Ed School Screening and Beyond, Dept.

I applied to George Mason School of Education in the fall of 2005. They had a special program for people in the workforce called the “Career Changer” program. It was aimed at people like me who wanted to get into teaching after having been out in the working world. In my case, I had been out in the working world for over 30 years, and was preparing for a career of teaching after I retired which would be five years from then.

In an effort to make it look like it was a special hard-to-get-in program, the school held a meeting in which the applicants had to go through a series of interviews, and then prepare a writing sample. At the introductory session, a woman addressed the candidates, as we were called and said in her opening remarks that “School is not your father’s classroom anymore.” Holding up an index finger, she then declared what it was: “Inquiry-based!” I might have been one of the few in the crowd who knew what those words meant. I knew that many of the people in that room would be swayed to the “not your father’s classroom” standard.

She went on, mostly about what to expect in teaching and then made a plea for getting a masters in education. “Research shows that teachers who only have a teaching credential tend to leave the profession after three years, but those with masters degrees stay the course.” I resolved not to do that, and don’t regret it.

One of the events of the evening was being interviewed by various ed school professors who asked us questions such as “What do you see yourself doing in five years?” which were probably designed to see how well we could bullshit and sound sincere. The capstone event was having to prepare a writing sample. We were sent to a room with computers and told to write an essay on a topic whose theme I can’t recall, but it was some broadly bland theme like “What is the Value of an Education?” and we had to come up with 1500 words. I think I wrote something along the lines of “Lack of education results in falling prey to things like “inquiry-based classrooms are better than direct instruction”– but phrased a bit more diplomatically. I even included references.

A few weeks later I received my acceptance letter which reminded me of those promos one receives in the mail notifying you that you may have already won one million dollars in some sweepstakes competition. I spent the next four years taking one class per semester at night, and finishing up my student teaching in California, after having secured a tentative agreement with George Mason that they would cooperate with Cal Poly–which they tried to get out of. I hadn’t counted on the people who had been so agreeable to the idea leaving the college or taking new positions within it. My new advisor reluctantly went along with it because I had the foresight to get the agreement in writing.

About the only thing of value I got out of ed school was hearing the professors’ stories of their days as teachers–it was a lot more useful than the textbooks we had to read, or the 3 hour night-time sessions. One class on “Methods of Math Teaching” consisted of the teacher stating a particular category (like “attributes of effective teaching”) and asking the class to come up with ideas. She would write down the ideas, filling the white-board with a list, and then would proclaim “Good list!” There were other things that filled the three hours, such as students reporting out on an article that was assigned. I recall reporting out on an article on doing away with grades, and using “standards-based” descriptions of students’ performance. The strange thing about it was that in reporting the article, I was presenting the side of the author and essentially selling the idea to my classmates, who–I’m pleased to say–were having none of it. Maybe I meant for that to happen–yeah, it was “intentional”. (The edu-word “intentional” hadn’t yet achieved its current status of popularity at that time.)

In another class (“Literacy in the Content Area”) many of the students were already student-teaching. I worked full time so I couldn’t do that. One assignment was for students to report out on a technique they had used to teach a particular topic. A student who taught social studies talked about how he taught the unit on civil rights, and in particular the Freedom Riders. He recounted how the civil rights workers had their bus torched in Mississippi, but the next day, got on another bus and continued their brave journey. It was indeed an inspiring story, but now his assignment (this was for his class of eighth graders) was to have the class pretend they were Freedom Riders, and their bus had just been torched. Those who chose to continue the ride were to stand on one side of the room; those who chose not to, would be on the other side. Our class now did this activity.

In deciding whether I would stay on the bus or decline, I considered my own situation: I have a wife and child, so that if I were to die, they would be left in the lurch. So I was one of two people who declined and stood on one side of the room while the rest of the class stood on the other. I felt guilty and ashamed; I could only imagine how eighth graders would feel doing such an activity. I thought the activity was inappropriate and that one could simply ask students to think about what they would do if they were a Freedom Rider and leave it at that. No need to put them at risk for being called “racist”, although now-a-days this has probably become de rigueur in classrooms. Maybe this student was way ahead of his time.

All in all, the most useful skill I took with me from ed school was how to make it look like I was going along with the party line, while doing what I felt was right for the students. And fortunately I retired just before having to participate in PD devoted to checking one’s privilege and admitting one’s white fragility became education’s shiny new thing.

Traditional Math (12) Translating from Words into Algebraic Expressions (7th Grade)

This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. (Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!)

1. Introduction/Opening Monologue

Translating English into math is an important skill and is basic to solving all word problems. They have done this to a limited degree using arithmetic methods which I use in my opening monologue to the day’s lesson. I start off asking them to tell me how to write the numerical equation for various statements.

I quickly give a worked example to ward off students giving me the answer to the problem, though I can assure you that no matter how many warnings and instructions you give, that will occur: “The cost of ten yo-yos if each costs three dollars.”  What I’m looking for is the numerical equation, not just the answer; in this case it’s 10 x 3 = 30.

Now I give the statements, telling students to write it down in their notebooks or on whiteboards:

“The number of students on three buses if each bus holds twenty two students”  Answer:  3 x 22 = 66

“The amount of money Nina earned if she mowed the lawn for $15 and walked the dog for $4”.  Answer: $15 + $4 = $19

“The number of students in each group if fifteen students are divided into five equal groups”  Answer: 15 ÷ 5 = 3 or  15/5 = 3

The process of writing statements numerically is then extended to expressing statement in algebraic terms in which variables are used.

2. Translating into algebra

Warm-Ups. For this lesson I typically include warm-ups that provide a preview to what we will be doing, and serve as a segue to the day’s lesson. Examples include:

  1. Write the numerical expression for 5 more than 10. (Answer: 10+5 or 5 + 10)
  2. Write the numerical expressions for 5 less than 10. (Answer 10 – 5)
  3. Write as an expression: Seven times x (Answer: 7x)

Defining the Variable. Variables will represent unknown quantities. If I say “some number”,  since we don’t know what that number is, it can be represented by a letter. Taking a problem from one of the warm-ups, I’ll point out that if I say “five more than ten” we write that as 5 + 10. I’ll ask: “Suppose I say ‘five more than some number’.  How would I write that?” There is general consensus reached quickly that it is 5 + x, which can also be written as x + 5.

I give them numerical forms first and then extend that to variable.  “Five times a number” (5x)

“A number divided by 3”.  I’m looking for x/3, but if I get x ÷ 3, I accept it, and quickly point out that in algebra we write it in fractional form.

Now I start to use unknown quantities that name specific things, like “Two miles more than an athlete ran.”  If there is a stunned silence, I’ll ask what the unknown quantity is, and they’ll pick up that the answer is 2 + x.

“Be careful on this next one,” I’ll warn. “I’m betting that at least people will get this wrong.”  The problem: “5 less than some number.”  Usually many people will say 5 – x.  For those who get it right, I ask them to explain why they wrote it that way. The bottom line answer is five is being subtracted from a number. I advise at this point that if a statement is confusing, see how to write it using numbers, and I refer to one of the warm-up problems. For example “5 less than 10” is a no-brainer; we write it as 10-5.

This particular error is a common one that is the gift that keeps on giving. Some students will repeat the mistake through the year; so it is good to keep repeating it.

After working with addition and subtraction, I turn to multiplication and division, and ramp the problems up so they are doing both. Starting with “Three times a number”, and “The cost of some number of games of bowling at $4 per game, for example, and ending with “Four less than two times a number.”

More Problems.  After the students are fairly comfortable with the initial translations, I give more “story-oriented” problems, that increase with difficulty.

Here are some taken from Brown, et al (2000) that I like to use, and which I also include in homework worksheets. Students are to provide the expression in terms of the variable:

1, The Tigers had twice as many hits as the Yanks. If x = the number of hits by the Yanks, then ____ = number of hits by the Tigers.  (Answer: 2x)

2. The length of a rectangle is four times the width. If x = the width, then ____ = the length. (Answer: 4x)

3.  Mac is x years old. How old will he be next year? (x+1)

(This particular problem causes students difficulty because although they have been writing expressions in terms of a variable, some think that this problem is asking for a number.  I remind them that they are to express Mac’s age in terms of x.)

4. Trish is t years old. How old was she 7 years ago? (Answer t-7)

5. Karen will be m years old next year. How old is she this year? (Answer: m-1)

6. Pete worked 4 hours more than Quinn. Quinn worked 2 hours more than Rob.  If x= the number of hours Rob worked, then  ___ = the number of hours Quinn worked and ___ = the number of hours Pete worked.  (Answers: Quinn’s hours = x + 2;  Pete’s hours: x + 6

 

Reference: Brown, Richard G., Smith, G.D., Dolciani, M.P.; (2000). Basic Algebra; McDougal Littell, Illinois.

 

Traditional Math (11) Algebraic Expressions (7th grade)

This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. (Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!)

  1. Introduction

Students have had some exposure to solving equations in their earlier courses, having to solve problems such as 3 +n = 10, and 13 – n = 5. These are solved using the arithmetic properties of numbers using the relationships known as “number families” or “number bonds”.  These concepts view a number addition equation in three ways. For example 8 + 4 = 12 lends itself to two other related equations, namely 12 – 4 = 8 and 12 -8 = 4.

Faced with the problem 3 + n = 10, the number family approach teaches students that n can be expressed as 10 – 3, and therefore n = 7.  For a problem like 13 – n = 5, the student know the number family 5 + 8 = 13, and therefore 13 – 8 = 5. He can also approach the problem by exchanging the n and the 5 to obtain 13 – 5 = n.

In seventh grade, students are given an introduction to the basics of algebraic expressions and learn to solve simple equations using the tools of algebra. This represents a different approach than they are used to and is a much more powerful method that allows them to solve more complex equations and provides an approach for solving word problems.

