The real number line and the vocabulary that goes with it: positive, negative and zero; integers and whole numbers; graphing a number as a point; comparing two numbers by their positions rather than their digits; and plotting decimals and fractions on a line whose scale marks need not be integers.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 2 — Properties of Real Numbers
The Real Number Line
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 65-70 — the lesson these objectives are drawn from
Warm-up
You have used negative numbers outside mathematics for years. What is new is being asked to order them.
Discussion prompt
Two overnight temperatures are recorded: negative 8 degrees and negative 3 degrees. Which night was colder, and did you have to think about it?
Hint: Most people get this right instantly for temperature and hesitate on the same numbers written on their own.
Answer:
\[ -8 < -3 \]
Negative eight is colder, and almost nobody hesitates when the word temperature is present. Strip the context away and the same two numbers suddenly feel harder, because the digit eight is bigger than the digit three. The number line restores the thing that temperature gave you for free: a picture in which colder is further left.
Concept
The numbers used in this book are real numbers, and they can all be pictured as points on a single line. Points to the left of zero are the negative numbers, points to the right are the positive numbers, and zero is neither.
real number line — A line on which every real number is represented by exactly one point, with negatives to the left of zero and positives to the right.
That single picture settles every question about order in this chapter.
Figure (svg): A real number line from -6 to 6 with the negative side, zero and the positive side labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 65-65
Section
Section 1
Concept
Every real number is either positive, negative, or zero. Zero is genuinely a third case rather than a small positive number, and treating it as one causes trouble later.
Figure (svg): A real number line from -6 to 6 with the negative side, zero and the positive side labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 65-65 — the Real Number Line diagram
Picture it
One picture, and it is the answer to almost every question in this chapter.
Figure (svg): A real number line from -6 to 6 with the negative side, zero and the positive side labelled
Notice how much of the line is to the left of zero. Before this chapter you worked almost entirely on the right-hand half; from here on both halves are in play.
Worked example
Trivial once the vocabulary is fixed, and worth being certain about before anything harder.
\[ \text{Classify each as positive, negative or zero: } \; -7, \; 0, \; 4.5, \; -0.2, \; \tfrac{3}{8}. \]
Negative seven is negative
Why: It carries a minus sign and sits to the left of zero.
\[ -7\text{ negative} \]
Zero is neither
Why: It is the point the two sides are measured from, and it belongs to neither of them.
\[ 0\text{ neither} \]
Four point five is positive
Why: No minus sign, and it sits to the right of zero.
\[ 4.5\text{ positive} \]
Negative nought point two is negative
Why: A number can be negative and still be very close to zero; the sign decides, not the size.
\[ -0.2\text{ negative} \]
Three eighths is positive
Why: A fraction with no minus sign sits to the right of zero.
\[ \frac{3}{8}\text{ positive} \]
Figure (svg): The solution to Worked example name the category shown as a ladder of expressions, one row per algebraic move
\[ -7 \text{ neg}, \; 0 \text{ neither}, \; 4.5 \text{ pos}, \; -0.2 \text{ neg}, \; \tfrac{3}{8} \text{ pos} \]
Verify: check that the sign, not the size, decided every one
Why: Negative nought point two is far closer to zero than four point five is, yet it is the negative one. Size and sign are independent, and mixing them up is exactly what the number line is there to prevent.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 65-65
Sorting
Sign decides, not size. Sort each number.
Sort into buckets
Drop each number into the right column.
Two items here are very close to zero and one is very far from it, and all three are negative. Sign and size are independent, which is the whole reason the number line has two directions rather than one.
Worked example
This is Example 1 from the textbook. Two moves per number: how far, and which side.
\[ \text{Graph } -2, \; 0 \text{ and } 3 \text{ on a number line.} \]
Draw a line and mark zero
Why: Zero is the reference point, so nothing can be placed until it is drawn.
\[ \text{mark } 0 \]
Plot negative 2
Why: The digit says two units; the sign says to the left. Two units left of zero.
\[ -2\text{ plotted} \]
Plot 0
Why: Zero is the point already marked.
\[ 0\text{ plotted} \]
Plot 3
Why: Three units, and no minus sign, so three units to the right.
\[ 3\text{ plotted} \]
Figure (svg): The numbers negative 2, 0 and 3 plotted on a number line from -5 to 5
\[ -2, \; 0, \; 3 \text{ plotted left to right} \]
Verify: read the three points back from left to right
Why: Reading the plotted points left to right gives negative two, zero, three — which is their order from smallest to largest. If a plotted point is out of that order, its side or its distance is wrong.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 65-65
Trap
\[ \text{Is } 0 \text{ positive?} \]
Answer yes, because it has no minus sign
Why: Everything with no minus sign so far has been positive, so the absence of a sign reads as positive.
