Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
Grade 7
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evaluate expressions that involve whole numbers and decimals, including expressions
that contain brackets, using order of operations;

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add and subtract fractions with simple like and unlike denominators, using a variety of
tools (e.g., fraction circles, Cuisenaire rods, drawings, calculators) and algorithms;
demonstrate, using concrete materials, the relationship between the repeated
addition of fractions and the multiplication of that fraction by a whole number
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determine, through investigation, the relationships among fractions, decimals,
percents, and ratios
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solve problems that involve determining whole number percents, using a variety of
tools (e.g., base ten materials, paper and pencil, calculators)
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Operational Sense
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Proportional
Relationships
Grade 8
represent, compare, and order decimals to hundredths and fractions, using a variety of
tools (e.g., number lines, Cuisenaire rods, base ten materials, calculators);
select and justify the most appropriate representation of a quantity (i.e., fraction,
decimal, percent) for a given context
divide whole numbers by simple fractions and by decimal numbers to hundredths,
using concrete materials (e.g., divide 3 by ½ using fraction strips; divide 4 by 0.8 using
base ten materials & estimation)
use a variety of mental strategies to solve problems involving the addition and
subtraction of fractions and decimals (e.g., use the commutative property: use the
distributive property: 16.8 ÷ 0.2 can be thought of as (16 + 0.8) ÷ 0.2 = 16 ÷ 0.2 + 0.8 ÷
0.2, which gives 80 + 4 = 84);
solve problems involving the multiplication and division of decimal numbers to
thousandths by one-digit whole numbers, using a variety of tools (e.g., concrete
materials, drawings, calculators) and strategies (e.g., estimation, algorithms); using
fraction strips; divide 4 by 0.8 12 using base ten materials and estimation);
solve multi-step problems arising from real-life contexts and involving whole numbers
and decimals, using a variety of tools (e.g., concrete materials, drawings, calculators)
and strategies (e.g., estimation, algorithms);
use estimation when solving problems involving operations with whole numbers,
decimals, and percents, to help judge the reasonableness of a solution
Quantity
Relations
hips

Grade: 7/8 Timeline:

represent, compare, and order rational numbers (i.e., positive and
negative fractions and decimals to thousandths);
translate between equivalent forms of a number (i.e., decimals,
fractions, percents
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solve multi-step problems arising from real-life contexts and
involving whole numbers and decimals, using a variety of tools (e.g.,
graphs, calculators) and strategies (e.g., estimation, algorithms
use estimation when solving problems involving operations with
whole numbers, decimals, percents, integers, and fractions, to help
judge the reasonableness of a solution;
multiply and divide decimal numbers by various powers of ten
(e.g.,“To convert 230 000 cm3 to cubic metres, I calculated in my
head 230 000 ÷ 106 to get 0.23 m3.”)
solve problems involving addition, subtraction, multiplication, and
division with simple fractions;
represent the multiplication and division of fractions, using a variety
of tools and strategies (e.g., use an area model to represent ¼
multiplied by 1/3)
solve problems involving percents expressed to one decimal place
(e.g., 12.5%) and whole-number percents greater than 100 (e.g.,
115%)
solve problems involving percent that arise from real-life contexts
(e.g., discount, sales tax, simple interest)
Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
Grade: 7/8 Timeline:
Big Ideas (Marian Small, Big Ideas from Dr. Small Gr. 4-8 pp. 42 & 61)
Fractions:
1. Fractions can represent parts of regions, parts of sets, parts of measures, division or rations. These meanings are equivalent (…)
2. A fraction is not meaningful without knowing what the whole is.
3. Renaming fractions is often the key to comparing them or computing with them. Every fraction can be renamed in an infinite number of ways.
4. There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
5. Operations with fractions have the same meanings as operations with whole numbers, even though the algorithms differ.
Decimals:
1. Decimals are an alternative representation to fractions, but one that allows for modeling, comparisons, and calculations that are consistent with whole
numbers, because decimals extend the pattern of the base ten place value system.
2. A decimal can be read and interpreted in different ways; sometimes one representation is more useful than another in interpreting or comparing decimals or
for performing and explaining a computation.
Culminating Task / Problem:
Gr. 7: Students choose a recipe (of a three layered cake or jar) and find out how much they need, etc. to make to serve the class.
Gr. 8: Students find the area of each section / colour of a flag. (BW Math Common Assessment 2006-2007)
Formative Problems / Tasks Related to Culminating task:
Vocabulary
Numerator
Value
Hundredths
Day
6.
7.
1
8.