I like to start the unit by giving them a problem, and prefacing it with the following statement: “I’m going to give you a problem that you will think you know the answer to and you will probably be wrong.”  This serves as a challenge and a dare and also defuses the fear of making a mistake because they will want to prove me wrong. The problem is: “John and his sister have $110 between them. John has $100 more than his sister. How much do each of them have?”

Almost instantly hands are raised and students will call out confidently: “John has $100 and his sister has $10.”

I say that that is incorrect, because the problem says John has $100 more than his sister. If she has $10 how much will John have?”

They quickly figure out that he would have $110. “And what is the sum of $110 and $10?”  Seeing their error, some of the students then resort to a “guess and check” procedure, trying various combinations.  Someone will inevitably say “John has $90.”  I respond that if so, then his sister has $100 less. What is $100 less than $90?”  I cross my fingers that they remember how to work with negative numbers and one of the braver students will volunteer that it is -$10. Since you can’t possess a negative amount, we know that is wrong.  Eventually after enough guessing and checking they come up with John has $105 and his sister has $5.

While “guess and check” is a strategy that can solve problems, I point out that it took some back and forth before they came up with the right answer. “There is a way to get that answer on the first try,” I announce. That way, of course, is algebra. I tell them that they will learn to solve this problem and others using algebra in this unit.

  1. Writing and Evaluating Algebraic Expressions

Warm-Ups. Typical Warm-Ups for this lesson should incorporate past concepts.

  1. A 5 foot length of ribbon is cut into 2 ½ inch strips. How many strips are there? (Answer: 5 ft = 60 in, so 60 ÷ 2 ½ = 60 x 5/2 = 24 strips)
  2. -2 ½ x 3 2/5 (Answer: -5/2 x 17/5 =-17/2 = -8 ½
  3. (8/3)/(5/6) (Answer: 8/3 ÷ 5/6 = 8/3 x 6/5 = 16/5 = 3 1/5
  4. If you lose $2 every week for four weeks in a row, what is your loss after 4 weeks, and the fifth week you make $7, do you have a net gain or loss, and by how much? (Answer: -2 x 4 = -8; -8 + 7 = -1; a loss of $1.)
  5. What is 5 times n if n = -3? (Answer: 5 x -3 = -15).

Students have had experience using letters to represent numbers as discussed above.  Now we take it further with the goal of being able to represent English in terms of algebraic expressions. To do this students need to know the general rules of variables, and what a variable is.

A letter used to represent a number is called a variable. I liken it to a fill in the blank.  The sentence “I have __ apples” can have different meanings depending on the number that is used to fill the blank.  “I have x apples” does the same thing. The variable x in this case represents a changing—a fill in the blank type—number. The formal definition that I give to students is:

A variable is a letter or symbol used to represent an unknown value. Any letter can be used as a variable.

Addition and Subtraction. To be able to work with variables it is helpful for students to plug numbers in to various expressions that have variables, and to calculate the value of that expression. We start the process by looking at addition and subtraction.  I ask students if I have some unknown amount of apples, I can represent that by a variable.  I then ask students if I let n represent the unknown amount of apples and then want to represent 3 more than that amount, how would I write it?

I point to the definition of variable which states that the variable represents an unknown value. I will tell them: “In algebra, when we don’t know what the value is of a quantity, we represent it with a letter.”

Students are generally slow to respond to this but eventually someone will say the correct answer of n + 3.

I give more examples of this nature: “I have an unknown amount of apples and I give 5 away. How do we represent this?  x -5.

I will have them look at the first example of n + 3, and ask how many apples does the expression represent if n equals 5. I continue with other numbers: 100, 2,538, etc.  In the second example of x – 5, I might ask “What is the number of apples if x equals 10, 27, 5?” and so forth.

Now I might ask: “If I have an unknown amount of apples and call that amount m, and I get more apples of the same amount; how would I write this?”  They may hesitate a bit, but I am after m + m.

I now want to take it out of variables representing objects and just letting the variables represent “a number”.  I have an unknown number; I’ll ask how do I represent an unknown number? By now they’re in the rhythm of the questioning and will call out letters; usually x, but I want them to know x isn’t the only one they can use.  I’ll now ask I want to represent 5 more than that unknown amount. Next, with the x + 5 written on the board, I ask what is the value if x equals 4; then 3,  then 0,  then -5, -10 and so forth.

Finally, I’ll ask what happens if I have an unknown number and I add another different unknown number to it. How would I represent that?  I tell them to write it in their notebooks, or on a mini-whiteboard. I’m looking for two different variables added: a +b, x + y, and so on.

Like Terms. At this point I introduce some additional vocabulary: Term, like terms, and numerical coefficient.

When addition or subtraction signs separate an algebraic expression into parts, each part is a term. For example a + b consists of two terms, a and b.  Suppose I had 2a + 3b. Then 2a and 3b are terms.   The numerical part of a term that contains a variable is called the numerical coefficient of the variable. I will ask what the numerical coefficient is of 5x, of -24y.

If two terms have the same variable, or combination of variables, they are called “like terms”. I provide examples such as 2x and 34x, 8ab and 5ab.  Because the terms are the same, they can be added, just like we added x + x previously. This tends to be confusing at first, so I will liken it to “like objects”.  For example, if I have 2 apples and then get 3 more, I am adding 2 apples + 3 apples for a total of 5 apples. The variables can be thought of similarly.  So 3x – 2x + 5x, is the same as adding the numerical coefficients—that is 3 – 2 + 5—and adding the variable afterward: 10x. The summary statement I give them is:

To combine like terms that have variables, add or subtract the coefficients.

Multiplication. With the above as an introduction, I move on to multiplication. I ask students if I have 3 boxes and each box contains 5 apples, how would I calculate the total amount of apples?  They immediately know it is 3 x 5, but I write it as 5 + 5 + 5, and then next to it I write 3 x 5.  If each box has 7 apples, then similarly we have 7 + 7 + 7, which is written 3 x 7.

What if I don’t know the number of apples in each box, I’ll ask. How would I represent the total number of apples?

If x is the variable, then 3 boxes with an unknown amount of apples in each one can be represented by x + x + x.  How can we write this as a multiplication statement like before?  In algebra we represent it as 3x, meaning 3 times x.

After stating that, I ask students to tell me what 3x equals if x equals, 7, -3, 5/3, and so on, so they get the idea that values for the variable are substituted, and that 3x means multiplied by 3.

Division.  So far in previous lessons and discussions, whenever we’ve talked about fractions, we have mentioned that fractions are division.  So 5/2 is the same as 5 divided by 2; 2/3 is 2 divided by 3 and so on. I tell students that in algebra, we are going to represent division by a fractional form, rather than by using the symbols they have been using. The divide sign, ÷, is not used, particularly when using letters.  The expression x ÷ y is written as x/y. Similarly, x ÷ 3 is written x/3; 3 ÷ y as 3/y.

Examples for them to solve. I will write on the board various expressions with the values of the variables given for each problem, and ask them to find the value of each. These include expressions such as 5/x where x = 2.   5ab where a = -1, b = 2;  z + 2x where z = 5 and x = – 2, and so forth. I instruct them to leave improper fractions in that form; it is not necessary to convert them to mixed numbers.

Ending the Lesson. While the lesson may move fast, much of the information is new.  It is good to focus on these basics so they are familiar with combining like terms and being able to write expressions such as “three times some number” and “the sum of two different numbers” as a + b.  The next lesson will focus more on translating more complex English expressions into algebraic ones as well as the use of parentheses and order of operations.

Homework may include evaluation problems such as:

When y = 2, evaluate the expressions:

  1. y + 23 2. 6y            3.  8/y          4. 4 + y – 7

When m = 8 evaluate the expressions:

  1. 3m                       6. m/m                  7.  m x m x m

Traditional Math (10) Complex Fractions (7th Grade)

This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational. Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!

One thing that I almost never do is post the day’s “learning objectives” on the board. I find it sufficient to say to the class “Today we’re going to learn about…” and then say whatever it is we’re going to learn.  That seems to be enough.  There are occasions though when I will tell them what type of problem they will be solving at the end of the particular lesson.  I did this when teaching the topic of complex fractions to my class of seventh graders.

I announced that at the end of the lesson they would be able to do the following problem (which was a challenge problem that appeared in the JUMP Math teacher’s manual):

I was expecting to hear gasps and exclamations of “No way!” when a boy raised his hand and said “Oh, I know how to solve that.” He then narrated what needed to be done. He had certainly never seen this exact same problem before. He put together basic skills that he learned and saw how they fit together and solved a more complex problem—an example of knowledge transfer. Which is what this lesson is about, though most students will not be able to solve something like this straight off like this student did.

Warm-ups.  The warm-ups for this particular lesson should focus on what we have been doing, as well as some word problems:

1. (-2/7 + 5/14) ÷ 3/28   Answer:  1/14÷3/28 = 1/14 x 28/3 = 2/3

2. (2-3)/(2-7) Answer: -1/-5 = 1/5 

3.  -3/4 x 5 ¼  Answer:  -3/4 x 21/4 = -63/16 or -3 15/16

4.  How many 2/3 ounce servings are in a 5/6 ounce cup of yogurt?

          Answer: 5/6 ÷ 2/3 = 5/6 x 3/2 = 5/4 o 1 ¼ serving

Basic Lesson. Since we have just finished a lesson that covered fractional division, I will ask students to solve something like 2/3 ÷ 4/15 which they can do fairly readily.  They have learned in previous lessons that fractions are division. The fraction 6/2 is a division: 6 divided by 2. Similarly 2/3 is a division: 2 divided by 3. Therefore we can represent a fractional division problem as a “complex fraction”. The problem just given can be represented as:

Examples, Worked and Otherwise. We then practice rewriting complex fractions as ordinary fractional division problems, and solving them such as:

After a few of these, the problems can be more complicated:

And the solution to the first problem given (which my student solved without any lesson):

In general, my students enjoy complex fractions and look at them as puzzles. This is now something to add to the repertoire of problems to include on future warm-ups.

Traditional Math (9): (7th grade) Multiplying and Dividing Rational Numbers

This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational.

