6.1.3A Multiplication & Division
Overview
Students at this level model multiplication and division of fractions and connect these models to procedures for multiplying and dividing fractions. Place-value patterns are used to multiply and divide finite decimals by powers of 10. The relationship between decimals and fractions, as well as the relationship between finite decimals (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), is used to understand and explain the procedures for multiplying and dividing decimals. Common procedures are used to multiply and divide fractions and decimals efficiently and accurately. Problem solving with positive rational numbers is extended to include arithmetic with decimals, fractions, and mixed numbers. Students build on understanding that percents are ratios per 100 to solve problems in various contexts that require finding percent of a number or what percent one number is of another.
All Standard Benchmarks
6.1.3.1 Multiply and divide decimals and fractions using efficient and generalizable procedures, including standard algorithms.
6.1.3.2 Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions.
6.1.3.3 Calculate the percent of a number and determine what percent one number is of another number to solve problems in various contexts.
6.1.3.4 Solve real-world and mathematical problems requiring arithmetic with decimals, fractions and mixed numbers.
6.1.3.5 Estimate solutions to problems with whole numbers, fractions and decimals and use the estimates to assess the reasonableness of results in the context of the problem.
Benchmark Group A
6.1.3.1 Multiply and divide decimals and fractions using efficient and generalizable procedures, including standard algorithms.
6.1.3.2 Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions.
What students should know and be able to do [at a mastery level] related to these benchmarks
- Model multiplication with fractions and connect models of multiplication with fractions to procedures for multiplying fractions;
- Model division with fractions and connect models of division of fractions to procedures for dividing fractions;
- Use fractions, mixed numbers, and decimals to represent quotients in division of whole numbers;
- Recognize and use the place-value patterns in multiplying and dividing finite decimals by powers of 10;
- Use place value and their understanding of multiplication of fractions to justify procedures for multiplying finite decimals;
- Use place value and their understanding of representing quotients as fractions to justify procedures for dividing decimals;
- Recognize fractions, decimals, and percents as ways of representing rational numbers;
- Convert among fractions, decimals, and percents;
- Develop efficient, accurate, and generalizable methods for multiplying and dividing fractions and decimals;
- Estimate product and quotients of problems involving decimals, fractions, mixed numbers, and improper fractions;
- Solve problems requiring arithmetic with decimals, fractions, and mixed numbers.
Work from previous grades that supports this new learning includes:
- Multiply multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms;
- Divide multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms. Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or a mixed number, or a decimal;
- Estimate products and quotients of multi-digit whole numbers by rounding, using benchmarks, and place value to assess the reasonableness of results;
- Consider the context in which a problem is situated to select the most useful form of the quotient for the solution and use the context to interpret the quotient appropriately;
- Estimate solutions to arithmetic problems to assess the reasonableness of results;
- Solve real-world and mathematical problems requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships between operations, the use of technology, and the context of the problem to assess the reasonableness of results;
- Read and write decimals using place value to describe decimals in terms of groups from millionths to millions;
- Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts;
- Round numbers to the nearest 0.1, 0.01, and 0.001;
- Locate fractions on a number line, including mixed numbers and improper fractions;
- Represent equivalent fractions using fractions models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions;
- Apply the commutative, associative and distributive properties and order of operations to generate equivalent numerical expressions and to solve problems involving whole numbers.
NCTM Standards:
Work flexibly with fractions, mixed numbers, and decimals to solve problems
- Build on prior knowledge from previous grade levels and everyday life;
- Use of decimals and fractions includes measurements and comparisons;
- Ensure solid understanding of context when deciding among differing representations for equivalency and moving flexibly between them;
Understand and use the associative, commutative, and distributive properties to simplify computations with integers, fractions, and decimals
- Use mathematical properties to simplify many computations involving fractions and decimals;
- Use common procedures to multiply and divide fractions and decimals efficiently and accurately;
Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use
- Develop own methods of computation and sharing results with class;
- Explain why methods chosen and subsequent solutions are reasonable;
- Compare and evaluate personal method with traditional algorithms;
Understand and use the inverse relationship of multiplication and division to simplify and solve problems
- Be mindful of the decision of when to multiply or divide when working with fractions, mixed numbers, or decimals;
- Increase understanding of division as being repeated subtraction rather than just a rote procedure of invert and multiply;
- Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods
- Learn when an exact answer or an estimate is needed;
- Identify which computational method should be chosen;
- Evaluate reasonableness of solution;
- Increase mental computation and estimation.
