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Fractions, decimals, percentages | Stage 4 | Mathematics | Year 7 What percentage of your household's grocery bill is spent on you?

Summary of Substrands Duration: 4 weeks 2 days

S3 Fractions and Decimals (Review)

S4 Fractions, Decimals and Percentages (part)

Start Date:

Completion Date:

Teacher and Class:

Outcomes

Mathematics K-10

●MA4-1WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

●MA4-2WM applies appropriate mathematical techniques to solve problems

●MA4-3WM recognises and explains mathematical relationships using reasoning

●MA4-5NA operates with fractions, decimals and percentages

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Overview Key Words Suggested Assessment

Throughout this unit students will investigate the use of fractions, decimals and percentages. There are many situations in everyday life where things, amounts or quantities are measured in fractions, decimals and percentages or parts of a whole such as shopping, telling the time and cooking.

Fraction, equivalent, mixed numerals, denominator, numerator, proper, improper, reciprocal, simplify, multiple, convert, decimal, decimal point, tenths, hundredths, quantity, approximation, terminating, recurring, percentage, irrational number, root, pi, increase, decrease, rounding, unitary method, percentage, estimate

● Pretest on Stage 3 Fractions and Decimals● Open ended questions such as ‘Write down

everything you know about ½ ‘● Give the students an answer and ask them to write

what the question might have been

Generally, teachers should design specific assessment tasks that can be drawn from a variety of the following sources of information and assessment strategies:• student responses to questions, including open ended questions,• student explanation and demonstration to others,

• questions posed by students,

• samples of student work,

• student produced overviews or summaries of topics,

• investigations or projects,

• students oral and written report

• practical tasks and assignments,

• short quizzes

• pen and paper tests, including multiple choice, short answer questions and questions requiring longer responses, including interdependent questions ( where one answer depends on the answer obtained in the preceding part)

• open book tests

• comprehension and interpretation exercise

• student produced worked samples,

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• teacher/student discussion or interviews

• observation of students during learning activities including the student’s correct use of terminology

• observation of a student participating in a group activity

Rego Content Teaching, learning and assessment Resources

● Stage 4 - Fractions, Decimals and PercentagesStudents:

Compare fractions using equivalence; locate and represent positive and negative fractions and mixed numerals on a number line (ACMNA152)

■ determine the highest common factor (HCF) of numbers and the lowest common multiple (LCM) of numbers

■ generate equivalent fractions

The content of this unit is building upon the Stage 3 work on Fractions and Decimals. Give students a pretest in order to plan the unit effectively.

● Brainstorm the difference between the factors and multiples of a number. Use examples to show the meaning of highest common factor (HCF) and lowest common multiple (LCM).

● Use a real life example to introduce equivalent fractions. For example:

Jilly cuts a pizza into two equal pieces. Each piece is exactly half of the pizza. Her son wants to eat some pizza so she decides to cut one of the halves into two smaller, equal pieces. Now the son eats two quarters of the pizza and Jilly eats half a pizza. Students compare the amount that each person has eaten.

● Fraction file on Smartboard● Fraction maker on

Smartboard● Fractions/decimals/

percentages file on Smartboard

● http://www.bbc.co.uk/ bitesize/ks3/maths/number/fractions/revision/1/(BBC bitesize)

● http://www.mathsisfun.com/ fractions-menu.html

● http:// www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/years7_10/teaching/frac.htm

● http://www.bbc.co.uk/ bitesize/ks3/maths/number/fractions/revision/3/

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■ write a fraction in its simplest form

■ express improper fractions as mixed numerals and vice versa

● Discuss two ways of finding equivalent fractions:

1. Multiplying or dividing both the numerator and the denominator by the same number as long as the number is not zero. For example:

2. Drawing diagrams. For example:

Equivalent fractions for ❑❑ are shown below.

Relate the simplifying of fractions to equivalent fractions. For

example ❑❑

is the same as ❑❑● .● Show examples, starting with smaller numerators and

denominators. Lead students to discover that dividing both the numerator and denominator by the HCF will minimise the steps.

