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1. Use number lines or other manipulatives to teach integers.

2. Build integer understanding without manipulatives

3. Use real life examples when modeling math to make topic relatable

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I plan to study effective teaching strategies for teaching seventh grade students how to add and subtract negative numbers.  I’m researching this topic because I want to gain a better understanding of whether teaching integers using manipulatives is better than the usual teaching strategy of lecture and independent practice problems

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Andrews, E. (2015). Stanford neuroscience research identifies more effective way to teach abstract math concepts to children. Stanford News.  Retrieved March 22, 2017, from http://news.stanford.edu/2015/07/06/symmetry-math-schwartz-070615/

Bellonio, J. (2012). Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete. Yale-New Haven Teachers Institute. Retrieved March 22, 2017, from http://teachersinstitute.yale.edu/curriculum/units/2001/6/01.06.12.x.html

Bornstein, D. (2011).  A Better Way to Teach Math. New York Times. Retrieved March 22, 2017, from https://opinionator.blogs.nytimes.com/2011/04/18/a-better-way-to-teach-math/

Cope, L. (2015). Math Manipulatives: Making the Abstract Tangible. Delta Journal of Education    5(1), 10-19

Finely, T. (2015).  The Nuts and Bolts of Explicit Modeling. Edutopia. Retrieved March 22, 2017, from https://www.edutopia.org/blog/nuts-and-bolts-explicit-modeling-todd-finley

Ginsburg, David, (2011). Combining Integers Instead of Adding and Subtracting Them. Education Week Teacher. Retrieved March 22, 2017, from http://blogs.edweek.org/teachers/coach_gs_teaching_tips/2011/11/combining_integers_instead_of_adding_and_subtracting_them.html?_ga=1.14499570.493073241.1490310858

Ginsburg, David, (2012). Procedural Fluency: More Than Memorizing Math Facts. Education Week Teacher. Retrieved March 22, 2017, from http://blogs.edweek.org/teachers/coach_gs_teaching_tips/2012/07/procedural_fluency_more_than_memorization.html?qs=teaching+strategies+with+integers

Musriani A., & Mustamin (2014). Improving the Learning Result of the Integer Number Operation Using Card Model. Journal of Education and Learning, 8(1) 23-28.

Reylan, J. R., Marcelino, J., Mortel, R., & Lapinid, M. R. (2014). Activity-Based Teaching of Integer Concepts and its Operations. Retrieved March 21, 2017, from http://www.dlsu.edu.ph/conferences/dlsu_research_congress/2014/_pdf/proceedings/LLI-II-016-FT.pdf

Rodgers, L. (2012). Laurie Rogers: In Defense of Using the Number Line to Teach Negatives. Education News. Retrieved March 22, 2017, from http://www.educationnews.org/k-12-schools/laurie-rogers-in-defense-of-using-the-number-line-to-teach-negatives/

Webster, J. (2016).Introducing Integers and Rational Numbers to Students with Disabilities. Thought Co.  Retrieved March 22, 2017, https://www.thoughtco.com/introducing-integers-and-rational-numbers-3110484

Young, L., & Booth, J. (2015).  Student Magnitude Knowledge of Negative Numbers. Journal of Numerical Cognition, 1(1), 38–55

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Improving the Learning Result of the Integer: Number Operation Using Card Model

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Improving the Learning Result of the Integer: Number Operation Using Card Model

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Musriani and Mustamin (2014) conducted classroom action research on teaching students how to add integers with card models. They noticed students were having difficulty performing addition and subtract of integers and created the card model which consisted of (+) and (-) signs on the cards. The cards were used as manipulatves. They determined that student learning improved based on their methods of teaching integers with cards. They did not see an increase in test scores of the students who were not taught with cards.

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Teaching integers to students with Disabilities

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Teaching integers to students with Disabilities

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Webster (2016) asserts that student with disabilities need to have an intellectual foundation built using manipulatives and visuals in order to be prepared to do operations with negative numbers or apply algebraic understanding of integers to algebraic equations. He feels a number line is the best manipulative to use when teaching students about integers

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Positive (or natural) and negative numbers can confuse students with disabilities. Special education students face special challenges when confronted with math after 5th grade. They need to have an intellectual foundation built using manipulatives and visuals in order to be prepared to do operations with negative numbers or apply algebraic understanding of integers to algebraic equations. Meeting these challenges will make the difference for children who might have the potential to attend college.

Integers are whole numbers, but can be whole numbers both greater than or less than zero. Integers are easiest to understand with a number line. Whole numbers that are greater than zero are called natural, or positive numbers. They increase as they move to the right away from the zero. Negative numbers are below or to the right of the zero. Number names grow bigger (with a minus for "negative" in front of them) as they move away from the zero to the right. Numbers growing larger, move to the left. Numbers growing smaller (as in subtraction) move to the right.

Common Core Standards for Integers and Rational Numbers

Grade 6, the Numbers System (NS6)Students will apply and extend previous understandings of numbers to the system of rational numbers.

NS6.5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

NS6.6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

NS6.6.a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (-3) = 3, and that 0 is its own opposite.

NS6.6.b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

NS6.6.c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

Understanding Direction and Natural (positive) and Negative Numbers.

I emphasize the use of the number line rather than counters or fingers when students are learning operations so that practice with the number line will make understanding natural and negative numbers much easier. Counters and fingers are fine to establish one to one correspondence but will become crutches rather than supports for higher level math.

The pdf number line here is for positive and negative integers. Run the end of the number line with positive numbers on one color, and the negative numbers on another. After students have cut them out and glued them together, have them laminated. You overhead or write on board markers (though they often stain the laminate) to model problems like 5 - 11 = -6 on the number line.

I also have a pointer made with a glove and a dowel, and a larger laminated number line on the board, and I call one student to the board to demonstrate the numbers and jumps.

Provide lots of practice. You "Integer Number Line" should be part of your daily warm up until you really feel that students have mastered the skill.

Understanding the Applications of Negative Integers.

Common Core Standard NS6.5 offers some great examples for applications of negative numbers: Below sea level, debt, debits and credits, temperatures below zero and positive and negative charges can help students understand the application of negative numbers. The positive and negative poles on magnets will help students understand the relationships: how a positive plus a negative moves to the right, how two negatives make a positive.

