Mathematics Unit Strand(s) & Areas: NS & Numeration: Fraction, Decimals & PercentsGrade: 7/8 Timeline:

Grade 7 / Grade 8
Quantity Relationships /
  • represent, compare, and order decimals to hundredths and fractions, using a variety of tools (e.g., number lines, Cuisenairerods, base ten materials, calculators);
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  • represent, compare, and order rational numbers (i.e., positive and negative fractions and decimals to thousandths);

  • select and justify the most appropriate representation of a quantity (i.e., fraction, decimal, percent) for a given context
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  • translate between equivalent forms of a number (i.e., decimals, fractions, percents

Operational Sense /
  • 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 ÷ 0.2can 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) andstrategies (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);
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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, and percents, to help judge the reasonableness of a solution
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  • use estimation when solving problems involving operations with whole numbers, decimals, percents, integers, and fractions, to help judge the reasonableness of a solution;

  • evaluate expressions that involve wholenumbers and decimals, including expressions that contain brackets, using order of operations;
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  • 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.”)

  • 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;
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  • solve problems involving addition, subtraction, multiplication, and division with simple fractions;

  • demonstrate, using concrete materials, the relationship between the repeated addition of fractions and the multiplication of thatfraction by a whole number
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  • 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)

Proportional Relationships /
  • determine, through investigation, the relationships among fractions, decimals, percents, and ratios
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  • 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 that involve determining whole number percents, using a variety of tools (e.g., base ten materials, paper andpencil, calculators)
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  • solve problems involving percent that arise from real-life contexts (e.g., discount, sales tax, simple interest)

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 / Denominator / Product / Factor / Dividend / Divisor / Quotient / Sum / Difference / Proportion
Value / Representation / Equivalent / Common / Models / Percent / Fraction / Decimal / Place value / Tenths
Hundredths / Thousandths / Rounding
Day / Big Idea &
Learning Goal / Minds On / Action / Consolidation / Assessment
1 /
  1. Big Idea –
  2. 1 - Fractions can represent parts of regions, parts of sets, parts of measures, division or rations. These meanings are equivalent
  3. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
Learning Goal – We are to represent proportions in different ways. / Students are given a fraction and are asked to represent it as many ways as they can (one per sticky note). / 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. / Congress / Gallery Walk -
How do you know these are the same?
Why do we need different representations?
What connections can you make to real life?
Focus – different ways to represent (pictures, decimals, fractions, percent)
Summarize – Different types of representations can represent the same amount
2 / Big Idea-
  1. 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.
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) / Students brainstorm to determine where do they see decimals, fractions & percents in real life / 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
At the end of the action – have students justify their choice for ____ (whichever will be the focus of consolidation)
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? / Congress – Choose 1 -3 of the most “controversial” representations. Have students defend their choice of different representations in the different contexts
Prompt students to get the words “operations, comparing, visualizing”
Summarize –
Decimals – better for performing operations
Percent – better for comparing to a whole amount
Fractions – better for visualizing a quantity / 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

3 / Big Idea –
  1. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
  1. 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 & orderproportions in different formats (percent, decimal, fraction, etc.). This is important because all formats are used in real life. / Students are given a piece of paper with a value on it. They must find their equivalent value. / 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. / 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
4 / Big Idea –
  1. 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.
  2. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
Learning Goal – We are learning to compare fractions with different denominators. / Review previous day’s activities – Ask “Why do we need to compare fractions?” Brainstorm / Turn & Talk / Provide students with manipulatives etc. Choose one of the fractions and show as many equivalent values ½, 2/5, 7/8, 3/4, 7/10 / Bansho
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
/ Exit Card – Choose a fraction that we haven’t talked about today. Show two methods for finding an equivalent fraction.
5 / Big Idea –
  1. 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.
  2. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
Learning Goal – We are learning to compare fractions with different denominators / 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 / Hand in work sheet
6 / Big Idea –
  1. 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 & subtract fractions with uncommon denominators / Demonstration / Video – Use a measuring scoop (1 cup) to add water to a larger measuring cup (4 cupper).
Turn & Talk: How would you represent this mathematically?
Do the reverse: How to represent? / Partner:
Option 1: How is adding fractions like adding whole numbers? How is it different?
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? / Gallery Walk
Summarize –
  1. 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

7 / Big Idea –
  1. 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.
  2. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
Learning Goal – We are learning to addsubtract fractions with uncommon denominators / 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. / Bansho/Congress/ Gallery Walk
Summarize: Different strategies for combining fractions
  1. Add whole numbers, then add fractions with common denominators
  2. Turn to improper fractions & add
  3. Draw pictures – is it precise?
  4. Use manipulatives
Discuss most efficient way to achieve the calculations / Exit Card / Math Journal:
Would the same strategies we found for adding fractions work for subtracting fractions? Explain with an example.
8 / Big Idea –
  1. 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.
  2. 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 & subtract fractions with uncommon denominators / Practice Day. / Mini Quiz
9 / Big Idea –
  1. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
Learning Goal – We are learning to multiplya whole number and a fraction / Which picture best represents multiplying? Turn & talk to your partner. Share for a few minutes. / How is multiplying a fraction and a whole number like multiplying two 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? / Gallery Walk
Summarize –
  1. Answer: When multiplying +ve whole numbers, product is bigger, when multiplying with a fraction, product is smaller than the whole number
  2. Operations – repeated addition, groups, “of”
  3. Models – can still use number lines, pictures, strips/counters

10 / Big Idea –
  1. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
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? / 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?
11 / Big Idea –
  1. 1 - Fractions can represent parts of regions, parts of sets, parts of measures, division or rations. These meanings are equivalent (…)
  2. 2 - A fraction is not meaningful without knowing what the whole is.
  3. 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.
Learning Goal – We are learning to use a denominator of 1 for a whole number when it is useful in operations. / 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?
 x 1 = 3
 4 4 / Summarize
  1. A whole number can have a denominator of 1
  2. ¾ could be in lowest terms (lowest equivalent fraction)
  3. Strategy for looking at the product compared to the factors (product is bigger than factor B, so factor A must be bigger than product)

12 / Big Idea –
  1. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
Learning Goal – We are learning to multiply fractions effectively & efficiently in every day mathematical problems. / Practice Day – Multiplying Fractions / Mini Quiz
13 / Big Idea –
  1. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
Learning Goal – We are learning to divide whole numbers by fractions / Watermelon demonstration
What mathematical operation can be demonstrated with this watermelon. Demonstrate suggestions. / How many of a simple fraction are in a total of 5 watermelons? Justify your answer.
Prompts
Simple fraction – what would be a simple fraction to you?
How can you show your thinking?
What mathematics are involved here? / 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.
14 / Big Idea –
  1. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
Learning Goal – We are learning do divide fractions by fractions / 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. / Summarize
  1. Invert & multiply is an efficient strategy for dividing fractions
  2. The order of a division questions change the meaning of the question

15 / Big Idea –
  1. 4 - There are multiple models and/or procedures for comparing and computing fractions, just as with whole numbers.
Learning Goal – We are learning to divide fractions effectively & efficiently in every day mathematical problems. / Practice day / Mini Quiz
Summative Assessment – Fraction Flag (See Bluewater Math Common Assessment, Grade Eight 2006 – 2007, Task #5)
DECIMALS
Day / Big Idea &
Learning Goal / Minds On / Action / Consolidation / Assessment
16 / 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 represent numbers using place value. / 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? / How can you arrange these gas prices to show what the range of gas pricesis 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.) / 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)
17 / 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. / Which number is greater?
0.34 or 0.43
Why is it greater?
Which number is greater?