Adjustment

NLVM: equivalent fraction manipulative

● http://nlvm.usu.edu/en/nav/ frames_asid_105_g_3_t_1.html?from=category_g_3_t_1.html

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■ place positive and negative fractions, mixed numerals and decimals on a number line to compare their relative values● interpret a given scale to

determine fractional values represented on a number line (Problem Solving)

● choose an appropriate scale to display given fractional values on a number line, eg when plotting thirds or sixths, a scale of 3 cm for every whole is easier to use than a scale of 1 cm for every whole (Communicating, Reasoning)

To extend, some students can simplify fractions with larger numerators and denominators.

● Use diagrams to introduce the concept first. For example, if you divide pizzas into quarters, how many pizzas will make up 11 quarters?

Conversely, if you have 15 quarters, how many pizzas will you have?

● Introduce the number line with whole numbers first. Discuss placing fractions and decimals on the number line. Start with a scale with missing fractions such as:

Construct a number line using string or wire in the classroom and divide it into simple multiples. Extend it into other multiples. For

example, start with a number line with a scale of ❑❑● and then extend to quarters.

AdjustmentAsk students to plot a number on the number line that is not a multiple of the denominator given. For example on the number line

above ‘Where would you plot ❑❑ ‘?

Fraction Tutorial


● String or wire● Fractions cards

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● Solve problems involving addition and subtraction of fractions, including those with unrelated denominators (ACMNA153)

■ add and subtract fractions, including mixed numerals and fractions with unrelated denominators, using written and calculator methods

● Revise the addition and subtraction of fractions with the same denominator. Use diagrams to assist if necessary.

● For fractions with different denominators, use equivalent fractions to change into the same denominator or use the method below:

Point out that this method may not be the shortest method but it will always give the correct solution.

● With mixed numerals, show two methods:

1. Changing both fractions to improper fraction. For example:

Fraction Quiz Program● http://www.bbc.co.uk/

bitesize/ks3/maths/number/fractions/revision/7/

NLVM: Adding fractions interactive.

● http://nlvm.usu.edu/en/nav/ frames_asid_106_g_3_t_1.html?from=category_g_3_t_1.html

● Magnetic Fractions

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● recognise and explain incorrect operations with fractions, eg explain why

(Communicating, Reasoning)

● interpret fractions and mixed numerals on a calculator display (Communicating)

■ subtract a fraction from a whole number using mental, written and calculator methods,

eg

2. Adding the whole numbers first and then adding the fractional parts. For example:

● Demonstrate the steps for adding and subtracting fractions on the calculator. Get students to check their answers for non-calculator questions on the calculator.

● Students could use pictorial methods or other methods of adding and subtracting to explain incorrect methods.

● Students could represent problems using diagrams or placing whole number over one and using a written method.

AdjusmentShow struggling students how to use calculators to add subtract and reduce fractions

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● Multiply and divide fractions and decimals using efficient written strategies and digital technologies (ACMNA154)

■ determine the effect of multiplying or dividing by a number with magnitude less than one

■ multiply and divide decimals by powers of 10

■ multiply and divide decimals using written methods, limiting operators to two digits

● compare initial estimates with answers obtained by written methods and check by using a calculator (Problem Solving)

■ multiply and divide fractions and mixed numerals using written methods● demonstrate multiplication of a

fraction by another fraction using a diagram to illustrate the process (Communicating, Reasoning)

● explain, using a numerical example, why division by a fraction is equivalent to multiplication by its reciprocal (Communicating, Reasoning)

● Discuss examples where a fraction or decimal is multiplied by a number less than one. Is the solution greater or smaller?

● Multiply decimals by powers of 10 using written methods. Lead students to find a rule to multiply by powers of 10.

● Show students written methods to multiply and divide decimals. Discuss estimates as a way to check answers.

