examples of division problems that produced a quotient that is a positive number. d. If the quotient is a positive number, what must be true about the signs of the dividend and divisor?

Rules for Dividing Two Integers:

• A quotient is negative if the divisor and the dividend have _____ signs.
• A quotient is positive if the divisor and the dividend have ____ signs.

Exercise 2: Is the Quotient of Two Integers Always an Integer Is the quotient of two integers always an integer? Use the work space below to create quotients of integers. Answer the question and use examples or a counterexample to support your claim. Conclusion: Every quotient of two integers is always a rational number, but not always an integer.

Exercise 3: Different Representation of the Same Quotient Are the answers to the three quotients below the same or different? Why or why not? a) -14 ÷ 7 b) 14 ÷ -7 c) -(14 ÷ 7)

Exercise 4: Fact Fluency—Integer Division

Lesson 12 Problem Set

• Find the missing values in each column:

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M and D Integers

Multiplying and dividing integers

Here you will learn strategies on how to multiply and divide integers, including using visual models as well as using the number line.

Students will first learn about integers in 6th grade math as part of their work with the number system and expand that knowledge to operations with integers in the 7th grade.

What are multiplying and dividing integers?

Multiplying and dividing integers is when you multiply or divide two or more integers together to give a product or quotient that can be either positive or negative.

You can multiply and divide integers using visual models or a rule.

Do you notice a pattern or rule?

Rule for multiplying integers:

• If the integers have the same sign , the product will be positive .
• If the integers have different signs , the product will be negative . Multiply the absolute values and make the answer negative.

Rule for dividing integers:

• If the integers have the same sign , the quotient will be positive .
• If the integers have different signs , the quotient will be negative . Divide the absolute values and make the answer negative.

Common Core State Standards

How does this apply to 6th grade math and 7th grade math?

• Grade 6: Number System (6.NS.C.6) Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
• Grade 7: Number System (7.NS.A.2) Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

[FREE] Multiplication and Division Check for Understanding (Grade 4, 5 and 7)

Use this quiz to check your grade 4, 5 and 7 students’ understanding of multiplication and division. 10+ questions with answers covering a range of 4, 5 and 7 grade multiplication and division topics to identify areas of strength and support!

How to multiply and divide integers?

In order to add and subtract integers using counters:

• If the integers have the same sign, the product or quotient is positive.  If not, go to step 2.

If the integers have different signs, the product or quotient is negative.

Find the product or quotient.

Multiplying and dividing integers examples

Example 1: multiplying integers with the same sign.

Multiply: (-4) \times(-12)= \, ?

• If the integers have the same sign, the product or quotient is positive. If they don’t go to step 2. 

-4 and -12 have the same sign so the product is positive.

2 If the integers have different signs, the product or quotient is negative.

Integers have the same sign.

3 Find the product or quotient.

(-4) \times(-12)=48

Example 2: multiplying integers with different signs

Multiply: (-13) \times 8= \, ?

If the integers have the same sign, the product or quotient is positive.  If not, go to step 2.

-13 and 8 do not have the same sign.

The integers have different signs so the product is negative.

Find the product or the quotient.

(-13) \times 8=-104

Example 3: dividing integers with the same sign

Divide: \cfrac{(-18)}{(-3)}= \, ?

-18 and -3 have the same sign so the quotient will be positive.

\cfrac{(-18)}{(-3)}=6

Example 4: dividing integers with different signs

Divide: -120 \div 3= \, ?

-120 and 3 do not have the same sign.

-120 and 3 have different signs so the quotient will be negative.

-120 \div 3=-40

Example 5: multiplying integers word problem

From sea level, a submarine descends 25 \, ft. per minute (-25 \, ft.).

After 6 minutes, the submarine’s distance can be modeled by (-25) \times 6 = \, d , where d is the submarine in relation to sea level.

How far below sea level is the submarine?

-25 and 6 do not have the same sign.

-25 and 6 have different signs so the product will be negative.

-25 \times 6=-150

Because the submarine is descending, after 6 minutes it will be 150 feet below sea level.

Example 6: dividing integer word problems

On a certain winter day, the temperature changed at a rate of -4 degrees Fahrenheit per hour.

After a specific amount of time, the change in temperature was -36 degrees Fahrenheit, which is modeled by (-36) \div (-4) = \, h, where h represents the amount of hours.

How long did it take for the change in temperature to be -36 degrees Fahrenheit?

-36 and -4 have the same sign so the quotient is positive.

The integers have the same sign.

(-36) \div(-4)=9

Teaching tips for multiplying and dividing integers

• Multiplying and dividing integers are foundational skills for Algebra 1. Using manipulatives helps students formulate conceptual understanding.
• Have students identify the patterns with multiplying and dividing integers so that they can figure out the rules on their own.
• Although practice integer worksheets have their place, have students practice problems through digital games or scavenger hunts around the room to make it engaging.
• Reinforce essential vocabulary such as dividend, divisor, quotient, factors,  and product.

