Author

David MacIsaac (SRSB)

Title

Dividing Fractions with Pattern Blocks

Grade Level

8,9

Subject Area

Mathematics

Overview of unit/lessons/activities (assumptions of prior knowledge/learning)

Students should learn outcomes involving fractions by using concrete materials. This will provide ideas for using manipulatives (pattern blocks) to learn division of fractions.

Correlations to ICT and curriculum outcomes

ICT Integration Outcomes:

• explore curriculum concepts under study using specialized software; measuring, sampling and recording equipment; and computer-based simulations, with teacher assistance (PTS 9.2)
• explore the curriculum through a wide range of print and electronic forms; accessing and processing information by means of the specialized techniques associated with the technology they select (PTS 9.3)
• select appropriate measuring and recording devices and/or software to collect data, discover patterns of change over time, solve problems and make logical decisions based on their investigations; with teacher assistance (RPSD 9.1)

Math (Grade 8)

• Divide fractions concretely, pictorially, and symbolically (B8)
• Estimate and mentally compute products and quotients involving fractions (B9)

Projected timeline for preparation and for carrying out activities

6 one hour lessons

Equipment Requirements: (computers, software, etc)

Pattern blocks
4 computers - access to a computer lab would be nice but not necessary.

Teaching materials provided (Blacklines, worksheets, templates, teacher materials)

http://lrt.ednet.ns.ca/PD/BLM/pdf_files/pattern_blocks/all_blocks.pdf

http://lrt.ednet.ns.ca/PD/BLM/table_of_contents.htm all blackline masters(math)

Pattern block black line masters may be found here. An overhead set of pattern blocks would be helpful. Students can make their own using the above template. A zip lock bag to store the students manipulatives (one for each student) is helpful. Have some extra sets on hand for students who “forget” theirs at home.

Overhead blocks are available here:
https://w3apps.ednet.ns.ca/nssbb/search_alr_list.asp

Magnetic blocks are useful too. They can be obtained here: https://w3apps.ednet.ns.ca/nssbb/search_alr_list.asp

Resources available for teacher/student use (websites, references, etc)

Software: “Understanding Math“ provides students with self-directed lessons in most of the mathematics strands, and addresses concepts from grades 4 to 12. In the section “Understanding Fractions”, the division of fractions is covered using pattern blocks to teach the process. “Understanding Math” is available through the NS School Book Bureau, if your school does not already have it installed on computers. Remember that we are looking for understanding, not memorization of a rule! The process is important here. Let your students struggle!

Some helpful websites are: (not all address division of fractions)

National Library of Virtual Manipulatives:
http://nlvm.usu.edu/en/nav/index.html

Understanding Fractions(excellent overview for teachers/parents)
http://mathcentral.uregina.ca/RR/database/RR.09.95/hanson4.html

Fun Fractions
http://math.rice.edu/~lanius/Patterns/add.html

Worksheets
http://www.superkids.com/aweb/tools/math/fraction/basic.sht

Saskatchewan Ministry of Education – Model Unit: Fractions in Action (grades 6 and 7)
http://www.sasklearning.gov.sk.ca/docs/midlmath/fraction.html

Neufeld – Lessons using Understanding Fractions
http://www.neufeldmath.com/worksheets/fractions/

Lesson Planet’s Fraction Lessons (Note: use of this site requires a purchased subscription)
http://www.lessonplanet.com/search/newsearch?media=lessons&keywords=fraction

NCTM Illuminations – Pattern Block Fractions
http://illuminations.nctm.org/LessonDetail.aspx?ID=U113

Java applet for pattern blocks
http://www.arcytech.org/java/patterns/

Fraction activities for various grades (different manipulatives, including pattern blocks)
http://www.wou.edu/~burtonl/courses/math395/Jensen.Martin.McArthur.McClatchey.pdf

Fraction story Information and blackline masters for teaching fractional concepts
http://www.nsa.gov/teachers/es/frac38.pdf

Teacher reading (interesting article)
http://www.ncolr.org/jiol/issues/PDF/3.4.2.pdf

Detailed instructions for each activity or lesson (teacher notes, activity information, learning strategies, teacher role, student roles)

I believe that the way that we teach dividing is important. We need to teach our students correct math language.

