# Ordering proper and improper fractions

Purpose

The purpose of this series of lessons is to develop understanding of the relative size of fractions and where these fit with whole numbers on a number line.

Specific Learning Outcomes
• Understand and use the < and > relationship symbols.
• Explore the relative size of unit fractions.
• Recognise fractions as numbers which ‘fit’ between whole numbers on a number line.
• Consolidate understanding of the relative size of fractions and where they fit with whole numbers on a number line.
• Operate with understanding with proper fractions, improper fractions and mixed numerals.
• Recognise that when comparing the size of fractions the size of the whole matters.
Description of Mathematics

As students have been making equal parts with materials, they have come to recognise that the more equal parts that a region is cut into, the smaller those pieces are.

The concepts of size and order have been implicit rather than explicit. In this unit the focus is on having students work with regional representations so they can make practical comparisons of the size of parts and can picture the fraction in this way. It is a considerable conceptual shift for them to then see the symbol as a number in its own right and to appreciate that these numbers fit within our whole number numeration system. This transition to seeing where fractions fit between whole numbers develops an increasing awareness and appreciation of some of the number properties of fractions. With reference to the Teaching Model, the move here is to greater abstraction.

In locating numbers on a number line there is a natural recognition that fractions do not simply exist between 0 and 1 they are not simply parts of one whole. This leads to the use of mixed numerals (whole numbers and fractions) and exploring their relationship with improper fractions, demands the use of number operations, in particular multiplication and division. As these different symbolic representations of fractions are applied to fractions of sets, the demand for an understanding of and competency in applying these number operations increases.

Whilst students should continue to use words (oral and written), pictures and materials in their exploration of fraction, the appropriate use of symbolic recording must also be emphasised. The relationship of equality is more familiar to students who will have used the = sign regularly. However, it is recognised that insufficient attention is currently given to developing students’ understanding and use of the other relationship symbols, < and >. Comparing and ordering fractions is an obvious context in which to express relationships using this notation.

In any work with fractions reference to the whole (1) is critical. The question must be asked, for example, “Half of what?” Understanding that the size of the whole matters, is a fundamental idea that must underpin any proportional/ relational thinking.

These ideas are presented in five sessions however, as they include complex concepts, they can be extended over a longer period of time. A number of games are included. Whilst these are introduced and used within sessions to consolidate ideas, they can also be added to the class or group independent activities.

#### Links to the Number Framework

Stages 5-6

This unit supports teaching and learning activities in the Student Fractions e-ako 5 and 5+, 6 and 6+ and complements the learning activities in Book 7 Teaching Fractions, Decimals and Percentages.

Required Resource Materials
• Half a slice of bread and butter, by Peter Durkin
• An apple
• Plastic knife
• Counters
• Paper strips
• Glue sticks
• Scissors
• Number line 0-100
• Two coloured pegs
• Plastic beans
• Unifix cubes
• Poster paper
• Blank dice
Activity

#### Session 1

The purpose of this session is to consolidate understanding of fraction symbols, to introduce the < and > relationship symbols and to explore the relative size of unit fractions. Students review their understanding that the more pieces (parts of regions) something is cut into, the smaller the pieces will be.

Activity 1

Begin by playing Fraction Bingo (Attachment 1).
(Purpose: to consolidate recognition and correct reading of fraction symbols).
A bingo card and 10 coloured counters are given to each student in the group/class. The nominated caller reads numbers clue cards and students cover the matching number on their card with a counter. The first player to fill their card calls “Bingo”.

Activity 2

1. Read Half a slice of Bread and Butter.
If this is unavailable: take one apple, cut it and share it between two students. Have each student invite a friend to share their half. Have each student cut their half in half again (now quarters). Have each of the four students invite another friend each and cut their quarter in half again (now eighths). Repeat once again (sixteenths). Predict what would happen if this was repeated once more (thirty seconds).

2. Discuss what is happening to the number and size (thickness) of the pieces as this is happening. (The sixteenth will be nearly see-through because it is so thin). Have a student record the fractions they have made on the class/group chart.

3. Have the students in pairs talk about what is happening and decide on a summary statement to record on the chart. For example: ‘The more pieces that one (thing) is cut into the smaller the fractions will be. As the fractions get smaller the denominator (number of parts) gets bigger.’

Activity 3

1. Distribute MM 7-7 (Fraction Strips) to pairs of students and make a long strip of paper, glue sticks and scissors available.  Have students discuss the fraction strips and challenge them to add (draw in) two more rows of fractions on page 2 (thirtieths and sixtieths).