Easy mistakes to make

• Mixing up the rules for multiplication and division For example, when multiplying integers with the same sign, you get a negative product, and when dividing integers with the same sign, you get a negative quotient.
• Mixing up the rules of multiplication and division with addition and subtraction For example, applying the rule of addition of integers to (15) \div(-3)=5 where the absolute value of 15 is greater than the absolute value of 3 so the answer must be positive.

Related multiplication and division lessons

This multiplying and dividing integers topic guide is part of our series on multiplication and division. You may find it helpful to start with the main multiplication and division topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other topic guides in this series include:

• Multiplication and division
• Multiplication
• Multiplying multi-digit numbers
• Long division
• Negative numbers
• Negative times negative
• Multiplying and dividing rational numbers
• Dividing multi-digit numbers
• Multiplicative comparison

Practice multiplying and dividing integers

1. Multiply: (6) \times(-2)=\text { ? }

Using the rule for multiplying integers, 6 and -2 have different signs, so the product is negative.

(6) \times(-2)=-12

You can check your answer with counters.

6 groups of -2 counters is -12 counters

2. Multiply: (-15) \times(-3)= \text { ? }

Using the rule for multiplying integers, -15 and -3 have the same sign, so the product is positive.

(-15) \times(-3)=45

3. Divide: (-52) \div(4)= \text { ? }

Using the rule for dividing integers, -52 and 4 have different signs, so the quotient is negative.

(-52) \div(4)=-13

4. Divide: \cfrac{(-72)}{(-9)}= \text { ? }

Using the rule for dividing integers, -72 and -9 have the same sign, so the quotient is positive.

\cfrac{(-72)}{(-9)}=8

5. Multiply: (-190) \times(-10)= \text { ? }

Using the rule for multiplying integers, -190 and -10 have the same sign, so the product is positive.

(-190) \times(-10)=1900

6. A deep sea diver descends at a rate of 10 feet per minute below sea level. The diver descends at this rate for 8 minutes, which can be modeled by 8 \times (-10) = d, where d is how far the diver is below sea level. After 8 minutes, how far did the diver descend?

Using the rule, the signs of the numbers are different, so the product is negative.

8 \times(-10)=-80

The diver descended to -80 feet.

Multiplying and dividing integers FAQs

When it comes to sets of numbers, whole numbers are 0 and positive whole numbers. The set of integers includes all negative whole numbers, 0, and all positive whole numbers.

Multiplication and division of integers help when simplifying algebraic expressions and solving equations.

Yes, positive numbers are to the right of 0 and negative numbers are to the left of 0.

The positive sign does not necessarily need to be written in front of a number. For example, +5 is the same as 5. The positive sign is understood.

The next lessons are

• Types of numbers
• Rounding numbers
• Factors and multiples

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Multiplying and Dividing Integers: Advanced Lesson

With this lesson, students will learn about positive and negative numbers and learn the rules for multiplying and dividing by them.

Included with this lesson are some adjustments or additions that you can make if you’d like, found in the “Options for Lesson” section of the Classroom Procedure page. One of the optional additions to this lesson is to add another activity. For this activity, students will flip over two playing cards and multiply (red for negative, black for positive). Students collect the cards if they get the right answer, and the student with the most cards at the end wins.

Description

Additional information, what our multiplying and dividing integers lesson plan includes.

Lesson Objectives and Overview: Multiplying and Dividing Integers introduces students to working with positive and negative integers. The included manipulative is a great way for students to visualize the concept and practice solving problems. This lesson is for students in 6th grade.

Classroom Procedure

Every lesson plan provides you with a classroom procedure page that outlines a step-by-step guide to follow. You do not have to follow the guide exactly. The guide helps you organize the lesson and details when to hand out worksheets. It also lists information in the orange box that you might find useful. You will find the lesson objectives, state standards, and number of class sessions the lesson should take to complete in this area. In addition, it describes the supplies you will need as well as what and how you need to prepare beforehand. There are no supplies needed for this lesson. To prepare for this lesson ahead of time, you can copy the handouts.

Options for Lesson

Included with this lesson is an “Options for Lesson” section that lists a number of suggestions for activities to add to the lesson or substitutions for the ones already in the lesson. An optional addition to this lesson is to add another activity. For this activity, students will flip over two playing cards and multiply (red for negative, black for positive). Students collect the cards if they get the right answer, and the student with the most cards at the end wins. If some of your students have trouble multiplying or dividing, they can use a calculator.

Teacher Notes

The teacher notes page includes a paragraph with additional guidelines and things to think about as you begin to plan your lesson. This page also includes lines that you can use to add your own notes as you’re preparing for this lesson.

MULTIPLYING AND DIVIDING INTEGERS LESSON PLAN CONTENT PAGES

Integers (multiplying and dividing).

The Multiplying and Dividing Integers lesson plan includes one page of content. The lesson begins by explaining how to multiply or divide integers with the same sign. To do this, you multiply or divide the number regularly. The answer will always be positive. Next, the lesson explains how to multiply or divide integers with different signs. You also multiply the integers regularly, but in this case, the answer will always be negative. The lesson includes an example for each scenario.