Common Core State Standards (CCSS):
5BTB (Number And Operations In Base Ten) Perform operations with multi-digit whole numbers and with decimals to hundredths.
5NTB.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
5NF (Number And Operations--Fractions) Use equivalent fractions as a strategy to add and subtract fractions.
5NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.
5NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1
5NF.7.c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.
6NS (Number System) Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
6NS.3 Fluently multiply and divide multi-digit decimals using the standard algorithm for each operation.
6RP (Ratios And Proportional Relationships) Understand ratio concepts and use ratio reasoning to solve problems.
6RP3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
6RP3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
7NS (Number System) Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.
Misconceptions
Student Misconceptions and Common Errors
- Students may believe that multiplication always results in a larger number, while division always results in a smaller number.
- Students struggle to make meaning of problems involving fractions, such as [math]\frac{1}{2}\times 1\frac{3}{4}[/math] and [math]5 \div\frac{1}{4}[/math], making it difficult to estimate solutions and assess reasonableness of results.
- When comparing two rectangular area models, students may not recognize equivalencies. For example, the multiplication of [math]\frac{1}{2}[/math] and [math]\frac{3}{4}[/math] can be shown in different ways.
Although both show the result is [math]\frac{3}{8}[/math], students may believe the second model results in a larger number since it has 6 parts shaded compared to 3 in the first model.
- Students are sometimes confused when finding a fraction of a number. It helps to use the words "[math]\frac{1}{3}[/math] of 12" rather than "[math]\frac{1}{3}\times[/math] 12."
- Students who lack sufficient experience with grid or area models involving multiplication and division of fractions may misapply "the invert and multiply" rule by not inverting the second fraction, or inverting the first fraction, or inverting both fractions.
- When using the standard algorithm for division, students may ignore 0s in problems involving multi-digit dividends where the 0 is in the middle. For example, students may treat 40.2 ÷ 6 as 42 ÷ 6.
- When using the standard algorithm for multiplying decimals, students may determine the number of decimal places in the answer by counting the decimal places to the left of the decimal point instead of the right. For example, students may believe that 18.6 x 5.9 = 10.974, thinking that 3 decimal places are needed rather than 2.
- When placing the decimal point using the standard algorithm, students may begin counting from the left side of the product instead of the right. For example, students may understand that four decimal places are needed, but believe that 7.91 x 0.72 = 5695.2.
Vignette
In the Classroom
In the following vignette, students show the use of fraction bars and a number line to solve a problem requiring division of a mixed number.
Problem: You have [math]7\frac{2}{3}[/math] pounds of peanuts. You want to put them into 3 bags, putting the same amount into each bag. How many pounds of peanuts should you put into each bag?
Teacher: Student A, I see you used fractions bars to solve this problem. Can you show your solution?
Student A: Sure. I know that I have to share [math]7\frac{2}{3}[/math] pounds equally among 3 groups. I think of one fraction bar as one pound, so I laid out 7 pounds and 2 one-third pounds.
It's easy to see I can share 2 pounds with each of the three groups, but I have [math]1\frac{2}{3}[/math] pounds left. I broke the one pound into 3 equal one-third pound pieces.
Then I put 1 one-third pound in each group, and I also have 2 one-third pieces left to split up.
Now I can break each one-third into 3 equal pieces. Since one-third equals 3 one-ninths, then 2 one-third pounds equals 6 one-ninth pounds.
Now I can share 2 one-ninth pounds with each group.
Teacher: How much is in one group?