Students already know the rules about how to multiply and divide fractions. They also have learned the rules for multiplying negative numbers. Now these two skills come together so that students are doing problems like 2/3 x -5/6 and -4/5 ÷ 5/12.

Because they are familiar with these rules and procedures, I use this lesson to review the basics of fraction multiplication and division, as well as clear up misconceptions that occur along the way.

1.    3/5  ×10/4   Answer:  3/2

2.  -3 × -15  Answer: 45

3.   24/40  ÷11/7  (Reduce to lowest terms)  Answer:  3/5  ×7/11  =  21/55 

4. (-10 × 5) – 45 Answer: -50-45 = -95

5. -4/5  +5/9   Answer:   (-36+25)/45  = -11/45 

Teaching the Procedure.  The teaching of this procedure typically goes very quickly and students find it accessible and easy.  Starting with problem 1 of the warm-up, it is then easy to turn the problem into -3/5  ×10/4, and -3/5  ×-10/4  so that students can now apply the rules for multiplying negative fractions. Similarly, fractional division is handled the same way. Thus, variations to problem 3 using negatives can be introduced: -24/40  ÷11/7, and -24/40  ÷ -11/7.

Cancelling, Reducing to Lowest Terms, and Misconceptions. What becomes clear with these examples is the extent to which students cross cancel and reduce (i.e., simplify) fractions prior to the multiplication step.  Some students’ answer to problems like  3/5×5/17   is  15/85   which takes more time and effort to multiply and reduce (simplify) to lowest terms.  It takes much less effort to cancel the 5’s before multiplying, obtaining the answer of  3/17.

Other students may remember to cross cancel, but neglect to simplify fractions such as  24/40  , or  10/4. There is a mistaken notion that when multiplying fractions, one can cancel diagonally, but not within the same fraction. I like to put that misconception to rest quickly.  In algebra they will need to get used to doing both.

And although we call it “cancelling”, some mention should be made that what we are doing when reducing fractions, or cross cancelling, is dividing. The fraction 10/4 is equal to 5/2 because both the numerator and denominator are divided by 2 to obtain the equivalent fraction. Similarly the fraction 5/5 is equivalent to 1/1. Another misconception to put to rest is that a fraction like  5/5 does not equal zero. It seems very obvious that it is not, but I have seen students in algebra mistake the variable version of this —e.g., 2x/x  — as zero. Also, there is the common mistake of trying to cancel (i.e., divide) things that cannot be divided such as in (5+x)/(5+y), where students will think they can divide the 5’s. It’s a continual struggle and not just a one-off solution, but good to start putting some mistakes to rest right now.

Multiplication by Reciprocals. Another habit to instill is recognizing that multiplication of reciprocals equals one.  A problem like 4/5 x 5/4 obviously equals1. A problem like  4/5  ×35/28  is less obvious—at first, until one realizes  35/28  is the same as  5/4. Students who are not in the habit of reducing fractions before multiplying will be making an easy problem harder.   -4/5 x 35/28 is -1.

More importantly, however, is that students see that multiplication by a reciprocal equals one. Conversely, they also should be familiar with one divided by any fraction equals its reciprocal; i.e.,  1/(a/b)   =  b/a  . These properties will be revisited in the next unit on simple equations, since equations in the form  a/b x = c are solved by multiplying both sides of the equation by the reciprocal of  a/b  . This procedure is also a key to proving why the “invert and multiply” rule for fractional division works as it does for common fractions. 

Word Problems. Word problems that rely on multiplication and division of fractions are an important application of this topic. This should be taught as a separate topic on a different day. Students will need to recognize when a problem calls for division or multiplication. To scaffold these, it is helpful to give equivalent problems using whole numbers. For example, “How many 2 ft boards can be sawed from a 10 ft board?” Students will see immediately that solving the problem requires division: 10÷2. A problem like how many 1 1/2 ft boards can be sawed from a 15 ft board is then solved using the same structure: 15÷1/12.

Also, some problems require interpretation of a mixed number answer when the answer has to be a whole number. For example: Nina can carry 16 lbs. How many 1 1/2 books can she carry? One can divide 16 by 1 1/2 to get the answer of 32/3 or 10 2/3. But the answer must be a whole number since we are not dealing with fractions of a book. So the answer is that she can only carry 10.

1. The temperature is currently 0 degrees F. The temperature increases 2 1/2 deg F each hour. What will the temperature by 3 hours from now? Three hours ago?

Answers: Three hours from now is solved by 3 x 2 1/2: 3 x 5/2 = 15/2 or 7 1/2 degrees. Three hours ago would be -3 x 2 1/2 = – 7 1/2 or 7 1/2 degree decrease.

2. The temperature is currently 0 deg F. The temperature decreases 1 1/2 deg F each hour. What was the temperature 3 hours ago?

Answer: -3 x -1 1/2 = -3 x -3/2 = 9/2 or 4 1/2 degree higher.

3. 4/5 of a lasagna are shared by 3 people. What fraction of the lasagna does each person eat?

Answer: 4/5 ÷ 3 = 4/5 x 1/3 = 4/15

4. Sam rides his bike at 5 1/2 miles per hour. How far does he ride in 6 hours?

Answer: 5 1/2 x 6 = 11/2 x 6 = 33 miles

5. Julie rides her bike at 4 1/2 miles per hour. What fraction of a mile does she ride in 6 minutes?

Answer: Students must express 6 minutes in fraction of an hour: 6/10 or 1/10. Equation is 4 1/2 x 1/10 = 9/2 x 1/10 =9/20 of a mile

6. For problem 5, what distance is that in feet?

Answer: Students must know that there are 5,280 feet/mile. Then the answer is given by 9/20 x 5280 = 2,376 feet.

7. How many 1/2 cup servings are in 3/4 of a cup of yogurt?

Answer: 3/4 ÷ 1/2 = 3/4 x 2 = 6/4 = 1 1/2 servings. (Fraction of a serving is permissible here!)

8. Jim has 3/5 lb of dry pasta. He needs 3/16 lb of dry pasta to feed each person. How many people can he feed?

Answer: 3/5 ÷ 3/16 = 3/5 x 16/3 =16/5 or 3 1/5, so he can feed 3 people.

Traditional Math (8): Addition and Subtraction of Rational Numbers

This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational.

My wife and I once took dancing lessons to learn how to do the jitterbug and other steps. There was a basic dance step which we learned during the first session. In subsequent sessions we combined the basic moves with things like bending our knees, twirling around, and so on—adding more moves on top of the main moves. In the end, the foundational dance steps had served as the gateway to more involved moves which, to casual observers, appeared to be complicated, but to us was a mixture of mastered steps and moves.

Students are doing something similar math-wise when, after learning and (we hope) mastering the basics of working with negative integers, they apply this to working with negative fractions. Initially, they started with gains and losses, and determining whether, say, a loss of $10 and a gain of $4 represented a good or bad day. In other words, they learned to compute -10+4. In so doing they saw intuitively that there was a loss of $6, represented as -6.

The original steps students learned for adding and subtracting with negative integers now take on variations. Namely, students now learn to work with negative rational numbers (i.e., fractions and decimals). The next steps are solving problems like -3/7 + 2/9, -15 2/3 + 8 3/5, and -2.34 + 2.099.  To prevent overloading of information, mixed numbers and decimals should be taught on a separate day than common fractions.

Prior Knowledge. In addition to knowing how to work with negative integers, students must have knowledge of and fluency in determining order of fractions and decimals. Students are given problems in which they must order fractions and decimals from least to greatest or vice versa, as well as plotting the rational numbers on a number line. A typical problem might be to order from least to greatest the following: -2 5/6, -3 3/5, 1.7, -2.25, 8/9, 2/3.

Warm-Ups for Lesson on Common Fractions.  Warm-ups should review ordering fractions, addition and subtraction with negative integers, and addition and subtraction of fractions. Typical problems may include:

  1. 3/7 ____ 5/9   (Fill in blank with < or >)
  2. –15 +23 = ___
  3. 1.2 = ___/100
  4. 3/5 – 7/20
  5. 2/3 + 5/8 – 3/12

Scaffolding the Procedure for Common Fractions. The scaffolding strategy is to take what students know how to do, namely working with negative integers and extend that procedure as shown by the examples below:

Negative integers:   a)  7 – 12, b)  -5 – 7,  c) -10 + 15

Fractions with same denominator:   a) 7/20 – 12/20 = (7-12)/20 = -5/20 = -1/4

Putting the two integers together over the denominator now makes the problem into one they’ve seen before, namely 7-12.  Students should now do other examples independently.

b)  -5/8 – 7/8 = (-5-7)/8 = -12/8 = -1 ½

c) -10/17 + 15/17 = (-10+15)/17 = 5/17

More than two fractions:  -2/13 +5/13 – 6/13 -3/13 =

(-2+5-6-3)/13 = -6/13

Fractions with unlike denominators: a) -1/3 – 7/15  b) 3/7 -2/3  c) -7/8 – 5/12

Continue to place the integers together over the common denominator. As before, work through one example together; students should then work the others independently.

a) (-5-7)/15 = -12/15 = -4/5

b) (9-14)/21 = -5/21

c) (-21-10)24= -31/24

Warm up for Lesson on Mixed Numbers and Decimals. Warm-ups should review ordering mixed fractions and decimals, addition and subtraction with negative integers, and addition and subtraction of fractions. Typical problems may include:

  1. 3 5/8 ___ 3 2/3 Indicate whether > and < goes in the blank.
  2. 0.23 __ 0.099  Indicate whether > or < goes in the blank.
  3. 2 5/7 + 3 3/14
  4. -5/8 + 3/20

Scaffolding the Procedure for Mixed Numbers. The goal here is to extend students’ understanding of adding and subtracting mixed numbers that are positive to ones that include negative values.

Review of Adding and Subtracting by Wholes and Fractions:  Students are familiar with problems such as 15 2/5+5 1/5. These can be solved by first adding the whole numbers and then the fractions: 20 +3/5 = 20 3/5.

