Fraction Values and Changing Wholes

At a Glance What: Understand that as the defined Student Probe whole change so does the name for that Figure A Figure B fractional piece. Common Core Standards: CC.4.NF.1 Extend understanding of fraction equivalence and ordering. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) 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. (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and

100.) Name the fractional part of Figure A for each of the Matched Arkansas Standard: AR.5.NO.1.1 following colored pattern blocks: blue rhombus, green (NO.1.5.1) Rational Numbers: Use models triangle, red trapezoid. and visual representations to develop the

concepts of the following: Name the fractional part of Figure B for each of the ---Fractions: parts of unit wholes, parts of a following colored pattern blocks: blue rhombus, green collection, locations on number lines, trangle, red trapezoid. locations on ruler (benchmark fractions),

divisions of whole numbers;

Figure C ---Ratios: part-to-part (2 boys to 3 girls),

part-to-whole (2 boys to 5 people);

---Percents: part-to-100 List and explain how you found the Mathematical Practices: fractional amount for each of the colored Attend to precision. pattern blocks using Figure C as your Look for and make use of structure. whole: blue rhombus, green triangle, red Who: Students who think that individual trapezoid. fractional pieces have a specific name Answers: regardless of the whole. (Such as a green Figure A: triangle is always 1/6.) • Blue Rhombus: 1/3 of the yellow hexagon Grade Level: 4 • Green Triangle: 1/6 of the yellow hexagon Prerequisite Vocabulary: None • Red Trapezoid: 1/2 of the yellow hexagon Prerequisite Skills: None Figure B: Delivery Format: individual, small group • Blue Rhombus: 2/3 of the red trapezoid Lesson Length: 15-30 minutes • Green Triangle: 1/3 of the red trapezoid Materials, Resources, Technology: Pattern • Red Trapezoid: 3/3 of the red trapezoid blocks Student Worksheets: Pattern Block Charts 1, 2, 3 and Summarization Chart

Figure C: • Blue Rhombus: 2/9 of the yellow figure • Green Triangle: 1/9 of the yellow figure • Red Trapezoid: 1/3 of the yellow figure

Most students should be able to correctly name the parts for Figure A. It is with Figure B and Figure C that special attention needs to be placed when looking at students’ responses. Explanations for Figure C need to be closely examined when determining level of understanding.

Lesson Description The lesson is intended to help students develop an understanding of the relationship that exists between different patterns blocks when the defined whole is changed. Students will use a particular set of patterns blocks to find fractional amounts for a specific pattern block designated as the whole. There are multiple opportunities for students to record the information in a table for reference throughout the lesson. It is the intent of the lesson to provide enough repeated exposure to the patterns and relationships that are formed between the part and the whole that students will be able to make some generalizations about what happens when the whole is changed but the pattern block stays the same.

Rationale Fraction manipulatives can be useful tools when introducing and helping students to establish a conceptual understanding of fractions. With all the benefits that these tools offer they can also lead students to a superficial understanding of fractions if not addressed properly. Students may think that a particular pattern block or fractional piece is always 1/3 or always 1/4 because of its use within that set of manipulatives. It is the focus of this particular lesson to put an emphasis on the relationship between the part and the whole that exists. Not only does this lesson focus on part to whole relationships but also part to a changing whole. Giving students opportunities to explore and disprove these misconceptions will help to place the emphasis on the concept instead of the tool.

Preparation Provide students with pattern blocks (hexagons, trapezoids, triangles, and rhombi) and copies of the handouts “Pattern Block Charts 1, 2, and 3” and the “Summarization Chart.

Lesson

The teacher says or does… Expect students to say or If students do not, then the do… teacher says or does… 1. Let’s review the names of Students should be able to If students do not know the the pattern blocks we are state the names of the names, write the names and a going to use today in our following pattern blocks: description on the board or a lesson. yellow-hexagon chart to revisit throughout the blue-rhombus course of the lesson. red-trapezoid green-triangles 2. Take one of the yellow Students should be able to If students are not able to find hexagons from your pattern do this in three different all three ways, the teacher blocks. We are going to ways (using the blocks will need to directly model designate this block as our provided). how to find/cover the whole for this activity. 2- red trapezoids hexagon with the pieces. 3-blue rhombi Using the yellow hexagon as 6-green triangles your whole, find all the ways that you can cover the hexagon using only one type of block. 3. Now that we have covered Students should be focusing If students are having the hexagon all the possible on how many pieces of the problems finding the fraction different ways, identify the same size does it take to name for any of the smaller fraction name of each of cover the yellow hexagon. blocks, use the chart title the blocks used (trapezoid, Size of the pieces Pattern Block Chart 1 to help blue rhombus and triangle) (denominator) is an students see the connection in relation to the whole we important concept to make between the pieces, the determined. sure students have a firm whole, and the fractional grasp. value. Record the information for If students continue to each of the pieces in struggle with the idea of Pattern Block Chart 1. naming a fractional amount, this could mean that students need additional work before attempting to master the concept of “changing wholes”.

The teacher says or does… Expect students to say or If students do not, then the do… teacher says or does… 4. NOTE: After students have Students should see that if If students are not making the identified the fraction they cover the blue rhombus connection between the new names for each of the with the green triangles it whole and the new fractional pieces used to cover the will only take 2 blocks piece represented by the yellow hexagon, students instead of 6. green triangle, the teacher will repeat the process with may want to record the blue rhombus as the The green triangle is now information for the Blue whole. 1/2 of the designated whole, Rhombus in the table labeled the blue rhombus; however Pattern Block Chart 2. (This is If our new whole is changed earlier it was 1/6 when using used when the blue rhombus from the yellow hexagon to the hexagon as the whole. is the whole. a blue rhombus, how does that affect the fractional Students will record that it names of the other pattern takes 2 triangles to cover blocks? the blue rhombus and the fractional value is 1/2. Record the information for the green triangle on the table titled Pattern Block Chart 2. 5. Why has the fraction name Student should state that If students do not state that for the triangle changed? the number of pieces the size of the designated needed to cover/fill the whole has changed, ask the region has changed so students what was different therefore the label for that about the second problem piece must change. (what did we change?)