Denominator
Representation
Thousandths
Product
Equivalent
Rounding
Big Idea &
Learning Goal
Big Idea –
1 - Fractions can represent parts of regions, parts
of sets, parts of measures, division or rations.
These meanings are equivalent
Factor
Common
Minds On
Students are given a
fraction and are asked to
represent it as many ways
as they can (one per sticky
note).
Dividend
Models
Divisor
Percent
Quotient
Fraction
Action
Group of 4 – How could show the common
methods for representing fractions?
Students work with a group (4) to organize
their post notes in different categories.
Sum
Decimal
Difference
Place value
Consolidation
Congress / Gallery Walk How do you know these are the same?
Why do we need different
representations?
What connections can you make to real
Proportion
Tenths
Assessment
Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
9.
3.
4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
life?
Focus – different ways to represent
(pictures, decimals, fractions, percent)
Learning Goal – We are to represent proportions
in different ways.
Summarize – Different types of
representations can represent the same
amount
Congress – Choose 1 -3 of the most
“controversial” representations. Have
students defend their choice of different
representations in the different contexts
Big Idea1 - Decimals are an alternative representation to
fractions, but one that allows for modeling,
comparisons, and calculations that are consistent
with whole numbers, because decimals extend the
pattern of the base ten place value system.
Big Idea –
10. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
4.
3
4
Students brainstorm to
determine where do they
see decimals, fractions &
percents in real life
Learning Goal – We are learning to represent
proportions in different ways. This is important
because some representations are better for
specific contexts (fractions, decimals, percents)
2
Grade: 7/8 Timeline:
Students are given a piece
of paper with a value on
it. They must find their
equivalent value.
2 - A decimal can be read and interpreted in
different ways; sometimes one representation is
more useful than another in interpreting or
comparing decimals or for performing and
explaining a computation.
Learning Goal – We are learning to compare &
order proportions in different formats (percent,
decimal, fraction, etc.). This is important because
all formats are used in real life.
Big Idea –
11. 3 - Renaming fractions is often the key to
Here are some real life examples of when we
see proportions in our daily lives. How might
you decide when a specific representation is
better?
Examples: cooking / baking, shopping
(discounts, prices = don’t tell students),
gratuities / tipping, banking, weight, tools,
sports statistics, sharing, party planning, grades
Prompt students to get the words
“operations, comparing, visualizing”
At the end of the action – have students justify
their choice for ____ (whichever will be the
focus of consolidation)
Decimals – better for performing
operations
Percent – better for comparing to a whole
amount
Fractions – better for visualizing a
quantity
Prompting questions:
- What real life context can you connect to?
- Where have you seen this?
- Why is this the best choice?
- What makes it better than the other
choices?
- What would using the other choices
possibly look like?
How would you order these proportions?
Justify your strategy
Students are given a copy of the values given
to the class. Students must order them in order
of least to greatest.
Summarize –
Reflection –
Present students
with 3 pictures –
“Choose a picture
to represent in
three different
ways. Explain &
justify which is the
most effective
representation.”
Examples:
1. Map of
Canada
2. Mona Lisa
3. Bar of Music
4. Sports field
Bansho - strategy for comparing values
(i.e. drawing pictures, converting to
equivalent fractions, converting to
percents, converting to decimals)
Summarize – Strategies for:
Convert to decimals
Convert to fractions
Draw pictures
Convert to percent
Review previous day’s
activities – Ask “Why do
Provide students with manipulatives etc.
Choose one of the fractions and show as many
Bansho
Exit Card – Choose
a fraction that we
Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
comparing them or computing with them. Every
fraction can be renamed in an infinite number of
ways.
12. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
we need to compare
fractions?” Brainstorm /
Turn & Talk
equivalent values ½, 2/5, 7/8, 3/4, 7/10
Turn & talk – explain to
your partner how to find a
common denominator to
compare fractions.
Practice working with comparing fractions
using different strategies, especially common
denominators
Demonstration / Video –
Use a measuring scoop (1
cup) to add water to a
larger measuring cup (4
cupper).
Partner:
Gallery Walk
Option 1: How is adding fractions like adding
whole numbers? How is it different?
Summarize –
Learning Goal – We are learning to compare
fractions with different denominators.
5
Big Idea –
13. 3 - Renaming fractions is often the key to
comparing them or computing with them. Every
fraction can be renamed in an infinite number of
ways.
14. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
Grade: 7/8 Timeline:
Summarize different strategies for
comparing fractions
1. Common denominators *** focus on this strategy to prepare
for + and - fractions
2. Fraction strips
3. Number line
4. Counters
5. Grids
haven’t talked
about today. Show
two methods for
finding an
equivalent
fraction.