This causes real trouble in Chapter 6, where a solution set of positive numbers and one of non-negative numbers differ by exactly one point — and that point is zero.
Zero is neither positive nor negative.
Treat zero as its own third case
Why: It is the point the other two categories are defined relative to, so it cannot belong to either.
The habit to build now: whenever a rule mentions positive numbers, ask separately what it says about zero. The answer is often different.
Elimination
Four statements about the number line. Only one is correct.
Eliminate the wrong options
Which statement is true?
Survives elimination: A
Why: Zero is the reference point from which both directions are measured, so it belongs to neither. Option B is worth dwelling on: even if zero were positive, there would still be no smallest positive number, which is a fact about the real numbers worth meeting early.
Notation
Three short symbols, and a convention behind each one.
Annotate
On: \( -2, \quad 0, \quad 3 \)
The distinction between negative as a property and minus as an operation is worth building now. Lesson 2.4 rewrites every subtraction as an addition, and that rewrite is impossible to follow if the two words are used interchangeably.
Prediction
Commit before you compute.
Predict first
A submarine descends to 40 metres below sea level, then rises 15 metres. Where is it relative to sea level?
Correct: 25 metres below, so negative 25.
\[ -40 + 15 = -25 \text{ metres} \]
Why: Descending forty puts the submarine at negative forty; rising fifteen moves it fifteen units to the right, to negative twenty-five. It is still below sea level, so the answer is still negative — rising moves you rightwards along the line, and fifteen units of rightward movement from negative forty does not reach zero.
Section
Section 2
Concept
The scale marks on a number line are equally spaced and represent the integers. An integer is negative, zero, or positive. Zero and the positive integers together are called the whole numbers.
integer — One of the numbers the equally spaced scale marks represent: the negative integers, zero, and the positive integers.
\[ \ldots, -3, -2, -1, \quad 0, \quad 1, 2, 3, \ldots \]
So every whole number is an integer, but not every integer is a whole number — the negatives are the difference.
Figure (svg): The integers shown as evenly spaced scale marks, with negative integers, zero and positive integers labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 65-65 — the paragraph defining integers and whole numbers
Picture it
The dashed box marks the whole numbers inside the integers.
Figure (svg): The integers shown as evenly spaced scale marks, with negative integers, zero and positive integers labelled
The whole numbers are the right-hand half of the integers plus zero. That containment goes one way only, which is the point worth remembering.
Worked example
Classifying carefully now saves confusion when Chapter 12 adds more categories.
\[ \text{Classify } \; 5, \; -3, \; 0, \; 2.5, \; -\tfrac{1}{2} \text{ as integers, whole numbers, both or neither.} \]
Five is a positive integer, so it is both
Why: It lands on a scale mark and it is not negative.
\[ 5\text{ both} \]
Negative three is an integer but not a whole number
Why: It lands on a scale mark, but whole numbers start at zero and go right.
\[ -3\text{ integer only} \]
Zero is both
Why: It lands on a scale mark and it is explicitly included among the whole numbers.
\[ 0\text{ both} \]
Two point five is neither
Why: It sits halfway between two scale marks, so it is not an integer at all.
\[ 2.5\text{ neither} \]
Negative one half is neither
Why: Also between scale marks, and negative into the bargain.
\[ -\frac{1}{2}\text{ neither} \]
Figure (svg): The solution to Worked example integer, whole number, both, or neither shown as a ladder of expressions, one row per algebraic move
\[ 5 \text{ both}, \; -3 \text{ integer}, \; 0 \text{ both}, \; 2.5 \text{ neither}, \; -\tfrac{1}{2} \text{ neither} \]
Verify: check the containment in both directions
Why: Every number classified as whole was also classified as an integer, which is what nesting requires. And negative three is an integer that is not whole, which shows the containment does not reverse — exactly one example is enough to establish that.