Assign students in groups the task of making a visual chart to illustrate the point being made: perhaps for altitude, a cross cut showing Death Valley or the Dead Sea next and it's surroundings, or a thermostat with pictures to show whether people are hot or cold above or below zero.

Coordinates on an XY Graph

Students with disabilities need lots of concrete instruction on locating coordinates on a chart. Introducing ordered pairs (x,y) i.e. (4, -3) and locating them on a chart is a great activity to do with a smart board and a digital projector. If you don't have access to a digital projector or EMO, you might just create a xy coordinates chart on a transparency and have students locate the dots.

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Introducing INtegers and Rational Numbers to Students with Disabilities

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Positive (or natural) and negative numbers can confuse students with disabilities. Special education students face special challenges when confronted with math after 5th grade. They need to have an intellectual foundation built using manipulatives and visuals in order to be prepared to do operations with negative numbers or apply algebraic understanding of integers to algebraic equations.

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4th grade students learn integers with manipulatives

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4th grade students learn integers with manipulatives

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Andrews (2015) details how students are taught to add integers using symmetry and a magnetic strip. Andrews also notes that "neuroscientists have confirmed that the same region of the brain that assesses physical magnitudes is also key to comparing the magnitudes represented symbolically by numerals." Students used the symmetry approach to to better understand topics they hadn't studied yet such as  making sense of negative fractions and solving pre-algebraic problems.

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Building on new discoveries about how the brain grapples with abstract mathematics, researchers at Stanford Graduate School of Education have developed a classroom strategy for teaching children the often baffling concepts surrounding negative numbers.

The new strategy recruits the brain's use of visual symmetry to make sense of the physical world, and it could have profound implications for the way elementary schools teach math.

Using symmetry appears to have helped not just in teaching children about negative numbers but in improving their ability to solve higher-level math problems they haven't seen before.

"Learning about negative numbers is one of the first times that kids learn about abstract numbers - it's a gateway to more abstract learning," said Jessica Tsang, a Stanford researcher and lead author of a new study with Daniel Schwartz and Kristen Blair of Stanford and Laura Bofferding of Purdue University.

The study marks the latest and most concrete result of a five-year project that bridges the gap between new insights from neuroscience and the testing of new classroom teaching tools for fourth-graders in the San Francisco Bay Area. It is published in the current issue of the journal Cognition & Instruction.

Researchers have long thought that the brain harnesses and adapts "perceptuo-motor" capacities to make sense of abstract ideas.

For the purposes of making sense of math, the most widely studied of these capacities has been the ability to compare physical magnitudes such as size. Over the past 15 years, neuroscientists have confirmed that the same region of the brain that assesses physical magnitudes is also key to comparing the magnitudes represented symbolically by numerals.

Three years ago, the Stanford researchers proposed that the brain actually re-purposes several additional capabilities, such as sequencing and ordering, to solve math problems. In this project, the researchers focused on the brain's ability to process visual symmetry.

In 2012 the researchers found that most adults identified the midpoint between a negative number and a positive number more easily if the integers were more symmetric about zero.

For example, people were faster computing the mid-point of 6 and -4 than 8 and -2, even though both pairs of digits were displayed in the same locations on a video screen. Moreover, using functional magnetic resonance imaging, the authors found that visual symmetry-coding regions of the brain became more active for the more symmetric pairs.

Given that the adult brain recruits symmetry circuits for conceptual math problems, the researchers hypothesized that helping young students engage this native ability would improve their mathematics learning.

For the new study, the authors created a guide and tools that incorporate ideas about using symmetry to teach negative numbers. As intuitive as it might seem for people to rely on symmetry as a way to make sense of the world, few curricular materials now used for teaching math make explicit use of it.

The researchers then tested the tools with students and compared the results to those of other students who were taught with the existing approaches.

One of the new classroom teaching tools was a specially designed hands-on manipulative device. Students worked with a magnetic plastic strip that was numbered. To solve the problem 3 + -2, students attached three magnetized blocks to the right of zero and two blocks to the left of zero. The manipulative further included a hinge at zero, the point of integer symmetry. Students folded the two sides together, and the number of extra blocks on either side gave the answer, in this case +1. The hinge at zero helped students recruit their native abilities with symmetry, and the numbers on the little platform helped them coordinate the sense of symmetry with the symbolic digits.

After four hours of instruction spread over three weeks, the researchers documented encouraging results.

First, the children quickly applied the hands-on lessons with symmetry to build new strategies for solving abstract problems involving symbolic digits. When the symmetry students were asked to add up a string of negative and positive numbers, for example, many used a balancing strategy to simplify the problem.

Some paired up negative and positive numbers that would add up to zero and cancel each other out. Others clustered the positive and negative integers on separate sides, which made the balance between the positives and negatives more apparent.

Second, students were likely to incorporate symmetry as an almost automatic part of their thinking. That's important, said Tsang, because many skills - such as decoding words in reading - are more effective when they become instantaneous and reflexive.

But the biggest surprise was on what educational researchers call "generativity" - the tendency of students to apply the ideas of symmetry on their own to problems they haven't encountered before.

As it turned out, students who learned to rely on symmetry didn't simply do better than other students on the material they had just been taught. They also did better on topics that they hadn't yet studied, such as making sense of negative fractions and solving pre-algebraic problems.

"The big difference was that the symmetry instruction enabled students to solve novel problems and to continue learning without explicit instruction," said Schwartz, who holds the Nomellini & Olivier Professorship in Educational Technology at Stanford. "By untangling how the brain comes to know mathematics, we helped with a major goal of education - putting children on an upward trajectory of future learning."
This research was funded by the National Science Foundation and the Marcus and Marianne Wallenberg Foundation.

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Stanford research identifies better way to teach abstract math concepts

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Building on new discoveries about how the brain grapples with abstract mathematics, researchers at Stanford Graduate School of Education have developed a classroom strategy for teaching children the often baffling concepts surrounding negative numbers. The new strategy recruits the brain's use of visual symmetry to make sense of the physical world, and it could have profound implications for the way elementary schools teach math.

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Study on negative and positive numbers with a number line

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Study on negative and positive numbers with a number line

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Young and Booth (2012) describes the two main components of negative numbers, direction and magnitude, are at the root of difficulties seen with negative numbers. Their study discovered that the 6th and 7th students understanding of negative and positive numbers with a number line were similiar.  They also believe that fluency in integers  is important for mathematical achievement especially algebra.