MULTIPLYING AND DIVIDING INTEGERS LESSON PLAN WORKSHEETS

The Multiplying and Dividing Integers lesson plan includes three worksheets: an activity worksheet, a practice worksheet, and a homework assignment. You can refer to the guide on the classroom procedure page to determine when to hand out each worksheet.

TIC-TAC-TOE ACTIVITY WORKSHEET

The activity worksheet asks students to use a tic-tac-toe table to reinforce the rules for multiplying or dividing integers. They will use the table included on the worksheet and will place positive signs on one diagonal and fill in the rest of the places with negative signs. They will then use the table to fill in various statements related to the lesson material.

SOLVING PROBLEMS PRACTICE WORKSHEET

For the practice worksheet, students will solve multiplication and division problems using what they learned during the lesson.

MULTIPLYING AND DIVIDING INTEGERS HOMEWORK ASSIGNMENT

The homework assignment asks students to fill in the tic tac toe board and answer question about it. They will also solve 12 multiplication and division problems.

Worksheet Answer Keys

This lesson plan includes answer keys for the activity worksheet, the practice worksheet, and the homework assignment.  If you choose to administer the lesson pages to your students via PDF, you will need to save a new file that omits these pages. Otherwise, you can simply print out the applicable pages and keep these as reference for yourself when grading assignments.

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Teaching Multiplying and Dividing Integers

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Multiplying and dividing integers is the perfect math concept – the computation is relatively simple, there are opportunities for hands-on learning with manipulatives, and there are an abundance of explanations that support the conceptual understanding. 

However, it is a little tricky! Be sure to read “ How to Teach Integer Operations ” where we cover the specifics on adding and subtracting integers, as well as some common misconceptions to avoid. 

Multiplying and dividing integers is taught as early as 6th grade (in Texas) but primarily introduced in 7th grade. Multiplying and dividing integers extends into rational numbers which means there is a lot to cover!

Tip #1: Start By Using Models or Manipulatives

“Tricks” definitely have their time and place in math, but building conceptual understanding is key for students to build that mathematical fluency. Since math is always progressing and building on itself, tricks can often be mixed up with other tricks. Anyone who has taught integers or fraction operations probably has experienced this first hand. (Keep, change, flip or keep, change, change?)

Modeling why like signs result in a positive answer and why unlike signs result in a negative answer is more likely to stick than having students only copy down the “rules.” Though I do think it is helpful to do that too! In fact, I kept an anchor chart with all of the rules posted throughout most of the school year.

Here is a helpful Google Slide Deck that I recommend using to introduce WHY a positive times a negative results in a negative product and WHY a negative times a negative results in a positive product. I used a number line to introduce this concept, but counters can work too.

Here is the slide deck for you to copy and use in your classroom! The animations are included, so you just have to click your mouse to make the arrows move.

If you prefer counters, here is how I would teach positive times a negative.

And a negative times a positive. One thing to note is since you can’t take away 4 groups of 2, you can introduce zero pairs. With the zero pairs, you now have 2s to “take away.”

And lastly, a negative times a negative.

Tip #2: Ask them to Come Up With the Rules Using Patterns

The great thing about math is that the rules are always supported by patterns.

When using the above table (a snippet from a student handout), it is important to start in the top left corner where students are familiar with those facts. Allowing students to see the patterns that create the rules really makes the content stick.

You could give students a list of numbers like the one below and ask them to make observations about what they see. If a student can’t remember a rule, they can recreate a list of multiplication facts, and then synthesize the rules on their own.

• -5*(3)= -15

Tip #3: Other Fun Ideas for Practice

• Give students a value they are trying to reach.  Provide sticky notes or cards marked with a variety of integers. Students match integers to equal the given value. Similar to using counters, this allows for students to practice their fluency but also to be flexible problem solvers.
• Our MTM Activities – Entire bundle , speed dating , scavenger hunt and multiplying rational numbers digital activity , dividing rational numbers digital activity
• Playing War using this idea from Mrs. E Teaches Math .
• Make sure to include opportunities for real-world situations. Money, football gains and losses, temperature, and sea level are things that give these numbers context. Context provides students with opportunities for application as well as verifying if an answer makes sense. 
• Make sure to buy and laminate these number lines . I would give my students dry erase markers to write on them.

Tip #4: Putting It All Together

In my experience, students are doing well and seem to really grasp when they are completing operations in isolation. Asking students to complete work where they are switching between the 4 operations or using them together can cause students to confuse what they have learned.

When that happens, ask students to draw a picture of what is happening. Send them back to the models because that is what they are there for – to make sense of the math. Don’t put the counters or number lines away just because you have already taught the models. 

Since many students are resistant to draw models when they think they’ve “got it,” I would remind students that drawing the models accounted for half of the work/grade/completion. Once students mastered the four operations on a summative assignment, I would give them permission to use the algorithm only.

What tips do you have for teaching multiplying and dividing integers?

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3.4 Multiply and Divide Integers

Learning objectives.

By the end of this section, you will be able to:

• Multiply integers
• Divide integers
• Simplify expressions with integers
• Evaluate variable expressions with integers
• Translate word phrases to algebraic expressions

Be Prepared 3.7

Before you get started, take this readiness quiz.