It must be called something Did we change the size of the different and this difference triangle or the size of the is found in the denominator. whole? It is important to make sure students see that when the size of the whole changed, the fractional value of the triangle changed as well.

The teacher says or does… Expect students to say or If students do not, then the do… teacher says or does… 6. What is the current The students will record the relationship between the fact that if the rhombus is rhombus and the hexagon the defined whole then the in our new problem? hexagon is more than one whole. Follow-Up question (for (This will go in the space further prompting if under the label “number needed) needed to cover the What is considered the whole”.) whole amount? In the appropriate space Record the information for under “Fraction Name for the red trapezoid on the each Block” students will table titled Pattern Block record that the value of the Chart 2. hexagon is 3 because it takes three of the blue rhombi to take make one yellow hexagon. 7. What is the relationship Students must use their Finding the fractional amount between the blue rhombus knowledge about the for the trapezoid is more (the defined whole) and the relationship between complex than the triangle and red trapezoid? triangles and rhombi to the hexagon. determine how many How many rhombi are the rhombi are the same as a The teacher may need to same sizes as a red trapezoid. revisit the relationship trapezoid? between the triangles and Students should explain that rhombus with the physical Record the information for 1 rhombus and 1 green model using pattern blocks. the red trapezoid on the triangle will cover a red table titled Pattern Block trapezoid. The teacher needs to focus Chart 2. students’ attention on the The students will record the fact that there is not a direct fact that if the rhombus is relationship between the the defined whole then the rhombi and trapezoid. red trapezoid is more than one whole. (This will go in the space under the label “number needed to cover the whole”.)

The teacher says or does… Expect students to say or If students do not, then the do… teacher says or does… 8. If 1 rhombus and 1 triangle Students should be focusing The teacher needs to model cover the red trapezoid, their arguments on the the thought process out loud what fraction name should following logic: for the student when be given to the red A triangle is ½ of a blue describing how to use the trapezoid? rhombus. So therefore 1 knowledge about triangles to whole rhombus and ½ of find the value for the another rhombus should trapezoid. yield 1 ½ rhombi.

In the appropriate space under “Fraction name for each Block” students will record that the value of the red trapezoid is 1 ½ because it takes one blue rhombus and half of another (1 green triangle) to take make one red trapezoid. 9. We will repeat the process The value of the yellow one more time. hexagon is two because the This time we will use the red trapezoid is worth one red trapezoid as our whole and it takes two to designated whole. make up a yellow hexagon.

What is the value for the yellow hexagon?

Use the pattern blocks provided to determine the fractional amounts of the other pattern blocks (triangle, rhombus, and hexagon). 10. What is the value of the Students should be focusing green triangle if the red on the green triangle. It trapezoid is our whole? takes 3 triangles to cover the red trapezoid, so Record the information for therefore each triangle is the red trapezoid on the worth 1/3. table titled “Pattern Block Chart 3”.

The teacher says or does… Expect students to say or If students do not, then the do… teacher says or does… 11. How can you use this It takes one blue rhombus If each green triangle is 1/3 of information (about the and a green triangle to cover the figure, then how many of green triangle) to help us a red trapezoid. the green triangles does it find the value for the blue take to cover a blue rhombus? rhombus if the red 1 whole blue rhombus is trapezoid is the newly equivalent to 2 green When putting all this defined whole? triangles. information together, what fractional value does this Each green triangle is worth produce for the red 1/3 of a red trapezoid. trapezoid?

So, 1 blue rhombus is 2/3. 12. Now that we have explored Students will complete the Why do the fractional values various different pattern chart and discuss the of the blocks keep changing blocks as our designated relationship that exists each time? whole, we will construct a between the parts and the “Summarization Chart” to changing whole in the chart. help solidify our learning Conversation should focus and provide us with a clear on the size of the piece in picture of how the values relation to the size of the for each pattern block whole. changed. (How many pieces do you see versus how many pieces does it take to cover/fill the area or region.)

Teacher Notes None

Variations None

Formative Assessment

Figure A

List and explain how you found the fractional amount for each of the colored pattern blocks using Figure A as your whole: blue rhombus, green triangle, red trapezoid, yellow hexagon.

References Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide - Response to Intervention in Mathematics. Retrieved 2 25, 2011, from rti4sucess: http://www.rti4success.org/images/stories/webinar/rti_and_mathematics_webinar_presentati on.pdf An Emerging Model: Three-Tier Mathematics Intervention Model. (2005). Retrieved 1 13, 2011, from rti4success: http://www.rti4success.org/images/stories/pdfs/serp-math.dcairppt.pdf Marjorie Montague, Ph.D. (2004, 12 7). Math Problem Solving for Middle School Students With Disabilities. Retrieved 4 25, 2011, from The Iris Center: http://iris.peabody.vanderbilt.edu/resource_infoBrief/k8accesscenter_org_training_resources_ documents_Math_Problem_Solving_pdf.html

Pattern Block Chart 1

Block Used as One Whole

Blocks Number Needed to Fraction Name of Cover Whole Each Block

Pattern Block Chart 2

Block Used as One Whole

Blocks Number Needed to Fraction Name of Cover Whole Each Block

Pattern Block Chart 3

Block Used as One Whole

Blocks Number Needed to Fraction Name of Cover Whole Each Block

Summarization Chart

Whole Whole Whole Whole