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Student Magnitude Knowledge of Negative Numbers | Young | Journal of Numerical Cognition

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Student Magnitude Knowledge of Negative Numbers

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Students learned more about integers from hands on activities

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Students learned more about integers from hands on activities

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A study by Reylan et al (2014) determined that more students passed the post test than pre test of integers when taught using activities(manipulatives) such as  chips and interactive games. They also assert  "activity-based teaching introduced by the teacher can improve students’ conceptual understanding, perception and procedural skills in integers" the study was based on 7th grade students in the Phillipines.  The students were interviewed and explained how they liked using games and activities to learn math concepts.

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Bellino (2012) discusses reasons for use of manipulatives, why teachers don't use manipulatives an various types of manipulatives. Types of learners and lesson ideas using manipulatives.
It's also stated that mathematics achievement increased when manipulatives were used. Another interesting finding was that  "whether adults or children, will master mathematical concepts and skills more readily if they are presented first in the concrete/pictorial or with symbols"

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01.06.12: Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete

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Research indicates that manipulatives are particularly useful in helping children transition from the concrete to the abstract level. It is key for the teacher, however, to select the appropriate activities/manipulatives to support the transition. The transition often reflects the developmental process. Further research, in a review of activity-based mathematics learning, indicates that mathematics achievement increased when manipulatives were used.

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Finley (2015) explains effects of explicit modeling of math in 8 steps these strategies will be used when teaching integers without manipulative to see the effects. Explicit modeling should be shown to students them provide them with sample problems to complete on their own. They should gradually be given more in depth problems to build understanding and be required to explain their steps to further understanding. Finley believes that after modeling students should be given many practice problems to complete independently then reviewed by the teacher.

1. Breaks down the concept or skill into critical features or elements

2. Clearly describes the concept or skill

3. Clearly models the concept or skill

4. Provides multi-sensory instruction (visual, auditory, tactile, kinesthetic)

5. Thinks aloud as she or he models

6. Models examples and non-examples

7. Works with cues (pointing, circling, or highlighting)

8. Provides high levels of teacher-student interaction

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Although many other methods of teaching are noteworthy -- such as case study, problem-based learning, and simulation -- many scholars agree that students need "careful, personal instruction with clear demonstrations in reading and writing", as well as in social studies, math, and science. (Note: NCTM membership is required to view this math link.)

What can instructional modeling teach?

"Most human behavior," according to Stanford professor Alfred Bandura, "is learned observationally through modeling." As evidence, watch Bandura's controversial Bobo Doll Experiments where children, after witnessing adults assault a wobble clown, follow suit.

Several instructional methods incorporate teacher modeling:

Cognitive apprenticeships: Showing students how to do something within an authentic context while making the thinking behind the procedure visible

Demonstrations: Planning, introducing a lesson, presenting, giving a performance, and summarizing

Worked examples: Solving a problem and writing out the steps (PDF, 197KB)

Think alouds: Modeling metacognition by stating aloud everything an instructor thinks while reading aloud (PDF, 11KB)

I detail other modeling approaches below.

Planning Tips

Modeling necessitates meticulous planning. Start with assessment to identify areas of greatest need. After identifying a skill or concept to model, anticipate areas that are most likely to confuse learners. If the skill is particularly difficult, try the following:

Narrowing the steps

Breaking the skill down into two or more chunks and modeling them over the next few days

Modeling with an easier text to teach point of view (like Jon Scieszka and Lane Smith's The True Story of the Three Little Pigs instead of Virginia Woolf's Mrs. Dalloway)

Preparing to model the procedure more than once, especially in a group including English-language learners and students with learning difficulties

Asking more questions by writing down "check for understanding" questions in advance to reduce the number of details you have to juggle

Right Before Modeling. . .

I generally do three things in front of students before modeling a skill.

1. Review the previous lesson.

Example: "Yesterday, we learned how to add integers. There were three things we practiced. Who can tell me what those three things were? Today we are going to __________."

2. Introduce learning goals.

Example: "I'm going to teach you to/about __________. This skill/concept will help you __________. By the end of this demonstration, you'll be able to __________."

3. Introduce steps (in writing) and check for understanding.

Example: "So in this demo, we'll do __________ first, then __________, and then __________. Is there anything you want me to clarify before I start? Bob, tell me what you'll be able to do at the end of this lesson and why."

Explicit Modeling

Explicit modeling (EM), supported by ample research, works best for modeling facts and procedures that require precision, such as math and grammar. In the eight steps of EM, summarized in this PDF (44KB), the teacher:

1. Breaks down the concept or skill into critical features or elements

2. Clearly describes the concept or skill

3. Clearly models the concept or skill

4. Provides multi-sensory instruction (visual, auditory, tactile, kinesthetic)

5. Thinks aloud as she or he models

6. Models examples and non-examples

7. Works with cues (pointing, circling, or highlighting)

8. Provides high levels of teacher-student interaction (Source: MathVIDS)

I yield to criticism that EM is teacher centered and teacher directed, two overused and maligned classroom paradigms. Furthermore, I agree that the goal for any instructor is to shift students into more agentive, self-directed behaviors.

What EM does effectively, however, is lighten extraneous cognitive load, or mental effort -- a whopping benefit of the method. Heavy loads impact completion (my mom tries to use Facebook every two months, but because I've neglected to help her master that skill step by step, she'll never rejoice at uploaded photos of my crockpot masterpieces).

The equation: EM = decreased cognitive load = task completion.

After Modeling

Allow plenty of time for students to review and practice soon after the concept or skill has been demonstrated. Studies suggest that students compare their work with an expert's (such as the teacher) to improve learning (PDF, 197KB).

One math instructor I know models how to solve a problem, then assigns a related problem on the board for everyone to complete independently. If only a handful of students complete the problem perfectly, he asks them to write down the steps they took and explain them in small groups.

Karen Lea writes: "Modeling means that the teacher does most of the work the first time, and then gradually the students do most of the work" -- a research-supported approach (PDF, 1.3MB) referred to as the gradual release of responsibility (GRR). After the instructor demonstration, the following occurs in GRR:

1. Shared experience: The class collaborates with the teacher to work through the skill.

2. Guided practice: Students apply the skill with the support of peers, often while being observed by the teacher.

3. Independent practice: Students apply the skill on their own.

Monica Baines models GRR in this video.