Student A: I see that I have 2 pounds plus [math]\frac{1}{3}[/math] plus two [math]\frac{1}{9}[/math] pounds. I use one-ninth fraction bars to help show that since 1 one-third is equal to 3 one-ninths, I have 5 one-ninths in all. I have 2 pounds and 5 one-ninth pounds in each group.
[math]7\frac{2}{3}\div 3=2\frac{5}{9}[/math] and [math]2\frac{5}{9}[/math] pounds of peanuts should go in each bag.
Teacher: Very interesting. You used fraction bars to show how to divide [math]7\frac{2}{3}[/math] into 3 equal groups. Thank you for sharing. Student B used a number-line model with fraction bars. Will you show us your solution?
Student B: OK. To find the number in each of 3 groups, we need to divide the distance on the number line from 0 to [math]7\frac{2}{3}[/math] into 3 equal sections and find out the size of each section. I started by drawing fraction bars on the number line to show [math]7\frac{2}{3}[/math].
As I looked at the number line, I was wondering what intervals I could use to break up the distance from 0 to [math]7\frac{2}{3}[/math] that would give me a total number of parts divisible by 3. First I tried breaking each one-unit interval into 3 one-third intervals. Here's what I did.
[math]7\frac{2}{3}=7+\frac{2}{3}=\frac{21}{3}+\frac{2}{3}=\frac{23}{3}[/math]
But 23, the number of intervals, cannot be divided evenly by 3, so using interval lengths
of one-third will not work. Next I tried sixths, since [math]\frac{1}{3}=\frac{2}{6}[/math].
[math]\frac{23}{3}=\frac{23}{3} \times \frac{2}{2}=\frac{46}{6}[/math].
But 46 cannot be divided evenly by 3 either, so sixths will not work. So then I tried intervals of length one-ninth.
[math]\frac{23}{3}=\frac{23}{3} \times \frac{3}{3}=\frac{69}{9}[/math].
Bingo! 69 can be divided by 3, so this will work. Then I divided my number line into intervals of length one-ninth. There will be 69 intervals between 0 and [math]7\frac{2}{3}[/math].
Now because 69 ÷ 3 = 23, I put marks to show the 3 groups of 23 intervals.
Teacher: So how much do you find for each group?
Student B: [math]\frac{23}{9}[/math] is the same as [math]2\frac{5}{9}[/math], so I agree with Student A that each group gets [math]2\frac{5}{9}[/math] pounds.
Student C: I have another strategy, and I didn't use fraction bars or a number line. May I share it?
Teacher: Of course.
Student C: I knew that [math]7\frac{2}{3}[/math] needed to be divided equally into 3 groups, so the problem can be written as [math]7\frac{2}{3}\div 3[/math].
[math]7\frac{2}{3}\div 3=\frac{23}{3}\div\frac{3}{1}=\frac{23}{3}\div(\frac{3}{1}\times\frac{3}{3})=\frac{23}{3}\div\frac{9}{3}=\frac{23\div 9}{3\div 3}=\frac{23\div 9}{1}=23\div 9=2\frac{5}{9}[/math].
Teacher: You solved the problem without any diagrams, but by using symbols. I noticed that you found a common denominator for your fractions before dividing. The interesting thing about dividing fractions with common denominators is that the division always results in a denominator of 1. When dividing fractions with common denominators, the answer is actually determined by the division of the numerators. You can see that in our example. The final step is 23 ÷ 9, a division of the numerators.
Student B: You mean that always happens when you use common denominators?
Teacher: Let's try it with another problem and see what happens. How about [math]\frac{5}{6}\div\frac{3}{4}[/math]. What's a common denominator?
Student B: 24. I don't know if it's the lowest common denominator, but I know it's a common denominator because I just multiplied the denominators. 6 x 4 = 24.
Teacher: You're right. Common denominators are common multiples, so let's use 24 as our common denominator.
[math]\frac{5}{6}\div \frac{3}{4}=\frac{5\times 4}{6\times 4}\div\frac{3\times 6}{4\times 6}=\frac{20}{24}\div \frac{18}{24}=\frac{20\div 18}{24\div 24}=\frac{20\div 18}{1}=20\div 18=1\frac{2}{18}=1\frac{1}{9}[/math]
Student D: Wow! I like that strategy, but I used the "invert the divisor and multiply" strategy.
