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– grade 6 • Teacher Guide

Problem Solving: Fluency With Decimals

LAUNCH (8 MIN) _____________________________________________________________ Before

• What steps would you take to solve this problem?

During • Which parts of the problem use multiplication? Which parts use addition?

After • How could you use the Distributive Property to solve this problem?

PART 1 (8 MIN) _______________________________________________________________ Kala Says (Screen 2) Use the Kala Says button to discuss how real-world measurements

often involve decimals and fractions. Coins need to be made to specification so that they are a uniform size for machines and so that counterfeits are more easily identified.

• A real-world measurement for a Lafayette Dollar might be 38.4 mm plus or minus 0.3 mm. Why do many real-world measurements have the phrase plus or minus as part of the measurement?

While solving the problem • After you write the equation, what is the next step?

After solving the problem • Which coin has a greater diameter? Does your answer confirm that comparison? • How else can you check your answer?

PART 2 (8 MIN) _______________________________________________________________ Before solving the problem

• Why would the carpenter want to know the answer to this question?

Kala Says (Screen 1) Use the Kala Says button to remind students that the problem-solving strategies and properties they learned earlier in the course also apply to problems with fractions and decimals. You may want to quickly review the properties of operations.

• Kala mentioned that the strategies and properties for whole numbers still work. What extra steps do you need to take since you are working with decimals?

PART 3 (8 MIN) _______________________________________________________________ While solving the problem

• What are the different forms that appear on the tiles? • How can saying each number or expression out loud help you classify them?

CLOSE AND CHECK (8 MIN) _______________________________________________ • Describe how the Commutative Property can help you evaluate the expression

$1.25 + $2.36 + $2.75. • How is solving equations with whole numbers similar to solving equations with decimals?


Problem Solving: Fluency With Decimals

LESSON OBJECTIVES 1. Convert between fractions and decimals. 2. Solve real-world and mathematical problems by writing and solving equations

of the form x + p = q for cases in which p, q, and x are all nonnegative rational numbers.

3. Solve real-world and mathematical problems by writing and solving equations of the form px = q for cases in which p, q, and x are all nonnegative rational numbers.

FOCUS QUESTION What kinds of problems can you solve using equations with decimals?

MATH BACKGROUND In this lesson, students take some of the skills they have developed throughout this topic—operating with decimals, converting between fractions and decimals, and comparing and ordering numbers—and apply them in new contexts.

The first context that students encounter is solving equations involving fractions and/or decimals. Students solve one-step equations using inverse operations, but the operations require working with either fractions or decimals and sometimes converting the numbers to the same form. They may need a review of operating with fractions, which they learned in the two topics prior to this one. In Part 3, students also determine whether numbers and expressions in several different forms are equivalent. They need to choose the form that is most appropriate and convert all numbers to that form.

This topic expanded students’ knowledge of basic operations and solving equations to include fractions and decimals. In the next topic, students will continue learning about new types of numbers as they encounter integers.

LAUNCH (8 MIN) ____________________________________________________ Objective: Solve problems involving operations with decimals.

Author Intent Students analyze and solve a problem involving multiplication and addition of whole numbers and decimals.

Questions for Understanding Before

• What steps would you take to solve this problem? [Sample answer: I would find the distance he runs each weekend day, and then find the total distances he runs during the week and during the weekend, and add those distances.]

During • Which parts of the problem use multiplication? Which parts use addition?

[Sample answer: Finding the distances run each weekend day uses multiplication, and finding the total distance run during the weekend and during the week uses multiplication. Finding the total distance run during the entire week uses addition.]

After • How could you use the Distributive Property to solve this problem? [Sample

answer: You can regroup the terms 1.75 × 5 + 1.75 × 2 × 2 as

Problem Solving: Fluency With Decimals continued


1.75(5 + 2 × 2) = 1.75 × 9. This shows that the runner runs 1.75 miles 9 times a week.]

Solution Notes The purpose of this activity is to present students with a straightforward problem involving multiplication and addition with decimals. How students organize the information in order to find the total number of miles may vary, and you can discuss how addition and multiplication each play a part in solving this problem.

