Integers 5: Adding different signed numbers.

Created Feb. 7, 2024 by Susan Jones

Part 1: Lesson Description

Lesson Title

1.5 Integers: Adding Signed Numbers with Different Signs. Part 1

Note: Integer operations show up again and again in algebra. Students may need more time with these concepts, and more practice using manipulatives and number lines, and then practice making the cognitive connections so that when they see -3 + -4 they automatically grasp that this indicates a loss of seven.
Students may also benefit from learning to separate the “adjective” of the negative sign next to a number and the “verb” of subtraction. The integer game here: http://www.estimation180.com/integers1.html can help with that, especially the “opposite card.”

This lesson may need to be broken up into two lessons. The text author teaches the integers for part of class period, and then teaches “number sense” — standard algorithms for arithmetic operations, which most students at that placement have not mastered — for the rest of the class. This lesson doesn’t happen in one day because it requires students to see a plus sign and … sometimes subtract.

This lesson pertains to “Section 1.5: Addition Positive and Negative Numbers.”

Download: Chapter one transitions integers_1.docx

as Open Document Format (draft)

Download: Chapter One Transitions Math April2016.odt

Google Doc link: Chapter One Transitions Math Google Doc

Key with Teacher's Notes and Comments:

Download: Chapter One Transitions Integers Key.docx

Google Docs Link: Chapter One Transitions Math (Integers) Answer Key - Google Doc

https://www.youtube.com/watch?v=nYdVKlyn1pE — not OER, but Public Broadcasting System - with references to the Mean Girls and Darth Vader. It integrates the “realistic” examples with symbols and number lines. It includes a preview of subtracting negative integers with a girl saying “I don’t *not* like him…” The comments note that some students thought it was not to their personal taste but many commenters said it was helpful.

https://illinois.pbslearningmedia.org/resource/muen-math-ns-addinverse/additive-inverse/#.WcKyLLJ96Uk Public Broadcasting System 1:35 video explaining “additive inverse” — and it is downloadable.

https://www.youtube.com/watch?v=wu5R6659j1Q — 2:45 video introducing “additive inverse” with gardening example, showing that negative seven shovels of dirt plus positive seven shovels of dirt gets you back to flat, zero.

https://www.youtube.com/watch?v=w7gGSZdvFWY — OER video (CC-BY) explaining adding signed numbers (and sticking to adding).
https://www.youtube.com/watch?v=rPJN8pTFWTw another OER explaining adding signed numbers on number line.

https://www.youtube.com/watch?v=hGVm2xs0HEA videos explaining different models for explaining adding and subtracting integers to students (audience: teachers)

Abstract

For many adult students, positive and negative integers are an example of when math “keeps changing the rules.” This is the fifth of six lessons guiding students in constructing the abstract concepts necessary to understand adding positive and negative integers.

This lesson will review previous knowledge about negative numbers and teach adding integers with different signs. This is a time when it’s important to stress to students that understanding this lesson will help them in many future lessons.

Learner Audience / Primary Users

This lesson is part of a structured sequence of multisensory, conceptual lessons building basic math concepts for students who struggle with number sense and arithmetic, especially figuring out what to do with practical problems. It is taken from Parkland College's "Transitions" course for students whose placement test scores do not qualify them for Pre-algebra. In 4 semesters, fewer that 5% of students have questioned whether their placement was valid. They recognize the need for more than a quick review of math.

Educational Use

Curriculum / Instruction
Professional Development College & Career Readiness Standards (CCRS) Alignment

Level: Adult Education
Grade Level: Middle school and up
Subject: Mathematics
Domain or Strand:

Number Sense
Primarily Math Practices 1, 2
Level D (because it includes negative numbers -- It is not assumed that students are competent with the first three levels.)
Standard Description:
Apply previous knowledge (see lessons 1.1 to 1.4) with practical examples of adding the same signed amounts and extension of that to the number line. In this lesson, students learn about adding different signed numbers.

Language

English

Material Type

Instructional material
Student Text
Student practice
Lecture Notes
Video

Learning Goals

The purpose of this lesson is for learners to be able to:

Understand “opposite” integers and that they add to become zero.
Establish that we’re essentially exploding our idea of “adding” so that now, we can perform “addition” and have a result that is smaller.
Add numbers of different signs.

Keywords

Adult Education
Integers
Math
Math Anxiety
Multisensory
Learning Disabilities
Developmental Math
CCRS
RI.6.2

Time Required for Lesson

15-20 minutes (though this is somewhat variable; for some students this will be “review,” and for others, a new application of new knowledge).

Prior Knowledge

Learners should have counting skills and have completed lesson 1.1 -1.4 This lesson takes the leap into the abstract “oppositeness” of adding a negative number… and having to subtract to get the result. This course is designed for students with minimal "prior knowledge" or math skills.

