Richland Parish School Board



Grade 6

Mathematics

Unit 3: Fractions, Decimals, and Parts

Time Frame: Approximately five weeks

Unit Description

The focus of this unit is on concepts and basic relationships of fractions and decimals. There is an emphasis on estimating outcomes prior to developing the computation algorithms that give the exact answers. The development of the concept of rate, ratio, proportion, and percent continues by representing and working with miles/hour, dollars/pound, miles/gallon, and other derived rates and percents.

Student Understandings

Students can compare fractions, decimals, and positive integers by the use of symbols. Students can identify place value to the ten-thousandths place. They can solve ratio, proportion, and percent problems with models and pictures. They can use rates to solve real-life problems.

Guiding Questions

1. Can students find the GCF and LCM of whole numbers?

2. Can students generate equivalent forms of fractions and decimals?

3. Can students utilize various strategies to work with rates and ratios such as mph, mpg, and dollars/pound?

4. Can students use models or pictures to solve problems involving ratio, proportion and percents with whole numbers?

MATH: Grade 6 Grade-Level Expectations and Common Core State Standards (CCSS)

|Grade-Level Expectations |

|GLE # |GLE Text and Benchmarks |

|Number and Number Relations |

|3. |Find the greatest common factor (GCF) and least common multiple (LCM) for whole numbers in the context of |

| |problem-solving (N-1-M) |

|4. |Recognize and compute equivalent representations of fractions and decimals (i.e., halves, thirds, fourths, fifths, |

| |eighths, tenths, hundredths) (N-1-M) (N-3-M) |

|13. |Use models and pictures to explain concepts or solve problems involving ratio, proportion, and percent with whole |

| |numbers (N-8-M) |

|Measurement |

|20. |Calculate, interpret, and compare rates such as $/lb., mpg, and mph (M-1-M) (A-5-M) |

|CCSS for Mathematical Content |

|CCSS # |CCSS Text |

|The Number System |

|6.NS.7b |Understand ordering and absolute value of rational numbers. |

| |b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write |

| |–3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC. |

|ELA CCSS |

|Reading Standards for Literacy in Science and Technical Subjects 6–12 |

|RST.6-8.4 |Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific |

| |scientific or technical context relevant to grades 6–8 texts and topics. |

Sample Activities

Activity 1: Fraction Vocabulary Awareness (CCSS: RST.6-8.4)

Materials List: Fractions BLM, pencil

Before beginning the unit, have students complete a vocabulary self awareness chart (view literacy strategy descriptions). Provide students with the Fractions BLM. Do not give students definitions or examples at this point.

|Word |+ |( |– |Example |Definition |

|Greatest Common Factor | | | | | |

|(GCF) | | | | | |

|Least Common Multiple | | | | | |

|(LCM) | | | | | |

|Denominator | | | | | |

|Numerator | | | | | |

|Equivalent Fractions | | | | | |

|Ratio | | | | | |

|Proportion | | | | | |

|Percent | | | | | |

Ask students to rate their understanding of each word with either a “+” (understands well), a “(” (some understanding), or a “–” (don’t know). During and after completing the activities throughout this unit, students should return to the chart to fill in examples and definitions in their own words. The goal is to have all plus signs at the end of the unit. After all the students have completed the chart, have them share their examples and definitions with each other to check for accuracy.

Activity 2: Greatest Common Factor (GLE: 3)

Materials List: squares tiles, Grid Paper BLM, Greatest Common Factors BLM, pencil

Have students work in groups to create all possible rectangular arrays for the numbers 12 and 28 using square tiles or the Grid Paper BLM. Have them list the dimensions (factors) of the arrays for each number and circle all of the factors that the 2 numbers have in common. Ask which is the largest or greatest common factor (4). Continue with other pairs of numbers. Help students determine the greatest common factor (GCF) of the pair of whole numbers by comparing the arrays and selecting the two largest common dimensions. Students should select the arrays 3 by 4 and 4 by 7 because they have the largest dimension of 4 in common. Four is the greatest common factor of 12 and 28.

