Negative Numbers and Subtraction Are Not the Same Thing

One of the frustrating things about teaching math occurs when students encounter unnecessarily confusing math symbols. This week the 6th graders have been struggling with understanding how to use the subtraction operation with negative numbers. Subtraction and negative numbers share the identical notation but are very different concepts.

Subtraction is an action, what mathematicians call an operation.  For instance 6 - 2 = 4 means start with the number 6 then do the action of removing 2 objects. The remaining number is equal to 4.

This is represented by the symbol -.

A negative number is the opposite of a number in the sense that under addition the two cancel each other exactly. A negative number is a thing, a place on the number line, not an action.  The opposite of 3 is -3 because 3 + -3 = 0.

Negative numbers are distinguished from positive with the symbol -.  See the problem?

Subtraction is a verb.  Negative numbers are nouns.  But the math language is the same. Imagine how awful the English language would be if nouns and verbs of related concepts were the same words and we all had to figure out which one was in use by the context of the sentence. For example what if "baker" and "baking" were combined into one word "bakering". "John is baking" would be indistinguishable from "John is a baker" because both would read "John is bakering". It would drive everyone nuts.


Our poor 6th graders have to confront this noun/verb conflation in expressions such as 6 - (-2). 6 - (-2) means "start with 6, now take away (minus sign) the opposite of 2 (-2)". Taking away the opposite of 2 can be a tricky idea.

Black chips are positive 1s and red chips are negative 1s.  The circled red chips represent physically removing or subtracting -4.

I have found that one great way of teaching tricky ideas is by using manipulatives - physical objects that kids can touch and, well, manipulate that stand in for mathematical ideas.  This week we have been using poker chips and drawings of circles with + and - in the middle to visualize this relatively complex combination of concepts.  The picture above is an illustration of 2 - (-4) = 6.  The black chips are positive 1s the red negative 1s.  A black chip and a red chip together make a "zero pair".  So the picture starts with 4 zero pairs of black chips and red with 2 unpaired black chips left over.  Hence, the number 2.

Taking away (-4) is represented by physically removing 4 red chips.  Now there are no more zero pairs, just 6 black chips by their lonesome.

By Friday I got the sense that the ideas were beginning to click.

Now on to 6 x (-2) and how that can be understood as repeated addition:  (-2)  + (-2) + (-2) + (-2) + (-2) + (-2) = -12.


On Edit:  Look out 7th and 8th graders because this exact problem is found in square roots.  There is an action called taking a square root and there is a number concept with the same name and notation.  Nice job mathematicians, nice job.

Standards Based Grading is Coming

Standards based grading is a change from the grades with which we parents and teachers are familiar.  In regular grading the scores for various quizzes and tests are weighted and average over a semester to form an aggregate percentage.  Percentage cutoff values then determine a letter grade - 90% for an "A", 80% for a "B" and so forth.  What we see is that our son or daughter got an "A" or a "B".  Recently I have become uneasy with how little information is conveyed by a letter grade.  If someone gets a "B" in Algebra 1 do they understand polynomials?  What about absolute values?  Inequalities? The variation in understanding across concepts that all math students possess is hidden by a simplistic letter grade.  There must be a better way.  Of course I wouldn't be writing this if there wasn't.  There is.

Standards based grading is a type of assessment in which data on student understanding from quizzes and tests is organized into important skills. Student data remains relatively disaggregated. Assessments are focused on critical mathematical domains in the curriculum and specific skills within the domain are laid out for the students before assessments take place.

Here is one example of what this looks like in practice for a 4th grade classroom.

In this case the specific skills are link to the Common Core State Standards.  There are a few new things to notice.  First, what you see is a relatively detailed list of skills rather than entries like "Unit 1 Test".  Second, the evaluation is related to levels of understanding.  In principle it is a qualitative measure but in math often it is tied to a quantitative outcome (percentage on a quiz).

With standards based assessment a student and parent receives an inventory of strengths, weakness and material already mastered or at least covered so far.  This may or may not be in addition to an overall course grade.

