The purpose of an Estimation Exploration is to practice estimating a reasonable answer based on experience and known information. Students can identify fractions represented by the shaded portions in tape diagrams in which unit or non-unit fractions are marked. To estimate the shaded parts in an unmarked tape, students may rely on the size of benchmark fractions—, , or —and partition those parts mentally until it approximates the size of the shaded part. They may also estimate how many copies of the shaded part could fit in the entire diagram.
Launch
Groups of 2
Display the image.
“What is an estimate that’s too high? Too low? About right?”
1 minute: quiet think time
Activity
“Discuss your thinking with your partner.”
1 minute: partner discussion
Record responses.
The whole diagram represents 1. What fraction of the diagram is shaded?
Make an estimate that is:
too low
about right
too high
Student Response
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Advancing Student Thinking
Activity Synthesis
Consider asking:
“Is anyone’s estimate less than ? Is anyone’s estimate greater than ?”
“Based on this discussion does anyone want to revise their estimate?”
Activity 1
Standards Alignment
Building On
Addressing
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
In this activity, students examine number lines that have been partitioned into smaller and smaller parts. They see that this strategy can be used to generate many equivalent fractions and to verify if two fractions are equivalent.
Students encounter fractions with 5, 10, 15, and 20 for the denominator. Working with multiples of a number (in this case, 5) allows students to notice structure in how the partitioning of a part on a number line relates to equivalent fractions (MP7). (Students will not be assessed on fractions with denominator 15 or 20.)
If desired and logistically feasible, consider enacting Andre’s reasoning with one or more human number lines.
Place a strip of masking tape or painter’s tape, at least 25 feet long, on the floor of the classroom or a hallway.
Ask a student to stand on each end of the tape. They represent 0 and 1.
“How can we partition this line into fifths?” (Position 4 students on the tape, spaced apart equally between 0 and 1.)
Ask each student on the line to say the number they represent. (0, , , , , 1) Give each student a sign with their fraction. (Consider distinguishing the sign for with a different color.)
Invite 5 students to join the line, each person standing exactly in the middle of two others.
“What fraction do you represent now?” Ask every student to say the number they represent now. (0, , , , . . . 1) Give the student representing another sign showing .
Repeat to represent twentieths:
Ask 10 additional students to each join a space between two students representing tenths. (The students representing fifths and tenths should stay in place.)
Ask students to say aloud the number they represent now. (0, , , , . . . 1)
Give the student representing another sign showing .
“Can you explain why the student representing also ends up representing and ?”
If time allows, extend the activity to include fifteenths, with 2 additional students standing between each fifth.
MLR8 Discussion Supports. During partner work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner. Advances: Listening, Speaking
Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Record responses on a display and keep visible during the activity. Supports accessibility for: Social-Emotional Functioning, Attention
Launch
Groups of 2
Read the first two sentences of the task as a class.
If possible, consider creating a human number line as outlined in the Activity Narrative.
Activity
“Think quietly for a minute about the first problem. Then, discuss your thinking with a partner and work on the second and third problems together.”
1 minute: quiet think time
7–8 minutes: partner work time
For the last problem, monitor for students who:
Extend each number line past 1 whole to show , , , and .
Skip-count by the number in each numerator 6 times—by 2 to get to , by 3 to get to , and by 4 to get to .
Multiply the numerator (6) and the denominator (5) by 2, 3, and 4 to get , , and .
Activity Synthesis
Invite a student to share their explanation of Andre’s strategy (first problem). Ask the class if they agree with the explanation or if they would change it or explain it differently.
Ask another student to show how the number lines could help us see if two fractions are equivalent (second problem).
Select previously identified students to share their reasoning for the last problem. Record their reasoning.
Solicit some initial impressions on how the strategies are alike and different, but save further comparisons for future lessons.
Activity 2
Standards Alignment
Building On
Addressing
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
In this activity, students continue to use the idea of partitioning a number line into smaller increments to reason about and generate equivalent fractions. Through repeated reasoning, students begin to see regularity in how the process of decomposing parts on a number line produces the numbers in the equivalent fractions (MP8). The task encourages students to think of the relationship between one denominator and another in terms of factors or multiples (even if they don’t use those terms which connect to work in a previous unit).
Partitioning a number line into smaller parts becomes increasingly inconvenient when the denominator gets larger. As students begin to think about the relationship between tenths and hundredths, they see some practical limitations to using a number line to find equivalent fractions and are prompted to generalize the process of partitioning. (Students are not expected to draw a full number line with 100 parts.)
Launch
Groups of 2
Read the first problem as a class. Ask students to think quietly for a moment about whether what Priya wants to do can be done. Then discuss with their partner.
30 seconds: quiet think time
1 minute: partner discussion
Activity
“Take about 7–8 quiet minutes to work on the task. Afterwards, discuss your responses with your partner.”
7–8 minutes: independent work time
2–3 minutes: partner discussion
For the first problem, monitor for students who partition the lines by:
Guessing and checking.
Reasoning multiplicatively (3 times what number gives 9, 10, or 12?) or in terms of multiples (Is 9, 10, or 12 a multiple of 3?).
Reasoning in terms of division (9 divided by 3 is what number?) or in terms of factors (Is 3 a factor of 9, 10, or 12?).
For the second and third problem, monitor for students who find equivalent fractions for by:
Partitioning the number lines into smaller increments quantifying the new number of parts.
Finding multiples of 1 and 10, and using this strategy to write an equivalent fraction with denominator 100.
Priya wants to find fractions that are equivalent to , other than . She wonders if she can find equivalent fractions with denominators 9, 10, and 12.
Can it be done? Use number lines to show your reasoning.
Represent on a number line. Then find 2 fractions that are equivalent to . How would you use the number lines to show that they are equivalent to ?
Can you find an equivalent fraction for with 100 for the denominator? Explain or show your reasoning.
Student Response
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Advancing Student Thinking
If students conclude that the answer to the last question is “No, it can’t be done,” consider asking:
“How did you find a fraction that is equivalent to one-tenth?”
“Into how many equal parts should each one-tenth section on the line be split to get hundredths? How do you know?”
Activity Synthesis
Invite previously identified students to share their strategies for answering the first two problems.
If not done by students in their explanations, consider asking students to revoice their reasoning in terms of factors and multiples.
See Lesson Synthesis.
Lesson Synthesis
“Today we used number lines and partitioning to help us write equivalent fractions and to tell if two fractions are equivalent.”
“How can number lines help us find equivalent fractions for ?” (We can draw a number line showing tenths, and then partition the tenths into 2 parts, 3 parts, 4 parts, and so on.)
“There were times in the lesson when some of you chose not to use the number lines to find equivalent fractions or to tell if two fractions were equivalent. Why was that?” (Sample responses: It was not necessary. It would take too long to draw all the tick marks. We could skip-count, reason about the numbers mentally, or find multiples of the numbers in the fraction.)
“In upcoming lessons, we’ll continue to develop our strategies for finding equivalent fractions and checking if two fractions are equivalent.”
Standards Alignment
Building On
3.NF.A.1
Understand a fraction as the quantity formed by 1 part when a whole is partitioned into equal parts; understand a fraction as the quantity formed by parts of size .