Have students find the greatest common factor of 5 and 20 without creating arrays. Begin by having students write all the different factors that multiply to get each number.

|5 |= 1 ( 5 |

| |The factors of 5 are 1, 5. Five is a prime number. |

|20 |= 1 ( 20; 2 ( 10; 4 ( 5 |

| |The factors of 20 are 1, 2, 4, 5, 10, 20 |

Have students circle all of the factors that both 5 and 20 have in common.

|5: | 1, 5 |

|20: | 1, 2, 4, 5, 10, 20 |

The highest number that both sets of factors have in common is the GCF. In this case, the GCF is of 5 and 20 is 5.

Distribute the Greatest Common Factors BLM. Have students work together or independently to practice finding the GCF. Discuss the answers as a class.

For additional practice with finding the greatest common factors, the students can visit the following website:



This website reviews the steps for finding the GCF and then provides practice problems with feedback to the students.

Activity 3: Least Common Multiples (GLE: 3)

Materials List: Least Common Multiples BLM, pencil

Write 5 and 8 on the board or overhead. Have the students make a list of multiples for each number.

5 – 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85

8 – 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88

Have a student go to the board and circle all the numbers that are in both lists. Tell students that the numbers that are circled are common multiples of 5 and 8. The smallest number of these common multiples is called the least common multiple. Point out to students that they cannot list all of the multiples because the pattern continues forever.

Have students try the following example on their own.

Find the least common multiple of 5 and 6. Allow students time to try to find the LCM and then discuss the solution as a class.

5 – 5, 10, 15, 20, 25, 30, 35, 40

6 – 6, 12, 18, 24, 30

The smallest number they both have in common is 30, so 30 is the LCM of 5 and 6. Provide additional examples for students if needed.

Distribute the Least Common Multiples BLM. Have the students work together to solve the problems. Discuss the solutions as a class.

Activity 4: Using Prime Factorization to Find LCM (GLE: 3)

Materials List: Finding LCM BLM, pencil

Begin the activity by providing two numbers. Have students divide each number by the smallest possible prime factors until the quotient is 1. For example, given the numbers 12 and 15:

12: 12 ÷ 2 = 6, 6 ÷ 2 = 3, 3 ÷ 3 = 1

15: 15 ÷ 3 = 5, 5 ÷ 5 = 1

The prime factorization is taken from the divisors:

12 has prime factors of 2, 2, 3; therefore, the prime factorization of 12 is 2 × 2 × 3.

15 has prime factors of 3 and 5; therefore, the prime factorization of 15 is 3 × 5.

To find the least common multiple, students should do the following:

1. List the prime factors of each number.

12 = 2 × 2 × 3

18 = 3 × 5

2. Find the common factors.

12 = 2 × 2 × 3

18 = 3 × 5

3. Multiply the common factor(s) and the extra factors to find the LCM.

3 × 2 × 2 × 5 = 60

Present the following problem:

George has piano lessons every 10 days and Anna has lessons every 8 days. George and Anna both had piano lessons today. How many days will it be until they both have piano lessons on the same day again?

To find the least common multiple, students should do the following:

1. List the prime factors of each number.

10 = 2 × 5.

8 = 2 × 2 × 2.

2. Find the common factors.

10 = 2 × 5.

8 = 2 × 2 × 2.

3. Multiply the common factor(s) and the extra factors to find the LCM.

2 × 5 × 2 × 2 = 40

The LCM is 40. It will be 40 days when they both have a piano lesson on the same day.

Distribute the Finding LCM BLM. Have students work with a partner to solve. Discuss the answers as a class.

Activity 5: Applications of GCF and LCM (GLE: 3)

Materials List: GCF/LCM Application BLM, paper, pencil

Have students apply the GCF and LCM in real-life situations. Use the discussion strategy, Think Pair Square Share (view literacy strategy descriptions). This strategy helps improve student improve learning and remembering by participating in a discussion about a given topic. After being given an issue, problem, or question; students are asked to think alone for a short period of time, and then pair up with someone to share their thoughts. Have pairs of students share with other pairs, forming, in effect, small groups of four students.

