Introductory Ac,vity Process (Review) Checking for basic concepts of frac2ons and Language Resource: Divide an A4 sheet of paper into two equal parts along the longest length. Cut to separate into two strips. • Ask students to fold the paper into four parts. • Ask students to use the second strip and show a different
fold that has four equal parts? Ask can you fold this strip to show me how much of the paper I would get if I got a quarter of it?
• Two Quarters? • Three quarters of it? • Glue examples into learning journal and demonstrate
fracGonal notaGon Revise how fracGons are wriHen. ExplanaGon of the terms Denominator and Numerator to represent the fracGon. Ask students why these demonstraGon fracGons are called ‘proper’ InteracGve website that assist students with placing fracGons On a pizza hHp://www.mathsisfun.com/numbers/fracGons-‐match-‐words-‐pizza.html On a number line.hHp://www.mathsisfun.com/numbers/fracGons-‐match-‐frac-‐line.html
IdenGfying fracGons on a number line.hHp://www.ixl.com/math/grade-‐3/fracGons-‐on-‐number-‐
lines
Australian Curriculum Year 4 ACMNA077 InvesGgate equivalent fracGons used in contexts. Key Idea Understanding that two fracGons are equivalent when they represent the same amount of the whole, and that there are several ways to represent the same quanGty. Terms and Defini,ons • Denominator – the boHom of a fracGon that
represents the number of equal parts in to which the whole has been divided.
• Numerator – the top number of a fracGon that represents the number of equal fracGon parts.
• Vinculum – the line. • Proper Frac,on – when the numerator is
smaller than the denominator. • Equivalent Frac,ons have the same value, even
though they may look different. • Equivalent Frac,ons represent the same number
or quanGty. (even though they may look different).
Resources • Paper Strips, Coloured pencils /crayons • FracGon Rods, FracGon fans • FISH Kit
Diagnos,c Assessment Have students sit in groups. Give each group several ‘sandwiches’ of the same size drawn on sheets of paper: Ask students to take a half a sandwich each. Note if students: -‐ Accept two of the quarters as half a sandwich. -‐ Think that their piece is a half. -‐ Think that each of the halves has exactly the same amount of bread. Some students have a strong connecGon between half and two and believe that you can only have halves if the whole is divided into exactly two pieces. Students may not see that the rectangle is the same as the triangle which is the same as two quarters …. Guiding QuesGons • What concepts does the student need to be retaught? • What concepts does the student need to pracGce? • What concepts is the student ready to be introduced
to? Ac,vity Process-‐Equivalent Frac,ons Resources: • Coloured pencils, • Three strips of paper.
u Ask students to imagine that they are a chocolate bar. Instruct the children to fold one strip in half (Model) and colour one half.
u Then have students fold the second strip into quarters and the third strip into eighths. (Stress the importance of accuracy).
Ques2ons to explore equivalence: 1. What part of the first strip did you colour? 2. What part of the 2nd and 3rd strip could you colour to
show the same fracGon?