Connect Your Learning Move to the Connect Your Learning screen. Start a conversation about the equation or equations students wrote in the Launch. Most likely, students used several equations such as 1.75 × 5 = 8.75 and 8.75 + 7 = 15.75. Ask if it is possible to write one long equation that is equal to the final answer, 15.75. Challenge students to write it.

PART 1 (8 MIN) ______________________________________________________ Objective: Identify equivalent fraction and decimal expressions and numbers.

Author Intent Students review how to solve an equation of the form x + p = q, this time where constants p and q are not whole numbers. They model a real-world situation using an equation of the form x + p = q, where one of the constants is a fraction and the other is a decimal. They solve the equation by subtracting a fraction from a decimal.

Instructional Design In the Intro, students see how to solve an equation using subtraction as the inverse operation of addition. Point out that the steps are the same as those students learned in previous topics, but the numbers involved are fractions and decimals. On Screen 2, show students the animation to introduce the problem. Call on students to write an equation that models the problem on the whiteboard and then solve it.

Questions for Understanding Kala Says (Screen 2) Use the Kala Says button to discuss how real-world

measurements often involve decimals and fractions. Coins need to be made to specification so that they are a uniform size for machines and so that counterfeits are more easily identified.

• A real-world measurement for a Lafayette Dollar might be 38.4 mm plus or minus 0.3 mm. Why do many real-world measurements have the phrase plus or minus as part of the measurement? [Sample answer: It is not possible to make items that are all exactly the same size. If a measurement is described as 38.4 ± 0.3 mm, then all coins must measure between 38.1 mm and 38.7 mm.]

While solving the problem • After you write the equation, what is the next step? [Sample answer: You need

to subtract 113

5 from each side. You can also start by rewriting both numbers in

the same form.]

After solving the problem • Which coin has a greater diameter? Does your answer confirm that comparison?

[The problem says that the Lafayette Dollar is longer than the Washington Dollar. My solution agrees with this statement.]



• How else can you check your answer? [Sample answer: If you subtract the two numbers, you should get a difference of

113

5 .]

Solution Notes The provided solution uses a Know-Need-Plan organizer to help students analyze the given information and model it as an equation. Part of the plan is to decide whether to solve the equation by writing both numbers as fractions or as decimals. Since the provided solution solves the equation using both forms, show students one way and open the provided solution for the other.

The key to writing an equation, using either fractions or decimals, is knowing which diameter is smaller. The problem gives the difference, and adding that difference to the smaller number yields an equation, as does subtracting the difference from the larger number.

Once students write an equation, make sure they know they can solve it using the same pattern that they use to solve an equation involving whole numbers.

Differentiated Instruction For struggling students: If students have trouble working with fractions, let them solve the problem first using decimals and then challenge them to get the same result using fractions. You can do a similar activity for those who struggle with decimals. This process builds confidence for students to choose the method that is best for a situation rather than the one they personally prefer.

For advanced students: Help students model the situation with a Words-to-Equation organizer. Remind them that there are several equivalent equations they can write that all model the same situation. Hint that the Intro used an equation of the form x + p = q.

Got It Notes While the Example asks students to write and solve an equation that models a real-world situation, this problem asks which of two equations models a given situation.

If you show answer choices, consider the following possible student errors:

Students who think that only one of the two equations can model the situation and do not realize that the two equations are equivalent may choose A or B.

PART 2 (8 MIN) ______________________________________________________ Objective: Solve real-world and mathematical problems by solving equations of the form x + p = q using equivalent equations.

ELL Support On the Student Companion page for the Part 2 Got It, there are two tasks for students to complete and discuss:

• Use a graphic organizer, such as Know-Need-Plan, to organize the information given in the problem.

• Check for the following: Why did you choose the organizer? How does the organizer help you solve the problem? How does the organizer help you explain your solution?

Beginning and Intermediate Have students use the Know-Need-Plan organizer. Provide partial responses for each of the organizer boxes and have students fill in the missing parts.



Advanced Have students use more than one graphic organizer and complete each one. Then have students discuss the strengths and weaknesses of each organizer.

Author Intent Students model a real-world situation with an equation of the form px = q, where constants p and q are decimals. They solve by dividing decimals and must remember to move the decimal point in both the dividend and divisor before using long division.