Present the following problems to the class one at a time. Have the students solve the problems independently and then pair up with partners to discuss their solution. Have each pair of students join another pair of students to discuss their solutions. Have groups share their solutions with the class.

1. Ms. Nguyen’s class purchased 45 pencils and 30 erasers to put in care packages. They want each package to contain the same number of pencils and the same number of erasers. What is the greatest number of packages they can make? 15 packages

2. The class is having a party. The students have voted to have hot dogs that come in packages of 10, with buns that come in packages of 8. What is the least number of packages of hot dogs and buns that you would need to buy to have the same number of each? 40 of each item: 4 packages of hot dogs and 5 packages of buns.

Have students discuss which problem involved finding the GCF and which problem involved finding the LCM.

Distribute the GCF/LCM Application BLM. Have students work in groups of 2 or independently to solve the problems. After discussing the solutions to the problems above, have the students create a text chain (view literacy strategy descriptions). Put students in groups of four. On a sheet of paper, ask the first student to write the opening sentence for the math text chain.

For example, a student might begin the chain with this sentence.

You are making cherry pies for a bake sale.

The student then passes the paper to the student sitting to the right, and that student writes the next sentence in the story, which might read:

Piecrusts are sold in packages of 3, and pie filling is sold in 4-can packages.

The paper is passed again to the right to the next student who writes the third sentence of the story.

What are the least number of piecrusts and cans of filling that you can buy to have the same number of each, and how many packages of each should you buy?

The paper is now passed to the fourth student who must solve the problem and write out the answer. The other three group members review the answer for accuracy.

Answer: 12 pie crusts and cans of pie filling. You would need to buy 4 packages of pie crust and 3 packages of pie filling.

This activity allows students to use their writing, reading, and speaking skills while learning and reviewing the concept of division. When text chains are completed, be sure students are checking them for accuracy and logic. Groups can exchange text chains to further practice reviewing and checking for accuracy.

Activity 6: Fraction Strips (GLEs: 4; CCSS: 6.NS.7b)

Materials List: different colored cardstock, scissors, rulers, pencil, paper

Have students work in groups of 2 and give each member of the group a sheet of different colored cardstock. Instruct each student to make fraction strips by cutting the cardstock into 6 one-inch wide strips. Remind students to use a ruler to make the appropriate measurements. Ask students to fold each strip into one of the following fractions: halves, fourths, eighths, thirds, sixths, and twelfths and then write the fraction on each piece (each of the thirds will have [pic] on it).

|½ |½ |

| |

|⅓ |⅓ |⅓ |

| |

|¼ |¼ |¼ |¼ |

| |

|1/6 |1/6 |1/6 |1/6 |1/6 |1/6 |

| |

|⅛ |⅛ |⅛ |⅛ |⅛ |⅛ |⅛ |⅛ |

| |

|1/12 |

|Athlete |Time (seconds) |

|M. Ball |12.11 |

|M. Evans |12.15 |

|W. Fergusen |11.93 |

|E. Fielden |11.83 |

|T. Marlbrough |11.75 |

|J. Scott |11.72 |

|T. Teal |12.08 |

|N. Williams |12.02 |

Ask the following questions.

• Who finished the race 1st? (J. Scott)

• Who finished the race last? ( M. Evans)

• List the runners in order from fastest to slowest.( J. Scott, T. Marlbrough, E. Fielden, W. Fergusen, N. Williams, T. Teal, M. Ball, M. Evans)

• Write two comparison statements comparing two times. (Sample answer: 12.02 seconds is less time than 12.15 seconds.)

• Write your comparison statement from above using symbols. (Sample answer: 12.02 seconds < 12.15 seconds)

Discuss why if 12.02 is a faster time, is it less than 12.15? Students need to realize that in time, a smaller number means it took less time to finish.

Have students share their comparison statements with the class. Listen for accuracy.

Distribute the Swim Meet Results BLM and have students work with a partner to complete. Discuss the answers as a class.

.