Sorting
Remember that every whole number is also an integer.
Sort into buckets
Sort each number into the most specific category it belongs to.
The middle column contains only negatives, and that is not a coincidence — it is the definition. Whole numbers are the integers with the negatives removed.
Worked example
A question worth asking of every new collection of numbers you meet.
\[ \text{If you subtract one whole number from another, is the answer always a whole number?} \]
Try a case where the answer is comfortable
Why: Seven minus three is four, and four is a whole number.
\[ 7 - 3 = 4\text{ whole} \]
Try a case where the first number is smaller
Why: Three minus seven is negative four, which is not a whole number.
\[ 3 - 7 = -4\text{ not whole} \]
State the conclusion
Why: One counterexample settles it: subtracting whole numbers does not always give a whole number.
Ask the same question of the integers
Why: Any integer minus any integer is an integer, because the negatives are now included.
Figure (svg): The solution to Worked example is the collection closed shown as a ladder of expressions, one row per algebraic move
\[ 3 - 7 = -4 \quad \text{not a whole number, but an integer} \]
Verify: check the integer claim on the same example
Why: Three minus seven is negative four, which is an integer, so the example that broke the whole numbers does not break the integers. That is precisely the gap the negative integers were invented to fill, and it is why this chapter exists.
Error analysis
The student sorted five numbers into integers and whole numbers. Two entries are wrong.
Annotate
On: \( \begin{aligned} \text{integers: } & \; 5, \; 0, \; 2.5 \\ \text{whole numbers: } & \; 5, \; 0, \; -3 \end{aligned} \)
Both errors are at a boundary: one between integers and non-integers, one between whole numbers and negatives. Definitions earn their keep exactly at boundaries, which is why those are the cases to test yourself on.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Integers | Whole numbers | |
|---|---|---|
| Includes negatives? | Yes | No |
| Includes zero? | Yes | Yes |
| Smallest member | there is none | zero |
The bottom-left cell is the interesting one. The integers run away to the left without end, so no integer is smallest — and that is precisely what having a negative side means.
Counterexample
A single counterexample is enough to demolish a general claim.
Discussion prompt
A student claims that every integer is a whole number. Give one counterexample and explain why one is enough. Then state the claim that is actually true, in the correct direction.
Hint: You only need to find one integer that fails to be whole.
Answer:
Negative three is an integer, since it lands on a scale mark, and it is not a whole number, since whole numbers are zero and the positive integers. That single case is enough because the claim said every integer — and one exception refutes every.
The true statement runs the other way: every whole number is an integer. Nesting claims almost always hold in exactly one direction, and checking which direction is the whole content of the claim.
Edge cases
The interesting cases for any definition sit right at its boundary.
Discussion prompt
Zero is on the boundary between the positive and negative integers. Name every category from this lesson that zero belongs to and every one it does not, and say why zero is included among the whole numbers when it is not positive.
Hint: Read the definition of whole numbers word by word.
Answer:
Zero is a real number, an integer and a whole number. It is not positive and it is not negative.
It is included among the whole numbers because the definition names them explicitly as zero together with the positive integers, rather than as the positive integers alone. That is a choice, made because whole numbers are used for counting and counting starts at nothing rather than at one — and it is the reason zero has to be listed separately every time.
Section
Section 3
Concept
The point on a number line that corresponds to a number is the graph of that number. Drawing the point is called graphing the number, or plotting the point. Two pieces of information go into every plot: a distance and a side.
A number with a large digit and a minus sign is far from zero on the left, which is exactly what makes it small.
Figure (svg): The numbers negative 2, 0 and 3 plotted on a number line from -5 to 5
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 65-65 — the definition of the graph of a number and Example 1
Picture it
Distance from zero on the horizontal, side of zero from the sign.
Figure (svg): The numbers negative 2, 0 and 3 plotted on a number line from -5 to 5
Once the three points are down, their order is visible without any comparison being made. That is the entire argument for drawing the picture.