Translate the quotient of 20 20 and 13 13 into an algebraic expression. If you missed this problem, review Example 1.67 .

Be Prepared 3.8

Add: −5 + ( −5 ) + ( −5 ) . −5 + ( −5 ) + ( −5 ) . If you missed this problem, review Example 3.21 .

Be Prepared 3.9

Evaluate n + 4 when n = −7 . Evaluate n + 4 when n = −7 . If you missed this problem, review Example 3.23 .

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.

We remember that a · b a · b means add a , b a , b times. Here, we are using the model shown in Figure 3.19 just to help us discover the pattern.

Now consider what it means to multiply 5 5 by −3 . −3 . It means subtract 5 , 3 5 , 3 times. Looking at subtraction as taking away , it means to take away 5 , 3 5 , 3 times. But there is nothing to take away, so we start by adding neutral pairs as shown in Figure 3.20 .

In both cases, we started with 15 15 neutral pairs. In the case on the left, we took away 5 , 3 5 , 3 times and the result was − 15 . − 15 . To multiply ( −5 ) ( −3 ) , ( −5 ) ( −3 ) , we took away − 5 , 3 − 5 , 3 times and the result was 15 . 15 . So we found that

Notice that for multiplication of two signed numbers , when the signs are the same, the product is positive, and when the signs are different, the product is negative.

Multiplication of Signed Numbers

The sign of the product of two numbers depends on their signs.

Example 3.47

Multiply each of the following:

• ⓐ −9 · 3 −9 · 3
• ⓑ −2 ( −5 ) −2 ( −5 )
• ⓒ 4 ( −8 ) 4 ( −8 )
• ⓓ 7 · 6 7 · 6

Try It 3.93

• ⓐ −6 · 8 −6 · 8
• ⓑ −4 ( −7 ) −4 ( −7 )
• ⓒ 9 ( −7 ) 9 ( −7 )
• ⓓ 5 · 12 5 · 12

Try It 3.94

• ⓐ −8 · 7 −8 · 7
• ⓑ −6 ( −9 ) −6 ( −9 )
• ⓒ 7 ( −4 ) 7 ( −4 )
• ⓓ 3 · 13 3 · 13

When we multiply a number by 1 , 1 , the result is the same number. What happens when we multiply a number by −1 ? −1 ? Let’s multiply a positive number and then a negative number by −1 −1 to see what we get.

Each time we multiply a number by −1 , −1 , we get its opposite.

Multiplication by −1 −1

Multiplying a number by −1 −1 gives its opposite.

Example 3.48

• ⓐ −1 · 7 −1 · 7
• ⓑ −1 ( −11 ) −1 ( −11 )

Try It 3.95

• ⓐ −1 · 9 −1 · 9
• ⓑ −1 · ( −17 ) −1 · ( −17 )

Try It 3.96

• ⓐ −1 · 8 −1 · 8
• ⓑ −1 · ( −16 ) −1 · ( −16 )

Divide Integers

Division is the inverse operation of multiplication. So, 15 ÷ 3 = 5 15 ÷ 3 = 5 because 5 · 3 = 15 5 · 3 = 15 In words, this expression says that 15 15 can be divided into 3 3 groups of 5 5 each because adding five three times gives 15 . 15 . If we look at some examples of multiplying integers , we might figure out the rules for dividing integers .

Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

Division of Signed Numbers

The sign of the quotient of two numbers depends on their signs.

Remember, you can always check the answer to a division problem by multiplying.

Example 3.49

Divide each of the following:

• ⓐ −27 ÷ 3 −27 ÷ 3
• ⓑ −100 ÷ ( −4 ) −100 ÷ ( −4 )

Try It 3.97

• ⓐ −42 ÷ 6 −42 ÷ 6
• ⓑ −117 ÷ ( −3 ) −117 ÷ ( −3 )

Try It 3.98

• ⓐ −63 ÷ 7 −63 ÷ 7
• ⓑ −115 ÷ ( −5 ) −115 ÷ ( −5 )

Just as we saw with multiplication, when we divide a number by 1 , 1 , the result is the same number. What happens when we divide a number by −1 ? −1 ? Let’s divide a positive number and then a negative number by −1 −1 to see what we get.

When we divide a number by, −1 −1 we get its opposite.

Division by −1 −1

Dividing a number by −1 −1 gives its opposite.

Example 3.50

• ⓐ 16 ÷ ( −1 ) 16 ÷ ( −1 )
• ⓑ −20 ÷ ( −1 ) −20 ÷ ( −1 )

Notice that the signs were the same, so the quotient was positive.

Try It 3.99

• ⓐ 6 ÷ ( −1 ) 6 ÷ ( −1 )
• ⓑ −36 ÷ ( −1 ) −36 ÷ ( −1 )

Try It 3.100

• ⓐ 28 ÷ ( −1 ) 28 ÷ ( −1 )
• ⓑ −52 ÷ ( −1 ) −52 ÷ ( −1 )

Simplify Expressions with Integers

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

Example 3.51

Simplify: 7 ( −2 ) + 4 ( −7 ) − 6 . Simplify: 7 ( −2 ) + 4 ( −7 ) − 6 .