Advanced Degree of Difficulty

While instructors often monitor students as they struggle with a new problem, learners rarely watch instructors take the same risk, thereby perpetuating the myth of teachers' intellectual supremacy. In real time, while students observe, try formulating and attempting to answer questions related to an historical artifact, or try writing a poem from scratch in front of the class, or try testing a new scientific hypothesis.

The payoff? You'll model grit.

Finally, keep your expectations of students high. They may struggle, but will nevertheless benefit from clear modeling and your faith in their abilities.

Please share your own modeling tips.

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The Nuts and Bolts of Explicit Modeling | Edutopia

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Although many other methods of teaching are noteworthy -- such as case study, problem-based learning, and simulation -- many scholars agree that students need "careful, personal instruction with clear demonstrations in reading and writing", as well as in social studies, math, and science. (Note: NCTM membership is required to view this math link.)

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Procedural or conceptual strategies for teaching intgers

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Procedural or conceptual strategies for teaching intgers

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Ginsburg (2012)  believes teachers should "build procedural understanding through conceptual understanding whenever possible especially when adding and subtracting integers  .He believes students must know when, as opposed to just how, to use a procedure.  He's against using manipulatives to teach math  concepts. He wants students to stop applying algorithms without thinking of why they are using it. Teachers should model mathematics so students can make connections.

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Education Week - Coach G's Teaching Tips

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Procedural fluency or conceptual understanding--math educators have debated for years which is more important. I sided with conceptual understanding until my colleague Angela McIver helped me see the value of procedural fluency in terms of stamina. Like all of us, students have finite energy. The more energy they use for procedures, the less energy they have for problem solving. And the less energy they have for problem solving, the less likely they are to gain conceptual understanding. A lack of procedural fluency can therefore contribute to a lack of conceptual understanding.

The issue, then, isn't if we should target procedural fluency but rather how we should target it. And, how to do so while also targeting conceptual understanding, as called for by the Common Core State Standards (the "dual intensity" instructional shift I mentioned in my last post).

Of course, it's hard to help students achieve something if you don't know what it is. A lot of teachers think procedural fluency is synonymous with "knowing your math facts," but it's more than that, as defined in the National Research Council book, Adding It Up: Helping Children Learn Mathematics:

Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

Does procedural fluency involve memorizing multiplication tables and other facts? Sure, but it also involves thinking. Students must know when, as opposed to just how, to use a procedure. And they must not only be able to perform procedures accurately, but also flexibly and efficiently--again, think stamina.

To illustrate the thinking aspect of procedural fluency, I ask teachers at workshops how their students would solve the following problem: 7/11 = ?/121. Middle and high school teachers say most of their students would cross-multiply, which provides the correct solution of 77 (7 x 121 / 11). Elementary teachers say most of their students would use the equivalent fractions approach, which also provides the correct solution (121/11 = 11; 11 x 7 = 77). Which approach is more efficient? Well, which is easier: multiplying 7 x 121 or 7 x 11?

One reason students don't always choose the most efficient method is that they don't really choose any method. They just apply an algorithm without thinking. They know how to use procedures but not always when to use them. They perform procedures accurately but not always flexibly or efficiently. In other words, they lack procedural fluency.

This brings us back to the Common Core challenge of helping students develop procedural fluency while also helping them develop conceptual understanding. And here are a few suggestions for meeting this challenge:

Model for students (and/or allow them to discover on their own and from each other) multiple solution strategies (example: comparing fractions).

Establish connections between new topics such as proportions (and cross-multiplying) and previous topics such as equivalent fractions.

Build procedural understanding through conceptual understanding whenever possible (example: adding and subtracting integers).

Will these ideas ensure procedural fluency for students? No. They still need to know their facts in order to use procedures flexibly, accurately, and efficiently. In the above problem, for example, the equivalent fractions approach is only more efficient than cross-multiplying if students have memorized multiples of 11.

I'll address the facts-fluency connection further in a future post, since I don't have the stamina for it now. Check out Jo Boaler's recent Ed Week article, Timed Tests and the Development of Math Anxiety, in the meantime.

Image by Vlue, provided by Dreamstime license
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Procedural Fluency: More Than Memorizing Math Facts

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Procedural fluency or conceptual understanding--math educators have debated for years which is more important. I sided with conceptual understanding until my colleague Angela McIver helped me see the value of procedural fluency in terms of stamina. Like all of us, students have finite energy.

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Non-Manipulative strategy for adding and subtracting integers

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Non-Manipulative strategy for adding and subtracting integers

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Ginburg (2011) explains a teaching strategy to add/subtract integers without using maniupaltives or "gimmicks" as he put it. He displays a straight forward way making what looks like an X/Y table but instead it's a (+) -. Students add the numbers and whichever sum is larger that gets the sign

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Education Week - Coach G's Teaching Tips

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"Man, I didn't do nothing," students often said to me when I spoke with them about their behavior. "My point exactly," I replied. "If you didn't do nothing, you must have done something."

It was after one of those exchanges when it occurred to me that the English language might provide a better way of helping kids add and subtract integers (really all real numbers) than the many approaches--from rules to manipulatives to real-life illustrations such as temperature and money--I had tried up to that point.

I started by asking students, "If Fluffy is not not a dog, what is she?" And after a few minutes of kids guessing every animal but dog, a boy named Derrick said, "Wait, she's a dog." Other students gave him a funny look and said, "Can't you read? It says she's NOT a dog." But Derrick barked right back: "Can't you read? It says 'Fluffy is NOT not a dog,' which means she is a dog."

This exercise not only helped students finally get why subtracting a negative is the same as adding a positive, but also inspired an approach to adding and subtracting positive and negative numbers that's more effective than anything else I've tried or seen. What distinguishes this approach--which my students and I called "The System"--is that it involves combining numbers rather than adding or subtracting them, as illustrated here:

If you've read my prior posts where I excused Aunt Sally and crumpled FOIL, you know I'm no fan of gimmicks that undermine mathematics and can cause misconceptions. But that's not the case with "The System," which improves students' computation and comprehension. It's also a good lead-in to combining like terms in Algebra.