Instructional Design This problem includes a blank Know-Need-Plan organizer to get students planning how to solve the problem, including writing an equation and solving it.

Questions for Understanding Before solving the problem

• Why would the carpenter want to know the answer to this question? [Sample answer: The carpenter may need to make sure the hole is the right size for the fan.]

Kala Says (Screen 1) Use the Kala Says button to remind students that the problem-solving strategies and properties they learned earlier in the course also apply to problems with fractions and decimals. You may want to quickly review the properties of operations.

• Kala mentioned that the strategies and properties for whole numbers still work. What extra steps do you need to take since you are working with decimals? [You have to move the decimal points before you can divide decimals.]

Solution Notes Show the provided solution, which fills in the Know-Need-Plan organizer and separates the string of equivalent equations on the left with the steps of the division process on the right. When the solution process involves more complicated computations, they can be done on the side to keep your work organized.

Error Prevention Students may lose their place in solving the equation when they perform a more complicated calculation on the side. Encourage students to write themselves a brief note to help them keep their place in the solution. For example, after they write the equation

420.25

20.5 = 1w , they might write "Divide 420.25 by 20.5 to find w." Then, after

they perform the division, they can return to their note to remember why they divided decimals.

Got It Notes Unlike in the Example, in this problem students only identify the equation and do not have to solve it. Remind students that writing the correct equation is as important as performing the computations involved in solving it. Reveal the provided solution, which includes a Words to Equation organizer and consider having students solve the equation on their own even though the problem does not ask them to.

If you show answer choices, consider the following possible student errors:

Students who select A or C may have mistakenly concluded that Equation I does model the situation. Students who select A or D may have mistakenly concluded that Equation II does not model the situation.



PART 3 (8 MIN) ______________________________________________________ Objective: Solve real-world and mathematical problems by using the words to equations method to write and solve equations of the form px = q for cases in which p, q, and x are all nonnegative rational numbers.

Author Intent Given seven expressions, including fractions, decimals, and division statements, students determine which of them are equivalent to a given division statement.

Instructional Design Call on students to use the whiteboard to compare each number to the given division statement and drag the tile to the correct category. When all numbers and expressions have been classified, click the Check button. Work with students to find another way to compare any numbers that snap back to their original positions.

Questions for Understanding While solving the problem

• What are the different forms that appear on the tiles? [Some numbers are fractions, others are decimals. Some tiles have a division problem, and one has a division expression.]

• How can saying each number or expression out loud help you classify them? [Numbers or expressions that have the same word form must be equivalent. (The opposite is not necessarily true.)]

Solution Notes The purpose of this activity is for students to practice converting between fractions, decimals, and division statements. While a formal meaning of equivalent is not discussed in the activity, its practical meaning in this context should be clear. Encourage students to use multiple forms to compare numbers in this problem.

Got It Notes All three numbers or expressions in this problem are equivalent. If you show answer choices, consider the following possible student errors:

Students who choose B or D may not recall that moving the decimal points does not change the quotient. If students choose A or D, they may not recognize that the quotient 0.333... is equivalent to the fraction

13 .

CLOSE AND CHECK (8 MIN) _____________________________________

Focus Question Sample Answer You can use equations to solve problems that have decimal amounts and unknown values. This includes real-world situations that have measurements or amounts that are not whole numbers.

Focus Question Notes Look for students to explain that they can now solve problems that involve both fractions and decimals, because they can convert one form to the other and quickly compare the results.



Essential Question Connection • Describe how the Commutative Property can help you evaluate the expression

$1.25 + $2.36 + $2.75. [Sample answer: Study the expression to plan a solution pathway. You see that $1.25 and $2.75 have tenths and hundredths digits that are easy to add. Use the Commutative Property to reorder the expression and then add: $1.25 + $2.75 + $2.36 = $4.00 + $2.36 = $6.36.]

• How is solving equations with whole numbers similar to solving equations with decimals? [Sample answer: You undo operations in the same way. For example, subtraction always undoes addition and multiplication always undoes division. You can use the same properties of operations for both kinds of equations as well. Place value matters in each case, but you have to think about the decimal point in a different way when you perform operations with decimals.]

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