Use written methods of multiplying fractions supported by diagrams.

Start with simple fractions such as ❑❑❑❑

and ❑❑❑❑

● Extend to mixed numerals.

Show students how to use common factors to simplify fractions. Discuss the advantages of using this method.

● Explain how to find the reciprocal of a fraction, including proper fractions and mixed numerals.

● Use an example to show why the division of a fraction is


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■ multiply and divide fractions and decimals using a calculator

■ calculate fractions and decimals of quantities using mental, written and calculator methods● choose the appropriate

equivalent form for mental computation, eg 0.25 of $60 is

equivalent to of $60, which is equivalent to $60 ÷ 4 (Communicating)

equivalent to multiplication by its reciprocal. For example:

Jilly has a half of one of her pizzas left. Her son asked her how many quarters there were in her half. Jilly cut the pizza so that it was clear that one half divided into quarters gave two pieces or two quarters.

❑❑

❑❑

The answer could also have been found by changing the division sign to a multiplication sign and tipping the second fraction upside down.

❑❑

❑❑

❑❑

❑❑

● Demonstrate the steps for multiplying and dividing fractions using the calculator. Students could then practice on the calculator after solving problems using a written method.

● Demonstrate various examples writing the quantity with a denominator of 1.

● Discuss how the word of can mean the same as multiply. Use an example like the following:

Jilly might take ❑❑ of an hour (60 minutes) to cook the pizzas. A third of 60 minutes is

the same as ❑❑❑❑

.

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● Express one quantity as a fraction of another, with and without the use of digital technologies (ACMNA155)

■ express one quantity as a fraction of another● choose appropriate units to

compare two quantities as a fraction, eg 15 minutes is

of an hour (Communicating)

● Demonstrate problems using the same units first such as:

‘What fraction of 10 is 3?’.

● Extend this to quantities with different units such as:

‘What fraction of 1.5 kg is 700g?’.

Adjustment

● Pose problems such as ‘What fraction of 1.5 kg is 900 mg?’.

Adjustment (Fractions in general)

● For extension, students could explore Egyptian fractions. As well as expressing words and whole numbers with hieroglyphics, the ancient Egyptians devised a way of expressing fractions. They placed a ‘mouth’ (pronounced ‘er’) over their symbols for whole numbers. For example:

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● Explain how fractions are used in reading music. Students may look at things such as the lengths of crotchets, quavers and minims. Time signatures also use fractions.

● Round decimals to a specified number of decimal places (ACMNA156)

■ round decimals to a given number of decimal places

■ use symbols for approximation, eg or

● Discuss the usefulness of rounding. For example, if $128.768765 is displayed on a calculator, the answer would be meaningless as we only want two numbers after the decimal point.

.

AdjustmentWhat number might round to 2.7? What is the smallest number it could be?

Use calculators fixed decimal function for struggling students

● http://www.mathsisfun.com/ decimals-menu.html

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● Investigate terminating and recurring decimals (ACMNA184)

■ use the notation for recurring (repeating) decimals, eg ,

,

■ convert fractions to terminating or recurring decimals as appropriate● recognise that calculators may

show approximations to recurring decimals, and explain why,

eg displayed as (Communicating, Reasoning)

● Show students how to change a fraction to a terminating decimal. For example:

❑❑

can be calculated as follows:

● Then introduce fractions that will result in a recurring or repeating decimal. For example:

❑❑

can be calculated as follows:

❑❑❑̇❑̇

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● Connect fractions, decimals and percentages and carry out simple conversions (ACMNA157)

■ classify fractions, terminating decimals, recurring decimals and percentages as 'rational' numbers, as they can be written in the form

where and are integers and

■ convert fractions to decimals (terminating and recurring) and percentages

■ convert terminating decimals to fractions and percentages

■ convert percentages to fractions and decimals (terminating and recurring)● evaluate the reasonableness of

statements in the media that quote fractions, decimals or percentages, eg 'The number of children in the average family is 2.3' (Communicating, Problem Solving)

■ order fractions, decimals and percentages

Define Rational numbers

Show that fractions decimals and percentages are all forms of rational numbers

Create tables such as:

Fractions Decimals %

1/2 0.5 50%

1/4 0.25 25%

Use the Aust. Bureau of Statistics and media advertising to discuss misleading and reasonable statements.