Teacher: How would you use that strategy to solve [math]\frac{5}{6}\div\frac{3}{4}[/math]?
Student D: Like this: [math]\frac{5}{6}\div \frac{3}{4}=\frac{5}{6}\times \frac{4}{3}=\frac{20}{18}=1\frac{2}{18}=1\frac{1}{9}[/math]
I got the same answer.
Teacher: Why does that strategy work?
Student D: I don't know. It just does.
Teacher: Then let's see if we can understand why. First let's start with easier numbers. We know that 12 ÷ 4 = 3. One way to think of 12 ÷ 4 is that if 12 is 4 groups, then the quotient represents how many are in one group.
Since there are 3 in one group, then 12 ÷ 4 = 3. Using the same idea, you can think of [math]\frac{5}{6}\div\frac{3}{4}[/math] like this: if [math]\frac{5}{6}[/math] is [math]\frac{3}{4}[/math] of a group, then the quotient is how many in one group.
Student C: OK. I understand that. Then I predict that the answer to [math]\frac{5}{6}\div\frac{3}{4}[/math] is probably a little more than 1.
Teacher: Why do you say that?
Student: Because if [math]\frac{5}{6}[/math] represents only [math]\frac{3}{4}[/math] of one group, then the size of one group must be larger then [math]\frac{5}{6}[/math]. Since [math]\frac{5}{6}[/math] is pretty close to 1, I think adding another [math]\frac{1}{4}[/math] of a group will give you just a little more than 1.
Teacher: Good reasoning. If [math]\frac{5}{6}[/math] is [math]\frac{3}{4}[/math] of a group, then the quotient of [math]\frac{5}{6}\div\frac{3}{4}[/math] is the how many in one group.
Since [math]\frac{5}{6}[/math] represents [math]\frac{3}{4}[/math] or 3 parts out of 4 of one group, we can divide [math]\frac{5}{6}[/math] by 3 to find the size of one part. Then we can multiply the size of one part by 4 to find the size of one entire group. How can we divide [math]\frac{5}{6}[/math] by 3 to find the size of one part?
Student: Dividing [math]\frac{5}{6}[/math] by 3 is the same as finding [math]\frac{1}{3}[/math] of [math]\frac{5}{6}[/math], or [math]\frac{1}{3}\times\frac{5}{6}=\frac{5}{18}[/math]. That tells us the size of each of those 3 parts is [math]\frac{5}{18}[/math].
Teacher: I can check that the size of one part is [math]\frac{5}{18}[/math] by multiplying it by 3 to see if the result is [math]\frac{5}{6}[/math]. [math]\frac{5}{18}\times 3=\frac{5}{18}\times \frac{3}{1}=\frac{5\times 3}{18\times 1}=\frac{15}{18}=\frac{5}{6}[/math]. Yes, the size of one part is [math]\frac{5}{18}[/math].
But remember, there are 4 total parts in one group.
Student A: So we need to multiply by 4 to find the size of the whole group.
Teacher: Exactly. [math]\frac{5}{18}\times 4=\frac{5}{18}\times \frac{4}{1}=\frac{5\times 4}{18\times 1}=\frac{20}{18}=1\frac{2}{18}=1\frac{1}{9}[/math].
Student D: But how does that connect to "invert and multiply?'
Teacher: Let's look more closely. What we actually did was [math]\frac{1}{3}\times \frac{5}{6}\times \frac{4}{1}[/math]. Using the commutative and associative properties, I'm going to rewrite that as [math]\frac{5}{6}\times (\frac{1}{3}\times \frac{4}{1})[/math], which is [math]\frac{5}{6}\times \frac{4}{3}[/math].
Student D: There it is! [math]\frac{4}{3}[/math] is the reciprocal of [math]\frac{3}{4}[/math]. Dividing is the same as multiplying by the reciprocal of the divisor!