Required Resources

Text and paper and pencil.There are support videos that can be viewed online or downloaded for viewing with a device that can show MP4 videos (computer, tablet, telephone).

Lesson Author: Susan Jones using text by Kathy Wahl

download image

License: Creative Commons CC BY 4.0 license

Part 2: Lesson

Learning Objectives

By the end of this lesson, the ESL, ABE or GED learner should be able to:

Explain what additive inverses are (and/or show examples).
Identify that when adding numbers of different signs, subtraction is the action taken to get the solution.
Illustrate adding integers of different signs on the number line.
Read and solve practical problems involving adding signed integers of different signs. (The next lesson will mix different kinds of problems together.)

Lesson Topics

Key topics covered in this lesson include:

Adding numbers of different signs to each other.
Addressing common misconception that “addition means you have more.”
Expressing integer addition in different formats.
Practicing and reinforcing old and new concepts to gain confidence and fluency

This lesson is the initial entry into math for students whose math background is below what’s expected for adults. They may have been in special education settings in K-12. In our setting, they will have taken the math placement test and scored below the required score to place into “Pre-Algebra.” However, they need more than “test preparation’ and review. “

These students are usually degree- or certificate- seeking students who have significant gaps in their arithmetic and algebra skills.

Relevance to Practice

Understanding negative numbers is relevant because of their use in topics such as budgeting and science, as well as in required courses and placement tests.

Key Terms and Concepts

Note: many of these terms will be “familiar” to learners, but we will be using and applying them in less familiar ways.

Addition Integer Negative

Opposite Parentheses Positive

Number Line Origin

Instructional Strategies and Activities

Warm-Up (five minutes )

This should be adapted to the learners. These pages review the skills and concepts in previous lessons about integers.

Integer review

-Addition with same sign on number line review

Give students time to work on these and discuss with neighbor, then review with class

Introduction (3-4 minutes)

We’ve added numbers using images and stories and red and black counters. We’ve seen what this looks on a number line.

Today we’re going to venture into some new learning that might seem like it “breaks the rules.” If things aren’t making sense, please keep asking questions. The rules aren’t changing at all — but we’re doing more complicated things with numbers, so we need to figure out the rules for the complicated situations.

*** additive inverse.*** — possible introduction of the concept …

Ever wish you could just undo something? Start over?

Additive inverses do that. If I walk forward 3 steps… then walk backwards three steps — where am I?
**We’re still walking, though! So in Math, we aren’t taking those first steps *away* … we’re “adding” Backwards steps.

Yes, the final result would be the same as if we took them away.

If I heat water to make tea — I’m adding heat. If I add ice cubes — I’m adding… but the temperature is going down.

In math there are lots of different ways of writing down the same value. 5 + 5 and 6 + 4 have the same value. 3 + -3 and 3 - 3 have exactly the same value… zero.

If I owe my sister $20, how much do I have to pay her to be even?

How would we see that on a number line?

How would we write that as a problem?

Have students provide several examples of addition problems with additive inverses. If it’s appropriate, ask them what the additive inverse of “x” or “-x” would be and write

X + -x = 0 and -a + a = 0.

Fill in page 18 in text.

Presentation / Modeling / Demonstration

Time: 5 minutes

So! If numbers are opposite sign but the same distance from zero, adding them brings us *right back* to zero.

Things aren’t always exact, though!

Encourage students to figure out with their neighbors the computations on page 19, and then discuss as class. (Depending on the students, discuss with actions showing the forward and backward change in position.)

Guided Practice

Time: 15 minutes

Use text p. 20 -to have students practice illustrating addition of different -signed numbers on the number line and with Rs and Bs. .

Ask students what they notice about adding positive and negative numbers and work through page 21 to generate the rule that … even though it looks like it’s breaking the rules, at this higher level of math we’ve learned that … even though there’s a plus sign between the numbers, we subtract to find out the answer to our question.

So in early understanding of math, adding meant making something bigger, but now it means “putting two things together,” but … one of them might be going the opposite direction!

Give students time to discuss this. Complete examples 8 -12 in the text.

Evaluation

Time: 5-10 minutes (will vary)

https://docs.google.com/document/d/1pl8CPlgJKzAbuVPlj6_LEEphUPy47f-qgbFgMBO9RX4/edit?usp=sharing

Part 3: Supplementary Resources & References

Supplementary Resources

The concept in song:

https://www.youtube.com/watch?v=xweiQukBM_k opposites attract — two steps forward two steps back… (Paula Abdul)

https://www.youtube.com/watch?v=lTHCwU9rUdY one step forward two steps back (Desert Rose)

http://www.scholastic.com/teachers/top-teaching/2012/09/positive-approach-teaching-negative-numbers ideas for teachers for explaining negative numbers

References/Attribution Statements

Lessons and images are created by Sue Jones;

Transitions, Chapter One contributed by Kathy Wahl and Sue Jones of Parkland College to this project.