In some cases values must be carried such as 15 2/3+5 2/3. To obtain the total, the 4/3 is changed to the mixed number and then added to 20 for the final answer of

For subtraction, the same procedure applies: whole numbers subtracted and then the fractional portion. When we have problems where the second fraction is larger than the first, regrouping is necessary:

17 2/9-5 1/3   becomes 17 2/9-5 3/9.  This requires “borrowing”, in which 1 is borrowed from the 17. I have found the easiest way to explain this is by rewriting the problem as: 16+1+2/9-5 3/9

Then the “1” is written in terms of the common denominator. Since 1 can be written as 9/9, the problem can be rewritten as 16+9/9+2/9-5 3/9 which becomes 16 11/9-5 3/9.  This now allows us to subtract whole numbers, and fractions separately:  16-5=11;  11/9 – 3/9=8/9.  The answer is 11 8/9.

Using “Improper Fraction” Form:  The above method would be a complex way to solve problems that have negative numbers for reasons that will be shown.  A less confusing, though sometimes labor intensive method is to express the mixed numbers in improper fraction form. In the problem above 17 2/9 becomes 155/9;  5 3/9 becomes 48/9   (155-48)/9=107/9. Converting to a mixed number results in 11 8/9.

This method is generally easier to work with when there are negative numbers involved. For example, consider the problem 8 2/9 – 10 2/3: the problem becomes 8 2/9 – 10 4/9

8 2/9 – 10 2/3 = 8 2/9 – 10 4/9 = (74-94)/9

This is now solved as done with common fractions: -20/9, which can then be expressed as a mixed number: – 2 2/9

The problem can be solved using the “wholes and fraction” method, but students tend to find it confusing.  The above problem 8-10 + (-2/9) = -2 – 2/9 = -2 2/9.

Suppose, however, that the problem were 8 7/9 – 10 2/3.  Now it becomes 8 7/9 – 10 4/9, which in turn is expressed as (8-10) + (3/9).  The final form is -2 + 1/3, which is solved as -6/3 + 1/3 = -5/3 = -1 2/3.   Students need to keep track of a number of things—when is the fraction part added, and when is it subtracted. When it is added, -2 + 1/3 becomes – (2-1/3).

At this stage we want to keep things straightforward.  More advanced students may be given a few problems to be solved with the wholes and fraction method for extra credit.  In general, however, problems with mixed numbers use numbers small enough that the improper fraction method is the most efficient way to go.

For large numbers such as 130 5/18 – 231 7/22, a more efficient way to solve it would be to express the fraction portion as a decimal, so the problem becomes 130.278 – 231.318. Which brings us to the topic of working with decimal fractions.

Adding and Subtracting Decimals. Problems like 0.015 – 0.05 follow the same principles as integers. That is, we have to figure out whether there is a gain or a loss. This entails determining whether 0.05 is less than or greater than 0.015.  A common mistake is that students seeing the 15 in 0.015 and 5 in 0.05, assume that because 15 is greater than 5 that 0.015 is greater than 0.05. One way to avoid this mistake is to fill in the decimals with zeroes so they are now comparing 0.015 with 0.050.  It’s evident that 15/1000 is less than 50/1000. There will be therefore be a net loss: – (0.050-0.015) = -0.35.

Another technique to use is to ask “If something costs $0.015 per gallon and another costs $0.05 per gallon, how much does 1,000 gallons of each cost?”  $15 is clearly less than $50.

Students should be given practice with identifying which decimals are larger, along with the addition and subtraction problems.

Traditional Math (7): Rational Numbers (Seventh Grade)

This is the first part of a new sub-chapter in the chapter on Seventh Grade math, in what will be a book called Traditional Math, to be published by John Catt Educational.

Chapter 7.2: Rational Numbers

I recall when teaching an accelerated seventh grade math class, introducing the topic of rational numbers. When I had finished, a boy with unusual insight into math made the following observation: “Since they can contain whole numbers as well as fractions, rational numbers are deeper than just fractions”.  

I have kept his words in mind. Rational numbers can be confusing for students, particularly since they appear to be a fancy way of saying “fractions”. This notion is reinforced since most chapters on rational numbers focus on operations with fractions. Students are also confused by the notion of “irrational” numbers, since the topic is introduced before they know what square roots are, or what the number pi is all about.

This lesson provides the definition of rational numbers which is revisited later in the year.  It is also revisited in later courses as students understand more of the structure of the real number system.

Review of Decimals and Fractions. I like to begin with a quick review of decimals and fractions. The warm-ups for the day typically will have questions such as “Which is greater? 0.099 or 0.2?” and “What is the decimal equivalent of 1/3?”, “What is 2 1/5 as a fraction?”  “What is 13/5 as a mixed number?”

In previous grades they have worked with decimals and how to convert from fraction form to decimal form. Having said this, the students often act as if this is the first time they have worked with converting fractions to decimals. I therefore spend a few minutes on the first day going over how to express decimals as fractions, and vice versa. They should also know the decimal representations of certain fractions, such as 1/4, 1/5, 1/8, 3/8, 7/8, as well as the repeating decimal representations of 1/3, 2/3, and the ninths. (Some students will not know these; I therefore make one of the goals of this unit that students know these representations.

Rational Numbers. Because students are not yet fluent using symbols, starting with the formal definition of “rational number” will be a distraction. The definition is:

A rational number is a number that can be written as a/b where aand bare integers and b does not equal zero.

Instead, I begin with discussion and examples which will lead to the above definition of rational numbers. I first ask for examples of fractions. After the usual ones (e.g., 2/3, 5/6, etc.) I’ll ask whether -2/3, -5/6 are also fractions. Students will agree but there might be some hesitation. “What about 3/2, 5/3, 13/2. Are they fractions?”

Someone may point out that they are “improper fractions”. I like to make clear that what are referred to as “improper fractions” are still considered fractions. They can be expressed as mixed numbers as well, so I will show what those fractions are in that form, picking on students to do the conversions. In fact later in the year I make it known that unless directed to do so, students do not have to convert to mixed numbers; a fraction like 39/8 can be left in that form.

I then ask about numbers such as 6/3, 9/3, -25/5, -8/2,  4/1.  Are these fractions? There will be mixed responses. I point out that they’ve learned that 5/5, 6/6 and so on are “one whole” or the number one, and then ask if people have changed their minds. I’ll state “I think we can safely say that some fractions can be mixed numbers, and others can be whole numbers.” (Were I to leave it as an open question, there is bound to be one student who disagrees, which while interesting, takes away from the momentum and direction of the lesson.) I then plot the numbers we have on a number line. 

I will then give some examples of decimals: 0.333…,  0.75, -1.25, and plot those on the number line.  I’ll point out that as we learned from the warm-up exercises, we can express decimals as fractions as well.

Now I present the definition of rational numbers, pointing out that as we’ve seen we can express whole numbers, mixed numbers and decimals in fractional form. First, an informal definition: Numbers that can be expressed in a fractional form are called “rational numbers”. I’ll follow it up with the mathematical definition (given earlier in this section). I make it clear that rational numbers include fractions and whole numbers. They also include negative numbers. And like all numbers, they all can be located on a number line.

In the spirit of my unusual student who saw rational numbers as “deeper” than fractions, I now point out that the term “rational number” expands the concept of fraction, so that it includes numbers that are between two whole numbers as well as whole numbers themselves.

Checking for Understanding. I’ll ask how they know that, say, 9/3 is a rational number, looking at the definition. This is a worked example and I’ll ask whether nine is an integer, and three. Hearing agreement, I summarize that there are two integers in the form a/b.

I’ll give other examples, asking the same question but without the hints. Mixed numbers are included as examples, so students will need to make the leap and put them into fractional form; e.g., 1 2/3 is 5/3 to show conformity with the definition. Also included in the examples are decimals. Again, they must make a leap and put them into fractional form. 

I will ask whether decimals such as 0.333…., or 0.666… are rational numbers. Since I discussed repeating decimals at the beginning of the lesson, they should recognize at least one of these in fraction form. Then again, one learns to live with blank stares and temporary amnesia, and take it as part of the job of teaching.  In any event, the question serves as a lead-in to the next topic.

Terminating and Repeating Decimals. Students are familiar with the two types of decimals, terminating (such as 0.12, 1.2, and so on), and non-terminating repeating (such as 0.333…, 5.222…, 0.0101… and so on). 

Repeating decimals are those that will go on forever, but have a repeating pattern. Both the terminating and repeating decimals can be represented in fractional form, and are therefore, by definition, rational numbers. The decimal 0.333…. is 1/3, 0.0101… is 1/99.

Irrational Numbers. This brings us to the rather perplexing topic of irrational numbers. I once had a student who, thinking he was making a play on words, asked “If there are rational numbers, are there irrational numbers?”

He looked surprised when I told him that was an excellent question and in fact there are. An irrational number is one that cannot be expressed as a/b where a and b are integers.  In such cases, the number will be a non-terminating and non-repeating decimal. Non-repeating decimals do not have a pattern that repeats. For example, the decimal 0.01020304…if continued in this fashion does not repeat, and it is non-terminating. It is therefore an irrational number. There is no fraction that will produce such a decimal representation.

Students will have to take such claims on faith since the proof that irrational numbers are non-terminating and non-repeating is typically covered in college level math courses. Similarly, the number pi, which students may have heard about is also irrational and students must take it on faith that the famous sequence of numbers does not repeat.

Irrational numbers will be revisited with respect to square roots in the unit where square roots are discussed.

Division by Zero. The final discussion of this lesson is to talk about division by zero and why it is impossible. I start by asking students what 0/5 equals and then brace myself for the shock of that some students will not know this. We go over that zero divided by anything is zero because zero multiplied by any number is zero.  I then ask if 5/0 has an answer. Many will say “zero” to which I respond “So you’re saying that zero times zero equals five?”  I continue by asking whether other numbers will do and they quickly see that no number will work. This demonstrates why division by zero is impossible and I explain that it is considered (and called) “undefined”. For any fraction, zero cannot be in the denominator, because it represents division by zero, and that’s where I leave it.  This concept should be revisited at appropriate times during the course.