Teacher: Terrific! You've got it!
Resources
- From prior experiences with whole numbers, students may have developed the misconception that multiplication always results in a larger number. Using concrete models and pictorial representations to visualize multiplication as "repeated addition" helps students understand that multiplication is a scalar relationship, where one factor is multiplied by another. Scaling by a factor greater than 1 (repeating the addition more than 1 time) results in an increased number. Scaling by a factor less than 1 (repeating the addition a fractional number of times) results in a smaller number. The example 4 x 3 = 12 and 4 x [math]\frac{1}{2}[/math] = 2, where 4 is multiplied by scale factors 3 and [math]\frac{1}{2}[/math], can be used to illustrate this. Once students understand that multiplication does not always result in an increased number, the relationship between multiplication and division as inverse operations that "do" and "undo" each other can be used to show that division does not always result in a decreased number.
- It is essential to build on students' understanding of multiplication and division and connect previous experiences with whole numbers to fractions and decimals. Using simpler problems involving whole numbers is an effective strategy to help students make meaning of problems involving computation with fractions. The example below shows how the factors 4 and 3 can be used as a foundation for multiplying the factors 4 and [math]\frac{2}{3}[/math].
- Division has two common interpretations: measurement (or quotitive) and sharing (or partitive). Each interpretation has its associated language. Patterns in language used with dividing whole numbers can help develop understanding of division examples that involve fractions. The chart below describes different division examples, first using the quotitive interpretation and then using the partitive interpretation. A number line is used to represent each example. The number line can be a very efficient model for representing division. If students have used the number line to model fraction multiplication, they will have the experience necessary to connect the number line model of division of fractions back to the number line model of multiplication of fractions.
- Since division has different interpretations, students may use different types of models to represent different situations. Also the same model may be used in a different way depending on the context of the problem. In other words, a student might use fractions bars to model both a quotitive and a partitive division problem but the model will look different. As with multiplication, it is important to help students connect their prior understanding of division with whole numbers to models for representing division that can lead to formal symbolic procedures for dividing fractions.
- It is important to give appropriate context to problems and not teach computational skills in isolation. One possible context for [math]\frac{1}{2}\times 1\frac{3}{4}[/math] is the need to determine how much flour is needed to make [math]\frac{1}{2}[/math] batch of cookies that calls for [math]1\frac{3}{4}[/math] cup flour. Determining the number of people that will receive a portion of 5 candy bars that have been broken into [math]\frac{1}{4}[/math]s is a possible context for 5 ÷ [math]\frac{1}{4}[/math].
- When using rectangular area models, the same result may look different. For example, the multiplication of [math]\frac{1}{2}[/math] and [math]\frac{3}{4}[/math] can be represented in two ways.
The second model shown appears to have a larger area. Remind students that fractions represent [math]\frac{part}{whole}[/math] relationships. Since the [math]\frac{part}{whole}[/math] relationship shown in the second model is [math]\frac{6}{16}=\frac{3}{8}[/math], both models have the same result.
- Teachers' personal algorithmic knowledge of "invert and multiply" when dividing by fractions can interfere with the construction of a more complete understanding of the concept. It is essential for students to have multiple experiences with area and number line models connecting division of whole numbers to division of fractions that can lead to formal symbolic procedures for dividing fractions.
- Students should be introduced to strategies other "invert and multiply" when dividing by fractions. Some strategies include:
The Number Line Model for [math]3\frac{1}{2}\div \frac{1}{2}[/math] can be used to show that there are 7 groups of [math]\frac{1}{2}[/math] in [math]3\frac{1}{2}[/math] Therefore, [math]3\frac{1}{2}\div \frac{1}{2}=7[/math].
The Rectangular Area Model uses pictures to show how the division works. In the following example, an area model is used to show that there are 4 eighths in [math]\frac{1}{2}[/math]. Therefore, [math]\frac{1}{2}\div\frac{1}{8}=4[/math].