Hand in work
sheet
Learning Goal – We are learning to compare
fractions with different denominators
Big Idea –
15. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
6
Learning Goal – We are learning to add &
subtract fractions with uncommon
denominators
Turn & Talk: How would
you represent this
mathematically?
Do the reverse: How to
represent?
7
Big Idea –
16. 3 - Renaming fractions is often the key to
comparing them or computing with them. Every
fraction can be renamed in an infinite number of
ways.
17. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
Learning Goal – We are learning to add &
1.
Option 2: How is subtracting fractions like
subtracting whole numbers? How is it
different?
Prompts:
Would an example help you?
What part of the question can you compare?
How would you find the answer for these
questions?
Game:
Student A creates a mixed
number or improper
fraction, Student B writes
the equivalent mix
number / improper
fraction. Then switch and
continue (T chart)
Explain why 1 ½ + 1 1/3 has to be between 2 ¾
and 3.
Answer: Answers might have
fractions if +/- fractions, but will
always be whole if whole
number
2. Operations – subtracting is
taking away, adding is putting
together (doesn’t matter if
whole or fractions)
3. Models – can still use number
lines, pictures, strips/counters
Bansho/Congress/ Gallery Walk
Summarize: Different strategies for
combining fractions
1.
2.
Add whole numbers, then add
fractions with common
denominators
Turn to improper fractions &
add
Exit Card / Math
Journal:
Would the same
strategies we
found for adding
fractions work for
subtracting
fractions? Explain
with an example.
Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
subtract fractions with uncommon
denominators
Grade: 7/8 Timeline:
3.
4.
Draw pictures – is it precise?
Use manipulatives
Discuss most efficient way to achieve the
calculations
8
Big Idea –
18. 3 - Renaming fractions is often the key to
comparing them or computing with them. Every
fraction can be renamed in an infinite number of
ways.
19. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
Practice Day.
Mini Quiz
Learning Goal – We are learning to add &
subtract fractions with uncommon
denominators
Big Idea –
20. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
9
Which picture best
represents multiplying?
Turn & talk to your
partner. Share for a few
minutes.
Learning Goal – We are learning to multiply a
whole number and a fraction
How is multiplying a fraction and a whole
number like multiplying two whole numbers?
How is it different?
Gallery Walk
Prompts:
Would an example help you?
What part of the question can you compare?
How would you find the answer for these
questions?
1.
Summarize –
2.
3.
Big Idea –
21. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
10
Learning Goal – We are learning to
multiplying two fractions
What fraction of the
whole square is shaded?
What other fractions can
you show by colouring
different parts?
How would you represent half of a half? Would
it be the same for a half of a quarter?
Answer: When multiplying +ve
whole numbers, product is
bigger, when multiplying with a
fraction, product is smaller than
the whole number
Operations – repeated addition,
groups, “of”
Models – can still use number
lines, pictures, strips/counters
Bansho –
1. Money
2. Array (Area Model)
3. Music (?)
4. Math sentence ( ½ x ½ = ¼)
5. Pictorial (other than Area)
Summarize
1. Array – good visual
(demonstrate with two colours);
helps to understand the
meaning of multiplication
2. Sentence – organizing
information, quick & efficient
Reflection –
Consider the two
strategies of
multiplying
fractions. When
might you use the
different
strategies?
Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
Big Idea –
22. 1 - Fractions can represent parts of regions, parts
of sets, parts of measures, division or rations.
These meanings are equivalent (…)
23. 2 - A fraction is not meaningful without knowing
what the whole is.
24. 3 - Renaming fractions is often the key to
11
comparing them or computing with them. Every
fraction can be renamed in an infinite number of
ways.
What are the different
ways can you write 3 as a
fraction?
How many different numbers can fit in these
boxes to make it true?
 x1 =3
 4 4
Grade: 7/8 Timeline:
Summarize
1.
2.
3.
Learning Goal – We are learning to use a
denominator of 1 for a whole number when it is
useful in operations.
Big Idea –
25. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
A whole number can have a
denominator of 1
¾ could be in lowest terms
(lowest equivalent fraction)
Strategy for looking at the
product compared to the factors
(product is bigger than factor B,
so factor A must be bigger than
product)
Practice Day – Multiplying Fractions
Mini Quiz
12
Learning Goal – We are learning to multiply
fractions effectively & efficiently in every day
mathematical problems.
