Statistics and Probability for RIT under 215: reading and creating line plots with fractional scales, telling values apart from frequencies, and answering total, most-common and difference questions.
Subject: NWEA MAP Growth Math · 60 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
MAP Growth · Statistics and Probability · RIT under 215
Reading, creating and interpreting line plots whose scale is measured in fractions
Objectives
One MAP skill sits here, but it combines data handling with everything you know about fractions.
The one idea: a line plot is a number line with one dot per measurement, and the dots stack up where values repeat.
Section
Part 1
Concept
A line plot is a number line with a dot placed above each measured value. Repeated values stack.
Figure (svg): A line plot with whole number labels and stacks of dots above each
Line plot — A number line with one dot for each data value, stacked where values repeat. Also called a dot plot.
Concept
How many dots sit above a value tells you how often that value occurred.
Figure (svg): A line plot with one stack highlighted and its count read off
Frequency — How many times a particular value appears in the data.
Warm-up
Retrieve before being taught.
Discussion prompt
If a line plot has four dots above the number 3, what does that tell you?
Hint: One dot means one measurement.
Answer:
That the value 3 was measured four times.
Notice you did not need to be taught that — a line plot is designed to be readable on sight, which is exactly why it is used.
Pattern
Three steps, whatever the question asks.
Step one is the one students skip, and it is where the fraction scales cause trouble.
Intuition
Nothing about the plot changes when the labels become fractions.
Figure (svg): A line plot with the scale marked in quarters and dots stacked above several of them
A dot above three quarters means one measurement of three quarters, exactly as a dot above 3 means one measurement of 3.
Socratic
A list of numbers holds the same information.
Discussion prompt
What does a line plot show you at a glance that a list of measurements does not?
Hint: Think about what you notice first when you look at each.
Answer:
The shape of the data: which values are common, which are rare, and how spread out everything is.
A list of twenty numbers has to be read carefully; a line plot of the same twenty is understood in a second because repetition becomes height.
Definition probe
Get the vocabulary secure before reading data.
Sort into buckets
Sort each feature by what it represents.
Explain it to yourself
Separating value from frequency is the whole reading skill.
Discussion prompt
In your own words, what is the difference between the label under a stack and the number of dots in it?
Answer:
The label is what was measured — a length, a weight, an amount.
The number of dots is how many times that measurement happened. Confusing the two is the most common error in reading a plot.
Section
Part 2
Worked example
A line plot of pencil lengths shows 2 dots above 1/4, 3 above 2/4, 4 above 3/4 and 1 above 4/4. How many pencils were measured?
Figure (svg): A line plot with one stack highlighted and its count read off
Read the scale
Why: The scale is in quarters of an inch.
Count every dot
Why: 2 plus 3 plus 4 plus 1.
Add them
Why: The total is 10 dots.
\[ 2 + 3 + 4 + 1 = 10 \text{ pencils} \]
Verify: what was counted
Why: Every dot is one pencil, so the total number of dots is the number of pencils — not the total length, which would be a different calculation.
Intuition
These are two different questions, and MAP asks both.
Figure (svg): A line plot with each stack multiplied by its value to find a total
Worked example
Using the same plot, what length occurred most often?
Figure (svg): A line plot with one stack highlighted and its count read off
Find the tallest stack
Why: The stack above 3/4 has 4 dots, more than any other.
Read its label
Why: That stack sits above 3/4.
\[ \text{most common length} = \frac{3}{4} \text{ inch} \]
Verify: the answer is a value
Why: The answer is 3/4 of an inch, not 4. The question asked which length, so the answer must be a length rather than a count.
Trap
Asked which length was most common, a student answers 4, because the tallest stack has 4 dots.
The question asks for a length, so the answer must come from the scale, not from the dot count. The answer is 3/4 of an inch.
Check what kind of thing the question wants: a value comes from the labels, a count comes from the dots.
Prediction
Decide what kind of answer the question wants.
Predict first
A plot has 5 dots above 1/2 and 2 dots above 3/4. Which length was measured most often?
Correct: 1/2
The word length in the question tells you the answer must be a measurement, not a count.