The Common Denominator Model works by finding a common denominator for both fractions. This results in a whole number division problem involving only the numerators. For example: [math]\frac{3}{4}\div\frac{1}{5}=\frac{15}{20}\div\frac{4}{20}=3\frac{3}{4}[/math].
- A focused discussion of multiplication and division of decimals should begin with an understanding of the basic idea of equivalence between fractions and decimals. To expand their understanding of this equivalence relationship, students must also revisit their understanding of division:
- As students learn to divide in situations where the answer is not a whole number, they often encounter situations such as the following:
It is reasonable to take this computational work and say, "5 goes into 17 three times with a remainder of 2." However, one should not write 17 ÷ 5 = 3 r 2, since no meaning for the equality sign makes this statement true. Instead, the information given by the computation is 17 = 5 x 3 + 2. We can divide both sides of the equation by 5 and get:
[math]17\div 5=\frac{17}{5}=\frac{5\times 3+2}{5}=3+\frac{2}{5}=3\frac{2}{5}=3\frac{4}{10}=3.4[/math];
- When the result of whole-number division is not a whole number, the answer can be expressed in three ways:
- As students learn about different representations of quotients, they realize that when a whole number is divided by a whole number, three results are possible:
Sometimes the remainder is 0 and sometimes it is not.
If the answer is not a whole number, then sometimes the quotient is less than 1 and sometimes it is not (i.e., it is a "mixed" number).
If the answer is not a whole number, sometimes the quotient is a fraction in "lowest terms" and sometimes it is not.
- An important skill that helps students to understand how to multiply and divide decimals is the ability to apply place-value patterns when multiplying or dividing by powers of 10:
When multiplying by greater and greater powers of 10, the digits move to greater and greater place-value positions. When dividing by greater and greater powers of 10, the digits move to lesser and lesser place-value positions.
- When multiplying decimals, students will begin to see the relationship between the number of decimal places in the factors and the number of decimal places in the product through a variety of carefully crafted examples:
- When dividing a finite decimal number by another finite decimal number, the following guidelines reduce the work to dividing one whole number into another whole number:
If the divisor (denominator) is not a whole number, obtain an equivalent problem by multiplying both the dividend (numerator) and divisor (denominator) by the same power of 10, chosen so that the new divisor is a whole number.
[math]3.55\div 0.25=\frac{3.55}{0.25}=\frac{3.55}{0.25}\times\frac{100}{100}=\frac{355}{25}=355\div 25[/math].
Divide as you would normally divide if the divisor was a whole number, being careful to use understanding of place value to align the digits in the quotient with the appropriate place value in the dividend.
Place the decimal point in the quotient directly above the decimal point in the dividend.
- It is important to ask students to estimate products and quotients before performing calculations with fractions and decimals and then assessing those results for reasonableness. This strategy often helps students recognize errors and provides opportunities to address misconceptions. For example, students that ignore 0s in multi-digit dividends when using the standard algorithm for division can recognize that their estimate differs by a power of 10. Estimation will also help students who incorrectly position the decimal point recognize their errors.
- Although students at this grade level can use estimation and reasonableness to justify the process for finding decimal products and quotients, teachers should understand that the equivalence relationship between decimals and fractions is the basis of any efficient, generalizable procedure. The examples below show how the equivalence relationship between fractions and decimals can be used to verify standard procedures for multiplying and dividing decimals:
- Students who struggle to position the decimal point correctly when multiplying decimals may benefit from writing the decimals as fractions first, and then multiplying.
- Multiplying and dividing decimals and fractions can be difficult for students who are still struggling with basic multiplication facts. Consider allowing these students to use a multiplication chart to aid them through the problems, while encouraging them to master the facts.
Feeding Frenzy
In this activity, students multiply and divide a recipe to feed groups of various sizes. Students use unit rates or proportions and think critically about real world applications of a baking problem.
Multiplication of Fractions
Students are able to multiply and manipulate various fractions to create area models that represent any fractions needed to be compared.
Additional Instructional Resources
Literature connections
Students read and discuss "Beasts of Burden" in The Man Who Counted: A Collection of Mathematical Adventures by Malba Tahan. In this story, three brothers must divide their father's camels.