Activity 8: Same or Different? (GLEs: 4; CCSS: 6.NS.7b)

Materials List: index cards, paper, pencil

Use the SQPL strategy (view literacy strategy descriptions) to challenge the students to explore fraction and decimal equivalency. SQPL stands for Student Questions for Purposeful Learning and involves presenting the students with a statement that provokes interest and curiosity. Put the following statement on the board or overhead for students to read. “If I ate 3/5 of a pizza and you ate 0.55 of a pizza, then I ate more pizza.” Have students work with partners and brainstorm 2-3 questions that would have to be answered to prove or disprove the statement. Some questions might be the following: Were the pizzas we each ate the same size pizzas? What is 3/5 written as a decimal? What is 0.55 written as a fraction? Which is greater 3/5 or 0.55?

Have each pair of students present one of their questions and write this question on chart paper or the board. Give the class time to read each of the questions presented. Give pairs of students time to select the ideas that they would use to prove or disprove the statement.

Discuss how to change a 3/5 to a decimal to make the numbers easier to compare.

[pic] = 5 [pic] The fraction [pic] can be written as the decimal 0.6.

3.0

0

To change 0.6 to a fraction, identify the place value position of the last digit. Since the 6 is in the tenths place, the fraction equivalent would be[pic]. 6 and 10 are both divisible by 2, so the fraction would simplify to [pic]. Give the students several fractions and decimals to try.

Divide the class into two groups. One group will be the “fraction” group and the other the “decimal” group. Ask the “fraction” group to provide a fraction expressed in halves, fourths, fifths, tenths, or hundredths to the “decimal” group. Give the “decimal” group a specified time limit to provide an equivalent decimal representation of the fraction. Note that the “fraction” group could give [pic] instead of [pic] or [pic]instead of [pic]. Have the two groups decide if the decimal expression provided is equivalent to the original fraction. If the decimal is equivalent, then the “decimal” group earns 1 point. If not, the “fraction” group earns 1 point. Next, have the “decimal” group provide a decimal representation of a fraction to the “fraction” group. Allow the “fraction” group a specified time limit to provide an equivalent fraction in lowest terms. Again, ask the two groups to decide if the fraction is equivalent. If correct, the “fraction” group earns a point; if not, then the “decimal” group earns a point. After the “fraction” group provides five fractions, have the groups switch roles.

If an answer is determined to be correct, ask students to record each answer on an index card. Once complete, shuffle the index cards and then pass them out to individual students. Have students form a “human number line” across the classroom. Order the fractions and decimals with the fraction/decimal equivalents standing one in front of the other. As a whole group, discuss the placement of students. Note that students will be paired on the number line. Place the cards on the wall so that students can view them and their placement at the end of the activity. Keep the cards for future use.

Reread the opening SPQL statement and the questions the students generated. As a class, answer each question and decide if the statement is true.

Activity 9: Box Scores (GLEs: 4, 20; CCSS: 6.NS.7b)

Materials List: sports data, index cards

Depending on the season, use sports scores from the newspaper’s sports section or the websites listed below to get data to order decimals. For example, show the average yardage for rushes by different football players, rebounds for basketball players, on-base percentage or batting averages for baseball, times for track stats—dashes and pole vaults. Write each statistic or number on an index card. As a class, order the stats on a number line. An example of batting averages could be .222, .234, .245, .255, .266, .289, and so on. Help students understand how these data can be interpreted as rates. For example, a baseball player batting .250 gets a hit once every four times he bats, on average.

0.250 = [pic] [pic] = [pic]

After exploring the decimals, have students mix these index cards with those from the previous activity. Repeat the “human number line” with the additional cards. Because the number of cards may now surpass the number of students, discuss with students how to solve this problem: i.e., pair the equivalent fractions and decimals, and eliminate one-half of them.

New Orleans Saints Statistics



New Orleans Zephyrs Statistics



New Orleans Hornets Statistics



Activity 10: Ratios and Proportions (GLE: 13)

Materials List: Ratio Notes BLM, Ratio Practice BLM, Ratio Cards BLM, pencil

Distribute the Ratio Notes BLM to the students. Discuss ratios and proportions by completing the Ratio Notes BLM as a class.