We use the order of operations. Multiply first and then add and subtract from left to right.

Try It 3.101

8 ( −3 ) + 5 ( −7 ) −4 8 ( −3 ) + 5 ( −7 ) −4

Try It 3.102

9 ( −3 ) + 7 ( −8 ) − 1 9 ( −3 ) + 7 ( −8 ) − 1

Example 3.52

• ⓐ ( −2 ) 4 ( −2 ) 4
• ⓑ −2 4 −2 4

The exponent tells how many times to multiply the base.

ⓐ The exponent is 4 4 and the base is −2 . −2 . We raise −2 −2 to the fourth power.

ⓑ The exponent is 4 4 and the base is 2 . 2 . We raise 2 2 to the fourth power and then take the opposite.

Try It 3.103

• ⓐ ( −3 ) 4 ( −3 ) 4
• ⓑ −3 4 −3 4

Try It 3.104

• ⓐ ( −7 ) 2 ( −7 ) 2
• ⓑ − 7 2 − 7 2

Example 3.53

Simplify: 12 − 3 ( 9 − 12 ) . Simplify: 12 − 3 ( 9 − 12 ) .

According to the order of operations, we simplify inside parentheses first. Then we will multiply and finally we will subtract.

Try It 3.105

17 − 4 ( 8 − 11 ) 17 − 4 ( 8 − 11 )

Try It 3.106

16 − 6 ( 7 − 13 ) 16 − 6 ( 7 − 13 )

Example 3.54

Simplify: 8 ( −9 ) ÷ ( −2 ) 3 . 8 ( −9 ) ÷ ( −2 ) 3 .

We simplify the exponent first, then multiply and divide.

Try It 3.107

12 ( −9 ) ÷ ( −3 ) 3 12 ( −9 ) ÷ ( −3 ) 3

Try It 3.108

18 ( −4 ) ÷ ( −2 ) 3 18 ( −4 ) ÷ ( −2 ) 3

Example 3.55

Simplify: −30 ÷ 2 + ( −3 ) ( −7 ) . Simplify: −30 ÷ 2 + ( −3 ) ( −7 ) .

First we will multiply and divide from left to right. Then we will add.

Try It 3.109

−27 ÷ 3 + ( −5 ) ( −6 ) −27 ÷ 3 + ( −5 ) ( −6 )

Try It 3.110

−32 ÷ 4 + ( −2 ) ( −7 ) −32 ÷ 4 + ( −2 ) ( −7 )

Evaluate Variable Expressions with Integers

Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

Example 3.56

Evaluate 2 x 2 − 3 x + 8 when x = −4 . Evaluate 2 x 2 − 3 x + 8 when x = −4 .

Keep in mind that when we substitute −4 −4 for x , x , we use parentheses to show the multiplication. Without parentheses, it would look like 2 · −4 2 − 3 · −4 + 8 . 2 · −4 2 − 3 · −4 + 8 .

Try It 3.111

3 x 2 − 2 x + 6 when x = −3 3 x 2 − 2 x + 6 when x = −3

Try It 3.112

4 x 2 − x − 5 when x = −2 4 x 2 − x − 5 when x = −2

Example 3.57

Evaluate 3 x + 4 y − 6 when x = −1 and y = 2 . Evaluate 3 x + 4 y − 6 when x = −1 and y = 2 .

Try It 3.113

7 x + 6 y − 12 when x = −2 and y = 3 7 x + 6 y − 12 when x = −2 and y = 3

Try It 3.114

8 x − 6 y + 13 when x = −3 and y = −5 8 x − 6 y + 13 when x = −3 and y = −5

Translate Word Phrases to Algebraic Expressions

Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is product and for division is quotient .

Example 3.58

Translate to an algebraic expression and simplify if possible: the product of −2 −2 and 14 . 14 .

The word product tells us to multiply.

Try It 3.115

Translate to an algebraic expression and simplify if possible:

the product of −5 and 12 the product of −5 and 12

Try It 3.116

the product of 8 and −13 the product of 8 and −13

Example 3.59

Translate to an algebraic expression and simplify if possible: the quotient of −56 −56 and −7 . −7 .

The word quotient tells us to divide.

Try It 3.117

the quotient of −63 and −9 the quotient of −63 and −9

Try It 3.118

the quotient of −72 and −9 the quotient of −72 and −9

ACCESS ADDITIONAL ONLINE RESOURCES

• Multiplying Integers Using Color Counters
• Multiplying Integers Using Color Counters With Neutral Pairs
• Multiplying Integers Basics
• Dividing Integers Basics
• Ex. Dividing Integers
• Multiplying and Dividing Signed Numbers

Section 3.4 Exercises

Practice makes perfect.

In the following exercises, multiply each pair of integers.

