Unit Ordering Fractions: Day 1: Number Lines

MO 15 min

A 30 min

C/D 15 min

60 min

Math Learning Goals

Students will:

  • Connect the meaning of the denominator of a fraction to the number of equal partitions between 0 and 1 on a number line
  • Represent fractions on a number line, using benchmarks to check precise and approximate placements, in order to compare and order the fractions

Materials

  • smart notebook file: Number Lines
  • or
  • BLM 1.1

Minds On...

Whole Class Guided Activity

Construct a number line from 0 to 1 on the board. Tell students that they will be placing the fractions 12, 14, 24, 34, 38, 15, 1620, 23 on the number line. Have a student select one of the fractions to label on the number line in the appropriate location. Probe their understanding using the following prompts:

  • Why did you select that fraction to place?
  • What was your reasoning for placing it where you did?
  • What strategy might you use to decide if the fraction is closer to 0, 12 or a whole number?
  • What was ______ doing as s/ he figured out where to position 45? How was that reasoning helpful?
  • How is finding the halfway point helpful? Are there other benchmark fractions that were helpful to you?

Continue with different students until all fractions are placed. It may be necessary to remind students that they should be dividing the number line into the same number of spaces as the denominator and that it may be useful to consider benchmarks to help place unusual fractions.

Adapted from: Comparing and Ordering Number Lines:

Action!

Small Group Practice Activity

Using large pieces of paper,students create three same length number lines, labeled as indicated in BLM 1.1. Ask students to place the corresponding fractions on the number line. They must be prepared to justify their thinking as to why they placed it there. Probe student thinking with the following prompts:

  • How did you decide where it went?
  • Is there a unit fraction that would help you place this fraction more accurately?
  • Is your fraction more than 1 or less than 1?
  • Compare your number lines with a different group. Explain your reasoning why you placed the fractions where you did.
  • What strategies did you use to decide where the fraction should be placed?

Consolidate/ Debrief

Whole Class Anchor Chart

Create an anchor chart which highlights key properties of each number system (e.g., % means ‘out of 100’, 0.5 is read ‘five tenths’ so it is a fraction with a denominator of 10) and outlines strategies for comparison/conversion between fractions, percentages and decimals.

Individual Math Journal

In your math journal, draw a number line and place the given numbers on it. Justify why you placed them where you did. Be sure to use appropriate math language.

  • benchmark fractions (12,13,23,14,34, 1 whole)
  • denominator
  • numerator
  • decimals
  • proper fraction
  • improper fraction
  • greater than
  • less than
  • benchmark fractions

Home Activity or Further Classroom Consolidation

BLM 1.1: Number Lines

Label each of your number lines as shown below and place the corresponding fractions accordingly.

First number line

Show where these fractions are located and be prepared to explain your reasoning.

All students: 12, 23, 45, 14

Your choice: 1720, 318, 1635, 3692

Second number line

Show where these fractions are located and be prepared to explain your reasoning.

All students: 810, 38, 15 , 64

Your choice: 2983, 1318, 5792

Third number line

Show where these fractions are located and be prepared to explain your reasoning.

All students: 84, 104, 152, 5010, 12, 34

Your choice: 338

Unit Ordering Fractions: Day 2: Number Lines with Decimals

MO 10 min

A 40 min

C/D 10 min

60 min

Math Learning Goals

Students will:

  • Connect the meaning of the denominator of a fraction to the number of equal parts between 0 and 1 into which a number line is partitioned
  • Represent fractions on a number line, using benchmarks to check precise and approximate placements in order to compare and order the fractions

Materials

  • fraction strips
  • fraction towers/circles
  • base ten blocks
  • number lines
  • hundred board
  • money

Minds On...

Individual Exploration

Students use 0.5, 0.75, 0.25, 1.0, 34, 12, 1, 14 To match the fractions with their equivalent decimal representations. Probe student thinking using the following prompts:

  • Why did you put those two together?
  • How do you know they have the same value / or they represent the same amount?
  • Explain a connection to fair sharing, money or to a manipulative.

Pairs Activity

Students place the previous set of fractions and decimals on a number line labelled from 0 to 1. Probe student thinking using the following prompts:

  • What was your reasoning for placing the numbers there?
  • What strategy might you use to decide if the decimal card is closer to 0 or 1?
  • What was ____ doing as s/he figured out where to place 0.75? How did that reasoning help ____?
  • How is finding the halfway point helpful?

Comparing and Ordering Number Lines:

Action!

Pairs Activity

Consider whether it is more appropriate to pair students as: strong/strong and weak/weak, or weak/strong for this activity. Provide pairs with a copy of BLM 2.1. Circulate and ask probing questions where you detect misunderstanding, confusion, or lack of detail. Encourage early finishers to try a second choice in question 2. Decide whether or not there is value in whole class discussion of question 1.

Whole Class Malk Learning Activity

Ask an appropriate pair to present their thinking about question 2A, encouraging classmates to question statements and illustrations they do not understand. Repeat for 2B and 2C.

Note that all fractions in 2A and 2B are all equivalent, but not so in 2C.

Note that all fractions in 2A and 2B are all equivalent, but no so in 2C.
An implmentation trajectory for Math Talk Learning Community can be found here

Consolidate/ Debrief

Individual Exit Card

Record your response to the following prompt. You will be asked to hand this in before leaving. Frank now thinks that 14, 0.5 and 416 would all be placed in the same location on the number line. Do you agree or disagree? Explain your thinking in more than one way. This can be assessed using the following criteria. Does the student:

  • understand that the three numbers do not all represent the same amount?
  • convert between fractions and decimals?
  • use manipulatives and/or diagrams to support their thinking?
  • make connections to benchmarks and/or real life situations (e.g. money, 12?)?

