8: Integers
- Page ID
- 188462
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As future elementary school teachers, it’s important to understand that many students struggle with integer operations because these concepts challenge their existing ideas about how numbers work. For children who have learned that larger numbers always mean more, the introduction of negative numbers—values less than zero—can feel abstract and even nonsensical. Adding to the challenge, students often confuse the rules for adding and subtracting integers, especially when subtracting negatives, which may seem illogical without a solid conceptual foundation. Multiplying and dividing with negative numbers brings further difficulty, as the idea that two negatives make a positive can seem arbitrary. Unlike whole number operations, which often come with clear, concrete models, integer operations can be harder to visualize and internalize. That’s why it’s essential to use a variety of teaching strategies—like manipulatives, real-world analogies, games, and digital tools—to help students build a deeper and more intuitive understanding. These approaches allow each child to engage with the material in a way that makes sense to them, transforming what can be a confusing topic into one that is meaningful, memorable, and even fun.
Chapter Learning Objectives
SLO 8.1 Students will represent integers using models such as the integer number line and colored chips, and demonstrate addition, subtraction, and ordering of integers using these tools.
SLO 8.2 Students will analyze and apply the properties of integer operations, including addition, subtraction, multiplication, and division, to solve mathematical problems and verify relationships such as closure, commutativity, and distributivity.
SLO 8.3 Students will explain integer multiplication and division through repeated addition, patterns, and missing-factor approaches, and compute using negative exponents and scientific notation.
SLO 8.4 Students will evaluate relationships between integers, applying the properties of inequalities (e.g., transitivity, multiplication by a positive or negative number) and theorems such as additive cancellation and opposite of the opposite
Essential Questions
- How do negative numbers extend the number system to make subtraction and other operations more complete and meaningful in real-world contexts?
- In what ways do the properties of integer operations lay the groundwork for understanding algebraic reasoning and solving equations?"
Learning That Transfers
Students will be able to explain how extending the number system to include integers supports flexible problem-solving in both real-world and mathematical contexts, and apply the properties of integer operations to model and solve algebraic equations.
From Plus to Minus, the Golf Ball Reminds Us
NEED AUDIO FOR STORY
From Plus to Minus, the Golf Ball Reminds Us
One sunny day, with the breeze nice and cool,
Morgan and Grandpa set off with a tool—
A putter, some balls, and a pencil in hand,
To play mini golf at "The Green Wacky Land."
“Today,” Grandpa said, “we’ll keep score the right way,
With numbers that rise, and numbers that stray.
You’ll see how the numbers can go up and down,
Not just above zero, but also below ground!”
The first hole was tricky, with curves left and right,
Morgan got through with two strokes—what a sight!
Grandpa took four, with a laugh and a grin,
“Remember in golf, the low score will win!”
“So your score is two, and mine is four high,
But to show who is winning, let’s give this a try:
We’ll subtract my four strokes from yours—so neat!
That’s two minus four… hmm, that’s negative two, sweet!”
“Negative two? What’s that mean?” Morgan frowned,
“Are we digging below the score into ground?”
Grandpa just chuckled, “It’s numbers that dive,
They’re just as important to keep math alive!”
“Positive numbers go up—high and tall,
But negative numbers go down, just like fall.
They help us when scores need to measure below,
When you’re ‘under par’—it’s good math to know!”
At Hole Number Six, things started to swing,
Morgan made a birdie—one under—zing!
“That’s minus one, since you beat par by a bit,
A negative score shows you’re winning, that’s it!”
Grandpa had a bogey, one over, poor guy,
“Plus one for me,” he said with a sigh.
“Let’s show it on the line—see this chart here?
It helps us keep track, makes it all very clear.”
They counted up scores with each hole they played,
Positive, negative, swaps that they made.
Adding negatives pulled scores back to small,
Subtracting a negative? That’s adding, after all!
Morgan grinned wide, “So subtracting your loss,
Is like giving me points? Well, I’ll be the boss!”
“You’ve got it!” said Grandpa, “It’s patterns we find,
Opposites, adding, they all intertwine.”
At the last hole, the scorecard was done,
Morgan had negative five—she had won!
Grandpa had positive six on his side,
“But I had more fun!” Grandpa smiled wide.
On the way home, Morgan sat up with a grin,
“Negative numbers help show when we win.
Adding, subtracting, they tell quite a tale,
With numbers that climb and with numbers that sail.”
Grandpa just smiled, gave a wink and a clap,
“You’re thinking in math—what a marvelous map!
From scores to equations, from golf to the sky,
Integers help us all multiply!”