Big Idea –
26. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
13
Learning Goal – We are learning to divide
whole numbers by fractions
Big Idea –
27. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
with whole numbers.
14
28.
Watermelon
demonstration
How many of a simple fraction are in a total of
5 watermelons? Justify your answer.
What mathematical
operation can be
demonstrated with this
watermelon. Demonstrate
suggestions.
Prompts
Simple fraction – what would be a simple
fraction to you?
How can you show your thinking?
What mathematics are involved here?
How many different ways
can we divide our class?
Does the order of the fractions in a division
question matter? Justify your answer with an
example.
Learning Goal – We are learning do divide
fractions by fractions
Big Idea –
29. 4 - There are multiple models and/or procedures
for comparing and computing fractions, just as
15
with whole numbers.
Practice day
Gallery Walk
Summarize –
1. Dividing means putting into
groups
2. Quotient of a division question
with fractions is larger than both
the dividend & divisor
3. Invert & multiply is an efficient
strategy for dividing fractions
Reflection – What
does dividing by a
fraction mean?
Provide an
example.
Summarize
1. Invert & multiply is an efficient
strategy for dividing fractions
2. The order of a division questions
change the meaning of the
question
Mini Quiz
Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
Grade: 7/8 Timeline:
Learning Goal – We are learning to divide
fractions effectively & efficiently in every day
mathematical problems.
Summative Assessment – Fraction Flag (See Bluewater Math Common Assessment, Grade Eight 2006 – 2007, Task #5)
DECIMALS
Day
Big Idea &
Learning Goal
BIG IDEA- Decimals are an alternative
16
representation to fractions, but one that
allows for modeling, comparisons, and
calculations that are consistent with whole
numbers, because decimals extend the
pattern of the base ten place value system.
Learning Goal - We are learning to represent
numbers using place value.
BIG IDEA- Decimals are an alternative
representation to fractions, but one that
allows for modeling, comparisons, and
calculations that are consistent with whole
numbers, because decimals extend the
pattern of the base ten place value system.
17
Learning Goal – We are learning to use the place
value system to represent different decimal
numbers.
Minds On
Provide students with the
names of the different
place values.
How would you arrange
these values to make a
complete place value
chart?
Where would the decimal
go?
Which number is greater?
0.34 or 0.43
Action
How can you arrange these gas prices to show
what the range of gas prices is across the
globe?
Prompting Questions:
How do you know that ___ is greater than ___?
Does this show how you have compared?
Do you need all gas prices to show a range?
Are there areas to group? (geographically,
frequency etc.)
Students are given Place Value Score Sheet
(Appendix ___)
One deck of cards (10, K, Q removed) per pair.
Player A flips first card, and chooses which
place value to place it in (Ones, Tenths, etc.).
0.34 or 0.3
J are wild cards
Why is it greater?
Which is greater?
.021 or .2
Player B flips a card, and places her numeral in
any place value.
Play continues until all place values are filled
in.
Why is it greater?
Students complete 3 rounds (Battles) and add
their three numbers to find the total – who
won.
Assessment
Highlight:
Bansho (or Congress?):
Methods to compare gas prices
Place value
Number line
Charts
Graphs?
Summarize:
Place value to compare
numbers – look at the numeral
in the place value to compare
(greater, lesser)
Play Place Value War:
Why is it greater?
Which number is greater?
Consolidation
Exit Card:
Highlight
How did place value play a role
in this game?
How did the Wild Cards affect
the game?
Summarize:
Greater numerals in bigger place
values result in bigger numbers.
What if the object
of the game was
to have the lowest
final score?
Describe how that
would affect your
strategy.
Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
Grade: 7/8 Timeline:
After playing, pairs answer the question on
chart paper?
-
BIG IDEA- Decimals are an alternative
representation to fractions, but one that
allows for modeling, comparisons, and
calculations that are consistent with whole
numbers, because decimals extend the
pattern of the base ten place value system.
Learning Goal – We are learning to use the place
value system to represent different decimal
numbers.
Does the game depend more on
strategy or luck? How do you know?
Guess My Number:
Teacher chooses a
number (e.g. 21.684)
What number on the number line does this dot
represent?
Option #1
Students use math
vocabulary from day
before (place value) to ask
Yes/No questions of the
teacher to try and guess
the number.
13.45
13.46
Option #2
1.3
1.4
18
Highlight:
Exit Card:
Gallery Walk (choose 2 or 3 student
answers):
Is your dot worth more 13.453?
Why or why not?
Is it worth more than 13.4580?
Why or why not?
How did you figure out the
value the dot represents?