Home Activity or Further Classroom Consolidation

BLM 2.1: Decimals and Fractions on the Number Line

  1. Place the following numbers on the number line above.
    1. 0.9
    2. 0.2
    3. 0.4
    4. 910
    5. 710
  2. Choose one of the following questions, and answer it in the space below.
    1. Frank says that 612, 24, and 0.5 can all be represented in the same place on our number line. Do you agree/disagree? Why?
    2. Frank says that 912, 34 and 0.75 can all be represented in the same place on our number line. Do you agree/disagree? Why?
    3. Frank says that 0.666, 23 and 3245 can all be represented in the same place on our number line. Do you agree/disagree? Why?

Unit Ordering Fractions: Day 3: Number Lines - multiple number stystems

MO 20 min

A 30 min

C/D 10 min

60 min

Math Learning Goals

Students will:

  • reason as they place fractions and decimals on a number line
  • connect different representations of the same number
  • communicate their rationale, including their use of benchmark numbers

Materials

  • fraction strips
  • fraction towers/circles
  • base ten blocks
  • number lines
  • hundred board
  • money

Minds On...

Whole Class Matching Activity

Each student receives one card and circulates amongst classmates to match the numerical value of their representation to three other students in the room. Once completed correctly, each group will have a hundreds grid, fraction card, decimal number and hundredths card representing the same number.

Small Groups Discussion

Once groups have been formed, students share the card they have and everything they notice about their card. Sentence starters include:

  • I think my card belongs here because …
  • I see how my representation connects to _____’s card because …
  • My representation is the similar to _____’s because …
  • My representation is different than _____’s since …
  • Something I noticed about all four representations is …

Cards can be found on page 51 of The Guide to Effective Instruction (Vol. 6); 7 sets makes a class of 28 in groups of 4; see pg. 59-63 in the Guide for more details.

Action!

Pairs Activity

Provide students with a set of cards (BLM 3.1). Have them place each card on a number line. Inform them that they will be required to justify their reasoning.

Circulate to support and extend student understanding using the following types of questions:

  • Which number representations do you know and recognize?
  • What strategies could you use to place the other ones?
  • How can you use benchmarks to help you place some of the other ones?
  • Which representations show the same amount?
  • Is there a fraction that is close to 1?
  • Is there a fraction that is close to 0?

Provide students with a mixture of representations (fractions, decimals, hundreds grid, hundredths card) In pairs students place these on a number line- justifying their reasoning.

50100,12,5,80100,45,8,525(need to add other numbers)

Consolidate/ Debrief

Independent Math Journal

Show 6 different representations with two that don’t belong. Explain which representations go together and why.

Home Activity or Further Classroom Consolidation

BLM 3.1: Fractions on a Number Line

0.7 0.06 12 910 1516 0.01 0.4 23 1100 58 2929 0.95 68
12 23 34 910
0.33 0.59 1% 10%
0.45 90% 100% 80100
410 your choice: your choice: your choice:
12 23 34 910
0.33 0.59 1% 10%
0.45 90% 100% 80100
410 your choice: your choice: your choice:

Unit Equivalency in Fractions: Day 4:

60 min

Math Learning Goals

Students will:

  • Investigate relationships among fractions, decimals and percentages
  • reason about the connection between the intervals on the number line and the denominator

Materials

Minds On...

Individual Math Journal

Students respond to this prompt: “What is the relationship between fractions, decimals and percentages?”

Comparing and Ordering Number Lines:

Action!

Pairs Activity

Instruct students to draw two number lines and then place the numbers in the proper place.

  1. 0,1,50%,75%,48,0.50,0.25,40%
  2. 0,50%,2,1.05,1.10,13%,25%13,910,84%,1.9,0.6

If students are struggling it may be helpful to ask them what strategies they have used in previous lessons to order numbers and to determine equivalency between the different number systems (fractions, decimals and percentages).

Small Groups Exploration

Distribute envelopes containing Set A or Set B below of eight fractions in numerical form to pairs. Students decide how they will place their set of fractions on a number line. Remind students that they may need to make more than one number line to help show fractions that are equivalent.

[Set A: 24, 12, 13, 26, 34, 68, 510, 39]

[Set B: 86, 1 26, 1 13, 43, 58, 28, 14, 13]

Consolidate/ Debrief

Whole Class Anchor Chart

Create an anchor chart which highlights key properties of each number system (e.g., % means ‘out of 100’, 0.5 is read ‘five tenths’ so it is a fraction with a denominator of 10) and outlines strategies for comparison/conversion between fractions, percentages and decimals.

Whole Class Discussion

  • How did you decide whether to do one / two / three number lines?
  • How are your number lines like the fraction towers?
  • Which numbers were easy to place?
  • What did you find challenging?
  • What questions do you still have about equivalent fractions?

Home Activity or Further Classroom Consolidation

Students complete textbook questions that require them to convert between fractions, decimals and percentages.

Additional Equivalent Fractions Activities:

[Choose from these for students who need more practice, or an extension.]

Choose a fraction. Show as many equivalent fractions as you can, using the manipulatives provided.

What do you notice about the numerator and denominator in the fractions 35 and 1220?

You have a bag of marbles. There are 5 black and 5 white marbles in the bag. What is the probability of getting a black? Now, if you remove one black and one white (4 of each), what is the probability of pulling out a black? What about 3 of each? What do you notice?

Look at the two granola bars. Sue says that the two bars both show 14 of the granola bar is shaded. Mitchell says that the fractions are different. Who is right?

Jian threw his paper airplane 0.66 m and Sylvain threw his 23 of a meter. Whose airplane went the farthest?

(Other examples: 0.75 and 34; 0.2 and 210; 1.25m and 54)