Why: The tallest stack has 5 dots and sits above 1/2, so the most common length is 1/2. The 5 is how often it happened, not what was measured.
Discrimination
Sorting the question type first prevents the commonest error.
Sort into buckets
Sort each question by what kind of answer it needs.
Faded example
A plot has stacks of 3, 5, 2 and 4 dots.
Fill in the blanks
3 + 5 + 2 + 4 = 14
Why: Every dot represents one measurement, so adding all the stack heights gives 14 items measured in total.
Check
Solve it on paper before you click.
Check your understanding
A line plot shows 1 dot above 1/4, 4 dots above 2/4 and 2 dots above 3/4. How many measurements were taken?
Answer: A
Why: Each dot is one measurement, so add the stack heights: 1 plus 4 plus 2 is 7 measurements in total.
Error analysis
A student was asked the most common weight on a plot.
Annotate
On: \( \text{tallest stack has 6 dots above } \tfrac{1}{2} \;\Rightarrow\; \text{answer: 6} \)
Read the noun in the question — weight, length, amount — and make sure your answer is that kind of thing.
Analogy
A line plot is one of a family of displays you meet constantly.
Match the pairs
Why: A line plot sits between a tally and a bar chart: it keeps each measurement visible as its own dot while still showing the shape of the data at a glance.
Notation
The label under the line tells you what is being measured.
Annotate
On: \( \text{length in inches} \)
Read the axis label before the data — it tells you what kind of thing every answer should be.
Section
Part 3
Concept
Choosing the scale is the first decision when creating a plot, and the first thing to check when reading one.
Figure (svg): Two line plot scales compared, one in halves and one in quarters
If any measurement is in quarters, a scale marked only in halves cannot show it.
Intuition
The steps along a line plot are equal, exactly as on a number line.
Figure (svg): A line plot with the scale marked in quarters and dots stacked above several of them
So a scale in quarters runs 0, 1/4, 2/4, 3/4, 4/4 — not 0, 1/4, 1/2, 3/4, 1 with uneven gaps.
Worked example
Measurements are 1/2, 3/4, 1/4, 1/2 and 1. What scale should the line plot use?
Figure (svg): Two line plot scales compared, one in halves and one in quarters
Look at the denominators
Why: The values use halves and quarters.
Find a denominator that fits all
Why: Quarters works, since halves can be written as quarters.
Rewrite everything in quarters
Why: 1/2 becomes 2/4, and 1 becomes 4/4.
\[ \tfrac{2}{4}, \tfrac{3}{4}, \tfrac{1}{4}, \tfrac{2}{4}, \tfrac{4}{4} \]
Verify: every value has a tick
Why: All five values now land exactly on a quarter mark, so the scale holds the whole data set with nothing falling between ticks.
Prediction
Look at the denominators.
Predict first
Measurements are 1/2, 1/4, 3/8 and 5/8. What scale should the plot use?
Correct: eighths
Pick the smallest piece present in the data — that becomes the step size.
Why: The values use halves, quarters and eighths. Only eighths can hold all of them, since 1/2 is 4/8 and 1/4 is 2/8 but 3/8 cannot be written in quarters.
Sorting
Find the smallest piece in each set.
Sort into buckets
Sort each data set by the scale it needs.
The scale is set by the smallest piece in the data, never by the largest value.
Trap
For measurements including 3/8, a student draws a scale in quarters.
3/8 falls between 1/4 and 1/2, so it has no tick to sit on. The scale must be in eighths.
Check every value against the scale before plotting. If one falls between ticks, the scale is too coarse.
Faded example
Rewrite in quarters so everything lands on a tick.
Fill in the blanks
\tfrac2___ = \tfrac___}___
Why: Multiplying top and bottom by 2 turns a half into two quarters, which lands exactly on the second tick of a quarter scale.
Check
Solve it on paper before you click.
Check your understanding
Measurements are 1/4, 1/2, 3/4 and 1. Which scale fits them all with no value between ticks?
Answer: A
Why: The smallest piece in the data is a quarter, and every other value can be written in quarters: 1/2 is 2/4 and 1 is 4/4. So quarters holds all four.
Section
Part 4
Concept
Creating a plot is a fixed procedure once the scale is chosen.