Traditional Math (6): Negative Numbers for Seventh Grade (cont) — Multiplying and Dividing Negative Integers

This is the conclusion of the chapter on negative numbers for seventh grade. These chapters will eventually become a book on Traditional Math, to be published by John Catt Educational.

  1. Multiplication of Negative Integers

The multiplication of negative integers can be a confusing topic for students—particularly the rule that the product of two negative numbers equals a positive number. The main problem that students have is in seeing what multiplication by a negative number may mean.

I’ve found that providing examples of situations modeled by multiplication with negative numbers is effective in helping students understand the rules. I start with a review of negative and positive numbers in terms of changes in particular situations. That is, their daily experience with integers is how they describe things like temperature, electrical voltage, elevation above or below sea level, bank balances, and gains and losses.

Integers also can represent a change in the situation as they have seen with the “gains and losses” problems. That is, changes in money earned versus money lost, temperature increases and decreases.

To start, I ask students to describe various changes as positive or negative:

a) Ann gained 4 pounds in the last month. (4)

b) Jerome lost 14 pounds in a week. (-14)

c) Kathy lost $40 on a roller coaster ride. (-40)

d) Five minutes from now. (5)

e) Ten minutes earlier. (-10)

The next examples require a bit more thought; describe the change in terms of positive or negative:

f) The temperature changed from -3 degrees to 2 degrees.

g) The football team lost 5 yards on the first play and gained 10 yards on the second.

h) A bird was at 120 feet above the water to 30 feet above the water.

i) The water was turned on at 10:00 AM and turned off at 12 noon.

j) Janet finished the drive at 3:00 PM; she started at 1:00 PM.

Multiplication of two numbers in which one number is negative. Students know how to multiply positive numbers and know how to represent them as repeated addition. A problem like “Ted made $10 an hour for 3 hours; how much was his total pay?” is represented at 10 + 10 + 10 or 3 x 10. 

After showing the above problem I ask students how they would write the following problem using repeated addition: “Sonia lost 3 pounds for 2 weeks in a row; how much did she lose after two weeks?” (If students need a hint, I will ask how they would represent a loss of 3 pounds.) Students will generally know the answer intuitively and upon hearing the answer of -6, I write on the board:

(-3) + (-3) = 2 x (-3) = -6.

 Finding an example problem to represent (-2) x (3) is a bit more difficult since adding three -2 times does not make sense. Dolciani’s “Modern Algebra: Structure and Method” (1962) contains an example that I’ve used in seventh grade which provides meaning to the negative values.

The example is of a water tank with water flowing into it at the rate of three gallons per minute. I ask if the three gallons per minute is a positive or negative number. If it flows at three gallons for every minute, how much more water will there be in the tank after two minutes? Notice I am asking how “much more”, not what is the total amount of water. We are calculating the change in the amount of water in the tank and students are quick to give the correct answer of six. I now ask if the time, two minutes, is positive or negative.  Both numbers are positive and we can represent the situation as 2 x 3 or 6 gallons more water than what was there before. This suggests the rule they all know:

2 x 3 = 6:  Positive number x positive number gives positive number.

For the next scenario, I want to know how many gallons less was in the tank two minutes ago, if water is flowing into the tank at three gallons per minute. Intuitively, most students will know that there will be six gallons less, because the water in the tank is increasing for each minute. Therefore for each minute prior, there was three gallons less. In my experience students will shout out this answer. I ask how “2 minutes ago” is represented. I will hear someone, usually hesitant, saying “Negative two?” And that is correct. We know there will be six gallons less than there was, so we have:

(-2) x (3) = -6: Negative number x positive number gives negative number.

This example shows that the negative number can be the multiplicand (i.e., the number being multiplied) and the positive number the multiplier.

Now I tell the students to assume that the water is flowing out of the tank at the rate of three gallons per minute. I ask how we represent that, making sure they understand that it is -3, since it is representing a loss. The problem now becomes what is the change in gallons after two minutes. Students should recognize that the two minutes is a positive value.  I ask how we represent this situation as a multiplication statement. Since this is similar to the opening problem about losing 3 pounds per week for two weeks, students should that the answer is once again 2 x (–3) = -6. This time the -6 represents a loss of 6 gallons. This problems suggests

2 x (-3) = -6: Positive number x negative number gives negative number

The final example is a negative number times a negative number. Before I present the example, I ask if anyone knows whether the product will be negative or positive.  For those who say it’s positive I will ask why. Some explanations will be vague, but I’ve found that at least one person will extrapolate what we’ve done before. In such instance I then proceed with the example. If it doesn’t happen, it isn’t a problem. I just proceed and say something along the lines of “Let’s find out.”

In this case, the water is flowing out of the tank at the rate of three gallons per minute, represented as -3.  I want to know if two minutes ago (-2) there was more or less water in the tank, and by how much. At this point, most will know the tank held six gallons more water. I now write:

(-2) x (-3) =6: negative number x negative number gives positive number.

 At this point the rules are summarized:

  1. The product of two integers with different signs is negative.
  2. The product of two integers with the same sign is positive.

Admonition. After establishing the rules via the examples, I admonish students that the examples only suggest that these rules are true. There is a mathematical proof of these rules which I provide after they have learned about the distributive rule. For now, it’s all they need in order to get a sense of what’s happening when we perform these multiplications.

Nevertheless, there will undoubtedly be some students who will not understand how the examples work and why the rules are suggested by them. I tell these students that it will become clearer the more they work with such problems, but that for now should just follow the rules and “trust the math” that it is telling a true story.

Practice and extension. Students are now ready to work on guided practice problems, the first few of which summarize what we have just learned—problems such as (-5) x (-7), (-2) x (4), (-1)(5), and (-1)(5).  After the last two problems I will give them the following:  5 + (-1)(5), and -5 + (-1)(-5). The first problem becomes 5 + (-5), which goes back to what they have just learned about adding negative integers. It is not unusual that they will look at the problem as if they have never seen it before.  I ask them if they can remove the parentheses, and remind them that +(-) equals -.  The problem then becomes 5 -5, or zero.

Similarly -5 + (-1)(-5) becomes -5 + 5 which is also zero.  I will ask them to summarize what multiplying a number by -1 changes it into. There may be blank stares, and if so, I will remind them that 5 and -5 are called “opposites” which some have likely forgotten. It is not unusual for such lapses in memory, given the amount of new information that they are taking in.  But it is important to link what they have learned in this lesson with what they know about addition and subtraction rules. To that end, other problems may include: 5 – (-2)(5); -4 + (3)(-2), and so forth.

So for 5 – (-2)(5), the number (-2) is multiplied by 5 which yields -10. The resulting problem is then 5 – (-10). From this point, a common mistake will be forgetting to include the negative sign next to 10, and writing 5 -10. 

For these type of combination multiplication and addition/subtraction problems, it will be necessary to remind them of order of operations which they have had in sixth grade—multiplication operations are performed before addition or subtraction. Also, using parentheses to denote multiplication should be explained. This notation is easier to work with than the form 5 + (-1) x (5), since the parentheses make the order of operations more obvious. In addition, using parentheses for multiplication prepares students for algebraic notation. Later when they work with evaluating expressions like a + bc, by substituting numbers for a, b, and c, they will already have had experience with that form. Also, they will see that inan equation in the form 2x +4 =8, the 2x represents 2 multiplied by an unknown number.

Multiplying more than two numbers.  Before I set them loose on homework, I put up a problem and ask how I would solve it:  (-2)(3)(-5). Those who get it, I then ask to explain to the class. Such problems are broken down and solved by multiplying two numbers at a time; they above problem then becomes (-6)(-5), which is 30.  A problem like (-5)(2)(-2)(4)(-1) becomes (-10)(-8)(-1), which is -80.  After maybe two more, I will ask them if the number of negative numbers in the problem helps them determine whether the final product is negative or positive. I have them discuss that, leading them to see that an odd number of negatives will result in a negative product.

  1. Division of Negative Integers

It is definitely recommended that the warm-ups preceding today’s lesson include not only problems about multiplying negative integers, but also addition and subtraction. New information recently learned tends to eclipse older information. This can be seen by a common mistake that students will start making, and I’m sad to say will persist among some students well into the school year. That mistake is to conflate the rule stating that the product of two integers with the same sign is positive with the rule for addition of two integers with the same sign. Students will see a problem like -5 -5 and now in addition to making the mistake of saying it is zero, will then say it is positive 10. 

Warm-ups should include straight multiplication problems like (-2)(5) and (-1)(-4)(3), but also a problem like 2 – (-2), 2 + (-5), and the combination multiplication and addition/subtraction problems that get at the problem described above.  For example -5 – (-1)(-5), which becomes -5 – 5. 

Going over the warm-ups will then serve as a review of what has been covered so far. At this stage I have found myself saying “Why did I ever think this was going to be easy?” Some classes will get it more easily than others, I’ve found, but there are always relapses, forgetfulness, and it will be necessary to repeat the rules.

Division as inverse of multiplication. Division of negative numbers is a straightforward application of the rules for multiplying negative numbers. The rules are the same:

  1. The quotient of two integers with like signs is positive.
  2. The quotient of two integers with unlike signs is negative.

I start the lesson by asking what 2 x (-3) is.  Hearing -6, I then write 2 x (-3) = -6.  Since division is the inverse of multiplication we can divide the produce, -6 by either 2 or -3 to get the other factor. That is -6 ÷ (-3) = 2, and -6 ÷ 2 = -3.  In other words, when we divide -6 by 2 we are seeking a number which when multiplied by 2 yields -6.  That number has to be negative. I will ask the class why and then wait while an uncomfortable silence pervades. I would like someone to say “To get -6, 2 has to be multiplied by -3 because a positive number multiplied by a negative number is negative.” I will settle for something reasonably close, however. If someone says that -3 x 2 equals -6 I’ll go with it and give further examples. I don’t want to spend a lot of time trying to get them to say the right thing and then have them forget what it is we’re learning.