In my opinion the main advantages of standards based assessment are:

• students and parents have a clear picture of course expectations and required skills for the duration of the course
• 
  
• students know what they are supposed to be learning at any point in time
• 
  
• future teachers can look at the grade book and know what the students needs and accomplishments are prior to beginning instruction in the new year.

Above is a screenshot of the standards based grade book for 6th grade that I developed this week. The first page of the document shows five 6th grade skills.  The second page has my rubric for scoring each mathematical domain that is assessed.

I look forward to passing out standards based assessment forms to all students soon so that everyone knows what we are working on and what is expected.  I hope that by going to a standards based grading system parents and students will have a better understanding of both the courses I teach as well as what their students have learned.

What to ask your 8th grader

What were all those math problems doing taped up in the hallways?

Why was Dr. Sally surprised to be building a fort on Halloween with Brad Pitt?

And what does that have to do with polynomials?

New 6th Grade Homework Grid for this week and probably next

Here is the new grid.  You can always find my homework assignments by clicking on the recent homework assignments link on the right side of this page.

6th Grade

Mathematical Domain	IXL Tab	IXL Skills	Recommended Finish Date
****Updated****
Absolute Value and Number Line	6th	C.1 – C.4	September 29, 2014
Add and Subtract	6th	I.1, I.2, I.4 – I.8	September 29, 2014
Multiplication	6th	K.1, K.2, K.5, K.6, K.10, K.11, K.12	October 3, 2014
Equations and Inequalities	8th Grade	Variable Expressions T.1 – T.7 Single Variable Equations U.1 – U.4 NEW: Inequalities X.1 – X.5	October 3, 2014

Integer Multiplication and Addition Virtual Manipulatives

I could talk and talk and talk but we all know that lectures aren't that great of a way of learning.

Doing is much better.

So for the 6th graders out there please spend some time doing.  Here is one website that allows you to experience and explore adding and subtracting positive and negative numbers and another multiplying positive and negative numbers.  You should watch the little tutorial in the second link so that the visual model of multiplication makes sense.


I do NOT want you to NOT do these exercises.  Not not.  Whose there?  Math inverse.

Specific IXL skills

I have noticed that some middle school students, perhaps distracted by the recent beautiful fall days or perhaps the mystery of irrational numbers, are not practicing the assigned skills.  I am happy that they are practicing math!  But our classes are now much more focused on a set of skills and a particular math domain.

There are cases where individual students have approached me and ask to work on different skills based on their current achievement level.  But there are others who seems to accidentally working on the wrong skills.

I do this all the time myself so I empathize.  My family has lots of stories in which they tell me one thing and I quite conveniently hear quite another.

Nonetheless, I want to remind students and parents what our current focus is in each of the classes so we all get on the right track.

6th grade is currently working on single variable expressions and numbers with exponents.

7th grade is working on classifying real numbers, mathematical properties and order of operations.

8th grade is working on polynomials and variables with exponents.

Please check with your student that they are practicing the skills from the recent homework assignments page on this blog.

And forgive the occasional spacey forgetfulness of those of us who may be distracted by the mysterious nature of logic, math and fall colors.

Driving in a snow storm at night with kids fighting in the back seat

We've all been there, right?  Driving is usually what psychologists call an associative task.  Associative tasks are things we do like brushing our teeth, washing the dishes or driving in normal conditions.  We can simultaneously wash dishes and sing along with a familiar song on the radio because each task uses up almost no cognitive resources.  In fact, scientists have recently learned that habits are controlled by the basal ganglia and not the pre-frontal cortex, the part of brain involved with higher level reasoning. Habits just sort of happen.  We don't actively think about putting toothpaste on our toothbrush we just do it.  We are on autopilot.  Our brains are free to think other thoughts.

The other type of task is called a cognitive task.  Cognitive tasks are jobs or activities that require concentration.  Calculus.  A 30 foot putt for par.  Writing a research paper.  We can't sing a song and write a research paper.  At least I can't.