These are some points to emphasize with the students. A ratio is a comparison of two quantities. A ratio can compare a part to a part or a part to a whole. Ratios can be written in 3 ways, as shown in the table below.

| | | |

| | |5 to 8 |

|Part to Part | |5:8 |

| | |[pic] |

| | | |

| | |5 to 13 |

|Part to Whole | |5:13 |

| | |[pic] |

Discuss equivalent ratios using the following examples:

To make lemonade, lemon juice is to be mixed with water in a ratio of 1 to 4. This means that for every 1 part lemon juice, there will need to be 4 parts of water. If there were 2 cups of lemon juice, there would be 8 cups of water. The ratio of lemon juice to water in the example above was 1:4, but this ratio could be written as 100:400, or 20:80, or 5:20.

These ratios are equivalent because they have the same meaning – the amount of water is four times the amount of lemon juice.

| |

|To find equivalent ratios, multiply or divide both numbers of the ratio by the same number. |

|This is similar to finding equivalent fractions. |

For example:

[pic] = [pic] = [pic]

Discuss proportions using the following example. A proportion is an equation that shows that two ratios are equal or are proportional. You can determine if two ratios are proportional by deciding if they are equivalent ratios. [pic] is equivalent to[pic].

|Ratio – 3 of 5 rows |Ratio – 15 of 25 circles |Proportion |

| | | |

| | | |

| | |[pic] |

| | | |

| | | |

| | | |

| | | |

Another way to determine if two ratios are proportional is to use cross products. If the two ratios are proportional, the cross products must be equal. In the proportion above, 3 × 25 = 75 and 5 × 15 = 75. [pic] [pic] The ratios are proportional.

Have students determine if the following ratios are proportional or not.

1. [pic] 2 × 28 = 56; 7 × 10 = 70. The ratios are not proportional.

2. [pic] 5 × 36 = 180; 9 × 15 = 135. The ratios are not proportional.

3. [pic] 2 × 15 = 30; 3 × 10 = 30. The ratios are proportional.

4. [pic] 12 × 48 = 576; 24 × 38 = 912. The ratios are proportional.

Distribute the Ratio Practice BLM and have students work with partners to complete. Discuss the answers as a class.

Activity 11: Percents (GLE: 13)

Materials List: Percents BLM, pencil

Tell students that a percent is a ratio whose denominator is 100. The term percent means out of 100. The symbol % is used for percent.

In the example in Activity 10, one of the ratios is 15 of 25 circles. This is a part to whole ratio. To express this as a percent, set up a proportion. If 15 out of 25 circles are selected, how many circles out of 100 circles would be selected?

[pic] Since 25 times 4 equals 100, multiply 15 by 4 to make the problem proportional.

[pic] 60 out of 100 equals 60%. So 60% of the circles are selected.

Present the following part to whole ratios to students and have them work together to write each ratio as a percent.

1. Anthony got 8 questions out of 20 correct on his Science test. What percent did he get right?

[pic] [pic] 40 out of 100 equals 40%.

Anthony got 40% of the questions correct on his Science test.

2. The 6th graders are going on a field trip. 13 out of 50 students have turned in permission slips so far. What percent of the students have turned in permission slips?

13 of 50 [pic] 26 out of 100 equals 26%.

26% of the students have turned in permission slips.

3. John made 7 out of 10 free throw shots in the game last night. What percent of his free throw shots did John make?

7:10 [pic] 70 out of 100 equals 70%.

John made 70% of his free throw shots.

Part to part ratios must be written as part to whole ratios before they can be written as a percent.

Present the following problems to the class.

A. The ratio of boys to girls in our class is 3 to 2. What percent of the class are boys?

• First, write the ratio as a part to whole ratio. The whole is 5, so 3 out of 5 students are boys.

• Set up a proportion. [pic]

• Solve the proportion. [pic]

• 60 out of 100 equals 60%.