2. Have each student then cut off one of each different fraction piece and one 1 strip (top strip on each MM). Each student should turn their pieces number side face down, mix them up and then order them from smallest to largest, left to right, “guessing” the name of each piece and writing this on the blank side of paper piece before checking its name on the reverse side.
These can then be glued onto their own strip of paper, smallest on the left, largest piece (1) on the right hand end.

3. Draw attention to the fact that the smallest the fraction (1/60) piece the farthest away from 1.

Activity 4

1. On the class chart write the ‘is greater than’ symbol: >. Discuss this. Repeat with the ‘is less than’ symbol <. Review the meaning of the = symbol. Emphasise that each of these symbols expresses a relationship between the numbers on either side of the symbol. For example: 6 is greater than 4 can also be written 6 > 4, 6 is the same as 6 is also written as 6 = 6, and 4 is less than 6 can also be written 4 < 6.

2. Have students play in pairs Who has less and most? (Attachment 2)
(Purpose: to recognise the relative size of fraction symbols.)
Each student has one set of the Attachment 2 cards. These are shuffled and placed in a pile face down in front of each of the players who are sitting side by side. They place in front of them both the card saying “is less than: <”.
Player One turns over a card and decides which side of the symbol < card they will place their fraction. Once placed, the card cannot be moved.
Player Two turns over and places their card in the empty place. If the relationship sentence is true, Player Two takes the pair of cards, placing them face down in front of them.
Player Two then places a card on one side of the symbol card. Player One then turns over and places their card in the empty place. If it is true they keep the pair. If the expression is untrue the fraction cards are set-aside in discard pile.
For example:  Player One                        Player Two
This is not true. These cards are put in a discard pile to one side.  Player Two                         Player One
This is true. Player One keeps this pair of fraction cards.

The game continues till all fraction cards in the players’ individual piles are used.
The winner is the player with the most pairs.

#### Session 2

The purpose of this session is to have students consolidate their understanding of the relative size of fractions and to recognise them as numbers in their own right which ‘fit’ between whole numbers on a number line. Students are introduced to mixed numerals.

Activity 1

Introduce this session by playing Squeeze - Guess My Number with whole numbers on a number line.
(Purpose: to order two digit whole numbers on a number line.)
A peg is put at each end of a number line, for example, on 0 and 100. A student chooses a number between the pegs and writes it on a piece of paper. The rest of the students ask “less than” or “greater than” questions to find the mystery number. With each question, a peg is moved to eliminate numbers.
For example, if “Is it greater than 25?” is answered by “Yes”, then the zero peg is shifted up to 25 to eliminate all the numbers 25 and under. If “Is it less than 75?” is answered by “Yes”, then the top peg is shifted down to 75 to eliminate all the numbers 75 and over.
This continues until the mystery number is finally found by squeezing in from above and below.

Activity 2

1. Have the number line from Activity 1 In front of the students.
Write 1/2 on the class/group chart.
Pose the question: “Is this a number?”
Discuss (Yes).
Then pose, “Where does this number fit on our number line?” Have the students discuss this in pairs, agree on their decision, then have one student come up and show where 1/2 is located. (There may be misconceptions. For example, some students may suggest at 50 because it is halfway along the number line.).
Locate, and agree that 1/2 is halfway between 0 and 1. This can be difficult to see on a regular number line.
Explain to the students that they are going to ‘magnify’ this section (0-1) of the number line, and that they will create their own fraction number line.

2. Distribute A3 paper strips to each student.
Have them mark 0 and 1 at appropriate ends of the strip. Again refer to the whole number number line, highlighting the fact that their strip is the 0-1 section magnified. Discuss and have students fold to mark thirds. Have students now fold once (in half) and mark 1/2. Have them fold twice and mark 1/4 and 3/4. Have them fold three times and mark 1/8. Have them fold a fourth time, marking 1/16.
Finally with their number strip open have the fold the final 1/16 in half and mark 1/32.
Have them read out the numbers on their number strip from left to right.

3. Write on the class/group chart 1/18, 1/20, 1/80, 1/100 and have the students discuss where these would be located. (to the left of 1/8 and to the right of zero).
Ask where 1/1000 would be located, recognising that such a fraction would be too small to see on this number strip.
Expand this discussion to include the ‘infinitely small’ as appropriate.

4. Making sure that the 1-100 number line is visible, write 1 1/2, 1 1/4, 1 3/4 on the class chart and have students discuss their location on a number line.
Emphasise that fractional numbers fit between whole numbers on a number line.

5. Take three of the students’ number strips.
Mark two blank stickers with the numbers 2 and 3 and use these to join the number strips together making a strip 0-3. Glue onto card for strength.
Make adjustments to the number strip, adding 1 to all the fractions 1-2 and 2 to the fractions 2-3.