Driving on a sunny day on dry roads is an associative task for most adults. But once the snow hits driving turns into a cognitive task.

So what's this got to do with math? Some math tasks are associative mostly the ones that we have committed to long-term memory or algorithms that we have practiced so much that they have become second nature.

But some math tasks are cognitive.  It is my job to make sure that students are doing only one cognitive task at a time.

I am supposed to be the snowplow driver clearing the roads.  Scaffolding math problems is the educational analog of the snowplow.  Scaffolding provides answers or near answers for sub-tasks so that the only job remaining is the essence of the day's lesson.

Consider long division.

For a fifth grader, or for anyone, this is an incredibly complicated set of steps and calculations.  Each individual step seems to be governed by an arbitrary rule whose application requires a difficult calculation which is then followed by another arbitrary rule whose application requires another difficult calculation.  As you can see above this chain of "WHAAAAAAAT?" can go on for a number of steps. Imagine trying to do this if you are like most fifth graders and your single digit multiplication facts are still a little shaky.  It's a nightmare!

This week we are metaphorically clearing the roads of snow. We are scaffolding long division.  We are providing students enough support so they can fully focus on the logic and steps of the long division algorithm.  Below you will see part of a worksheet from 5th grade that gives students arrows, colors and a list of math facts to point the way down the previously hidden and slippery highway.  Blue numbers from above are used in the blue blocks below.  Yellow blocks have a similar correspondence.

When the intellectual clutter is cleared I hope that what they see before them is a crisp and cogent schema of the standard algorithm for long division.

Alternative Plans in Homework Assignments

I have noticed that some students are getting stuck and frustrated with the current homework assignments.  In response I am constructing alternative plans.  These plans meet the needs of the individual student for that particular skill.

You can find the grade by grade alternative plans in the homework tables found in the Recent Homework Assignment page.

I will also construct alternative plans if the assigned material is too easy.  For example if a 6th grade student is not challenged with Pre-Algebra number sense I will move them up to the corresponding curricula in Algebra.

So far I have only updated 5th grade with an alternative plan.  My intention is roll out more over the week as I continue to conference with individual students.

What to ask your middle school math student

This week the middle schoolers applied proportions to the 100 meter Olympic competition. As you can see in the picture above, Usain Bolt, in front, is substantially taller than his competitors.  We asked if this gives him a competitive advantage.  Then we asked if this is an unfair competitive advantage.

Unfair is the key word.  There is no objective metric for fairness (this makes economists nuts) so our discussion veered into territory that requires critical thinking (oh no!).  And the answer is not clear.  On the one hand Usain Bolt did nothing wrong or unethical to become tall.  He just has "tall" genes (Our new science teacher Kate is cringing right now if she is reading this).  But on the other hand there are sports that do norm by physical characteristics.  Wrestling has weight classes.  Most sports are divided by gender.

We also delved into some  numerical analysis and constructed the following counter-factual.  One hundred meters is 51 Usain Bolt body lengths long.  What if instead of running 100 meters all other competitors instead ran 51 body lengths.  Would Usain Bolt still have won?  Ask them.

This week middle schoolers will be running their own races with Karen in physical education.  They will record their times and their heights.  Will the same person who won the unadjusted race still be the winner if race distances were proportional to their height? We'll find out.

Who is really the fastest?  What is the proper and relevant benchmark?

If you are feeling especially adventurous you may want to extend the conversation to other important examples of proportionality.

Should taxes be proportional to income?  Are taxes like sales taxes and licenses plate fees fair?  Are they properly normed with respect to "ability to pay"?

And, of course, shouldn't the largest person in the family also get the largest piece of pie?  I know the other Dads at Kazoo School are with me on this one.

How to get the most out of studying

Homework Assignments Are Updated

The new 4th - 8th grade homework assignments for the week of September 22 - September 29 are out.

I will post the MathCounts homework later in the week.

The permalink for all homework is on the right.

But for your convenience I am posting it below as well.