Multiplication and Division of Decimals
This interactive lesson teaches methods for multiplying and dividing decimals using whole-number divisors.
reciprocal: the multiplicative inverse of a number; in other words, a reciprocal is a number that you multiply by so the resulting product equals 1. Example: The reciprocal of [math]\frac{3}{5}[/math] is [math]\frac{5}{3}[/math] because [math]\frac{3}{5}\times\frac{5}{3}=1[/math].
Reflection - Critical Questions regarding the teaching and learning of these benchmarks:
- What previous models and understanding of multiplication and division do students bring to my classroom?
- How can I use students' understanding of multiplication and division of whole numbers as a foundation for experiences with fractions and decimals?
- What strategies can be used to model multiplication and division of fractions and decimals?
- What strategies, other than standard algorithms, are students able to demonstrate for multiplying and dividing fractions and decimals?
- What evidence do I have that my students understand the relationship between decimals and fractions and recognize equivalencies?
- What evidence exists to show that students understand standard algorithms for multiplying and dividing fractions and decimals?
Materials - suggested articles and books
Unpacking a Conceptual Lesson: The Case of Dividing Fractions
This article uses pattern blocks and pictorial representations to demonstrate addition, subtraction, multiplication, and division of fractions. It offers a good description of the concepts behind "inverting and multiplying."
What do Students Need to Learn about Division of Fractions?
This article discusses the various ways fractions are divided and why students need to learn about fractions through dividing them. Several problems are presented.
NCTM A Research Companion to Principles and Standards for School Mathematics (Details about this resource can be found in the References section.)
Chapter 8, Conclusion to facts and algorithms as products of students' own mathematical ability, pp. 120-121.
Kaput, J. (1989). Linking representations in the symbol system of algebra. In Kieran, C. & Wagner, S. (Eds.). A research agenda for the learning and teaching of algebra. Hillsdale, NJ: Lawrence Erlbaum.
Kilpatrick, J., Martin, W., & Schifter, D. (Eds.). (2003). A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
South Carolina State Department of Education. Math Curriculum Standards, N.p., n.d. Web. 1 Apr. 2011. <http://ed.sc.gov/agency/Standards-and-Learning/Academic-Standards/old/cso/standards/math/>.
Minnesota's K-12 Mathematics Frameworks. (1998). St. Paul, MN: SciMathMN.
National Council of Teachers of Mathematics. (2010). Focus in grade 6 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
NJ Mathematics Curriculum Framework. The official web site for the state of New Jersey. N.p., n.d. Web. 1 Apr. 2011. <http://www.state.nj.us/education/frameworks/math/>.
Assessment
(DOK Level 1)
1. Multiply: 0.14 x 1.6
a) 2.240 b) 0.224 c) 0.0224 d) 0.00224
Answer: b
(DOK Level 1)
2. Divide: [math]3\frac{1}{2}\div \frac{1}{3}[/math].
Answer: [math]10\frac{1}{2}[/math]
(DOK Level 2)
3. The highest mountain on the moon is Mount Huygens. It is 5.5 about kilometers in height. Mount Everest, the highest mountain on earth, is about 8.8 kilometers in height. How many times taller Mount Everest than Mount Huygens?
a. 4.84 times b. 3.3 times c. 1.6 times d. 0.625 times
Answer: c
(DOK Level 2)
4. Andy wants to buy [math]3\frac{1}{3}[/math] cups of cashews. There are [math]\frac{5}{6}[/math] cup of cashews in each package. How many packages of cashews should Francis buy?
Answer: 4 packages
(DOK Level 3)
5. Describe and correct the error in the solution. Explain your reasoning.
Sample Answer:
The decimal point is in the wrong place. I estimated the product to be 20 (5 x 4). Since each factor has 1 decimal place, the product will have 2 decimal places. The correct answer is 18.62.