How do you know that there has
to be more than one reasonable
answer to the question
How many decimal
numbers are
between 1.2 and
1.2? How do you
know?
Prompting Questions
19
Big Idea - A decimal can be read and
interpreted in different ways; sometimes
one representation is more useful than
another in interpreting or comparing
decimals or for performing and explaining a
computation.
Learning Goal – We are learning to add and
subtract decimals without using a calculator.
Use the base 10 blocks to
show how to add
Show how you can use base 10 blocks to
complete:
345 + 138
Option 1:
3.45 + 1.38
And to subtract
Option 2:
3.12 – 1.78
312-178
On chart paper
Summarize:
Decimals can go on and on
forever (infinite)
Place value doesn’t end on the
whole side, or decimal side
Relate to PI (?)
Some decimals will repeat (i.e.
1/9)
How to represent repeating
decimals (bar, dot)
Gallery Walk
Highlight:
How does adding/ subtracting
decimals relate to
adding/subtracting whole
numbers?
Did the Base 10 blocks help?
How do they relate whole
numbers/ decimals (hundreds
becomes the whole)
Summarize:
Exit Card:
Provide example &
fill in the blank.
I can add
decimals….
Confidently,
sometimes, with
assistance, not at
all.
Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
Grade: 7/8 Timeline:
-
20
Big Idea - A decimal can be read and
interpreted in different ways; sometimes
one representation is more useful than
another in interpreting or comparing
decimals or for performing and explaining a
computation.
Learning Goal – We are learning to add and
subtract decimals without using a calculator
Big Idea - A decimal can be read and
interpreted in different ways; sometimes
one representation is more useful than
another in interpreting or comparing
decimals or for performing and explaining a
computation.
21
22
Learning Goal – We are learning to multiply
and divide decimals using mental math.
Big Idea - A decimal can be read and
interpreted in different ways; sometimes
one representation is more useful than
another in interpreting or comparing
decimals or for performing and explaining a
computation.
Learning Goal – We are learning to multiply
decimals by relating it to whole numbers.
Regrouping still occurs in both.
Line up the decimals for adding
/ subtracting.
You can use estimating to help
add decimals
I can subtract
decimals…
Confidently,
sometimes, with
assistance, not at
all.
Practice Day
How could you interpret
5 x 2.3
What happens to a number when you multiply
by a multiple of 10? When you divide by a
multiple of 10?
Prompt:
Visually
In life
Mathematically
Compare to whole
numbers
What happens to a number when you multiply
by 0.1, 0.01, 0.001….
What happens to a number when you divide by
0.1, 0.01, 0.001…
Why do we estimate?
When do we estimate?
If you have
12.8 x 4.4
How could you estimate the product?
Highlighting:
What patterns did you see in
multiplying and dividing?
How does multiplying fractions
compare to multiplying
decimals?
…. Dividing…?
Summarize:
When you divide by a decimal,
the number gets bigger
When you multiply by a decimal
the number gets smaller
You can simply move the digits
when multiplying or dividing by
units of 10.
Highlight:
Round factors to estimate the
product
Rounding numbers up or down
Relate to multiplying whole
numbers
Summarize
Multiply numbers without
decimal to get “target product”
Practice work
multiplying
decimals
(problems in text
or TIPS).
Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & Percents
Grade: 7/8 Timeline:
-
Big Idea - A decimal can be read and
interpreted in different ways; sometimes
one representation is more useful than
another in interpreting or comparing
decimals or for performing and explaining a
computation.
What does dividing mean?
Choose 2 numbers (one 2 digit, 1 one digit).
What does dividing by a
fraction mean?
How can you show where the decimal is in the
dividend and the divisor affects the placement
of the decimal in the quotient?
What might dividing by a
decimal mean?
23
Learning Goal – We are learning to divide
decimals by relating it to whole numbers.
24
25
Practice day: multiplying
& dividing decimals
Culminating Task – Pancakes for Poverty – See attached.
Prompting:
What if there were not decimals?
What if one number had a decimal?
What if the decimal was in a different
spot?
Can we use estimation?
How do we divide whole numbers?
What do you know about the answer
before you do any computations?
Use the product of the rounded
numbers to estimate product
Add decimal to the “target
product” to get close to the
estimated product
Highlight:
Relate dividing concept to whole
numbers
Using estimation to check the
reasonableness of an answer
Long division strategies
Summarize
Making the numbers easier to
manage, we can use mental
math strategies to find quick
answers
If the divisor is > 1, quotient is
smaller than dividend
If the divisor is <1, quotient is
bigger than dividend.