−4 · 8 −4 · 8

−3 · 9 −3 · 9

−5 ( 7 ) −5 ( 7 )

−8 ( 6 ) −8 ( 6 )

−18 ( −2 ) −18 ( −2 )

−10 ( −6 ) −10 ( −6 )

9 ( −7 ) 9 ( −7 )

13 ( −5 ) 13 ( −5 )

−1 · 6 −1 · 6

−1 · 3 −1 · 3

−1 ( −14 ) −1 ( −14 )

−1 ( −19 ) −1 ( −19 )

In the following exercises, divide.

−24 ÷ 6 −24 ÷ 6

−28 ÷ 7 −28 ÷ 7

56 ÷ ( −7 ) 56 ÷ ( −7 )

35 ÷ ( −7 ) 35 ÷ ( −7 )

−52 ÷ ( −4 ) −52 ÷ ( −4 )

−84 ÷ ( −6 ) −84 ÷ ( −6 )

−180 ÷ 15 −180 ÷ 15

−192 ÷ 12 −192 ÷ 12

49 ÷ ( −1 ) 49 ÷ ( −1 )

62 ÷ ( −1 ) 62 ÷ ( −1 )

In the following exercises, simplify each expression.

5 ( −6 ) + 7 ( −2 ) −3 5 ( −6 ) + 7 ( −2 ) −3

8 ( −4 ) + 5 ( −4 ) −6 8 ( −4 ) + 5 ( −4 ) −6

−8 ( −2 ) −3 ( −9 ) −8 ( −2 ) −3 ( −9 )

−7 ( −4 ) −5 ( −3 ) −7 ( −4 ) −5 ( −3 )

( −5 ) 3 ( −5 ) 3

( −4 ) 3 ( −4 ) 3

( −2 ) 6 ( −2 ) 6

( −3 ) 5 ( −3 ) 5

− 4 2 − 4 2

− 6 2 − 6 2

−3 ( −5 ) ( 6 ) −3 ( −5 ) ( 6 )

−4 ( −6 ) ( 3 ) −4 ( −6 ) ( 3 )

−4 · 2 · 11 −4 · 2 · 11

−5 · 3 · 10 −5 · 3 · 10

( 8 − 11 ) ( 9 − 12 ) ( 8 − 11 ) ( 9 − 12 )

( 6 − 11 ) ( 8 − 13 ) ( 6 − 11 ) ( 8 − 13 )

26 − 3 ( 2 − 7 ) 26 − 3 ( 2 − 7 )

23 − 2 ( 4 − 6 ) 23 − 2 ( 4 − 6 )

−10 ( −4 ) ÷ ( −8 ) −10 ( −4 ) ÷ ( −8 )

−8 ( −6 ) ÷ ( −4 ) −8 ( −6 ) ÷ ( −4 )

65 ÷ ( −5 ) + ( −28 ) ÷ ( −7 ) 65 ÷ ( −5 ) + ( −28 ) ÷ ( −7 )

52 ÷ ( −4 ) + ( −32 ) ÷ ( −8 ) 52 ÷ ( −4 ) + ( −32 ) ÷ ( −8 )

9 − 2 [ 3 − 8 ( −2 ) ] 9 − 2 [ 3 − 8 ( −2 ) ]

11 − 3 [ 7 − 4 ( −2 ) ] 11 − 3 [ 7 − 4 ( −2 ) ]

( −3 ) 2 −24 ÷ ( 8 − 2 ) ( −3 ) 2 −24 ÷ ( 8 − 2 )

( −4 ) 2 − 32 ÷ ( 12 − 4 ) ( −4 ) 2 − 32 ÷ ( 12 − 4 )

In the following exercises, evaluate each expression.

−2 x + 17 when −2 x + 17 when

• ⓐ x = 8 x = 8
• ⓑ x = −8 x = −8

−5 y + 14 when −5 y + 14 when

• ⓐ y = 9 y = 9
• ⓑ y = −9 y = −9

10 − 3 m when 10 − 3 m when

• ⓐ m = 5 m = 5
• ⓑ m = −5 m = −5

18 − 4 n when 18 − 4 n when

• ⓐ n = 3 n = 3
• ⓑ n = −3 n = −3

p 2 − 5 p + 5 when p = −1 p 2 − 5 p + 5 when p = −1

q 2 − 2 q + 9 q 2 − 2 q + 9 when q = −2 q = −2

2 w 2 − 3 w + 7 2 w 2 − 3 w + 7 when w = −2 w = −2

3 u 2 − 4 u + 5 3 u 2 − 4 u + 5 when u = −3 u = −3

6 x − 5 y + 15 6 x − 5 y + 15 when x = 3 x = 3 and y = −1 y = −1

3 p − 2 q + 9 3 p − 2 q + 9 when p = 8 p = 8 and q = −2 q = −2

9 a − 2 b − 8 9 a − 2 b − 8 when a = −6 a = −6 and b = −3 b = −3

7 m − 4 n − 2 7 m − 4 n − 2 when m = −4 m = −4 and n = −9 n = −9

In the following exercises, translate to an algebraic expression and simplify if possible.