4.1.4 Math Word Wall: frac2ons, part, whole, group, half/halves, thirds, sixth/sixths, quarter/quarters, eighth/eighths, tenth twelAh, numerator, denominators, comparing ( word bank of comparison language), grid/model, dividing , re-‐dividing, sharing, cuFng up , propor2ons., accurate
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DRAFT-‐This is a work in progress. MAG wriGng project 2012-‐2013
u Write concluding statement of equivalence: ½ is the same as….., ¼ is the same as ……,
u Give students the opportunity to discuss and jusGfy their reasoning
Ac,vity Process-‐Frac,on Family Walls With denominators 2,4,8,16 : 3, 6 , 12, 24 : 5, 10 and 100 Provide students with opportuniGes to explore the equivalence of fracGons using paper strips to build a fracGon wall . Provide each student with 5 strips of paper which are equal in length. Have students label one strip ‘one whole ‘ and then fold the other strips into halves, quarters, eighths and sixteenths and label each secGon on each strip with the representaGve fracGon symbol. Have students assemble their frac,on wall from the
whole to the smallest frac,on. Have the students repeat the process for other fracGons whole, thirds, sixths, , twelihs, twelihs whole , fiihs, tenths, twenGeths. combinaGon – whole, half, quarters, sixths, eighths InteracGve fracGon wall demonstraGng equivalence. hHp://56c2011.global2.vic.edu.au/games/fracGon-‐wall/
Extension and Varia,ons (opportunity to work with small groups who might need further instrucGon, pracGce or extension) 1. Number Lines are one model of fracGons
Provide students with strips of equal-‐length card and have them fold / mark it into fracGonal parts. Ask students to use a half, a third, a quarter and three quarters as reference points in order to be able to determine the size of a fracGon, and order and compare the fracGon numbers. Teacher presents fracGon cards to students asking quesGons such as: “Is 5/8 smaller or bigger than a half? What do you know about 4/8 that could help you ? (4/8 is a half). Encourage students to use strategies to order sets of fracGons with unlike numerators and unlike denominators; for example: 2/3: 4/5: 5/6: 9/10. Link to interacGve fracGon wall demonstraGng equivalence. hHp://56c2011.global2.vic.edu.au/games/fracGon-‐wall/ hHp://www.mathsisfun.com/numbers/fracGon-‐number-‐line.html 2. Folding Fold the strips into equivalent fracGons for thirds, fiihs, sixths, ninths and tenths. (AlternaGve equipment: paper rectangles, fracGon cakes, paHern blocks) Record results in learning journals. 3. Rolling Dice Label each face of a die with one of these fracGons 1/4, 2/6, 4/5, 1/6, 2/3, 6/10. Then label each face of the second die with 1/3. 2/12, 3/5, 8/10, 4/6, 2/8. Students take turns to roll the dice. Student records the two fracGons. To score a point the student has to decide whether the fracGons shown are equivalent or not
Ac,vity Process-‐Frac,ons of a Collec,on Provide students with a variety of objects. Have students find different fracGons of a collecGon and idenGfy which result in the same amount and which don’t. Example : Give a bag of 12 marbles, ask students to find a third, then two sixths, then a quarter, then four twelihs. • Have student idenGfy the numerator (how many
marbles are in each fracGon). • Ask why some of the different fracGons resulted in
the same number of objects. Ask students to explain why this happened.
• Have students state comparison statements about their collecGons of objects in terms of why is one half of the marbles more than one third of them.
Ac,vity Process-‐Frac,on Dominoes FracGon Dominoes allows for the consolidaGon of equivalent fracGon knowledge. Students could benefit from having a fracGon wall or fracGon or number line grid to refer to. Encourage the students to give reasons for their decisions. Students may need to refer to a fracGon wall. Source: First steps in Mathema2cs -‐ Number – Understand Frac2onal Numbers, 2010. Rigby: Port Melbourne. p. 133
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iPad Apps: Equivalent Frac6ons (Everyday MathemaGcs – McGraw-‐Hill) Frac6on Monkeys Frac6ons by Brai Screen Capture
A record of learning completed can be kept through screen captures. Thee are 2 methods available via the system • briefly press both the home and power buHon
at the same Gme -‐ the screen should then appear to flash and a picture of it saved to the Camera Roll album
• Tap that transparent icon of the round buDon and watch as it will unfold into dark square with four icons on it. The icon at the right of that square denoted as Device is virtual Device buHon.
4. Dividing. This task uses an area model of fracGons. Students draw a rectangle 6 cm by 4 cm and shade 3/4. The students make six copies. Then they subdivide the various copies “verGcally” to produce a range of equivalent fracGons, which they name. Add to student learning journal. Digital Learning-‐Scootle Cassowary Cost -‐ TLF-‐ID L86 Help a park ranger to arrange fencing in a wildlife sanctuary. Divide common geometric shapes into equal-‐sized secGons for keeping cassowaries. Group the enclosures to form a quaranGne zone for sick and injured birds. Then express divisions of the enclosures as fracGons. Frac6ons: Thirds, Sixths & Twelihs -‐ TLF-‐ID S5127 This interacGve resource is a game in which the student is required to answer quesGons relaGng to fracGons, including equivalent fracGons, and their representaGon. Frac6ons Equivalent: TLF-‐ID L3651 Manipulate a visual representaGon of a fracGon to find and name an equivalent fracGon. This object is one in a series of seven objects.