Use number lines to order fractions, decimals and percentages

AdjustmentUse the calculator to look for patterns when changing fractions to recurring decimals. Start with fractions with a denominator of 3 and 9.

More confident students can explore algebraic methods of changing recurring decimals to fractions. Start with more simple decimals such as ❑̇

Australian Bureau of Statistics


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● Investigate the concept of irrational numbers, including (ACMNA186)

■ investigate 'irrational' numbers, such as and ● describe, informally, the

properties of irrational numbers (Communicating)

Students can investigate the concept of

● by measuring the diameter and circumference of various circles.

Introduce the

● button on the calculator.

● Variety of circles (eg tin cans, glue stick, protractor)

● String● The pi song (utube)

http://www.youtube.com/watch?v=eDiSYp_51iY(different versions available)

● Find percentages of quantities and express one quantity as a percentage of another, with and without the use of digital technologies (ACMNA158)

■ calculate percentages of quantities using mental, written and calculator methods● choose an appropriate

equivalent form for mental computation of percentages of quantities, eg 20% of $40 is

equivalent to × $40, which is equivalent to $40 ÷ 5 (Communicating)

■ express one quantity as a percentage of another, using mental, written and calculator methods, eg 45 minutes is 75% of an hour

● Show the calculation of percentages of quantities using the fraction, decimal and calculator methods.

● Firstly, revise expressing one quantity as a fraction of another and changing a fraction to a percentage.

● Percentage calculator on Smartboard

● http://www.mathsisfun.com/ percentage-menu.html

● http://www.bbc.co.uk/ bitesize/ks3/maths/number/percentages/revision/3/

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● Solve problems involving the use of percentages, including percentage increases and decreases, with and without the use of digital technologies (ACMNA187)

■ increase and decrease a quantity by a given percentage, using mental, written and calculator methods● recognise equivalences when

calculating percentage increases and decreases, eg multiplication by 1.05 will increase a number or quantity by 5%, multiplication by 0.87 will decrease a number or quantity by 13% (Reasoning)

■ interpret and calculate percentages greater than 100, eg an increase from $2 to $5 is an increase of 150%

■ solve a variety of real-life problems involving percentages, including percentage composition problems and problems involving money● interpret calculator displays in

formulating solutions to problems involving percentages by appropriately rounding decimals (Communicating)

● use the unitary method to solve problems involving percentages, eg find the original value, given the value after an

● Start by exploring increasing and decreasing quantities by a given percentages in two steps.

● Encourage students to solve problems in one step. For example, when increasing a quantity by 7% you can multiply by 107% or 1.07 and decreasing by 7% is multiplying by 93% or 0.93.

● At this point explore increasing a quantity by a percentage greater than 100%.

● Discuss inflation, commission, holiday loading● Present examples of profit and loss in sales

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increase of 20% (Problem Solving)

● interpret and use nutritional information panels on product packaging where percentages are involved (Problem Solving)

● interpret and use media and sport reports involving percentages (Problem Solving)

interpret and use statements about the environment involving percentages, eg energy use for different purposes, such as lighting (Problem Solving)

● Ask students to collect a variety of food packages. Calculations can be made about the percentage of the product that contains fat, sugars, protein, carbohydrates and energy.Discuss the DI’s (daily intakes) for the various components.

● Students could also focus on a fast food company and the nutritional information they publish.

● Students could keep a diary about the usage of resources in their home such as electricity and water.‘Who uses the most water in your family?’Comparisons could be made with their classmates and other cultures.

● Food packages

● Newspapers, magazines

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Evaluation Adjustment

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