(DOK Level 4)
6. Create a model to prove that [math]2\frac{1}{2}\div\frac{1}{2}=5[/math].
Sample Answers:
Differentiation
FunwithFractions
This unit uses the set model to support students who struggle with basic fraction concepts and facilitates work with comparing and ordering and working with equivalency.
FunwithPatternBlockFractions
This unit uses the area model to support students who struggle with basic fraction concepts and facilitates work with comparing and ordering and working with equivalency.
FractionGame
This applet allows students to individually practice working with relationships among fractions and shares ways of combining fractions.
When students work with physical manipulatives, one major challenge is that the manipulation of multiple pieces, representing fractions, can be confusing for the student. This can cause the students to lose sight of the intended mathematical concept of the lesson. Having too many manipulatives prevents struggling students from connecting the mathematical concepts with their concrete representations (Kaput, 1989.) Using virtual manipulatives on a computer can assist with student understanding as they can bridge the physical to the abstract.
Multiplication of Fractions
This website allows students to select fractions and show a pictorial representation for the algorithm on the right. Both the horizontal and vertical axes are easily manipulated.
Provide a multiplication chart to assist students who are struggling with basic multiplication facts, while encouraging them to master the facts.
- Provide a graphic organizer to show the different ways division problems can be expressed.
- The language of division is especially tricky for English Language Learners, since it signals different situations. "How many in each group" signals partitive division, while "how many groups" signals measurement division. Provide a graphic organizer that shows models for partitive and measurement division and connects division of whole numbers to division with fractions.
- Provide a graphic organizer that show models connecting multiplication of whole numbers to multiplication of fractions.
- Provide a graphic organizer that connects multiplication of decimals and fractions.
- Provide a graphic organizer that connects division of decimals and fractions.
- Use graphic organizers such as the Frayer model shown below, for vocabulary development.
CalculatorRemainders
In this lesson, students develop a deep conceptual understanding of the relationship between remainders and the decimal part of quotients.
SoccerShoot-Out
Score by correctly multiplying or dividing fractions. Students need to reduce to simplest form to be correct. There are three levels and a Super Brain level that uses all 4 operations.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) | Teachers are: (descriptive list) |
using a variety of models, such as number lines and rectangular area models, to explore and make sense of processes for multiplying and dividing fractions. | connecting students' previous experiences with multiplying and dividing whole numbers to experiences with multiplying and dividing fractions. |
representing multiplication and division of fractions with a variety of models. | modeling a variety of strategies to multiply and divide fractions. |
multiplying and dividing fractions using a variety of strategies and assessing results for reasonableness. | asking students to estimate products and quotients of problems involving fractions and decimals and justify their results. |
understanding that the commutative property applies to multiplication of decimals and fractions, but not to division. | using models to demonstrate that multiplication is commutative for positive rational numbers, but division is not. |
understanding that division can have different meanings: "how many in each group," (partitive) and "how many groups" (quotitive). | using real-life applications to model both partitive and measurement division. |
exploring place-value patterns that result when decimals are multiplied and divided by powers of 10. | making place-value patterns that result when decimals are multiplied and divided by powers of 10, explicit to students. |
recognizing the relationship between the number of decimal places in the factors and the number of decimal places in the product when multiplying decimals. | carefully crafting examples that support students to learn the standard algorithm for multiplying decimals. |
writing decimal quotients and their fractional representations side-by-side so that they begin to see patterns. | using the equivalence between fractions and decimals to justify the standard algorithm for dividing decimals. |
having multiple opportunities to build meaning for generalizable procedures for multiplying and dividing with fractions and decimals. | allowing ample time for students to make sense of computational processes so that they can eventually apply them effectively in problem-solving situations. |
communicating their reasoning in writing, drawings, and conversation. | asking students to explain their reasoning in a variety of formats. |
Parent Resources
- Models for the Multiplication of Fractions
This website uses visual area models to multiply fractions. - ModelsfortheDivisionofFractions
This website uses visual area models to divide fractions. - MultiplyingDecimals
This video demonstrates the standard algorithm for multiplying decimals. - DividingDecimals
This video demonstrates the standard algorithm for dividing decimals.
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