60% of the students in our class are boys.

B. Billy attended 4 of the band rehearsals and missed 1. What percent of the rehearsals did Billy attend?

Billy attended 4 out of 5 rehearsals.

4:5 [pic] 80 out of 100 equals 80%.

Billy attended 80% of the rehearsals.

Distribute the Percents BLM and have students work with partners to complete. Discuss the answers as a class.

The professor know-it-all strategy (view literacy strategy descriptions) can be used to check for understanding. Cut apart the Ratio Cards BLM and place the cards in a box. Instead of forming groups, choose an individual student to come to the front of the class. Have the student pull a ratio card from the box and read it aloud. After the student reads the ratio, the class will ask him/her a variety of questions. For instance: Can you express your ratio as a part to part and as a part to whole ratio? Can you write your ratio in at least 3 different ways? Can your ratio be simplified? Can you write your ratio as a percent? The class should hold the student accountable for her/his answer. After a couple of questions, a new student should be asked to be the professor know-it-all and go to the front of the class to repeat the process.

Activity 12: Grocery Math (GLEs: 20)

Materials List: grocery ads, Grocery Ad BLM, Rates BLM, pencil, paper, calculators

Provide students with grocery ads from a current newspaper or the Grocery Ad BLM. Have students work in pairs with a specified task: Given $20 to go to the grocery store, buy a variety of fruits and vegetables.

Direct students to select a combination of at least four different fruits and/or vegetables, estimate the cost, estimate the tax at 10% and record the estimate total. Have the students calculate the total cost and present their findings to the class. The findings should include the estimated cost and the final cost. Sketch a rectangle to represent the $20. Divide the rectangle into approximate parts to show prices (i.e., if $5.00 is spent on apples,[pic] of the rectangle should be marked and labeled “apples”). Use the rectangle as a visual representation of the information.

Considering the purchases, have students respond to the following: For all items sold by the pound, do you have enough money to buy 3 lbs. of each? 4 lbs. of each? 5 lbs. of each? Use a calculator to determine the cost of each fruit/vegetable at 3, 4, and 5 lbs.

Tell students that a rate is a ratio in which the units of measure of the quantities are different. A rate shows how the quantities are related to one another. One rate on the Grocery Ad BLM is “3 lbs. for $1.69.” You are comparing the weight of the onions with the price. Have students write a rate for the cost of potatoes. (5 lbs./$2.99 or $2.99/5 lbs.) If one of the measures in a rate is 1 unit, the rate is called a unit rate. Unit rates allow you to easily compare rates. A unit rate on the BLM is shown for apples. One pound of apples cost $1.19 or apples cost $1.19 per pound. Have students find another unit rate on the BLM. (1 lb of potatoes for $2.99, 1 lb of tomatoes for 1.29, 1 lb of grapes for $0.99) Have students write a unit rate for potatoes and onions. ($0.59 per pound, $0.56/lb) Have students find the unit rate for corn and bell peppers. (Corn is $0.33 each. Bell peppers are $0.50 each.)

If apples, peaches, tomatoes, and grapes can be bought in three-pound bags for $3.99, decide if it is cheaper to buy each item by the bag or by the pound. Have students write everything as unit rates to compare the costs.

Distribute the Rates BLM and have students work with a partner to complete. Discuss the answers as a class.

Teacher Note: When computing 10% tax, mental math should be encouraged. Students should be able to explain how they arrived at the answer. Accept answers that show a clear understanding that “moving the decimal 1 place to the left” is because they are multiplying by .1 (one tenth or 10- hundredths).

Activity 13: Tangram Ratio (GLE: 13)

Materials List: Tangrams BLM, square sheet of paper for each student, scissors, pencil

A tangram puzzle is made up of seven pieces: 2 large right triangles, 1 medium triangle, 2 small triangles, 1 parallelogram, and 1 square. A large square can be formed using all 7 tangram pieces. Have each student make his/her own set of tangrams so that he/she can have a direct understanding of the relationships between the parts and the whole and among the pieces to one another. Simple directions for creating the tangrams are given below. Directions with visual representations are on the Tangrams BLM.