Frac6on Fiddle: Comparing Unit FracGons. TLF-‐ID L2802 This is also an iPad app Two kiwis each gobble up part of a worm. IdenGfy which bird ate the most. For example, decide whether one-‐third (1/3) is larger than one-‐quarter (1/4). Build the fracGon that each bird ate. Compare the fracGons on a number line. Check which fracGon is bigger. This learning object is one in a series of seven objects. Park Frac6ons: TLF-‐ID L126 Help a town planner to design two site plans for a park. Assign regions on a grid for different uses such as picnic tables, swings, sandpits or ponds. Use this tool to explore how to express fracGons and display them in different ways. Select boxes within the grid and view or enter corresponding fracGons and their equivalents.
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Background
FracGons arise naturally in everyday situaGons involving sharing, cusng up and proporGons. For example, the cup was one-‐third full, three quarters of the class walk to school.
FracGons between 0 and 1 describe parts of a whole.
There are two main ways to represent fracGons. • As markers on a number line. • Shading parts of a square, called an area diagram.
Research has shown that visual learning theory is especially appropriate to the aHainment of mathemaGcs skills for a wide range of learners. Understanding abstract math concepts is reliant on the ability to “see” how they work, and children naturally use visual models to solve mathemaGcal problems. They are oien able to visualize a problem as a set of images. By creaGng models, they interact with mathemaGcal concepts, process informaGon, observe changes, reflect on their experiences, modify their thinking, and draw conclusions. (Rowan & Bourne, 1994) Source:Times FracGon Module hHp://www.amsi.org.au/teacher_modules/fracGons.html
• Tap that virtual Device buDon and find the dark square with six icons on it. One of them pictured by three bright dots is virtual More buHon. • Tap that virtual More buDon and find the dark
square with four icons on it. As its name implies the icon denoted as Screenshot is for virtual buHon to print screen
• Tap that virtual Screenshot buDon to print screen of your iconic tablet
Contexts for Learning
The acGviGes on the Thinking Blocks site provide both guided instrucGon and independent pracGce.
hNp://thinkingblocks.com/tb_frac2ons/frac2ons.html
Real Life Experience: Problem: An area of the school oval measuring 20m x 10m is going to be redeveloped. You need to draw up a plan that shows the following: “One quarter of the area is for an adventure playground; two eighths is for a grassed sea2ng area, and four sixteenths will be developed into a sand play area and the remaining area is for a water play adventure area.
Digital Assessment: TLF-‐ID L9771 For students with a consolidated understanding of equivalent fracGons. Comparing Frac6ons: strategies: Assessment : Test your understanding of fracGons. Decide which one of a pair of fracGons is larger, or if they are the same. Choose which strategy you used to compare the two fracGons. Link to other MAGs Year 2 MAG_2.3.6 FracGon CollecGons Year 3 MAG_3.3.6 FracGons (2)
Draw a plan (model) to match these measurements. Your scale is : 1cm – 1m. Ensure each secGon is Gtled and contains details of its area in cm² and perimeters are also marked in cm. Inves,ga,on:-‐Pose these situaGons to students: SituaGon 1 Andrew said, “Three quarters equals six eighths!” Angela said, ‘Not always, it depends!’ Ask students to: a) explore the equivalence and explain how both students can be right. b) find a way to represent an equal and unequal representaGon using materials of their choice. (Underling idea is that for three quarters to be equivalent to six eighths, the wholes must be the same.) SituaGon 2 Jake said, “Two fourteenths is double one seventh,” Josie said, “No, it isn’t. They are the same size.” Who do you think is right? Draw a diagram to jusGfy your answer, then share the results with your group. Source: First steps in Mathema2cs -‐ Number – Understand Frac2onal Numbers, 2010. Rigby: Port Melbourne. p. 129 & 133.
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