Extend and Use Properties for RIT under 215: place value in whole numbers and decimals, standard and expanded form with decimals and fractions, reading decimal models, and composing and decomposing decimals in multiple ways.
Subject: NWEA MAP Growth Math · 61 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
MAP Growth · The Real and Complex Number Systems · RIT under 215
Standard and expanded form, decimal columns, and composing numbers in more than one way
Objectives
Seven MAP skills sit here, and all of them come from one chart.
The one idea: every column is ten times the column to its right, and that never stops being true.
Section
Part 1
Concept
The digit 3 can mean 3, or 30, or 300, or 3,000. What decides is the column it is written in.
Figure (svg): A place value chart for the number four thousand three hundred eighty two
Place value — The value a column carries. From the right: ones, tens, hundreds, thousands.
So the value of a digit is the digit multiplied by its column value.
Concept
Expanded form writes out what each digit is worth; standard form is the compact version.
Figure (svg): The number four thousand three hundred eighty two broken into four parts and added back together
Converting between them is just adding up, or splitting apart.
Warm-up
Retrieve before being taught.
Discussion prompt
In the number 5,271, what is the 2 worth? And the 5?
Hint: Say the value, not the digit.
Answer:
The 2 is worth 200 and the 5 is worth 5,000.
Notice you did not say two and five. Naming the value rather than the digit is exactly what these MAP items reward.
Pattern
Three steps, whether the number has a decimal point or not.
Counting out from the ones column, rather than from the left, is what keeps you right when a decimal point appears.
Socratic
It is worth knowing why this earns so much curriculum time.
Discussion prompt
Why can we write any number at all using only ten digits?
Hint: Think about how many different symbols you actually know.
Answer:
Because the position carries information as well as the digit. The same ten symbols mean different amounts in different columns.
Without place value you would need a new symbol for every quantity, which is exactly the problem Roman numerals have.
Definition probe
The digit is the same each time; only the column changes.
Sort into buckets
Sort each number by what the 7 in it is worth.
Worked example
Write 4,382 in expanded form.
Figure (svg): The number four thousand three hundred eighty two broken into four parts and added back together
Name each column
Why: From the left: thousands, hundreds, tens, ones.
Multiply each digit by its column
Why: 4 thousands is 4,000; 3 hundreds is 300; 8 tens is 80; 2 ones is 2.
Join with plus signs
Why: Write the four values added together.
\[ 4382 = 4000 + 300 + 80 + 2 \]
Verify: by adding back
Why: 4,000 plus 300 is 4,300, plus 80 is 4,380, plus 2 is 4,382 — the number we started with.
Faded example
Two terms are given; supply the rest.
Fill in the blanks
6259 = 6000 + 200 + 50 + 9
Why: The number is 6,259. The 2 sits in the hundreds column so it is worth 200, and the 9 sits in the ones column so it is worth 9.
Trap
Asked what the 3 is worth in 4,382, a student answers three.
The 3 sits in the hundreds column, so it is worth 300. The question asks for the value, not the digit.
Reading the column name aloud before answering — three hundreds — makes the value obvious.
Check
Solve it on paper before you click.
Check your understanding
In the number 8,507, what is the value of the digit 5?
Answer: A
Why: Counting from the ones column, the 7 is ones, the 0 is tens and the 5 is hundreds. So the 5 is worth 5 times 100, which is 500.
Ranking
The same digit, four different columns.
Put in order
Why: Each step to the right divides the value by ten, so the values are 6,000, then 600, then 60, then 6 — a factor of a thousand between the first and last.
Explain it to yourself
Knowing why it exists makes it much easier to write.
Discussion prompt
In your own words, what does expanded form show that standard form hides?
Answer:
It shows what each digit is actually worth, which standard form leaves you to work out from position.
That is why it is useful when learning to add and subtract — the values you are combining are written out explicitly.
Analogy
You already use place value fluently when handling cash.
Match the pairs
Why: Money is a decimal place value system with a pound as the whole. Ten pence is a tenth of a pound and a penny is a hundredth, which is exactly why prices are written with two decimal places.
