[IS.3 - All Students] [IS.4 - All Students] Remind students to bring their fraction, decimal, or percent examples found for Lesson 1. Separate students into groups of four, grouping differently than for the previous lesson. Ask students to describe how the fraction, decimal, or percent was used in the specific context in the example they brought. Give each student one minute to explain his/her example. Ask students who brought a decimal example to raise their hand. Randomly select several of these students to explain how the decimal or decimals were used in the example they selected. [IS.5 - Struggling Learners] Allow students to represent the decimal on the board and explain the place values (tenths, hundredths, thousandths). Provide at least one example for tenths, hundredths, and thousandths if they are not given by students. Discuss what the place values represent and relate them to the fractional values (i.e., 0.1 = 1/10, 1.01 = 1(1/100) ). Give students approximately 2–3 minutes to write each decimal example as a ratio in fraction form.
“Can any of our examples be simplified? If so, how can we do it?” [IS.6 - All Students] (Divide the numerator and denominator by the same factor, such as 2, if they are both even.)
Continue to work through the remaining examples, showing the correct ratio and the steps to simplify each to lowest terms.
“In our lesson today, we will continue to look at the relationship between fractional parts and decimals.”
Class Cake Activity
“Imagine that I have a cake and I need to divide it into 10 equal pieces, like this.” Illustrate a square cake cut into 10 strips and shade in the pieces used in the examples.
“If I ate one slice and gave my friend 2 slices, what ratio will we have eaten?” (3/10 or 3 to 10 or 3/7 or 3 to 7)
“What kind of fraction is that?” ( = part-to-whole, = part-to-part)
“How could we represent this ratio using a decimal?”(0.3, since the first decimal place value represents tenths and our ratio is 3 parts of the 10 parts in the whole cake, or if students say 3/7, then it is 3 parts eaten to 7 parts not eaten.)
“How can 3/7 and 3/10 give the same information?” (Both compare the number of parts eaten to something. The first fraction compares it to the part NOT eaten and the second fraction compares it to the total amount. You can get the total amount from the first fraction; just add 3 and 7 together.)
“What part-to-whole ratio will remain in the pan?” (7/10 or 7 to 10)
“How could we represent this ratio using a decimal?” (0.7)
Ask a student to explain why the decimal is 0.7. Give several more examples of numbers of slices eaten and remaining until the class understands the connection between the shaded sketch in tenths, the fractional ratio in tenths, and the decimal in tenths.
Next, illustrate the cake cut into 100 equally sized square pieces, use hundreds grids on transparencies to make it easier to demonstrate several examples (M-6-4-2_Hundreds Grid.docx). Shade in the number of pieces used in each example. “Suppose I cut the cake into 100 equal slices. If I ate 2 slices and gave 2 slices away, what ratio will we have eaten?” (4/100, some students may realize that 2/50 and 1/25 are also equivalent representations)
“How could we represent this ratio using a decimal?” (0.04, because the digit 4 is in the hundredths place and we have eaten or removed 4 parts out of 100, and shaded or removed 4 out of 100 parts on our cake sketch on the hundreds grid.)
“What fractional ratio and decimal represent the number of slices remaining in the pan?” (96/100 and 0.96, 96 unshaded or uneaten square pieces out of 100)
Ask a student to explain why the fraction 96/100 and the decimal 0.96 represent the situation. Give several more examples of number of slices eaten and remaining until the class understands the connection between the picture in hundredths, the fractional ratio in hundredths, and the decimal in hundredths.
“If I ate the same amount of cake (or the same ratio of cake eaten compared to uneaten), but the cake was originally cut into 200 slices, how many slices would I have eaten?”(4 slices, because each of the 100 slices would be split into 2 smaller slices, so 2 × 2 = 4 smaller slices equal in overall size or area to the original two pieces)
By adding more lines, demonstrate how the hundreds grid would be divided into smaller pieces to arrive at 200 pieces, even though the amount shaded does not change. This is a good opportunity to show that if both the numerator and denominator of the ratio 2/100 were multiplied by the factor 2, the resulting ratio would be 4/200.
“What if the cake were only sliced into 50 slices, how many pieces would I need to eat to have the same ratio of the cake?” (One slice; reasoning it out, each slice would be twice as large as the original pieces. Demonstrate the shading on the hundreds grid by highlighting lines to show the hundreds grids regrouped into 50 larger pieces with the same space shaded as it was for 2 small pieces out of 100, which will now be equivalent to one larger piece. Also discuss dividing numerator and denominator of the ratio 2/100 each by 2 to get the new ratio 1/50.)
Continue to emphasize the connection between the three representations: visual, fractional, and decimal.
Clarify any misconceptions before moving on to the Flower Garden Partner Activity.
Flower Garden Partner Activity
Assign each student a partner to complete this activity. Each pair will need a Flower Garden Activity sheet (M-6-4-2_Flower Garden.docx) and a variety of colored markers. Students will design a garden using four colors within a hundreds grid, representing 100 flowers in 100 equal sections in a garden plot. Each group will be allowed to select the specific number of each color flower to create its design (they need to select a different amount of each flower color; even numbers will work best in this application). Once the design is created, students will write a fractional ratio and the corresponding decimal for each flower color in their garden. They will also calculate the changes in the number of flowers needed to keep the same ratios if the number of places for flowers in the garden changed to 50 and to 200. Explain that students will be sharing their design and calculations with others in the class at the end of the work time. Provide either chart paper with grid lines or hundreds grids on transparencies with markers for students to use for presentation of their garden designs. Give students approximately 20–30 minutes to work.