So if your students have trouble adding and subtracting positive and negative numbers, you definitely won't want to not teach them "The System."

Image provided by GECC, LLC with permission
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Combining Integers Instead of Adding and Subtracting Them

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"Man, I didn't do nothing," students often said to me when I spoke with them about their behavior. "My point exactly," I replied. "If you didn't do nothing, you must have done something."

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Rodgers (2012) believes number lines is the best way to teach integers to students as opposed to non-real world word problem stories.  She cautions using number stories that don't represent actual numerical values such as portraying cheating and fighting as a negative numbers when telling a story.  Students have been learning with number lines and must understand it in order to use it correctly.Rodgers believes "without an understanding of the principles behind the number line, students can’t add and subtract negative numbers with proficiency. Without that capability, algebra, geometry and calculus will be beyond them.

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It's simple to teach mathematical positives and negatives to a child. It's been done successfully with the number line around the world, in private schools, homes, tutoring businesses and online. Unfortunately, many schools in America no longer teach the number line, don't teach it to mastery, or they cloud any fledgling understanding of it by emphasizing other, less-effective methods.

First, I'll explain the number line. Then I'll show you what's being emphasized in its place.

Traditional Math Method Used to Teach Negatives
The Number Line
A number line is a straight line with a series of real numbers listed at intervals. Typically, "zero" is a point in the middle, negative numbers are listed to the left of zero, and positive numbers are listed to the right of zero. Arrowheads are placed at each end to show that the line and numbers continue indefinitely. Each point is assumed to correspond to a real number, and each real number corresponds to a point. Like this:

( Comment: A number line is flexible and comprehensive. It can be drawn with integers, fractions, decimals, irrational numbers or a combination thereof. It can illustrate zero, positive and negative numbers, and small or large numbers. It provides an effective visual for adding and subtracting positive and negative numbers. It can be used to articulate changes in measurements, temperature, money, time or quantities. It's clear, efficient and functional. It's easy to draw, explain and understand.)

Naturally, people who love reform math prefer to not use the number line. Various alternate models are used that are not efficient and not comprehensive. Here are a few.

Reform Math Methods Used to Teach ... Uh ...

These models (with myriad variations) are common to reform curricula. The examples illustrated below are quoted (with permission) from materials originating from two public entities: Henrico County Public Schools and the Georgia Department of Education. The wording is theirs. Please notice that the Georgia materials are said to be based on the Common Core State Standards.
Story Model
In this model, children are given a story that supposedly illustrates a mathematical calculation or equation.

Example 1: "If you have spent money you don't have (-5) and paid off part of it (+3), you still have a negative balance (-2) as a debt, or (-5) + 3 = (-2)."

Example 2: "Getting rid of a negative is a positive. For example: Johnny used to cheat, fight and swear. Then he stopped cheating and fighting. Now he only has 1 negative trait so (3 negative traits) - (2 negative traits) = (1 negative trait) or (-3) - (-2) = (-1)"

( Comment: Stories and problems do help to apply math skills that have already been learned, but - as a teaching method - the Story Model provides no real mathematical understanding. The story in Example 2 also attributes adverse characteristics to negatives. In mathematics, a negative is simply a lessening of a quantity, not an adverse characteristic.)
Charged Particles Model
In this model, imaginary items are added to replace items that are already there: "When using charged particles to subtract, 3 - (-4) for example, you begin with a picture of 3 positive particles."

"Since there are no negative values to 'take away,' you must use the Identity Property of Addition to rename positive 3 as 3 + 0. This is represented by 4 pairs of positive and negative particles that are equivalent to 4 zeros."

"Now that there are negative particles, you can 'take away' 4 negative particles. The modeled problem shows that the result of subtracting 4 negative particles is actually like adding 4 positive particles. The result is 7 positive particles. This is a great way to show why 3 - (-4) = 3 + 4 = 7."

( Comment: How does a child make the leap from 3 particles to 11 particles? Where is the explanation of what's actually happening? As I was typing in the Charged Particles Method, my brain felt like it was melting. Brain cells began to give up and die. My daughter had to rescue me with chocolate.)

The Stack Model and The Row Model

In the Stack Model, students draw boxes on top of each other in "stacks" and then count them. In the Row Model, students draw boxes in "rows" and then count them. Subtraction for the Stack and Row methods means creating pairs, as in the Charged Particles Method, then "adding zeroes," crossing out items, and redrawing boxes over and over. This figure illustrates the solution to 3 - (-4) = ?

( Comment: The children will draw a lot of boxes, but they will not come to understand negatives.)
The Postman Model
In this model, a story is provided that supposedly leads students through understanding negatives. Children are to act out parts using manipulatives and props. The concept can be done with other scenarios, but this example from Henrico County School District is based on a mail carrier:

"A postman only brings financial mail. Sometimes she brings bad news, e.g., a bill for $5 = -5. Sometimes she brings good news - a check for $5 = +5. If she brings both you get two pieces of paper but zero dollars. You always start a zero with a cash drawer full of matching checks and bills that equal zero dollars. So if she brings me two checks for $5, no sweat, she helps me by $10, answer = +10. Similarly if she brings me 2 checks for $5 the result is 2 * -5 = -10. Now here is the tricky part: -2 * + 5 = ? Well the - sign means takes away from me. But if we start at zero how can she take anything away? This is where the cash drawer of matching checks and bills saves us. We just take away 2 checks and are left with 2 bills to pay. -2 * + 5 = -10. Similarly, if she takes away our bills, she helps us and the money we would have used to pay the bill can now be spent on bubblegum. -2 * + 5 = +10."

I think the second "2 checks for $5" is actually supposed to be two bills for $5. Please also remember that young students are the intended audience. The story suggests using a bag filled with Monopoly money and paperclips. Students pretend to deliver mail, cash checks at a bank and pay bills. At some point, however, the mail carrier makes a mistake, and the story goes on to say:

"If it was a check, that would be subtracting a positive. To get the check out of the bank, you would have to pay the bank (which would make you lose money). If it was a bill (taking away a negative), you can keep the money attached to the bill and give the bill back to the mail carrier. This would show that taking away a negative would give you more money."