The product of −3 −3 and 15

The product of −4 −4 and 16 16

The quotient of −60 −60 and −20 −20

The quotient of −40 −40 and −20 −20

The quotient of −6 −6 and the sum of a a and b b

The quotient of −7 −7 and the sum of m m and n n

The product of −10 −10 and the difference of p and q p and q

The product of −13 −13 and the difference of c and d c and d

Everyday Math

Stock market Javier owns 300 300 shares of stock in one company. On Tuesday, the stock price dropped $12 $12 per share. What was the total effect on Javier’s portfolio?

Weight loss In the first week of a diet program, eight women lost an average of 3 pounds 3 pounds each. What was the total weight change for the eight women?

Writing Exercises

In your own words, state the rules for multiplying two integers.

In your own words, state the rules for dividing two integers.

Why is −2 4 ≠ ( −2 ) 4 ? −2 4 ≠ ( −2 ) 4 ?

Why is −4 2 ≠ ( −4 ) 2 ? −4 2 ≠ ( −4 ) 2 ?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction

• Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
• Publisher/website: OpenStax
• Book title: Prealgebra 2e
• Publication date: Mar 11, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/prealgebra-2e/pages/1-introduction
• Section URL: https://openstax.org/books/prealgebra-2e/pages/3-4-multiply-and-divide-integers

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Chapter 2, Lesson 4: Multiplying Integers

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Combining Polynomials

4.1: Notice and Wonder: What Can Happen to Integers (5 minutes)

CCSS Standards

Building Towards

• HSA-APR.A.1

Routines and Materials

Instructional Routines

• Notice and Wonder

The purpose of this warm-up is to elicit the idea that integers can be combined in ways that result in integers, or in ways that do not result in integers. This will be useful when students experiment to find out which operations integers are closed under in a later activity. While students may notice and wonder many things about these images, the possible results of combining integers using each operation are the important discussion points.

Display the 4 equations for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

• \(7 \boldcdot 9 = 63\)
• \(7 + 9 = 16\)
• \(7 - 9 = \text-2\)
• \(\frac{7}{9} = 0.777 \ldots\)

Student Response

For access, consult one of our IM Certified Partners .

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the equation. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the difference between division and the other three operations—that is, that adding, subtracting, or multiplying integers results in other integers but the same is not true for division—does not come up during the conversation, ask students to discuss this idea.

4.2: Experimenting with Integers (15 minutes)

• MLR2: Collect and Display

The purpose of this activity is for students to experiment with performing operations on numbers to see which operations yield numbers of a different type than the ones they started with, and which ones do not. This introduces the idea of closure (although the word closure does not need to be introduced). Students do not have to find a complete mathematical proof for each statement they think is true, but they should construct an argument for it based on their observations (MP3).

Arrange students in groups of 2. Tell students that today, we’re going to see what happens to polynomials when we perform mathematical operations on them. We will start by experimenting with integers. Invite students to name some mathematical operations, and record them for all to see throughout the activity. If needed, remind students that an operation is something you can do to a number or a pair of numbers to get another number, like adding them or raising one to the power of the other.

Which of these statements are true? Give reasons in support of your answer.

• If you add two even numbers, you’ll always get an even number.
• If you subtract an even number from another even number, you’ll always get an even number.
• If you add two odd numbers, you’ll always get an odd number.
• If you subtract an odd number from another odd number, you’ll always get an odd number.
• If you multiply two even numbers, you’ll always get an even number.
• If you multiply two odd numbers, you’ll always get an odd number.
• If you multiply two integers, you’ll always get an integer.
• If you add two integers, you’ll always get an integer.
• If you subtract one integer from another, you’ll always get an integer.

Are you ready for more?

• If you add two rational numbers, you’ll always get a rational number.
• If you multiply two rational numbers, you’ll always get a rational number.
• If you divide two rational numbers, you’ll always get a rational number.

Pair groups together to briefly share one statement they agreed with and one statement they disagreed with.

After groups have shared with each other, here are some questions for discussion:

• “Was there anything that surprised you while you were thinking about each statement?” (I never noticed that adding two odd numbers always makes an even number. I wasn’t sure what could happen when two even numbers are multiplied, but now it makes sense that the result will always be even.)
• “If you have some odd numbers, what could you do to them to get an odd number? An even number?” (Multiply them. Add or subtract them.)
• “If you have some even numbers, what could you do to them to get an odd number?” (Average them, divide them by two, add 1 to their sum.)
• “If you have some integers, what could you do to them to get something that’s not an integer?” (Divide one of them by the other, average them.)

The purpose of the discussion is for students to understand that some operations on a type of number will produce numbers of that same type, but others will not. For example, performing multiplication on odd numbers always produces an odd number, but performing addition or subtraction on odd numbers will not produce an odd number. If students ask whether there is a word for this, tell them that another way to say this is “odd numbers are closed under multiplication.” If you have some odd numbers and you want to get another kind of number, you can’t do it by multiplying.

4.3: Experimenting with Polynomials (15 minutes)

• MLR8: Discussion Supports

Required Materials

• Pre-printed slips, cut from copies of the blackline master

The purpose of this activity is for students to experiment with adding, subtracting, and multiplying polynomials to see if they will always get a polynomial. As with the integers in the previous activity, it is not important for students to develop a mathematical proof of their answers, but they should find reasons to support their answers. They will share their reasons with others and critique each other’s arguments (MP3).