Worked example
A wider spread than Example 1, to make the scale decision real.
\[ \text{Graph } \; -4, \; -1, \; 0, \; 2 \text{ and } 5 \text{ on one number line.} \]
Find the smallest and largest values
Why: Negative four and five, so the line has to reach at least that far in both directions.
\[ -4\text{ to } 5 \]
Choose a line a little wider than that
Why: Running from negative five to six leaves the outermost points from sitting on the ends.
\[ -5\text{ to } 6 \]
Plot the negatives to the left of zero
Why: Four units left, then one unit left.
\[ -4\text{ and } -1 \]
Plot zero and the positives to the right
Why: Zero at the centre, then two units right and five units right.
\[ 0, 2, 5 \]
Figure (svg): The solution to Worked example graph five mixed numbers shown as a ladder of expressions, one row per algebraic move
\[ -4 < -1 < 0 < 2 < 5 \]
Verify: read the points back left to right
Why: The plotted order is negative four, negative one, zero, two, five — which is the numbers in increasing order. Reading back is the only check a graph needs, and it catches both a wrong side and a wrong distance.
Matching
Four numbers, four descriptions of where they sit.
Match the pairs
Why: Every position is described by two things: a count of units and a side. The digits give the count and the sign gives the side, and both have to be read before anything is drawn. Note that negative five is further from zero than negative two, which will matter in the next section when the two are compared.
Worked example
Guided Practice 3 in spirit. The scale is a decision, and making it first saves redrawing.
\[ \text{Graph } \; -20, \; -5, \; 15 \text{ and } 40 \text{ on one number line.} \]
Find the range the points span
Why: From negative twenty to forty, a span of sixty units.
\[ \text{span of } 60 \]
Choose a scale interval that gives a manageable number of marks
Why: Marks every ten give seven or eight of them across the whole span, which is readable.
\[ \text{step of } 10 \]
Draw the line from below the smallest to above the largest
Why: From negative thirty to fifty gives a little room at each end.
\[ -30\text{ to } 50 \]
Plot each point against the scale
Why: Negative twenty is two marks left of zero; forty is four marks right.
Figure (svg): The solution to Worked example choose the scale before plotting shown as a ladder of expressions, one row per algebraic move
\[ \text{scale marks every } 10, \text{ from } -30 \text{ to } 50 \]
Verify: check that every point lands where the arithmetic says
Why: Negative five sits halfway between the marks at negative ten and zero, and fifteen sits halfway between ten and twenty. Both are consistent with a scale of ten, so the scale was applied uniformly — which is the thing that can go wrong when the marks are not integers.
Trap
\[ \text{Graph } -6 \]
Count six units from zero and mark the point
Why: The digit is the loud part of the number, and counting is the active part of the task.
Counting six units to the right plots 6, not negative 6. The two points are twelve units apart, and everything built on the picture afterwards is wrong.
\[ \text{Graph } -6 \]
Read the sign first to fix the direction, then count
Why: Direction before distance means the count cannot go the wrong way.
A quick check: the point should be on the same side as every other negative you have plotted. A negative sitting to the right of zero is visibly out of place, which is another reason to plot several numbers on one line.
Faded example
Supply the distance and the direction.
Fill in the blanks
\text7 -7: \textleft ___ \text___ ___ \text___
Why: The digits give the distance and the sign gives the direction, and both are needed before a point can be drawn. Reading them in that order — sign first to fix the direction, then digits to count — is what prevents a negative from being plotted on the positive side.
Estimation
A number line runs from negative 10 to 10 with marks every 2 units.
Predict first
Roughly where does negative 7 sit?
Correct: Between the marks at -8 and -6, closer to neither — exactly halfway.
\[ -8 < -7 < -6, \quad \text{exactly halfway between} \]
Why: With marks every two units there is no mark at negative seven, so it sits exactly midway between negative eight and negative six. Option B plots the digit on the wrong side, which is the error the previous trap slide is about, and option C rounds to a mark rather than estimating between them.
Socratic
You can compare numbers without a picture. The picture still earns its place.
Discussion prompt
Give two things a plotted number line shows you that a written list of the same numbers does not. At least one should be about avoiding a specific error.
Hint: Think about what you can see all at once on a line but have to work out from a list.