Figure (svg): A line plot with the scale marked in quarters and dots stacked above several of them
Worked example
Ribbon lengths are 1/4, 2/4, 2/4, 3/4, 3/4, 3/4 and 4/4 metres. Draw the line plot.
Figure (svg): A line plot with the scale marked in quarters and dots stacked above several of them
Choose the scale
Why: All values are in quarters, so the scale runs in quarters.
Label the line
Why: Mark 1/4, 2/4, 3/4 and 4/4 at equal spacing.
Place the dots
Why: One above 1/4, two above 2/4, three above 3/4, one above 4/4.
Check the total
Why: 1 plus 2 plus 3 plus 1 is 7 dots.
\[ 1 + 2 + 3 + 1 = 7 \text{ ribbons} \]
Verify: against the data
Why: There were seven measurements listed and seven dots plotted, so nothing has been lost or double-counted.
Intuition
Counting your dots and comparing with the data list is the single best check on a plot you have drawn.
Figure (svg): A line plot with whole number labels and stacks of dots above each
A mismatch means a value was missed or plotted twice, and it is much easier to catch now than later.
Ranking
Put the procedure in order.
Put in order
Why: The scale must be chosen before the line is drawn, or values will not land on ticks. Checking last is what catches a missed measurement.
Step zero
One decision governs everything that follows.
Discussion prompt
You have a list of fractional measurements to plot. What must you decide before drawing the line?
Hint: Look at the denominators before touching a pencil.
Answer:
The scale — specifically the smallest piece present in the data, which becomes the step size.
Drawing the line first and choosing the scale afterwards almost always means redrawing it when a value falls between ticks.
Check
Solve it on paper before you click.
Check your understanding
Data: 1/2, 1/2, 1/4, 3/4, 1/2. How many dots go above 1/2 on the plot?
Answer: A
Why: The value 1/2 appears three times in the list, so three dots stack above the 1/2 mark. The other two measurements go above 1/4 and 3/4.
Error analysis
A student plotted the data 1/4, 1/2, 1/2, 3/4 on a scale of halves.
Annotate
On: \( \text{scale: } 0, \tfrac{1}{2}, 1 \quad \text{but data includes } \tfrac{1}{4}, \tfrac{3}{4} \)
Always set the scale from the smallest piece present, not from the values that happen to be easiest.
Real world
Any repeated measurement produces one naturally.
Discussion prompt
A class measures the length of their pencils to the nearest quarter inch. Why is a line plot a better display than a list of names and lengths?
Answer:
Because the question is usually about the group, not about individuals: what is typical, what is unusual, how spread out are they.
A line plot answers all three at a glance, while a list of names would have to be sorted first.
Invariant
However the plot is redrawn, one number does not change.
Step through it
What is identical across all three lines?
The total dot count is invariant and always equals the number of measurements. That is why counting dots is the standard check on a plot.
Section
Part 5
Concept
Finding a total amount means multiplying each value by its frequency, then adding the results.
Figure (svg): A line plot with each stack multiplied by its value to find a total
This is where the fraction addition from earlier decks comes back.
Worked example
A plot shows 2 dots above 1/4, 3 above 2/4 and 1 above 3/4. What is the total length?
Figure (svg): A line plot with each stack multiplied by its value to find a total
Multiply each value by its frequency
Why: Two lots of 1/4 is 2/4; three lots of 2/4 is 6/4; one lot of 3/4 is 3/4.
Add the results
Why: The denominators already match, so add the numerators: 2 plus 6 plus 3 is 11.
Simplify
Why: 11/4 is 2 and 3/4.
\[ \frac{2}{4} + \frac{6}{4} + \frac{3}{4} = \frac{11}{4} = 2\tfrac{3}{4} \]
Verify: the size is sensible
Why: There are six measurements, each under one unit, so a total under six is expected — and 2 and 3/4 fits comfortably.
Intuition
Questions about how much longer or shorter are subtractions between two values on the scale.
Figure (svg): A line plot with one stack highlighted and its count read off
The frequencies play no part in a difference question — only the labels do.
Worked example
On the same plot, how much longer is the longest ribbon than the shortest?