Activity 3

Play Squeeze - Guess My Number on this newly created class fraction number line. Have the class agree on the smallest acceptable fraction for the purposes of this game.

Activity 4

1. Have the fraction number strip 0-3 from 2 and 3 above visible to the students. Using fraction circle pieces, place in front of the students a regional representation of two thirds and of three quarters.
Write on the class/group chart:
“Which is closer to one, 2/3 or 3/4? Why?” Or “Which is near to making one whole circle, two thirds or three quarters? Why?”
Have the students discuss in pairs and come to an agreement on the answers to both parts of the question.
Share and discuss. (For example: “The more pieces there are, the smaller each piece is. As the denominator gets bigger, the smaller just one missing piece will be.”)

2. Write on the chart 11/12 and 7/8. Ask them to picture these, agree on which is closer to one and explain why. Show the regions and discuss further. Activity 5

Have students work in pairs to make a poster, with words, diagrams and symbols, explaining what they have learned in Session 2 about ordering fractions.
Ensure that students understand that their own classmates will be the audience for their posters and that the posters are to be displayed on the maths wall space.

See also the Trains activity from Numeracy Book 7.

#### Session 3

The purpose of this session is to have students consolidate their understanding of the relative size of fractions and where they fit with whole numbers on a number line. Students are formally introduced to improper fractions and mixed numerals which they make and order.

Activity 1

1. Create a fraction number strip to at least 5 by joining student fraction strips from Session 2, and pasting these on card. Discuss which fractions need to be adjusted and together agree where additional fractions can be added. For example, locate 3/8, 5/8 and 7/8, 1 3/8, 1 5/8 etc.

2. Display this where all can see it and have students count aloud in fractions of equal parts, taking turns around the group. For example: one half, one, one and a half, two, two and a half, three…. Or one eighth, two eighths, three eights...
Have one student place a coloured peg on each of the fraction numbers as the counting proceeds. Be sure to also count backwards with pegs being removed accordingly. For example: 5, 4 1/2, 4, 3 1/2, 3…
Finish by counting forwards in quarters and leave the pegs in place.

Activity 2

1. Have the 0-5 number strip (on the floor) in front of the students and make available at least 40 unifix cubes of one colour. Connect 4 cubes to make one ‘rod’. Explain to the students that this is 1(one) and that each coloured cubes is 1/4 of this 1 ‘rod’. Locate this rod below 1 on the number strip. Ask individual students to show with cubes and locate below the number strip, 1/4, 2/4, 3/4 and 1 1/4. Discuss then continue to model 1 1/2, 1 3/4 ……. To at least 3. “Read the models” aloud with the students, “1/4, 2/4, 3/4 ...

2. Count the number of quarters altogether in each of the models and have individual students record these as improper fractions on separate small cards and locate them below the cube models. 3. Read the improper fraction cards aloud together. Remove these, distribute them randomly to the students, then work around the group, having them relocate the cards in their correct places.

4. Have individual students collect a card of their choice, taking note of its position and the cubes at that position. Remove all of the cubes and repeat the card relocation, having students match mixed numerals with improper fractions without the support of the materials.

5. Write on the class chart: 5/4 and 1 1/4.
Ask, “Do these have the same value?” Discuss.
Write beside 5/4, improper fraction and beside 1 1/4 write mixed numeral.
Ask students to discuss these fraction names and suggest their own definition for each of these terms. Agree on the best definition and record this beside each term.
For example:
Proper fractions: fractions less than one
Improper fractions: fractions greater than one
Mixed numerals: whole numbers and fractions written together

Activity 3

1. Have students work in pairs. Give each pair the 2 sets of fractions from Attachment 3 and have students cut these into separate cards then work together to re-order each set, locating whole number and mixed numeral cards below the matching improper fraction as they lay them out.

2. Have students take 45 cubes and make models to match the set of thirds, beginning by making one (1) ‘rod’ of three (thirds).

Activity 4

Have the students play Stars to Mars: Fractions in order (Attachment 4).
(Purpose: to practice concepts of fraction order, mixed numerals and improper fractions)
To play you need the game board, star cards, a coloured counter for each player, 1 blank dice marked twice with 1, 2 and 3.
Players begin by becoming familiar with the game board. One game board is shared by the players who each place their coloured counter on EARTH.
The star cards are shuffled and placed face down in the Stars to Mars box.
Players take turns to roll the dice and follow directions.
If a player lands on a square with a star they pick up a star card, and answer any question posed to the satisfaction of the other players. The card is replaced at the bottom of the pile.
The correct roll must be made to reach the MARS.
The winner is the player to reach MARS first.