Answer:
First, the ordering is visible without any comparisons being made. A list of five numbers requires you to compare pairs; a plotted line shows the whole order at once, simply by reading left to right.
Second, it prevents the negative-digit error. On a line, negative eight is visibly further from zero than negative three and visibly further left, so the fact that it is smaller is something you see rather than something you have to remember. That single benefit is why the next section builds its whole rule on positions rather than digits.
Section
Section 4
Concept
On a number line, numbers to the left are less than numbers to the right, and numbers to the right are greater than numbers to the left. That single rule covers every comparison in the chapter, negatives included.
\[ -5 < -4 \qquad \text{and equivalently} \qquad -4 > -5 \]
Every comparison can be written two ways, and both say the same thing.
Figure (svg): Negative 5 and negative 4 plotted on a number line, showing that negative 5 is to the left and therefore less
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 66-66 — Example 2, Compare Integers
Picture it
The digit instinct says five beats four. The line says otherwise.
Figure (svg): Negative 5 and negative 4 plotted on a number line, showing that negative 5 is to the left and therefore less
Negative five sits to the left, so it is the smaller of the two. Among negatives the bigger digit means the smaller number, and the line is what makes that visible rather than something to memorise.
Worked example
This is Example 2 from the textbook. Plot first, then read.
\[ \text{Graph } -5 \text{ and } -4, \text{ then write two inequalities comparing them.} \]
Plot both numbers
Why: Five units left of zero and four units left of zero.
\[ -5\text{ and } -4\text{ plotted} \]
Read which is further left
Why: Negative five is further left, since it is further from zero on the negative side.
\[ -5\text{ is left} \]
Write the first inequality
Why: Left means less, so negative five is less than negative four.
\[ -5 < -4 \]
Write the second inequality
Why: The same fact from the other side: negative four is to the right, so it is greater.
\[ -4 > -5 \]
Figure (svg): Negative 5 and negative 4 plotted on a number line, showing that negative 5 is to the left and therefore less
\[ -5 < -4 \qquad -4 > -5 \]
Verify: test the claim with a temperature
Why: Negative five degrees is colder than negative four degrees, and colder means less. The everyday reading agrees with the number line, which is a good sign that the rule rather than the digits was applied.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 66-66
Ranking
Five numbers, mixed signs. Plot mentally before you order.
Put in order
Why: Split by sign first: the negatives are negative six, negative four and negative one, and among them the largest digit is the smallest number, so they run negative six, negative four, negative one. Then zero, then the single positive. Splitting by sign before ordering is faster and safer than trying to compare across the sign in one pass.
Worked example
Guided Practice 1 to 3. Plot each pair before writing anything.
\[ \text{Compare } \; -6 \text{ and } -2; \quad 2 \text{ and } -3; \quad -5 \text{ and } -7. \]
Compare negative 6 and negative 2
Why: Negative six is further left, so it is less.
\[ -6 < -2\text{ and } -2 > -6 \]
Compare 2 and negative 3
Why: A positive is always to the right of a negative, so two is greater.
\[ -3 < 2\text{ and } 2 > -3 \]
Compare negative 5 and negative 7
Why: Negative seven is further from zero on the left, so it is less.
\[ -7 < -5\text{ and } -5 > -7 \]
Notice which pair was easiest
Why: The mixed-sign pair needed no thought at all: every positive beats every negative.
Figure (svg): The solution to Worked example three comparisons from guided practice shown as a ladder of expressions, one row per algebraic move
\[ -6 < -2, \quad -3 < 2, \quad -7 < -5 \]
Verify: check each against the digit instinct and note where they disagree
Why: For the two negative pairs, the digit instinct gives the wrong answer both times — it says six beats two and seven beats five. For the mixed pair it happens to give the right answer. That pattern is exactly why negatives need the line and positives do not.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 66-66
Trap
\[ \text{Compare } -8 \text{ and } -3 \]
Note that 8 is bigger than 3, so conclude that -8 is bigger than -3
Why: Years of practice with positive numbers has made digit size and number size the same thing.
\[ -8 > -3 \quad \text{(wrong)} \]
Negative eight is a deeper hole than negative three. The digit records how far from zero, and on the negative side further from zero is further down.
\[ -8 < -3 \]
Plot both and read left to right
Why: Negative eight is eight units left of zero and negative three is three units left, so negative eight is further left.