Figure (svg): A line plot with one stack highlighted and its count read off
Find the extremes
Why: The largest value with a dot is 4/4 and the smallest is 1/4.
Subtract
Why: 4/4 minus 1/4 is 3/4.
\[ \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \]
Verify: only values were used
Why: The dot counts were never involved, because a difference in length depends only on the two lengths themselves.
Prediction
Decide before calculating.
Predict first
To find how much longer the longest item is than the shortest, what do you use?
Correct: only the two extreme values on the scale
Frequencies matter for totals and for most-common questions, but never for a range or a difference.
Why: A difference in length compares two lengths, so it uses the leftmost and rightmost values that have dots. How many dots sit above them is irrelevant.
Matching
Four common question types, four different methods.
Match the pairs
Why: Two of these use only the dots, one uses only the labels, and one uses both together. Identifying which before calculating is most of the skill.
Faded example
Three dots above 1/4 and two above 3/4.
Fill in the blanks
3 \times \tfrac9___ + 2 \times \tfrac______ = \tfrac______ + \tfrac______ = \tfrac___}___
Why: Three quarters plus six quarters is nine quarters, which is 2 and 1/4 in total.
Check
Solve it on paper before you click.
Check your understanding
A plot shows 2 dots above 1/2 and 3 dots above 1/4. What is the total amount?
Answer: A
Why: Two lots of 1/2 is 1, and three lots of 1/4 is 3/4. Adding gives 1 and 3/4 in total.
Elimination
Some answers can be discarded on size alone.
Eliminate the wrong options
A plot has 4 dots, all above values under 1. Which could be the total amount?
Survives elimination: e1
Why: Four measurements each under 1 must total somewhere between 0 and 4, so only 2 and 1/2 is possible. Bounding the answer rules out three options with no arithmetic.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about line plots is false?
Survives elimination: t2
Why: The tallest stack tells you the most common value, not the largest. The largest value is the rightmost one that has any dots above it, however few.
Explain it
A classmate keeps answering with the stack height.
Discussion prompt
How would you fix this for them?
Answer:
Get them to underline the noun in the question — length, weight, how many — before looking at the plot.
If the noun is a measurement, the answer comes from the labels. If it is how many, the answer comes from the dots. Underlining makes the choice automatic.
Missing information
Not every plot question can be answered.
Discussion prompt
A line plot has dots but no labels along the line. What questions can you still answer, and what can you not?
Answer:
You can still say how many measurements there were, and which value was most common relative to the others.
You cannot give any actual measurement, total or difference, because those all need the scale. Without labels the plot shows shape but no quantities.
Estimation
A rough total checks your exact answer.
Predict first
A plot has 8 dots, all above values between 1/4 and 3/4. Roughly what is the total?
Correct: about 4
Multiplying the dot count by a typical value is a fast way to bound the answer.
Why: Eight measurements averaging about 1/2 each gives roughly 8 times 1/2, which is about 4. The true total must lie between 2 and 6.
Commit first
Decide your answer and your confidence before revealing.
Predict first
A plot shows 1 dot above 1/4, 2 above 2/4 and 1 above 4/4. What is the total amount?
Correct: 2 1/4
Note that 9/4 and 2 1/4 are the same number — MAP may list either form.
Why: One quarter, plus two lots of 2/4 which is 4/4, plus 4/4 gives 1/4 plus 4/4 plus 4/4, which is 9/4, or 2 and 1/4.
Exit ticket
One item that tells you whether the deck landed.
Predict first
A line plot shows 3 dots above 1/2 and 1 dot above 3/4. How many items were measured, and which length was most common?
Correct: 4 items, and 1/2 was most common
Why: There are 3 plus 1, which is 4 dots and therefore 4 items. The tallest stack has 3 dots and sits above 1/2, so 1/2 is the most common length.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw a line plot with a scale in quarters and at least three stacks of dots. Label which part gives a value and which gives a frequency, and write the four question types with the method for each beside them.
If you can draw this from memory, every line plot item becomes a reading exercise followed by ordinary fraction arithmetic.
Recap
One skill, but it draws on everything.
Underline the noun in the question first: if it names a measurement, answer from the scale; if it says how many, answer from the dots.
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