Similarly if we divide 6 by -3, the number that is multiplied by-3 to yield 6 has to be negative. Why? Because the product of two negative numbers is positive.

I will then provide examples for the class to work on and include combination problems such as -10÷2 -3. Again, they will need to be reminded of order of operations; i.e., division operations occur before addition or subtraction.

The bottom line rules are exactly the same as multiplication: for two numbers which can be paraphrased into something succinct enough to fit on a bumper sticker: Like signs: positive. Unlike signs: negative.

Despite the fact that the rules are ultimately the same, students will still get confused. It is a matter of repetition and practice. Also, it is essential to include all types of problems that involve computations with negative numbers, not only in warm-up questions but in quizzes and tests given throughout the year.

I conclude this sub-chapter by providing examples of warm-up questions that I have given after covering all the topics discussed in this entire chapter.

  1. -3 x 3
  2. (-5 + (-20) ) ÷ (-2)
  3. -60 ÷ (-10)
  4. (5-10) x (4-8)
  5. (5-(-4) ) ÷ (-3) = ? + 6

Problem 5 is a more challenging problem and it is admittedly a “front-loading” of the type of problem they will be solving later when we cover expressions and equations. Typically this problem opens up discussion when we go over the problems despite the hints I gave students when they are working the problems. 

Solving the left hand side results in 9 ÷ (-3) which equals -3.  We now have to find what value the question mark represents.

I will sometimes give them a simpler problem that they’ve seen before such as 9 = 7 + ?.  They will solve this quickly (usually—there are always exceptions!) and hearing the correct answer, I ask how they did it.  The usual answer is “Two added to seven is nine.”  Which is correct, though I’d much rather hear, “Nine minus seven equals two.”  If I don’t hear that, I will say it; something like, “So if 9 = 7 + 2, we can also say 9 – 7 = 2 and 9 – 2 = 7.”  This same pattern can then be applied to the problem at hand : -3 -6 = -9. 

 

 

 

 

Traditional Math (5): Negative Numbers for Seventh Grade (cont) — Subtracting Integers

This is a continuation of the chapter on negative numbers for seventh grade. The next installment will be on multiplying and dividing negative numbers.

Sub-Chapter 4: Subtracting Integers

This lesson provides a clarification and, for some, a revelation, that subtraction of two numbers is the addition of an additive inverse. In formal mathematical terms, x – y is defined as x+(-y) where -y is the additive inverse of y. They have seen this already with problems such as 4+(-10) on the number line, which they have learned is the same as 4-10. There is one important case that hasn’t yet been explored, which is subtracting a negative integer; e.g., 5-(-10). This particular case will be the new procedure that they learn; everything up to that point is a clarification of what has come before.

At least one of the warm-ups I’ve used for this lesson provides a segue to the subtraction of a negative number. Specifically:

The temperature yesterday was 4 below zero. Today it is zero. By how much did the temperature increase?

 

Adding the opposite. I start this lesson by asking the class to find 7 + (-4).  After this is done, I think ask them to tell me what 7 -4 equals.  I ask why we obtained the same answer and if I don’t hear something along the lines of “It’s the same problem” (which sometimes happens despite all my intentions) then I’m not afraid to say “Do you suppose that 7 +(-4) is the same problem as 7 – 4?”

At this point I disclose that whenever they have been adding negative integers, they are subtracting. Subtraction is really the addition of an additive inverse.  Stated more simply, subtracting a number is the same as adding the opposite of this number.  I quickly give an example: The problem 7 – 3 is the same as adding the additive inverse of 3, which is 7 + (-3).  I ask them to write it without brackets as we did in the previous lesson: 7 – 3.

Addition and subtraction are what are called inverse operations. The students have been using this fact for years, having been told that 5-2 is a number which when added to 2 equals 3.  That is 5 = 2 + x.  They have worked with number bonds or number families, so they are familiar with 2+3 = 5, 5-3 = 2, and 5-2 = 3.  When we have a problem like 4 – 10 (warm up problem 3), we are finding what number added to 10 equals 4.  They may have solved it using the “good day, bad day” technique, seeing first that there is a loss and it’s a loss of 6, or -6.  I then write 10 + (-6) and ask what it is equal to. They can see the answer is 4.

I remind them that this is how they were taught to check if their answer to a subtraction problem is correct. This may seem like new information even with reminders that this is what they have been doing the past few days starting with the “gains and losses; good day, bad day” technique. Therefore, more examples are necessary to reinforce the procedures so they are comfortable with doing such problems.  I then give them three or four problems to do and have them check the answers. Thus, to check that 3- 6 equals -3, the student would add 6 + (-3) to obtain 3, which checks.

 

What Subtraction Represents.  A problem like 10 – 4 can represent the loss of 4 things—those things being many different items such as money, weight, length and so on. For example, the question “If the temperature was 10 degrees and it decreased by 4 degrees, what is the resulting temperature?” is answered by subtracting 4 from 10. If, however, the question were “The temperature was 4 degrees and now it is 10 degrees. By how much did it change?”, the answer is still six, but the numbers represent different things.

In the first instance, six represents the new temperature after a decrease of four degrees. In the second it is the amount of increase in temperature from four to ten degrees.

Subtracting a Negative Integer. Now we come to subtraction of a negative integer such as 10 – (-4). The second model is what we use when we present subtraction of a negative integer in order to keep things straightforward for students, I limit the examples to finding the amount of change rather than what a loss of -4 represents.  (An example of a problem that asks what’s left after a loss of -4 would be: A person has $10 in his bank account after $4 has been deducted in error. The bank corrects this error by removing the debit of $4. This is done by subtracting the loss of $4: 10-(-4), which then becomes 10 + 4 or $14.  Even adults may find it confusing that ancelling a debt can be represented by the subtraction of a negative number. It is therefore highly likely that seventh graders will find the concept difficult. Since the goal is for   students to subtract negative numbers, it is far easier to explain the procedure using the “find the difference” model discussed above, rather than by the “find what’s left” model.)

I first point out how they answered the question of “If it were 4 degrees yesterday and 10 degrees today, what is the change in temperature?” I want students to see the form they used: Today’s temperature minus yesterday’s temperature.

Next, I ask: “If it were 4 degrees yesterday but 10 degrees today what is the change in temperature?” Using the form defined above, we obtain 10-4 which represents an increase of 6 degrees.

We are now ready to present the problem of 10 – (-4).  The question becomes: “If it were 10 degrees today, but -4 degrees yesterday, what was the change in temperature.” It can also be stated, “What was the increase in temperature from -4 to 10 degrees? The problem can now be written as 10 – (-4), which I leave up on the board.

Drawing a number line (either vertical or horizontal), I plot -4 and 10. As mentioned earlier, one of the warm-up problems for this lesson asked how much must the temperature increase from -4 degrees to reach zero. Although one would hope that students remember the warm-up problem, I usually have to remind them of it, as well as the answer: the temperature must increase 4 degrees to get to zero degrees from -4. That is, -4 + 4 equals zero.

Transferring this to the number line, it is apparent that zero to 10 degrees is an increase of 10 degrees, so we add 4 to 10—an increase of 14 degrees.

I then explain that rather than using a number line to calculate it, we can use the “add the opposite rule.  The problem 10 – (-4) becomes 10 added to the opposite of -4 or 10 + 4, which is 14.

Another example using depth under water illustrates subtracting a negative number as well. I scaffold the problem by first starting with all positive numbers: A bird was 5 feet in the air and flew up to 10 feet. What was the distance upward that it flew?  Students will easily see that it is “new height minus original height”, or 10 – 5.

The problem is now changed so that a bird dives underwater 5 feet to catch a fish, and then flies upward to a height of 10 feet above the water. What was the upward distance that it flew? I provide prompts, such as “How do we represent 5 feet underwater?” (-5) and “How is the problem written?”  10 – (-5).

I also want to show problems where there is a decrease. For example: “The temperature today is -5 degrees; yesterday it was 6 degrees. What is the change in temperature?”  The problem is written as -5 – 6, which equals -11, or a decrease of 11 degrees.

After these worked examples, students are now given four or five problems that require subtracting a negative. One or two problems will be word problems but the others are strictly numerical.

Vertical vs Horizontal Number Lines. Some teachers I have spoken with recommend using vertical rather than horizontal number lines. They have observed that when the number line is placed vertically, the students more easily grasped the idea of subtracting a negative. It appeared that up = positive, down = negative was a “natural” visualization as opposed to left = negative, right = positive. This is something to keep in mind if students struggle with this and other concepts—teachers can move back and forth between vertical and horizontal number lines as necessary.

Common error (again).  Students will continue making the mistake of seeing problems like -4 -4 as zero. Now there are three ways to adjust their thinking.

1) Gain/loss method: “I lost $4 and then I lost $4 more.”

2) Number line method: -4 -4 on the number line is (-4) + (-4).

3) The problem -4 – 4 is not the same as -4 – (-4). The latter equals zero because when one adds the opposite it becomes -4+4, and adding a number’s opposite will always equal zero.

Giving it a rest. Mastering the operations with negative numbers will be confusing for some students at first. Although these lessons continually refer to, build upon and reinforce the “gain/loss” procedure, some students may become overwhelmed by new information as well as the mathematical way of stating things. In particular, they are now learning to think of subtraction as the addition of an opposite number.

It is advisable to give students time to work with the newly learned procedures to ensure that they are comfortable with them and are achieving automaticity. Many textbooks, however, provide an additional topic such as showing pictorially how addition and subtraction work using circles where each circle represents a positive or negative unit integer. The purpose is to spotlight the conceptual underpinning of adding and subtracting negative integers. An example of how it works is shown in Figure 3.