Among negatives the bigger digit always means the smaller number. Say it as a temperature if the instinct fights back: eight below is colder than three below.
Two truths and a lie
Read each comparison carefully, and plot the pair if you hesitate.
Eliminate the wrong options
Which comparison is FALSE?
Survives elimination: D
Why: Negative nine is nine units left of zero and negative two is only two units left, so negative nine is further left and therefore the smaller of the two. The correct statement is that negative nine is less than negative two. This is the digit instinct in its purest form: nine looks like it should beat two, and among negatives it never does.
Discrimination
For each pair, decide whether comparing the digits gives the right answer.
Sort into buckets
Sort each pair by whether the digit instinct succeeds.
Prediction
Record low temperatures in six places, in degrees Fahrenheit.
Predict first
Which of these record lows is the coldest: -80, -61, -2, -70, -52, -23?
Correct: -80.
\[ -80 < -70 < -61 < -52 < -23 < -2 \]
Why: All six are negative, so the one furthest from zero is the smallest, and eighty is the largest digit among them. Plotted on a line, negative eighty sits furthest left and negative two sits furthest right, closest to zero — which is why negative two is the mildest of the six rather than the coldest.
Section
Section 5
Concept
Decimals and fractions can be graphed on a number line just as integers can. The scale marks do not have to be integers — they can be in units of 0.1, 0.5, 2, 5 or any other amount, and choosing a suitable one is part of the work.
Four sevenths is about 0.57, which places it just past halfway from zero to one.
Figure (svg): A number line with scale marks every tenth, showing negative 0.8 and four sevenths plotted
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 66-66 — Example 3, Graph Real Numbers, and the Study Tip on converting fractions
Picture it
The same line, rescaled to make the region between negative one and one usable.
Figure (svg): A number line with scale marks every tenth, showing negative 0.8 and four sevenths plotted
Nothing about the line changed except the labels on the marks. Choosing a scale is choosing which part of the line to look at closely.
Worked example
This is Example 3 from the textbook. The conversion is the move that matters.
\[ \text{Graph } -0.8 \text{ and } \tfrac{4}{7} \text{ on a number line.} \]
Notice that neither is an integer
Why: Both sit between scale marks on an integer line, so a finer scale is needed.
Convert the fraction to a decimal
Why: Four divided by seven is about 0.57.
\[ \frac{4}{7} = 0.57\text{ approx} \]
Choose scale marks in units of 0.2
Why: That gives readable marks between negative one and one, which is where both numbers live.
\[ \text{marks every } 0.2 \]
Plot each number
Why: Negative nought point eight is 0.8 unit left of zero; 0.57 is 0.57 unit right of zero.
Figure (svg): A number line with scale marks every tenth, showing negative 0.8 and four sevenths plotted
\[ -0.8 \text{ and } \tfrac{4}{7} \approx 0.57 \]
Verify: check that the fraction landed sensibly
Why: Four sevenths should be a little more than a half, since four is a little more than half of seven, and 0.57 is indeed just past 0.5. A rough sense of a fraction's size is a check on the division as well as on the plotting.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 66-66
Translation
Four fractions and their decimal values. Convert before you pair.
Match the pairs
Why: Every one of these is a division of the numerator by the denominator. Three of them terminate and one, four sevenths, repeats forever — which is why it is given as an approximation. For plotting, two decimal places is almost always enough, since a number line drawn by hand cannot distinguish finer than that anyway.
Worked example
Fractions, decimals and integers together, which is the realistic case.
\[ \text{Order } \; -1, \; 0.7, \; -\tfrac{1}{2}, \; \tfrac{3}{4}, \; -0.2 \text{ from least to greatest.} \]
Convert everything to decimals
Why: Negative one half is negative 0.5; three quarters is 0.75. The rest already are decimals.
\[ -1, 0.7, -0.5, 0.75, -0.2 \]
Split by sign
Why: Negatives: negative one, negative 0.5, negative 0.2. Positives: 0.7 and 0.75.