(Comment: There is so much wrong with this example, the explanation and the reasoning behind it, it's hard to know where to begin. The process is complicated; the financial philosophy is suspect. And what does that second paragraph even mean?)
The Balloon Model
In the Balloon Model, things are moved up (added) and down (subtracted). This concept can be done with anything that moves up and down (airplane, elevator, swimmer, etc.), but in the Balloon Model, sand bags represent negatives, and air bags represent positives. This illustration shows -3 + 4 = 1.

( Comment: The Balloon Model is a vertical number line, but it labors to be more relevant by using a balloon (or elevator or airplane or the sea). Thus, the concepts don't match up with what happens in "real life." The biggest problem: "Up" and "down" is not equivalent to "more" and "less."

Balloon Model: In real life, subtracting a sandbag lessens weight, while adding a sandbag adds weight. In the model, however, subtracting a sandbag somehow makes a quantity larger, while adding a sandbag makes it smaller. In the Building and Airplane models, nothing is added or subtracted to make a quantity larger or smaller; the movement up and down is mechanically driven. In the Sea Model, you could say that blowing air out of one's lungs (subtracting air) makes a person sink, although the change in weight would be negligible. But how does one add air under water?

Where in these models is infinity? Where are fractions and decimals? Where is zero located? At ground level? If so, are negatives below the ground? Since when have balloons and airplanes flown, elevators descended, and swimmers paddled below the ground?)

The (Hey, let's call them) "Net Changes" model

In reform curriculum Investigations in Number, Data, and Space (aka TERC), students are taught to think of "net changes," rather than addition and subtraction. Students use manipulatives to act out the "changes." They count things and keep tallies.

If you search the term "negatives" on the TERC Web site, you'll have a difficult time finding actual negatives, but games about thinking about negatives are explored. Actually, take a few minutes and skim through the 1st Edition units for TERC. This is the approach to K-6 math that has helped to kill off math proficiency in the United States.

( Comment: Looking through the TERC Web site, you can see the end of mathematics ... and possibly all of life as we know it. The school district in my city still uses this curriculum, despite my best efforts to change that.)
Summary
Each of the reform math models illustrated here is missing one or more critical concepts, such as zero, negatives, fractions, decimals, large numbers, and/or infinity. They don't adequately explain negatives, or show that the larger the negative number, the smaller the quantity. They don't handle all scenarios. Perhaps some are "fun" for a very short while; others are probably no fun at all. Time spent on these models wastes time, builds frustration and creates misunderstandings. Instead of clearly cementing concepts about addition, subtraction and negatives, the children are filling paper bags or talking about "negative" traits.
Without an understanding of the principles behind the number line, students can't add and subtract negative numbers with proficiency. Without that capability, algebra, geometry and calculus will be beyond them.
I don't know why so many people in public education seem not to respect the need for efficiency, effectiveness, sufficiency or student proficiency. Math is a tool, used in the "real world" to get a job done. Time is important; efficiency is vital; correct answers are critical. Those who can use math properly will be hired over those who can't. There is no time in the "real world" to discover methods, struggle with basic math, or constantly ask a "partner" for help. But math reformers seem to think they have all the time in the world. To them, math isn't about efficiency and correct answers; it's about struggle, failing, striving and playing games. Many reformers truly believe that if the teaching was efficient, the lesson failed. Thus, they're motivated to not just prefer the inefficient models, but to actually eliminate efficient models, to mock them and to label them as counterproductive.

Help your children gain a solid grounding in math by teaching them the traditional methods, such as long division, vertical multiplication and the number line. Traditional methods were developed and honed over thousands of years by very clever adults so that they would be efficient. More-efficient models will be developed, no doubt, but the reform models I've described are not better models.

Don't let anyone convince you that efficiency and effectiveness in math are unnecessary or counterproductive. People who actually use math (outside of a K-12 classroom) don't believe that.
Laurie H. Rogers has a bachelor's degree in mass communication and a master's in interpersonal communication, emphasizing the evaluation of argumentation and logic. In 2001, she founded Safer Child, Inc., a nonprofit child advocacy information resource. In 2007, she narrowed her advocacy to public education, and in 2010, she founded Focus on the Square™, a nonprofit organization dedicated to improving American K-12 education. Laurie is the author of the blog "Betrayed," located at http://betrayed-whyeducationisfailing.blogspot.com/. Her book Betrayed: How the Education Establishment Has Betrayed America and What You Can Do about It (Rowman & Littlefield Education, 2011) is now available from Amazon and Barnes & Noble. Besides serving on the executive committee for Where's the Math?, Laurie has a background in finance, journalism and child advocacy. She has volunteered in schools - tutoring children in literacy and math, and teaching chess, argumentation and knitting. She lives in Spokane with her husband, daughter and two cats. Contact Laurie Rogers at wlroge@comcast.net.

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In Defense of Using the Number Line to Teach Negatives

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It's simple to teach mathematical positives and negatives to a child. It's been done successfully with the number line around the world, in private schools, homes, tutoring businesses and online. Unfortunately, many schools in America no longer teach the number line, don't teach it to mastery, or they cloud any fledgling understanding of it by emphasizing other, less-effective methods.

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Bornstein (2011) ponders the question why is it acceptable for a person to say I'm not good at math but not acceptable for a person to say they are illerate? He explains how math comes more easily to some than others. Differences in student background and cognitive development is a factor however he believes teachers can improve student awareness in math. He suggests teachers ensure that students are constantly practicing the skill they are learning. He also encourages teachers to break steps down into smaller steps to see where the student confusion may lie. "Math is like a ladder if you miss a step, sometimes you can't go on" once students miss a step, their confidence level starts to deteriorate. When teaching integers he urges teachers to explain the numbers with a scenario and make sure they understand the concept before requesting an answer. This method showed a drastic increase in test scores and student understanding

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Is it possible to eliminate the bell curve in math class?

Imagine if someone at a dinner party casually announced, "I'm illiterate." It would never happen, of course; the shame would be too great. But it's not unusual to hear a successful adult say, "I can't do math." That's because we think of math ability as something we're born with, as if there's a "math gene" that you either inherit or you don't.

School experiences appear to bear this out. In every math class I've taken, there have been slow kids, average kids and whiz kids. It never occurred to me that this hierarchy might be avoidable. No doubt, math comes more easily to some people than to others. But the question is: Can we improve the methods we use to teach math in schools - so that everyone develops proficiency?