4^th Grade

Mathematical Domain	IXL Tab	IXL Skills	Recommended Finish Date
Name and Constructing Geometric Figures	4th Grade	P.1 – P.6 P.9, P.18	September 29, 2014

5^th Grade

Mathematical Domain	IXL Tab	IXL Skills	Recommended Finish Date
Number Theory	5th Grade	K.1 – K.7	September 29, 2014

6^th Grade

Mathematical Domain	IXL Tab	IXL Skills	Recommended Finish Date
Equations and Inequalities	8th Grade	Variable Expressions T.1 – T.7 Single Variable Equations U.1 – U.4 NEW: Inequalities X.1 – X.5	September 29, 2014

7^th Grade

Mathematical Domain	IXL Tab	IXL Skills	Recommended Finish Date
	Algebra 1	Continue doing: Numbers A.1 – A.5 Properties H.1 – H.3 NEW: Operations B.1 – B.7	September 29, 2014

8^th Grade

Mathematical Domain	IXL Tab	IXL Skills	Recommended Finish Date
	Algebra 1	Monomials Y.1 – Y.5 Polynomials Z.1 – Z.5 New  Polynomials Z.6 – Z.10	September 29, 2014

6th grade scope and sequence is ready for viewing

There is a link for all of the scope and sequence documents to the left right.  So far only 6th grade is done. But 6th grade is done!  Yes!

Saturday is for 6th grade scope and sequence

Today is the day that I am detailing the scope and sequence of 6th grade math.

I am not promising that I will have it ready for Monday. I am letting you know that this is getting 6+ hours of my time today. I have already spent a good deal of time on plans for 6th grade math so I am hopeful for finishing up a detailed plan soon.

Here is what to expect from all of my scope and sequence plans. The list below should give you a flavor of the format.



Mathematics Domain:  Solving basic equations and creating basic expressions

1. Skill:  Solving equations using addition and subtraction
2. Skill:  Solving equations using multiplication and division
3. Skill:  Creating expressions from real world scenarios that use addition and subtraction
4. Skill:  Creating expressions from real world scenarios that use multiplication and division

• Assessment:  Basic Equations and Expressions Test
• Target date for completion:  October XX, 2014

I expect each class to have around 20 mathematical domains and assessments.

Assessments are short (5-7) questions, are only administered when individual students are ready and may be retaken as often as students like in order to increase their grade and demonstrate mastery.

More on assessments and grading policies later.

Newsletter is out

Kazoo School parents please check your email for my first Kazoo Math newsletter.

I am keeping these short and sweet because I think that parents do not want a long dissertation. Newsletters are intended to be signposts pointing to upcoming announcements, due dates and developments. This blog will be the forum for details. Besides, if I need to write a 1000 word essay every time I publish a newsletter the reality is that they will come out infrequently. I want these to come out once a week.  In order to keep up that pace they need to be short.

Is this real math (Dragon Box 2 edition)?

The fourth and fifth graders are playing Dragon Box 2 as one of their stations on my 2 iPads.  Dragon Box 2 teaches algebra via an engaging and beautifully designed game.

So here's a screenshot showing what a level looks like that a fifth grader was working on today. And to be clear, this fifth grader completed the level meaning he solved the algebra problem.

The boring way to write this is 

Our fourth and fifth graders are capable of higher order thinking.

Middle School Homework Due Monday September 22, 2014

Work for 30 minutes per day (25 minutes for 6th grade) OUTSIDE OF CLASS Weds – Sunday.  Take one day off.

Get as far as you can.  Due Monday September 22, 2014.

6^th Grade

Tab in IXL	Domain	Skills
8th Grade	Variable Expressions	T.1 – T.7
8th Grade	Single Variable Equations	U.1 – U.4

7^th Grade

Tab in IXL	Domain	Skills
Algebra 1	Numbers	A.1 – A.5
Algebra 1	Properties	H.1 – H.3

8^th Grade

Tab in IXL	Domain	Skills
Algebra 1	Monomials	Y.1 – Y.5
Algebra 1	Polynomials	Z.1 – Z.4