Order the negatives, furthest from zero first
Why: Negative one, then negative 0.5, then negative 0.2.
\[ -1, -0.5, -0.2 \]
Order the positives and join the groups
Why: 0.7 then 0.75, and every positive comes after every negative.
Figure (svg): The solution to Worked example order a mixed list shown as a ladder of expressions, one row per algebraic move
\[ -1 < -\tfrac{1}{2} < -0.2 < 0.7 < \tfrac{3}{4} \]
Verify: check the two closest values separately
Why: Nought point seven and three quarters differ by only 0.05, so they are worth checking on their own: 0.75 is larger than 0.70, so three quarters comes last. When two values are close, converting both to the same number of decimal places settles it.
Trap
\[ \text{Graph } \tfrac{4}{7} \]
Place it near 4, because 4 is the visible whole number in it
Why: The numerator is the largest thing in the fraction, so the eye anchors on it.
Four sevenths is less than one. Plotting it near four puts it about seven times too far from zero.
\[ \tfrac{4}{7} \approx 0.57 \]
Divide the numerator by the denominator before plotting anything
Why: A fraction bar is a division sign, so the number it names is the quotient rather than either of its parts.
A quick sanity check first: is the numerator bigger or smaller than the denominator? Smaller means the fraction is between zero and one, which fixes the region before the division is even done.
Elimination
You need to graph 0.3, negative 0.7 and 0.55 on one line.
Eliminate the wrong options
Which scale is most suitable?
Survives elimination: A
Why: The three numbers span from negative 0.7 to 0.55, so a line from negative one to one with marks every tenth covers them with about twenty readable marks. The general rule is to make the scale interval roughly one tenth to one twentieth of the span you need to cover.
Estimation
Estimate before you divide.
Predict first
Between which two tenths does five ninths sit?
Correct: Between 0.5 and 0.6.
\[ \tfrac{5}{9} = 0.555\ldots, \quad \text{so } 0.5 < \tfrac{5}{9} < 0.6 \]
Why: Five is a little more than half of nine, so five ninths is a little more than a half — about 0.556. The last option is the fraction read upside down, giving nine fifths, which is a useful reminder that the numerator goes on top of the division as well as on top of the fraction.
Socratic
Fractions can be compared without converting. Converting is still usually the right move here.
Discussion prompt
Give one reason converting to decimals is the right approach when plotting, and one situation where comparing fractions directly would be better. Use a specific pair of fractions in each case.
Hint: Think about what a number line asks you to do that a comparison does not.
Answer:
For plotting, a decimal tells you directly how far along the line to go, which is what the task requires. Three sevenths and five ninths cannot be placed on a line until you know they are about 0.43 and about 0.56; comparing them as fractions tells you which is larger without telling you where either one goes.
Comparing directly is better when the fractions share a denominator or one is obviously bigger — three fifths against four fifths needs no arithmetic at all. It is also better when an exact answer matters, since a decimal that repeats has to be rounded, and a rounded value can make two close fractions look equal when they are not.
Comparison
Fill the blanks from memory before you scroll back. The bottom row is the one that costs marks.
Comparison matrix
| Two positives | Two negatives | |
|---|---|---|
| Which is further from zero? | the bigger digit | the bigger digit |
| Which is further left? | the smaller digit | the bigger digit |
| Which is the greater number? | the bigger digit | the smaller digit |
The top row is identical in both columns, which is why the digit instinct feels reliable. The bottom row reverses, and that reversal is the whole difficulty of this lesson.
Pattern
Whether the question says graph, compare or order, the same five moves cover it.
Step three is deliberately sign-before-digits. Reading the digits first is how a negative ends up plotted on the positive side.
Check
Vocabulary. Read the definitions carefully.
Check your understanding
Which of these is an integer but NOT a whole number?
Answer: A
Why: Negative three lands on a scale mark, so it is an integer, but the whole numbers are zero together with the positive integers only. Every negative integer is an integer that is not whole, and that is exactly the difference between the two collections.
Check
Comparing negatives. Plot the pair if you hesitate.
Check your understanding
Which statement is true?
Answer: A
Why: Negative twelve is twelve units left of zero and negative five is only five units left, so negative twelve is further left and therefore the smaller of the two. Among negatives the bigger digit always means the smaller number.