Looking at current math achievement levels in the United States, this goal might seem out of reach. But the experience of some educators in Canada and England, using a curriculum called Jump Math, suggests that we seriously underestimate the potential of most students and teachers.

Peter Bregg

"Almost every kid - and I mean virtually every kid - can learn math at a very high level, to the point where they could do university level math courses," explains John Mighton, the founder of Jump Math, a nonprofit organization whose curriculum is in use in classrooms serving 65,000 children from grades one through eight, and by 20,000 children at home. "If you ask why that's not happening, it's because very early in school many kids get the idea that they're not in the smart group, especially in math. We kind of force a choice on them: to decide that either they're dumb or math is dumb."

Children come into school with differences in background knowledge, confidence, ability to stay on task and, in the case of math, quickness. In school, those advantages can get multiplied rather than evened out. One reason, says Mighton, is that teaching methods are not aligned with what cognitive science tells us about the brain and how learning happens.

In particular, math teachers often fail to make sufficient allowances for the limitations of working memory and the fact that we all need extensive practice to gain mastery in just about anything. Children who struggle in math usually have difficulty remembering math facts, handling word problems and doing multi-step arithmetic ( pdf). Despite the widespread support for "problem-based" or "discovery-based" learning, studies indicate that current teaching approaches underestimate the amount of explicit guidance, "scaffolding" and practice children need to consolidate new concepts. Asking children to make their own discoveries before they solidify the basics is like asking them to compose songs on guitar before they can form a C chord.

Mighton, who is also an award-winning playwright and author of a fascinating book called "The Myth of Ability," developed Jump over more than a decade while working as a math tutor in Toronto, where he gained a reputation as a kind of math miracle worker. Many students were sent to him because they had severe learning disabilities (a number have gone on to do university-level math). Mighton found that to be effective he often had to break things down into minute steps and assess each student's understanding at each micro-level before moving on.

Take the example of positive and negative integers, which confuse many kids. Given a seemingly straightforward question like, "What is -7 + 5?", many will end up guessing. One way to break it down, explains Mighton, would be to say: "Imagine you're playing a game for money and you lost seven dollars and gained five. Don't give me a number. Just tell me: Is that a good day or a bad day?"

Courtesy of Mary Jane Moreau

Separating this step from the calculation makes it easier for kids to understand what the numbers mean. Teachers tell me that when they begin using Jump they are surprised to discover that what they were teaching as one step may contain as many as seven micro steps. Breaking things down this finely allows a teacher to identify the specific point at which a student may need help. "No step is too small to ignore," Mighton says. "Math is like a ladder. If you miss a step, sometimes you can't go on. And then you start losing your confidence and then the hierarchies develop. It's all interconnected."

Mighton saw that if he approached teaching this way, he could virtually guarantee that every student would experience success. In turn, the children's math anxiety diminished. As they grew more confident, they grew excited, and they began requesting harder challenges. "More than anything, kids love success," he says, "and they love getting to higher levels, like in a video game."

As the children experienced repeated success, it seemed to Mighton that their brains actually began to work more efficiently. Sometimes adding one more drop of knowledge led to a leap in understanding. One day, a child would be struggling; the next day she would solve a problem that was harder than anything she'd previously handled. Mighton saw that if you provided painstaking guidance, children would make their own discoveries. That's why he calls his approach "guided discovery."

Courtesy of Mary Jane Moreau

The foundation of the process is building confidence, which Mighton believes should be the first goal of a math teacher. Confidence begets attention, which begets rich learning. "I've never met a teacher who will tell you that a student doesn't need to be confident to excel in school," explains Mighton. "But I've never seen a math curriculum that follows the implications of that idea rigorously." Math is well-suited to build confidence. Teachers can reduce things to tiny steps, gauge the size of each step to the student and raise the bar incrementally.

When math is taught this way, surprising things happen.

Consider some of Jump's results. It's been used for four years in the public schools in Lambeth, one of the most economically depressed boroughs of London, England. Teachers placed into Jump the students who were struggling most in math. Among the 353 students who entered the program in fifth grade, only 12 percent began at grade level. Most were at least two grade levels behind and the vast majority were not expected to pass England's grade six (KS2) national tests. But 60 percent did.

In rural Ontario, Jump was recently evaluated in a randomized controlled study involving 29 teachers and about 300 fifth-grade students (controlled studies of math programs are rare). Researchers from Toronto's Hospital for Sick Children and the Ontario Institute for Studies in Education arranged for a control group of teachers to use their district's standard curriculum while another group used Jump. Each set of teachers was given two days of training relevant to the materials they would be using.

In five months, researchers found substantial differences in learning. The Jump group achieved more than double the academic growth in core mathematical competencies evaluated using a well known set of standardized tests. (The study has not yet been published.) "Kids have to make pretty substantial gains in order to see this kind of difference," explained Tracy Solomon, a developmental psychologist in the Research Institute at the Hospital for Sick Children who is the study's lead author. "It's impressive over a five-month period."

Solomon believes that the key to Jump's effectiveness is the way it "breaks math down to its component parts and builds it back up." And she notes that this "flies in the face of the way math is typically taught."

Interviewing teachers and principals, I have heard numerous stories of results like these. At times, they seem hard to reconcile with our assumptions about math. Isabel Grant, principal of the General Wolfe Elementary School, in Vancouver, British Columbia, has seen Jump produce impressive results in two schools where it has been used by a variety of teachers. Schools in British Columbia evaluate students based on whether they meet expectations for learning outcomes. "Teachers who used Jump were suddenly finding that they had all of their kids in the 'fully meeting expectations' category," Grant told me. "It was such a foreign experience. It doesn't typically happen when we're teaching science or language arts. And they were kind of at a loss. 'What do we do about this?'"

Another example is Mary Jane Moreau, who teaches at the Mabin School, an independent school in Toronto that does not screen students based on academic ability. Moreau, an experienced educator, dabbled with Jump for a year and started to see progress among her students, so she decided to immerse herself in the philosophy. "I was used to getting a bell curve in the past," she told me, "but what I started seeing was all the kids getting between 90 and 100 percent on tests, and within months, they were all getting between 95 and 100 percent."