Section
Part 2
Concept
Moving right, each column is one tenth of the one before. That does not stop at the ones column.
Figure (svg): A place value chart extending past the decimal point into tenths and hundredths
| column | value | as a fraction |
|---|---|---|
| tenths | 0.1 | 1/10 |
| hundredths | 0.01 | 1/100 |
| thousandths | 0.001 | 1/1000 |
Intuition
A hundred square makes a two-place decimal countable.
Figure (svg): A hundred square with thirty seven cells shaded to show zero point three seven
Each small square is one hundredth, and each full column of ten is one tenth.
Worked example
A hundred square has 37 squares shaded. What decimal is illustrated?
Figure (svg): A hundred square with thirty seven cells shaded to show zero point three seven
Name what one square is worth
Why: The grid is one whole cut into 100, so each square is one hundredth.
Count the shaded squares
Why: 37 squares are shaded.
Write it as a decimal
Why: 37 hundredths is written 0.37.
\[ 0.37 = \frac{37}{100} \]
Verify: by columns
Why: Three full columns of ten make 3 tenths, and 7 singles make 7 hundredths, giving 0.37 — the same answer read a different way.
Prediction
The chart works exactly as it did for whole numbers.
Predict first
In the number 36.47, what is the 7 worth?
Correct: 7 hundredths
Always count outward from the decimal point, never from the end of the number.
Why: Counting right from the decimal point, the 4 is tenths and the 7 is hundredths. So the 7 is worth 7 hundredths, or 0.07.
Definition probe
Locating the column is the whole skill.
Sort into buckets
In the number 5.284, sort each digit by its column.
Trap
Asked what the 4 is worth in 0.04, a student answers forty, because it looks like the tens column.
Past the decimal point the columns shrink. The 4 is in the hundredths column, so it is worth 4 hundredths, or 0.04.
Columns get bigger to the left of the point and smaller to the right. The point is the mirror, and the ones column is the centre.
Check
Solve it on paper before you click.
Check your understanding
In the number 12.605, what is the value of the digit 6?
Answer: A
Why: The first digit after the decimal point is the tenths column, and that is the 6. So it is worth 6 tenths, or 0.6.
Error analysis
A student was asked to read 0.5 and 0.05 aloud.
Annotate
On: \( 0.5 = \text{five hundredths}, \quad 0.05 = \text{five tenths} \)
Counting columns outward from the point, one at a time, prevents this swap entirely.
Step zero
One habit removes every column-naming error.
Discussion prompt
You need the value of a digit in 20.407. Where do you start counting, and in which direction?
Hint: Find the landmark that never moves.
Answer:
Start at the decimal point and count rightwards one column at a time: tenths, hundredths, thousandths.
Never count from the end of the number, because trailing digits shift everything. The point is the fixed landmark.
Notation
Models come in more than one form, and the key changes with them.
Annotate
On: \( \text{grid of }100 \;\Rightarrow\; \text{each cell} = 0.01 \)
The number of pieces in the whole is the denominator, and that is what fixes the decimal place.
Section
Part 3
Concept
Expanded form works past the decimal point exactly as it did before it.
Figure (svg): The decimal three point four seven written as a sum of ones, tenths and hundredths
Each digit is written as its own value, and the values are added.
Worked example
Write 3.47 in expanded form.
Figure (svg): The decimal three point four seven written as a sum of ones, tenths and hundredths
Name each column
Why: 3 is ones, 4 is tenths, 7 is hundredths.
Write each digit's value
Why: 3 is 3; 4 tenths is 0.4; 7 hundredths is 0.07.
Join with plus signs
Why: Add the three values.
\[ 3.47 = 3 + 0.4 + 0.07 \]
Verify: by adding back
Why: 3 plus 0.4 is 3.4, plus 0.07 is 3.47 — the number we started with.
Intuition
MAP also asks for expanded form written with fractions rather than decimals.
Figure (svg): The decimal three point four seven written as a sum of ones, tenths and hundredths
A tenth is 1/10 and a hundredth is 1/100, so 3.47 is 3 plus 4/10 plus 7/100.