Figure 3. Pictorial approach for subtracting integers, using circles

In my experience, when students are still trying to master the procedures for adding and subtracting negative integers, additional approaches may confuse more than enlighten. Some students may not be ready to absorb the yet another pictorial approach, particularly when they have adjusted to the pictorial approach using the number line.

The pictorial approach can be given later when, in the estimation of the teacher, students appear proficient and comfortable working with negative integers. I tend to not present it, particularly if I see that students are successful in working with negative numbers. Ultimately it is a judgment call, based on a teacher’s experience with teaching it, as well as their understanding of how it works and how and when to present it.

Traditional Math (4): Negative Numbers for Seventh Grade 

This is the first part of the unit on negative numbers for seventh grade.  It includes a general introduction, gains and losses, and adding negative numbers.  Tomorrow will be subtraction, and the next day multiplication and division.

In my Math 7 classes, the beginning of the school year begins with the unit on integers, which includes operating with negative numbers. A new teacher in a new classroom, with new school supplies coupled with a topic students haven’t had before often has the same allure and excitement as that of a shiny new toy that holds great promise for many exciting and life-changing hours.

Students view me, their new teacher, just as they do their brand new school supplies including the graph paper notebook that I hand out on the first day. New notebooks (particularly graph paper notebooks) hold the promise of being filled with information that will make them smart. After the first few lessons, it has not been unusual for me to hear from some parents that their kids have remarked “This is the first time I’ve really understood math.” 

As things become more complex, the feeling of newness and promise fades. Students go from saying “I finally understand it” to “I hate math” sometimes in the span of less than a week’s time. It is also not uncommon for students to understand and carry out a procedure perfectly the day it is introduced, only to have totally forgotten it the next day, with some students asking “When did we learn this?”

Nevertheless, there is good news. Based on what they have been doing in the lower grades with differences of quantities and computing changes in weight, amounts of money and so forth, students at this point know what losses and gains are and how to compute them. We build on this prior knowledge and intuition so that they are able to express and compute quantities in terms of negative numbers.

The first lessons build upon what they have learned previously. As the topic becomes more complex and students become confused, teachers can and should refer back to some of the introductory techniques as a way to underscore that what is being taught today is building on what they perfectly understood just a few days previous.

It is also important to continue refreshing the procedures for operating with negative numbers throughout the year, to ensure what has been mastered stays that way. Continued repetition and practice helps to lock in the procedure and ensure automaticity.

Some aspects of negative numbers will seem abstract to students to the extent that a procedure may not make logical sense or seem counter-intuitive. The mathematician John Von Neumann once said “In mathematics you don’t understand things. You just get used to them.” Like many things in mathematics, after experience and practice with a procedure and concept, what was once alien becomes familiar. The familiarity eventually allows students to see the concept as reasonable and accept it—and they may even wonder why they ever found it confusing. At that stage, it is not unusual for students to say “I understand it now.”  This is particularly true with the topic of negative numbers.

General Overview

The general arc of progression on this topic is:

1. Introduction to negative numbers: Number line, order, and absolute value

2. Gains and losses

3. Adding negative integers

4. Subtracting integers

5. Multiplying and dividing negative integers

1.  Introduction to negative integers

This unit focuses on the mathematical operations of addition, subtraction, multiplication and division with negative numbers. Some students have may have learned the operations with negative integers in sixth grade, depending on what textbook was used, and/or the inclinations and goals of their sixth grade teacher. For those students, this unit will be a review. Others may have had an introduction to what negative numbers are, but not operations with them.

I start this unit with a review of what all students may know about negative numbers. This review is generally not included in seventh grade textbooks and focuses on concepts with which they are familiar. It is an overview of what negative numbers represent, using examples such as temperature (degrees below zero), or depth (feet underground or under water). I generally take two days to do this review and overview.

On the first day, we cover the general concepts of gains and losses, and their representation. A gain of $10 can be represented as +$10 or $10. A loss is represented as -$10. A descent of  feet can be represented by the number -50. A drop in temperature of 10 degrees: -10

Some students will observe that you cannot possess a negative amount of anything. If no one makes this observation, then I do. It is worth mentioning that negative numbers can be a comparison, or a relative amount. For students just learning about negative numbers, it is a new way of expressing comparisons and changes. For example, students will readily answer the problem “If it was 60 degrees yesterday and 40 degrees today, what is the change in temperature?”

I will give students this problem and upon hearing the correct answer of twenty degrees, I ask if it is twenty degrees more or less. Hearing “less”, I then point out that it is a loss of twenty degrees. Since they have learned that a loss can be expressed as a negative number, I then ask if I can say that the change is -20 degrees.

Negative numbers can also be used to indicate direction, or relative position.  I will ask: “In a football game, if your team lost ten yards, how would you indicate that using a sign?” Of course I want them to say -10 and they usually will. The negative number tells us about the position of the ball relative to where it started.

Number Line. The discussion about negative numbers and direction directly relates to the number line, so I introduce it at this point. Students have seen number lines before, but now we look at it with respect to negative numbers.

 There are two simple principles for number lines that I state:

1. Negative numbers are numbers that are less than, or to the left of zero; positive numbers are to the right of zero.  

2. The bigger the number, the farther it is to the right. The smaller the number, the further it is to the left.

A number left of another number is less than that number: 5 < 7,  – 7 < -5.  A number right of another number is greater than that number.  -2 > -8. 

I might ask “If it is -20 degrees today and it was -30 yesterday, which day was colder?” It is obvious that -30 will be colder, and by plotting the points on a number line on the board, students easily see that -30 is to the left of -20.

I will show two numbers to the right of zero on the number line; say 5and 7. When we have two positive numbers, the number furthest from zero is the greatest number.  I will then show -5 and -7.  They will quickly identify -7 as the furthest from zero when asked. “Is this number the greatest number of the two?”  They will see that the opposite is true. When comparing two negative numbers, the greater number is the one closest to zero. 

Absolute Value. Students may have heard about absolute value in sixth grade, but now it is presented in more detail. I will have students draw a number line and plot two points on it; say -4 and 4. I may also other pairs, using different colors.  I will ask if -4 and 4 are the same distance from zero, and similarly for the other pairs.  These numbers are called opposite numbers.  I will ask them then to give me some opposites: What is the opposite number of 100? of -50? of 25? 

I will then pose a situation in which we say that the numbers represent a football team’s loss on a play, and positive numbers a gain on a play.  The point zero represents the point at which the play originated.   

A loss of 10 yards on a play is represented as -10. Looking now at the opposite pairs I have on the board, I will say that these represent losses and gains on a play. Although a loss of 4 yards can be represented as -4, the distance itself is 4 yards—the distance from the starting point, expressed as a positive number. To make it plainer, suppose someone is wearing a fit-bit and paces out the 4 yard loss. How many yards will show up on the fit-bit?  Will it be -4?  No, it will be a positive number.

Distance is always a positive number. We call the distance from zero on a number line (or, in general, the distance from the starting point) the “absolute value” of a number indicated by two vertical bars; e.g.,   So whether a number is to the left or right of zero, their distance from zero is always expressed as a positive amount. I have them do some examples at this point, mixing in yesterday’s discussion about how to determine whether a number is greater or lesser than another, with today’s discussion of absolute value:

Examples: Find the opposite number:  -3, 5, -500,

Find the greater number:  -3, |-6|; -2, -10; -2, |-2|; |-54|, |53|

This lesson provides a lead in to the next day’s discussion on net gains and losses.

2.  Gains and Losses

This lesson introduces students to the concept of adding negative integers. The approach for this lesson comes from JUMP Math and is a very effective way of introducing students to the concept of adding negative integers through the concept of gains and losses—without students realizing that that’s what they’re doing. After the first few minutes of working with the problems, it is amazing to see them doing intuitively what they will be doing when learning the formal rules for adding negative integers.

When I first started using the technique from JUMP, I mistakenly thought “This is going to be easier than I thought.” As it turns out, while it did make things easier, it wasn’t a slam dunk. Students will still get confused, and will look at what they learned with this technique as something that happened in the distant past and no longer applies. Which is why this particular lesson must be continually brought into subsequent lessons as a reminder that what they did intuitively is what they will continue to be doing. It’s just that the formal rules appear as a different entity.

Opposite Integers. We start with a review of opposite integers. A loss of 5 pounds is represented as -5, and a gain as +5, or just 5. Does a person who loses 5 pounds and then gains 5 pounds end up weighing more, less or the same as their starting weight? Students will generally agree that they will end up with the starting weight. 

Money is an easy and effective way to work with gains and losses. I start by having  students say what the integer is that represents a gain or loss and then state what the opposite integer is. For example, the integer representing a gain of $6 is +6 (or simply 6), and its opposite is -6. I note that while the initial number has the dollar sign attached, when it is expressed as an integer it is without the unit.  A loss of $7 is -7, and its opposite is +7, or 7. (Later in the lesson they will be instructed to write positive numbers without the plus sign.)

Identifying overall gains or losses. Writing +7 – 4 on the board, and explaining that the numbers represent money, I ask was more gained or lost? If there is a net gain, we call it a good day; a net loss is a bad day. They will see immediately that it was a good day. When asked how much was gained, students are quick to tell me $3.

Writing -4 + 4, I ask was anything gained or lost? Nothing was gained or lost, so zero represents “no change”.

We continue with examples. For each one I ask how they came up with the answer. For the problem -6 + 2, there is a loss of $4, (written as -4), which they will explain they derived by subtracting two from six to get a loss of 4.

I paraphrase what they’ve done:

“You have more of a loss than a gain. That means you have more negative numbers than positives. So we write it as an everyday subtraction problem with the signs reversed: 6 – 2. We get positive 4, but since it is a loss, we write it as -4.” 

Nothing fancier than that for now. Students will operate intuitively with the exercises in this lesson.