While pairs of students are working, move around the room monitoring progress and offering guidance to pairs who are having difficulty. Use questioning to get students to think about aspects of the problem they may not be incorporating. Encourage students to make any necessary corrections or clarifications and to simplify their ratios when appropriate.
Allow students 3–4 minutes to present. During presentations, continue to ask questions and encourage students to adjust and make corrections on their garden project.
Note to instructor: Since garden designs and flower counts will vary, you may ask a specified group of students or the remainder of the class to help check ratio accuracies during presentations. There is no answer key provided for this activity since numbers and designs will vary.
Partner Ticket Activity
Each pair of students will complete the Lesson 2 Partner Ticket (M-6-4-2_Lesson 2 Partner Ticket.docx and M-6-4-2_Lesson 2 Partner Ticket KEY.docx). Provide approximately 5–6 minutes to work.
Extension:
Use these suggestions to tailor this lesson to meet the needs of your students during the unit and throughout the year.
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Routine:Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. During this lesson the following terms should be entered into students’ vocabulary journals: factor, hundredths place, and tenths place. Keep a supply of vocabulary journal pages (M-6-4-1_Vocabulary Journal.docx) on hand so students can add pages as needed. Verbal language that describes the relationship of fractions to decimals and decimals to benchmarks (i.e., less than half, greater than half) should be included in the activities in this lesson. Throughout the school year, point out instances of decimals and their meanings as they are used in mathematics examples and your own life experiences, as well as fractions or ratios in context that could be converted to decimals. Ask students to bring up examples that they see in the news or other courses during the school day. Take time to discuss the use and meaning of the fractions and decimals in each particular context. Continue to emphasize the relationship between the visual form (hundreds grid or other sketches), fraction form, and decimal form in a variety of contexts that are being discussed.
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Small Group: Fractions to Decimals Activity: Use this activity for students who need additional practice changing fractional ratios to decimals or decimals to fractions. Emphasize that each fraction can be changed to an equivalent form with a denominator which is a multiple of 10 (10, 100, or 1000) to easily write each as a decimal. Focus on the fact that the factor needed to adjust the denominator to a multiple of 10 must also be multiplied or divided in the numerator to create an equivalent fraction, as this is often overlooked by students. Provide students with the Fractions to Decimals Activity Sheet (M-6-4-2_Fractions to Decimals.docx and M-6-4-2_Fractions to Decimals KEY.docx). Also have available several hundreds charts, which can be shaded to make the visual connection between the fractions and decimals for your visual learners (M-6-4-2_Hundreds Grid.docx).
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Expansion: Garden Challenge Activity: Use this activity for students who show proficiency and thorough understanding of the Flower Garden Partner Activity. The directions and setup are the same as the Flower Garden Activity but the flower garden design is provided. Students will begin by writing the ratio and decimal for each color in the original garden pattern. They will move on to calculate the number of each flower color needed to keep the same color ratio as was found in the original garden pattern when the total number of flowers used is changed to 150, 250, and 325 flowers. Provide each student or pair of students with a Garden Challenge sheet (M-6-4-2_Garden Challenge.docx and
M-6-4-2_Garden Challenge KEY.docx).
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Technology Connection: Quiz Yourself: If computers are available for students’ use, have individuals or pairs of students practice fraction and decimal relationships using Fraction Quiz online. Students will be given a fraction or decimal and will be asked to give the other form. Fractions need to be stated in simplest form. Place values for repeating decimals, such as 1/3, need to be given with three digits (i.e., 0.333). It may be helpful to go over a few strategies with students, such as using mental math to simplify 12/15 to 3/5 and then 3/5 to 6/10, in order to convert it to the decimal 0.6. Students should begin on level 1, but some students may be comfortable moving up to level 2 or 3 after some practice. If a question is answered correctly, “Correct!” appears on the screen. If a question is answered incorrectly a dialog box appears in the upper left corner explaining the correct answer. In either case, to move on to a new question, students must select the “ready” button. When students are ready to move to a more difficult level or are out of time, they can push “Score” to see a summary of their correct and incorrect responses. The Web site is:
http://www.shodor.org/interactivate/activities/FractionQuiz/
When students get to the welcome screen, instruct them to select the following options:
This activity relates the use of decimals to planning the division of a collection represented as a grid that is 10 by 10 or a 100-unit square. The emphasis of the activities is on the 100-unit square as the whole representing the size of a cake and a garden plot. The first part of the activity demonstrated that 1/10 represented one piece out of ten slices in a whole cake. Then 1/100 of the whole unit was shown to be equivalent to 1/100 of the cake. In the next activity one flower represented 1/100 of a whole garden plot that contained 100 flowers. The activities asked the students to partition the whole and represent the parts in fractional forms that represented tenths or hundredths. This created the background for students to use the same models to represent decimals and understand the equivalence between fractions with denominators that are powers of 10 and decimal notation. Students were asked to extend these concepts by making calculations of part-to-whole relationships of larger and smaller sizes, which kept the ratio of parts the same as in the original scenario. The relationship between physically partitioning a whole into tenths or hundredths, fractions with denominators of 10 or 100, and decimals in tenths or hundredths was a focus throughout this lesson. These relationships lead into the percent concepts explored in the next lesson.