Worked example
Write 3.47 in expanded form using fractions.
Figure (svg): A place value chart extending past the decimal point into tenths and hundredths
Name each column as a fraction
Why: Ones is 1, tenths is 1/10, hundredths is 1/100.
Multiply each digit by its column
Why: 3 times 1, 4 times 1/10, 7 times 1/100.
\[ 3.47 = 3 + \frac{4}{10} + \frac{7}{100} \]
Verify: against the decimal version
Why: 4/10 is 0.4 and 7/100 is 0.07, so this is the same sum as before written another way.
Translation
Every decimal column has a fraction twin.
Match the pairs
Why: The number of decimal places tells you the denominator: one place is tenths, two is hundredths, three is thousandths. The last pair shows that a trailing zero does not change the value.
Fill the middle
Write 5.62 with fractions.
Fill in the blanks
5.62 = 5 + \frac100___ + \frac______}
Why: The 2 sits in the hundredths column, so it is 2 hundredths, written 2/100. The 6 is in tenths, giving 6/10.
Check
Solve it on paper before you click.
Check your understanding
Which is 4.09 in expanded form?
Answer: A
Why: The tenths column holds a 0 and the hundredths column holds the 9, so the only non-zero parts are 4 and 0.09.
Comparison
Each row is one number in three forms.
Comparison matrix
| standard | expanded (decimals) | expanded (fractions) |
|---|---|---|
| 2.5 | 2 + 0.5 | 2 + 5/10 |
| 0.68 | 0.6 + 0.08 | 6/10 + 8/100 |
| 7.03 | 7 + 0.03 | 7 + 3/100 |
The three columns say the same thing; MAP simply asks for whichever one it feels like.
Elimination
Three of these misplace a digit.
Eliminate the wrong options
Which is the correct expanded form of 6.05?
Survives elimination: x1
Why: The 0 sits in tenths and the 5 in hundredths, so the only non-zero parts are 6 and 5 hundredths, giving 6 plus 0.05.
Real world
Money and measurement both punish a misplaced column.
Discussion prompt
A price is written as 4.5 pounds. Why might someone reading it as four pounds fifty be right, and reading it as four pounds five be wrong?
Answer:
The 5 is in the tenths column, so it is five tenths of a pound, which is fifty pence.
Reading it as five pence would put the 5 in hundredths, which would be written 4.05. The column, not the digit, carries the meaning.
Section
Part 4
Concept
The obvious split is not the only one. MAP asks for the unusual splits deliberately.
Figure (svg): Two different decompositions of the number one point three
1.3 is 1 plus 0.3, but it is also 13 tenths, and also 0.9 plus 0.4.
Worked example
Write 1.3 in three different ways.
Figure (svg): Two different decompositions of the number one point three
The column way
Why: 1 whole plus 3 tenths, so 1 plus 0.3.
The all-in-one-column way
Why: One whole is 10 tenths, so altogether there are 13 tenths.
A regrouped way
Why: 0.9 plus 0.4 also makes 1.3.
\[ 1.3 = 1 + 0.3 = 13 \times 0.1 = 0.9 + 0.4 \]
Verify: all three add correctly
Why: 13 tenths is 1.3, and 0.9 plus 0.4 is 1.3, so all three descriptions name the same number.
Intuition
Seeing 1.3 as 13 tenths is exactly what makes subtraction with borrowing possible.
Figure (svg): Two different decompositions of the number one point three
A number that can only be seen one way is a number you cannot regroup.
Sorting
Several look different but name the same number.
Sort into buckets
Sort each expression by whether it equals 2.4.
Recognising many forms of the same number is what the compose-and-decompose skill is testing.
Prediction
Converting a whole into a column is the key move.
Predict first
How many tenths are there in 3.5?
Correct: 35
This is the same regrouping used when borrowing in decimal subtraction.
Why: Each whole contains 10 tenths, so 3 wholes are 30 tenths. Adding the 5 tenths already there gives 35 tenths in total.
Faded example
Express 4.2 entirely in tenths.