Check
Plotting a fraction. Convert first.
Check your understanding
Where does the fraction three eighths sit on a number line?
Answer: A
Why: Three divided by eight is 0.375, which sits between zero and one and a little closer to 0.4 than to 0.3. A fraction whose numerator is smaller than its denominator always lands between zero and one, which fixes the region before any division is done.
Real world
A bank statement shows balances of 120, negative 45, negative 200 and 15 dollars over four weeks, where a negative balance means an overdraft.
Discussion prompt
Order the four balances from worst to best and say which week the account was in the most trouble. Then explain why somebody looking only at the digits might name the wrong week, and what everyday phrase makes the right answer obvious.
Hint: Think about what deeper in overdraft means as a position on the line.
Answer:
\[ -200 < -45 < 15 < 120 \]
The worst week is the one at negative 200, since it lies furthest left. Somebody comparing digits alone might see 200 as the biggest number in the list and call it the best week, which is the digit instinct applied to a negative.
The everyday phrase that fixes it is deeper in debt. Being two hundred dollars overdrawn is worse than being forty-five overdrawn, and worse means further left. Money and temperature are the two contexts where almost nobody gets negatives wrong, and borrowing that intuition for the bare numbers is the most reliable trick available.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Is negative 100 greater than or less than negative 1?
Correct: Less, because it lies further to the left.
\[ -100 < -1 \qquad \text{a gap of } 99 \text{ units} \]
Said as a temperature: a hundred degrees below zero is very much colder than one degree below zero.
Why: Negative one hundred sits a hundred units left of zero while negative one sits only one unit left, so negative one hundred is much further left and therefore much smaller. The gap between the two is ninety-nine units, so this is not a marginal case — and the digit instinct gets it exactly backwards, which is why it is worth testing yourself on the extreme version.
Explain it
They can add and subtract confidently and have never had to decide which of two negative numbers is bigger.
Discussion prompt
In no more than four sentences, and without using the word negative more than twice, explain why negative eight is less than negative three. Then give them one everyday situation that makes the answer obvious, and say why that situation helps.
Hint: The situation should be one where they already have the right intuition without any mathematics.
Answer:
A usable answer: picture a line with zero in the middle. Numbers get bigger as you move right and smaller as you move left. Eight below zero is eight steps left; three below zero is only three steps left. So eight below is further left, and further left means less.
Temperature is the situation to use. Nobody argues that eight degrees below zero is colder than three degrees below, and colder is exactly what less means here. It helps because their existing intuition about cold already encodes the number line, so nothing new has to be believed — only recognised.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Vocabulary is fixed by testing yourself at the boundaries: zero, one negative integer, and one number with a fractional part. Graphing is fixed by reading the sign before the digits, every time. Comparing negatives is fixed by saying the pair aloud as temperatures until the instinct rewires. Fractions are fixed by dividing before plotting, and by first asking whether the numerator is smaller than the denominator. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
Draw one long number line across the middle of a page running from negative ten to ten, with the negative side, zero and the positive side labelled. Plot on it one negative integer, one positive integer, zero, one negative decimal and one fraction you have converted to a decimal, labelling each point with both its original form and its decimal value. Above the line, write the definitions of integer and whole number and shade the part of the line the whole numbers occupy. Below the line, write one pair of negative numbers with both inequalities comparing them, and beside that pair write the wrong answer the digit instinct would give. Finally, in the margin, write the single sentence that settles every comparison in this lesson.
The shaded region for the whole numbers should start at zero and run rightwards, and it should include the zero mark itself. If your shading starts at one, check the definition again.
Recap
Five things, and the third one is the one that has to overwrite an instinct rather than fill a gap.
| If the question says | Your first move is |
|---|---|
| Graph these numbers | Read each sign before counting units |
| Which is greater | Ask which is further right |
| Write two inequalities | Say the same fact from each side |
| Graph a fraction | Divide to get a decimal first |
| Order from least to greatest | Split by sign, then order each group |
Lesson 2.2 stays on the same line and asks a different question about it: not which side a number is on, but how far from zero it sits. That distance has its own name and its own notation, and it is what makes speed different from velocity.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.1 The Real Number Line §2.1, pp. 65-70 — everything on these slides traces back here
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