She decided to see if the results would transfer to the standardized Test of Mathematical Abilities. Moreau teaches the same cohort of kids in fifth and sixth grades. Each September, for four years, the students wrote the test. From 2006 to 2007, the class percentile average jumped from 66 percent to 92 percent. From 2008 to 2009, with a new cohort, it increased from 54 percent to 98 percent.

Notably, the bell curve of the students' scores shifted to the right and narrowed - which is to say that the performance differences between the "slow" kids and the "whiz" kids began to fade away. Moreau encouraged her sixth-grade students to enroll in the Mathematica Pythagoras contest, which attracts only five percent of Canadian students, most of whom would be deemed "gifted" in math. All but one did. For each group, 14 out of 17 students beat the contest average.

Moreau is a dedicated teacher - and she has the benefit of small classes - but, even so, she hadn't seen results like this before. And it troubled her to think of students she had taught who didn't have the opportunity to learn math this way. "When I think about what we've been doing for years when we could have been doing something else," she told me, "I feel like I have to run so hard on this because I'm coming to the end of my career. But if I don't help to change attitudes, I'll feel like a criminal."

Jump is a modest outfit. Mighton has a staff of 10 to create materials and conduct teacher trainings. With decisions about math curriculum highly politicized, it's difficult for a small group to influence the debate. Big textbook companies and paid math consultants have a big say - and big investments - in what gets used.

It will take independent-minded educators to use Jump and see if its results can be replicated in more classrooms and schools. It's hard to imagine what society might look like if we could undermine the math hierarchies that get solidified in grade school. These patterns tend to play out across society, often in negative ways. Wasn't it the whiz kids who invented financial derivatives and subprime mortgages? And how many adults got themselves into hot water with their mortgages because, at bottom, they didn't really understand the risks?

Even deeper, for children, math looms large; there's something about doing well in math that makes kids feel they are smart in everything. In that sense, math can be a powerful tool to promote social justice. "When you have all the kids in a class succeeding in a subject, you see that they're competing against the problem, not one another," says Mighton. "It's like they're climbing a mountain together. You see a very healthy kind of competition. And it makes kids more generous to one another. Math can save us."

On Friday, I'll reply to comments, explain how Jump has helped one teacher to conquer her own math fears, and I'll get into some more details about how the program works - including the vital role of bonus questions.

﻿ Join Fixes on Facebook and follow updates on twitter.com/nytimesfixes.
David Bornstein is the author of "How to Change the World," which has been published in 20 languages, and "The Price of a Dream: The Story of the Grameen Bank," and is co-author of "Social Entrepreneurship: What Everyone Needs to Know." He is the founder of dowser.org, a media site that reports on social innovation.

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Is it possible to eliminate the bell curve in math class? Imagine if someone at a dinner party casually announced, "I'm illiterate." It would never happen, of course; the shame would be too great. But it's not unusual to hear a successful adult say, "I can't do math."

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What's the minimum number of students that should be in a focus group?

What if I don't get enough consent forms?

Is there a specific amount of time for data collection?

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Research  by Delta State University on using manipulatives to teach math

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Research  by Delta State University on using manipulatives to teach math

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Cope (2015) states that math should be taught using multiple representations such physical, pictoral and virtual.  She quotes a study by Piaget who hypothesized "that children were not mentally mature enough to grasp abstract mathematical concepts if their teachers only presented the concepts in writing (using words, numbers and symbols)". Thus making sure manipulatives are used. Cope also discuss studies against using manipulatves because students were using them as toys and not having an understanding of the concepts they were trying to replicate with the manipulatives. The study concludes that positive effects in student understanding of teaching with manipulatives provided students with concrete understanding of the skill

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Math Consultant Debra Feldman believes manipulatives should be used as a teaching strategy to help students learn the concept

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Teacher Allison Krasnow teaches 8th grade Pre-Algebra she believes "you can't be successful in Algebra if you don't have a real mastery in integers"

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Math.7.NS.A.1b

Common core State Standards

Math: Math

7: Grade 7

NS: The Number System

A: Apply and extend previous understandings of operations with fractions

1b:
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

Download Common Core State Standards (PDF 1.2 MB)

|

Math.7.NS.A.1d

Common core State Standards

Math: Math

7: Grade 7

NS: The Number System

A: Apply and extend previous understandings of operations with fractions

1d:
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

Download Common Core State Standards (PDF 1.2 MB)

Thought starters

1. How does the use of number lines help students visualize integer operations?

2. How does this approach support long-term conceptual understanding as compared to rote memorization of rules?

3. Why is it important to use both horizontal and vertical number lines?

School Details

Willard Middle School
2425 Stuart Street
Berkeley CA 94705

Data Provided By:

Teachers

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This 7th grade math teacher aims to ensure long-term conceptual understanding of integer operations knowing that students often stumble when only taught to follow the rules. She realizes that the use of number lines is essential for mastery and consistent with what students will do in Algebra I.Her students visualize the number values and operations on a number line helping them easily add integers of any size.

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Video on students using chips and coming up with rule for subtracting integers

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Math.7.NS.A.1c

Common core State Standards

Math: Math

7: Grade 7

NS: The Number System

A: Apply and extend previous understandings of operations with fractions

1c:
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

Download Common Core State Standards (PDF 1.2 MB)

|

Math.7.NS.A.1d

Common core State Standards

Math: Math

7: Grade 7

NS: The Number System

A: Apply and extend previous understandings of operations with fractions

1d:
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

Download Common Core State Standards (PDF 1.2 MB)

Thought starters

1. How do manipulatives deepen understanding of integer operations even for students who find the traditional algorithm easy?

2. Take time to think about what it means to "add the opposite" and how your students can understand this using zero pairs and two-colored counters.?

School Details

Columbia Secondary School
425 West 123rd Street
New York NY 10027

Data Provided By:

Teachers

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Zero Pairs, Manipulatives, and a Real-World Scenario

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Sabrina Joseph from Columbia Secondary School teaches 7th grade math in Harlem, New York. She demonstrates and explains integer subtraction with a real world scenario about warming up and cooling down soup and connects this example to the use of two-colored chips and ordered pairs.

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Colton provides Best practices on integer lessons for all students including ELL students with workshop materials given by the department of education

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Integer Cheat Sheet, lists rules for adding, subtracting, multiplying and dividing integers