This is also a good opportunity to remind students of the wide variety of expressions that are polynomials. For example, the sample polynomials that students can use in their experiments include one with a square-root coefficient. Students will work more with roots in later lessons.

Tell students that they will experiment with polynomials in the same way they experimented with integers in the last activity. If needed, briefly remind students what counts as a polynomial. Here are some questions about polynomials for discussion if needed:

• “Can the variable have negative powers?” (No, whole numbers only.)
• “Is a single number like 10 a polynomial?” (Yes, the power on the variable is 0.)
• “Do the coefficients have to be integers?” (No.)

Then, display the first two questions from the task statement for all to see. After quiet think time, informally poll the class and record the total number of “yes” votes next to each question.

Arrange students in groups of 2. Assign each group 1 of the questions to focus on. Tell students that their job is to decide what the answer to their question is, and to find reasons that support their answers. Distribute a set of pre-cut slips of polynomials to each pair of students. Students can test these polynomials to see if performing their operation on them will always result in a polynomial, or they can write their own polynomials to test. When a pair thinks they know whether they’ll always get a polynomial, they should find reasons that support their answer.

Once a group has at least one argument to support their answer, partner them with another group and tell them to take turns sharing their reasoning while the other group listens and works to understand.

Monitor for students who give especially clear justifications or use clear diagrams to share during the whole-class discussion.

Here are some questions about polynomials. You and a partner will work on one of these questions.

• If you add or subtract two polynomials, will you always get a polynomial?
• If you multiply two polynomials, will you always get a polynomial?
• Try combining some polynomials to answer your question. Use the ones given by your teacher or make up your own polynomials. Keep a record of what polynomials you tried, and the results.
• When you think you have an answer to your question, explain your reasoning using equations, graphs, visuals, calculations, words, or in any way that will help others understand your reasons.

Anticipated Misconceptions

Students may mistakenly believe they have found an example that proves the answer to one of the questions is “no,” because either they did not start with two polynomials, they made mistakes in calculating, or they do not see that the result is a polynomial. They may catch errors when sharing with the other group. Misunderstandings about the definition of “polynomial” may be useful to bring up during the whole-class discussion, so not all such errors need to be corrected during the activity itself.

The main takeaway students should have from this activity is an understanding of some reasons why polynomials are closed under addition, subtraction, and multiplication. Revisit the poll questions about polynomials. Ask students to raise their hand if they think the answer is “yes,” and record the total. Invite any students who have changed their minds to say why. For each question, ask at least one previously identified pair to share their work.

Lesson Synthesis

A key idea of this lesson is that integers and polynomials are both closed under addition, subtraction, and multiplication. Students have seen that integers are not closed under division, so this is a good time for them to wonder about what happens when one polynomial is divided by another, although they will not learn how to divide polynomials until later in the unit. In this lesson, students did a lot of practice performing arithmetic on polynomials, so any efficient strategies should be highlighted. Here are some questions for discussion:

• “What is something you found out that was surprising?” (I didn’t know that adding two odd numbers would always give you an even number. I wasn’t sure if multiplying two polynomials would always make another polynomial, so I was surprised to find out that it does.)
• “What was difficult about doing arithmetic on polynomials? How did you deal with that difficulty?” (Multiplying polynomials is kind of messy. I wrote down each piece separately and circled the like terms so I could keep track of what to add.)
• “What do you think would happen if we divided one polynomial by another? Would we always get a polynomial?” (I don’t think so, because dividing integers doesn’t always give you an integer. If we divide \(x^2\) by \(x^3\) , we get \(x^{\text-1}\) , and that’s not a polynomial.)

4.4: Cool-down - Mind the Gaps (5 minutes)

Student lesson summary.

If we add two integers, subtract one from the other, or multiply them, the result is another integer. The same thing is true for polynomials: combining polynomials by adding, subtracting, or multiplying will always give us another polynomial.

For example, we can multiply \(\text-x^2 + 4.5\) and \(x^3 + 2x + \sqrt7\) to see what happens. We’ll need to use the distributive property, and there are a lot of ways to keep track of the results of distribution when we multiply polynomials. One way is to use a diagram like this:

Then we can find the product by adding all the results we filled in. This diagram tells us that the product is \(\text-x^5 + 2.5x^3 - \sqrt{7}x^2 + 9x + 4.5\sqrt7\) , which is also a polynomial even though there are square roots as coefficients! No matter what polynomials we started with, multiplying them would give us a polynomial, because we would have to multiply each part of each polynomial and then add them all together. Adding or subtracting polynomials also gives us a polynomial, because we can combine like terms.

When thinking about polynomials, it is important to remember exactly what counts as a polynomial. Any sum of terms that all have the same variable, where the variable is only raised to non-negative integer powers, is a polynomial. So some things that might not look like polynomials at first, like -34.1 or \(7.9998x\) , are polynomials.