#### Session 4

The purpose of this session is to have students consolidate their understanding of the proper fractions, improper fractions and mixed numerals by working between these using sets and recording their findings.

Activity 1

1. Begin the lesson by reading together conclusions drawn in Session 3 in which the students worked with linear (number line) and regional (cubes) fraction concepts.

2. Explain to the students that they will be working with proper fractions, improper fractions and mixed numerals and sets of beans.
Have student pairs each take a handful of coloured beans between them and make writing material available.
Write on the class chart: In Jack’s Garden Shop beans come in packets (or sets) of six. Write a mixed numeral showing how many packets could Jack make with the beans you have.
Have a pair of students record their mixed numeral on the class chart and have them explain how they worked this out.
Record their explanation: For example: Joe and Manu took 23 beans. They wrote correctly 3 5/6 . “We made groups of six out of the beans. That was three packets and 5 beans left, so we wrote 3 and 5/6.”
Ask the students to write the improper fraction that have the same value as the mixed numeral. (In Joe and Manu‘s case it’s 23/6)

3. Repeat this with several students. Ask the student pairs to write an explanation of how to change an improper fraction (23/6) to a mixed numeral (3 5/6). Discuss and record their suggestions. Summarise formally by concluding that the numerator is divided by the denominator.

4. Have one student in each pair add another handful of beans to their pile. They should count these and write the total on their paper.
Pose several more examples of Jack’s bean packets with a different number each time.
Student pairs must record the mixed numeral and the improper fraction for each example and record in their preferred way how they worked this out.

5. Have students return their beans.
Pose examples on the class chart giving the number of beans and size of packets.
Once again, student pairs must record the mixed numeral and the improper fraction for each example and record in their preferred way how they worked this out.

Activity 2

1. Make beans available to the students should they need to use them. Pose several problems in which the fraction of the set (part) and total number of beans is given. Students need to work out the number in a set (whole) and the number of sets the total of beans will make, recording this as a mixed numeral.
For example: 3 beans is 1/3 of a packet. How many packets can be made with 17 beans? If 1/3 is 3 then 3/3 is 9. 17/ 9 is 1 8/9.
Pose further probems 2 beans is 1/6 of a packet. How many packets can be made with 17 beans? 4 beans is 1/4 of a packet. How many packets can be made with 17 beans? 5 beans is 1/2 of a packet. How many packets can be made with 17 beans?
Have students pair share their results, explaining their methods.

2. Have students make up problems for their partner to solve.

Activity 3

Have students review the game Who has less and most? from Session 1 and the symbols <, > and =.
Make beans available and have them play Bean Buster (Attachment 5).
(Purpose: to compare the relative size of fractions of sets, using the symbols <, > and =).

Activity 4

Conclude this session by having the students write in their maths diary, maths book or record together on the class/group chart, what they have learned about fractions of sets, proper fractions, improper fractions and mixed numerals.

#### Session 5

The purpose of this session is to develop the understanding that when fractions are being compared we must ensure that the fractions refer to the same whole (1).

Activity 1

Write on the class chart 1/2  > 1/4 .
Have the students read and discuss the truth of this relationship statement.
Pose the question, “Can you think of a time when this may not be true?” Discuss. The students may or may not suggest that ‘ if the half or a quarter are fractions of a different thing or amount that might make it untrue.’

Activity 2

1. Provide the student pairs with copies of the first page of Attachment 6, scissors, glue and blank poster paper.
Have them fold their poster paper in half.
Challenge them to use the images, or any of their own drawings, to make a poster on half of their paper, showing when 1/2 > 1/4 (same whole) and when 1/2 < 1/4 (different whole).
For example: Students cut 1/4 of any of the bigger geometric shapes and show this is bigger than 1/2 of the smaller shapes. (1/2 < 1/4)

2. Have students share their posters. Record on the class/group chart, “When we are comparing fractions we must first check the size of the whole. We should always ask half of what? Or a fraction of what?” Have students add this to their posters.

3. Provide students with copies of the second page of Attachment 6. Have them cut these out, pasting them on the other half of their poster paper, and show with drawings, symbols and words examples of when the statement about a fraction of a set would be true and when it would be untrue.
For example: Both students have 12 books altogether so the scenario is true because the whole amount is the same.
Mia, the girl, has 12 books altogether and Mapua has 28 books. His 1/4 (7 books) is more than Mia’s 1/2 (6 books) Here 1/2 > 1/4 is untrue because the whole amount is different.
Have students pair share their work.
Display the students’ work.

Activity 3

Conclude this session by making a bullet point review on the class/group chart of key points learned about ordering fractions.

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