Fold and cut a square sheet of paper by following these instructions:

1. Fold the square in half diagonally, unfold, and cut along the crease into two congruent triangles.

2. Take one of these triangles. Fold in half, unfold, and cut along the crease. Set both of these triangles aside. These are the 2 large triangles in the tangram set.

3. Take the other large triangle. Lightly crease to find the midpoint of the longest side. Fold so that the vertex of the right angle touches that midpoint, unfold and cut along the crease. You will have formed a middle-sized triangle and a trapezoid. Set the middle-sized triangle aside with the two large-sized triangles.

4. Fold the trapezoid in half, unfold, and cut. To create a square and a small-sized triangle from one of the trapezoid halves, fold the acute base angle to the adjacent right base angle and cut on the crease. Place these two shapes aside.

5. To create a parallelogram and a small-sized triangle, use the other trapezoid half. Fold the right base angle to the opposite obtuse angle, crease, unfold, and cut.

6. You should have the 7 tangram pieces: 2 large congruent right triangles

1 middle-sized triangle

2 small congruent triangles

1 parallelogram

1 square

7. The pieces may now be arranged in many shapes. Try recreating the original square.

After a quick review of the terms area and ratio, have students determine the ratio of the area of each piece to that of the other pieces by comparing the sizes of the pieces. For example, students should determine the ratio of a small triangle’s area compared to the medium triangle. Next, ask students to write the ratios of each of the tangram pieces to the whole (the completed puzzle). As an example, students should find that the ratio of a large triangle to the large square (the completed puzzle) to be 2 to 4, which reduces to a ratio of 1 to 2. This ratio compares the area of a large triangle to the area of the square (the completed puzzle). Have the students use the ratios to write proportions. When students complete the activity, have them rewrite the ratios of the area of single pieces to other pieces or to the whole. Have them write the ratios as fractions and as percents.

Some sample answers are:

large triangle to the whole figure 1:4 = [pic]= 25%

large triangle to large triangle 1:1 = [pic] = 100%

small triangle to parallelogram 1: 2 = [pic] = 50%

small triangle to square 1:2 = [pic] = 50%

Activity 14: Vacation Math (GLE: 20)

Materials List: Vacation Math BLM, Internet access, maps, or atlases, paper, pencil, calculator

We’re going on road trip! Distribute the Vacation Math BLM. On the BLM, students will find the distance to a destination, estimate the time it will take to get to the destination, and will find mph and mpg for the trip. Allow students, in groups of 2, to use the Internet, maps, or atlases to locate the distance from home to a destination of their choice. Have students use the Internet or newspaper car ads to find the fuel economy of the vehicle of choice. After students complete the Vacation Math BLM, have the groups share the information about their trips.

Sample Assessments

General Assessments

• Observe individual and group work throughout the unit.

• Have students create portfolios containing samples of experiments and activities.

• Facilitate small group discussions to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask might be:

o How did you get your answer?

o What are the key points or big ideas in this lesson?

o How would you prove that?

o What do you think about what ___ said?

o Do you agree with your group’s answer? Why or why not?

o How would you convince the rest of us that your answer makes sense?

• Have students create learning logs (view literacy strategy descriptions) using such topics as:

o The most important thing I learned in math this week was…

o Explain today’s lesson to a student who was absent today.

• Have students submit a written reflection to the following question as a Performance Task Assessment of the unit:

o When comparing two decimals such as 0.36 and 0.349, how can you decide which decimal represents the larger number?

Activity-Specific Assessments

• Activity 6: Have students write a story using fractions appropriately, making at least five comparisons about the numbers by using the symbols , or =.

• Activity 8: Have students bring a copy of a recipe from home. Have them write the amount of each ingredient in fraction, whole number, and or mixed number formats and convert the values to decimal numbers.

• Activity 13: Have students create a poster with original definitions and/or pictures to convey the meanings in order to demonstrate understanding of the math terms covered such as: ratio, area, triangle, trapezoid.

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