Fill in the blanks
4.2 = 42 \text___
Why: Four wholes are 40 tenths, and adding the 2 tenths already present gives 42 tenths altogether.
Check
Solve it on paper before you click.
Check your understanding
Which of these is NOT equal to 0.6?
Answer: A
Why: 0.06 has the 6 in the hundredths column, making it six hundredths, which is ten times smaller than 0.6. The other three all equal six tenths.
Error analysis
A student wrote 2.07 in expanded form.
Annotate
On: \( 2.07 = 2 + 0.7 \)
A zero in a decimal column is not a placeholder to ignore — it fixes where every later digit sits.
Section
Part 5
Concept
Every skill in this deck is a different question asked of the same place value chart.
Figure (svg): A place value chart extending past the decimal point into tenths and hundredths
If you can draw the chart and place the digits, you can answer all seven.
Matching
MAP phrases place-value items in several ways.
Match the pairs
Why: The first two look almost identical but want different answers — one a value and one a name. Reading which is wanted is worth a mark on its own.
Discrimination
The commonest wasted mark on this topic.
Sort into buckets
Sort each answer by what kind of thing it is.
Estimation
Place value alone answers this, without any calculation.
Predict first
Which is larger: 0.4 or 0.35?
Correct: 0.4
More digits does not mean a bigger number — 0.35 has more digits but is smaller.
Why: Compare the tenths first: 0.4 has 4 tenths and 0.35 has only 3. Since the tenths differ, nothing further needs checking, so 0.4 is larger.
Two truths and a lie
Two of these are sound; one is not.
Eliminate the wrong options
Which statement about place value is false?
Survives elimination: t2
Why: 0.35 has more digits than 0.4 but is smaller, because the tenths column decides first. Digit count says nothing about size — only the leftmost differing column does.
Explain it
A classmate thinks the decimal point is the centre of symmetry.
Discussion prompt
They believe the column right of the point should be called oneths. How would you correct this?
Answer:
The centre is the ones column, not the point. Going right from ones you get tenths, hundredths, thousandths.
There is no oneths column because the ones column is already the mirror line. Pointing at the chart settles it faster than any explanation.
Missing information
Not every question is answerable as asked.
Discussion prompt
A problem says: what is the value of the digit 6? What do you need before you can answer?
Answer:
The whole number the 6 sits in, because its value depends entirely on its column.
A digit on its own has no value beyond itself — place value is a property of position, not of the symbol.
Commit first
Decide your answer and your confidence before revealing.
Predict first
How many hundredths are there in 0.4?
Correct: 40
This regrouping is exactly what lets you compare 0.4 with 0.35 by putting both over hundredths.
Why: One tenth is 10 hundredths, so 4 tenths is 4 times 10, which is 40 hundredths. Written out, 0.4 equals 0.40.
Exit ticket
One item that tells you whether the deck landed.
Predict first
In the number 45.68, what is the digit 8 worth?
Correct: 8 hundredths
Why: Counting right from the decimal point, the 6 is tenths and the 8 is hundredths. So the 8 is worth 8 hundredths, or 0.08.
Invariant
Regrouping moves digits between columns without changing the number.
Step through it
What is identical on all three lines?
The value is invariant; only which column holds it changes. That is exactly what makes borrowing legal.
Edge cases
The columns do not stop at thousandths.
Discussion prompt
What would the column after thousandths be called, and what would it be worth?
Hint: Apply the divide-by-ten rule one more time.
Answer:
Ten-thousandths, worth 0.0001, or one ten-thousandth.
The pattern continues forever: each new column is a tenth of the one before. MAP below RIT 215 stops at thousandths, but the chart itself has no end.
Connect it up
Your handwriting, one page, from memory.
Draw it
Draw a place value chart from thousands down to thousandths, with the decimal point marked and the ones column highlighted as the centre. Write one example number across it, and beside it show the same number in expanded form with decimals and with fractions.
If you can draw this from memory, all seven skills in this deck are covered.
Recap
Seven skills, one chart.
A zero inside a decimal is never decoration — it fixes where every digit after it sits.
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