For example:

One whole

One third

If we say “one divided by one third some students may be confused.” However, if we say “how many groups of one third are needed to make one whole?”, the majority of students will understand. (Hopefully everyone, especially using pattern blocks). I would suggest that every student construct one whole using their pattern blocks.

Students are used to grouping from earlier grades so this should be understandable to them.

Two thirds

Following the line of questioning above, we would ask:

“How many groups of one third are needed to make two thirds?” Have the students construct this using manipulatives, even though it seems easy.

We need to do enough easy examples with our students to make sure that they understand how to read (interpret) the question before moving on.

“How many groups of two thirds are needed to make one third?” Tricky!

Some students will see that they need one half of a group. It may take longer for some others .

This represents using pattern blocks.(one representation)

Now it gets harder, but the line of questioning is the same. Remember that we want understanding, not memorization. Don’t teach the algorithm but encourage your students to find a “shortcut”. Don’t let them share the shortcut yet!

We want to ask,” How many groups of one half are needed to make seven twelfths?”

You need to build seven twelfths using halves. Students should quickly be able to see that the answer lies between one and two. Using pattern blocks is very helpful here. Students will see that they need one whole group of one half (six twelfths) plus a twelfth to make seven twelfths.

Questioning is really important here for the remaining piece (the twelfth). Students need to ask “what part of one half is one twelfth?” Using the blocks they will see one sixth! So now they need to put it together .This is time consuming, can be painful but when they get it, they get it!

Some may use a common denominator idea here! Whatever promotes understanding. We don’t all learn the same way or at the same pace. This is much easier with the pattern blocks in front of you.

We ask: how many groups of one half are needed to make one and three fourths?

Have the students model . It should look like this.

Students will see that they have three groups of one half and part of another group. We need to ask: What part of one half is the left over piece. Hopefully, they will see that they have one half of one half le

ft over. Therefore, our answer is:

You will note that I have not introduced the algorithm, which should be the very last part of your lessons. Try to encourage your students to make connections to the algorithm. With enough examples, they will discover it.

Student products expected

Students should be able to:

1. Explain to you (orally or in writing) what they are doing.
2. Demonstrate with the blocks how to solve a problem.
3. Draw a solution.

Samples(including teacher notes, assessment information, student work, if available)

https://w3apps.ednet.ns.ca/nssbb/search_alr_list.asp

The binder from which this lesson is drawn is available through the Nova Scotia Book Bureau. The students need to learn the division using the pattern blocks first. Then use the technology to support your teaching. You may wish to group your students and have different centers available in your class.

For those unfamiliar with pattern blocks, the basic set consists of 6 shapes:

In recent years, fraction blocks have become available to supplement the basic pattern blocks. The additional shapes are:

the pink double hexagon (one whole)

and the black chevron.

For our purpose, studying fractions, we will be using the double hexagon, the hexagon, the chevron, the trapezoid, the blue rhombus and the triangle.

After you have a sense that your students have some understanding of dividing fractions using pattern blocks, I suggest that you set up centers in your classroom with various levels of difficulty.

One center can be your 4 computers where students can use Understanding Math or one of the web sites above.

Logistics (organization, grouping, management issues, access to technology)

Make sure that you have more than enough pattern blocks. Taking a day to make and color them can be done in art. Don’t ever make the students divide without the pattern blocks. They will stop using the blocks when they are ready.

Assessment information (e.g., rubrics for products and/or process)

See above

Possible extensions

Once students attain an understanding of the basic concept, try dividing using improper and mixed fractions.

Adaptations for students requiring additional support

All students can do this to some extent. A possible adaptation to consider is the time needed to use the pattern blocks. Students will learn at different rates.