I point out again that if the first number is positive, like +7-8, we don’t have to write the positive sign.  If we write 7 – 8, it is assumed that the 7 is +7, and is positive. Additional examples help get them used to this, although it may take longer than you would like before there will still be no more blank stares when the “+” sign is omitted from the first number.

Two gains or two losses.  I will write on the board -5 -3=__? and ask if it is a good day or a bad day. If the response is stunned silence, I will state the problem as “I lost $5 and then I lost $3. Good or bad day?”  They will immediately see it is “bad”. I’ll then ask for the overall loss while keeping my fingers crossed that they give me the right answer. They usually do. It is an overall loss of $8, so -8 would be what is written in the blank.

After writing +2+2 =__ students see that the overall gain is $4, so 4 (or +4) is written in the blank. This will be revisited in the next day’s lesson and stated as a rule, that adding two positive numbers results in a positive number, and adding two negative numbers, results in a negative number.

A common error is to interpret two losses of the same number as zero; i.e., – 7 – 7 is mistakenly thought of as 7 – 7.  This mistake will come up repeatedly, and the remedy that I have taken is to remind students of what it represents in terms of two losses: “I lost $7 and I lost $7 more; how much did I lose in all?”

Adding more than two gains and losses.  Now we up the ante a bit, with a problem like +3-4-5. 

This can be solved sequentially. That is, the first two numbers are evaluated: +3-4 which students will know is  -1. Then we are left with -1-5 which students will know is -6.  An easier way is to add the total gains, then add the total losses. In this way we get +3 -9. It is an overall loss of 6 or -6.

Other examples: 2-5-4+8-3.  Adding the gains (and remembering that 2 is the same as +2), we get 2 +8; the losses are -5-4-3.  Total gains equal 10, and total losses are 12, so we have 10-12, for a total loss of 2, or -2. 

For homework I assign problems from JUMP Math (see Figure 1). Problems can also easily be constructed as a worksheet.

Figure 1: Gain/Loss problems from JUMP Math from AP workbook 7.1, Common Core edition, 2015; Toronto; (printed with permission)

  1. Adding Negative Integers on a Number Line

The next day’s lesson now represents what we did with gains and losses on the number line. Also, no matter how much they were on track during the previous day’s lesson, that was yesterday, and today is entirely different.

Because this is a new representation, students may think that the number line method is to be used for some problems, and gains and losses for another.

 It is therefore important to tell students that the number line method is a way to look at what was happening in the previous lesson when we worked with gains and losses. The warm up problems for the day should therefore focus on some of the gains and losses problems, as well as opposite integers:

If a football team gains 8 yards on the first play, how would we write that?  (8)  If they lose 10 yards on the second play, do they have an overall gain or loss?  (loss) By how much? (loss of 2 yards or -2)

The above warm-up, written as a gain/loss problem would be 8-10. I like to use a warm-up problem to segue to the day’s lesson. Using the above problem, students now do the problem on a number line. Rather than writing it as 8-10, however, we write it differently: (+8) + (-10).  The purpose is to emphasize that we are adding a loss of 10 yards.

The rules for showing this on the number line are kept simple: The first number, 8, is marked on the number line. To add a positive number draw an arrow to the right the specified number of units.  To add a negative number we draw an arrow moving left the specified number of units. 

For the above problem we would draw an arrow with a length of 10 units going left from the starting point of 8. It ends at -2 which is the answer, as shown in Figure 2:

Figure 2: Number line representation of (+8) + (-10)

Students then do various problems using the number line, including problems where opposites are added such as (+4) + (-4). I generally allow about 10 minutes for this guided practice. Included among the problems are the sum of two positive numbers and the sum of two negative numbers.

After it appears that students have the knack of doing problems on the number line (with full recognition that such appearance may be like the mirage of water on a highway that disappears as you approach it), I select a few number line problems that they have done and have them write the problems without the parentheses. For example, the problem (+7) + (-12), is the same as +7 – 12 (or writing the first number without the positive sign, 7-12). I repeat the explanation that this is the same as adding a gain of 7 and a loss of 12.  To help them write these problems without the parentheses, the following mnemonic, which I write on the board, proves useful:

 ++ = +, and +(-) = -;  Examples:   +(+5) = 5; +(-5)=-5

Students now rewrite the selected problems without brackets, and solve the problem  as they did yesterday with the “gain/loss” problems. Having them do this reinforces and builds upon their prior success in the previous lesson. Each problem must agree with what they obtained using the number line. If it doesn’t, we find out why. This part of the lesson leads to a formal summary of the rules for adding negative numbers and which I have made copies for gluing into their notebooks:

Summary of the Rules for Adding Negative Numbers.

  1. The sum of two positive numbers is positive.

     Example: I gained 3 lbs last week and 2 lbs this week. Total gain is 3 + 2 =5.

          2.The sum of two negative numbers is negative.

    Example: I lost 3 lbs last week and 2 lbs this week. Total loss is -3-2 = -5

  1. Adding integers with different signs: informal rule

Since students have been working with gains and losses to add negative numbers, the rule can be stated informally in terms of what they have been doing in this and the previous lesson:

Determine whether the sum represents a gain or a loss. Find the difference between the numbers. If it’s a loss, then give the answer a negative sign. If it’s a gain, it will have no sign, since no sign means positive.

Example:  Our team gained 4 yards and then lost 6 yards. Are we ahead or behind and by how much? The sum is represented as 4 – 6. The amount of loss is greater than the amount gained, so there is an overall loss, calculated as 6 – 4. The loss of 2 is written as -2.

In addition to the informal summary above, I explain to students that on the board is the formal rule for what they have just learned: Subtract the lesser absolute value from the greater absolute value. Then use the sign of the integer with the greater absolute value.

This is illustrated with an example:

Example:  -5 + 3.  The absolute values of the two numbers are 5 and 3. The integer with the greater absolute value is -5; since the sign is negative, the answer is –(5-3)= -2 

The formal rule will make more sense at a later time after they have sufficient experience with these type of problems. For now, I work with the informal.

Common errors. The error of thinking of -5 – 5 as zero will persist. Since they have been working with number lines, I show students who make this error what -5 -5 looks like on the number line, and then show +5) + (-5) and (-5) + (5)

Overthinking and the Lead In to Subtraction. I once had a student ask “If you can’t have a negative amount of anything then how can you add -10 to something?” I explained that you don’t physically have -10 of something; you are representing a loss of 10, just like we did in the gain and loss problems. “How can you add a loss?” This is an example of overthinking. For such questions, it’s good to remind them of the gain/loss problems they did on the previous day. 

Von Neumann’s quote about understanding in math and getting used to things applies here. The silver lining is that such confusion can be exploited in the next lesson in which subtraction of an integer is defined as addition of the additive inverse of the integer.

Figure 2: Number line representation of (+8) + (-10)

Students then do various problems using the number line, including problems where opposites are added such as (+4) + (-4). I generally allow about 10 minutes for this guided practice. Included among the problems are the sum of two positive numbers and the sum of two negative numbers.

After it appears that students have the knack of doing problems on the number line (with full recognition that such appearance may be like the mirage of water on a highway that disappears as you approach it), I select a few number line problems that they have done and have them write the problems without the parentheses. For example, the problem (+7) + (-12), is the same as +7 – 12 (or writing the first number without the positive sign, 7-12). I repeat the explanation that this is the same as adding a gain of 7 and a loss of 12.  To help them write these problems without the parentheses, the following mnemonic, which I write on the board, proves useful:

 ++ = +, and +(-) = -;  Examples:   +(+5) = 5; +(-5)=-5

Students now rewrite the selected problems without brackets, and solve the problem  as they did yesterday with the “gain/loss” problems. Having them do this reinforces and builds upon their prior success in the previous lesson. Each problem must agree with what they obtained using the number line. If it doesn’t, we find out why. This part of the lesson leads to a formal summary of the rules for adding negative numbers and which I have made copies for gluing into their notebooks:

Summary of the Rules for Adding Negative Numbers.

  1. The sum of two positive numbers is positive.

     Example: I gained 3 lbs last week and 2 lbs this week. Total gain is 3 + 2 =5.

          2.The sum of two negative numbers is negative.

    Example: I lost 3 lbs last week and 2 lbs this week. Total loss is -3-2 = -5

  1. Adding integers with different signs: informal rule

Since students have been working with gains and losses to add negative numbers, the rule can be stated informally in terms of what they have been doing in this and the previous lesson:

Determine whether the sum represents a gain or a loss. Find the difference between the numbers. If it’s a loss, then give the answer a negative sign. If it’s a gain, it will have no sign, since no sign means positive.

Example:  Our team gained 4 yards and then lost 6 yards. Are we ahead or behind and by how much? The sum is represented as 4 – 6. The amount of loss is greater than the amount gained, so there is an overall loss, calculated as 6 – 4. The loss of 2 is written as -2.

In addition to the informal summary above, I explain to students that on the board is the formal rule for what they have just learned: Subtract the lesser absolute value from the greater absolute value. Then use the sign of the integer with the greater absolute value.

This is illustrated with an example:

Example:  -5 + 3.  The absolute values of the two numbers are 5 and 3. The integer with the greater absolute value is -5; since the sign is negative, the answer is –(5-3)= -2 

The formal rule will make more sense at a later time after they have sufficient experience with these type of problems. For now, I work with the informal.

Common errors. The error of thinking of -5 – 5 as zero will persist. Since they have been working with number lines, I show students who make this error what -5 -5 looks like on the number line, and then show +5) + (-5) and (-5) + (5)

Overthinking and the Lead In to Subtraction. I once had a student ask “If you can’t have a negative amount of anything then how can you add -10 to something?” I explained that you don’t physically have -10 of something; you are representing a loss of 10, just like we did in the gain and loss problems. “How can you add a loss?” This is an example of overthinking. For such questions, it’s good to remind them of the gain/loss problems they did on the previous day. 

Von Neumann’s quote about understanding in math and getting used to things applies here. The silver lining is that such confusion can be exploited in the next lesson in which subtraction of an integer is defined as addition of the additive inverse of the integer.